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Einstein Studies
Editors: Don Howard John Stachel
Published under the sponsorship of the Center for Einstein Studies, Boston University

Volume 1: Volume 2: Volume 3: Volume 4: Volume 5:
Volume 6:

Einstein and the History of General Relativity Don Howard and John Stachel, editors
Conceptual Problems of Quantum Gravity Abhay Ashtekar and John Stachel, editors
Studies in the History of General Relativity Jean Eisenstaedt and A. J. Kox, editors
Recent Advances in General Relativity Allen 1. Janis and John R. Porter, editors
The Attraction of Gravitation: New Studies in the History of General Relativity John Earman, Michel Janssen, and John D. Norton, editors
Mach's Principle: From Newton's Bucket to Quantum Gravity Julian B. Barbour and Herbert Pfister, editors

Julian B. Barbour
Editors

Herbert Pfister

Mach's Principle:
From Newton's Bucket to Quantum Gravity

Birkhauser Boston • Basel • Berlin

Julian B. Barbour College Fann South Newington Banbury, Oxon OXl5 4JG England

Herbert Pfister Universitat Ttibingen Institut fUr Theoretische Physik D-72076 Ttibingen Gennany

Library of Congress In-Publication Data

Mach's principle : From Newton's bucket to quantum gravity / edited by

Julian B. Barbour, Herbert Pfister.

p.

em. -- (Einstein studies ; v. 6)

Includes bibliographical references and index.

ISBN 0-8176-3823-7 (alk. paper). -- ISBN 3-7643-3823-7 (pbk.

alk. paper)

1. Mach's principle. 2. General relativity (Physics)

I. Barbour, Julian B. II. Pfister, Herbert, 1936- . III. Series.

QC137.M33 1995

95-22539

530.1--dc20

CIP

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98765432 I

CONTENTS

Contents

v

Acknowledgments

viii

1. Introduction and Historical

General Introduction H. Pfister and J.B. Barbour

1

Mach before Mach J.B. Barbour

6

Mach's Principle before Einstein J.D. Norton

9

Mach's Criticism of Newton and Einstein's Reading of

Mach: The Stimulating Role of Two Misunderstandings

H. -H. von Borzeszkowski and R. Wahsner

58

Einstein's Formulations of Mach's Principle C. Hoefer

67

General Discussion: What is The Machian Program?

91

2. Nonrelativistic Machian Theories

Introduction

107

Selected Passages: Mach, Poincare, Boltzmann

109

Absolute or Relative Motion? B. Friedlaender

114

On Absolute and Relative Motion A. FtJppl

120

Motion and Inertia W Hofmann

128

On the Relativity of Accelerations in Mechanics

H. Reissner

134

The Possibility of Fulfillment of the Relativity Requirement

in Classical Mechanics E. SchrtJdinger

147

Weber's Law and Mach's Principle A.K.T. Assis

159

A Relative Newtonian Mechanics D. Lynden-Bell

172

3. General Relativity as a More or Less Machian Theory

Introduction

179

Selected Passages on Machian Ideas A. Einstein

180

Wheeler-Einstein-Mach Spacetimes J. Isenberg

188

Comments on Initial Value Formulation D.R. Brill

208

General Relativity as a Perfectly Machian Theory

J. B. Barbour

214

A Closed Universe Cannot Rotate D.H. King

237

VI Table of Contents

4. Other Formulations of Mach's Principle

Introduction

249

Direct Particle Formulation of Mach's Principle

J. V. Narlikar

250

Mach's Principle and the Creation of Matter F. Hoyle

262

The Integral Formulation of Mach's Principle D. Raine

274

Mach's Principle and Local Causal Structure U. Bleyer

and D. -E. Liebscher

293

5. Frame Dragging

Introduction

308

Absolute or Relative Motion? I. Friedlaender

309

On a Gyroscope Experiment to Measure the Rotation

Velocity of the Earth A. FtJppl

312

Dragging Effects near Rotating Bodies and in

Cosmological Models H. Pfister

315

Comments on Dragging Effects D.R. Brill

332

Dragging Effects near a Rigidly Rotating Disk of Dust

R. Meinel and A. Kleinwtichter

339

Dragging Effects and the Theory of Active Galactic

Nuclei V. Karas and A. Lanza

347

On the Interpretation of Dragging Effects in Rotating

Mass Shells J. Frauendiener

353

6. Experimental Status

Introduction

364

Testing Machian Effects in Laboratory and Space

Experiments C. Will

365

Dragging of Inertial Frames, Gravitomagnetism, and

Mach's Principle I. Ciufolini

386

Time Variation of Fundamental Constants: Bounds from

Local Data P.D. Sistema and H. Vucetich

403

Machian Effects in Physical Law and the Field Paradigm

of Modern Physics K. Nordtvedt

422

Table of Contents Vll

7. Critical Reflections

Introduction

436

Mach, the Expansion of the Universe, the Variation of

Inertial Mass, and Lense-Thirring W. Rindler

437

Mach's Principle and Theories of Gravitation

H.F.M. Goenner

442

Machian Ideas and General Relativity J. Ehlers

458

Reflections on Mach's Principle H. Bondi

474

8. Quantum Gravity

Introduction

478

The Higgs Field and Mach's Principle of Relativity of

Inertia H. Dehnen

479

Geometric Structures on Superspace D. Giulini

491

General Discussion: Time, General Relativity, and

Quantum Gravity

501

Names and Addresses of Contributors

527

Index of Different Formulations of Mach's Principle

530

Name and Subject Index

531

Acknowledgments
This volume and the conference at Tiibingen (July 1993) on which it is based would not have been possible without the very generous financial support of the Deutsche Forschungsgemeinschaft (DFG). We further thank the Landesregierung Baden-Wiirttemberg in Stuttgart and the Institut Dr. Friedrich Forster in Reutlingen for their contributions. The Max-Planck-House in Tiibingen and its manager, Mrs. M. HardersOpolka, supplied effective organization and the ideal congenial atmosphere for a conference of this size. Special thanks go to Mrs. C. Stiller for perfect secretarial work before, during, and after the conference, and to the local organization team U. Heilig, C. Klein, J. Klenk, and U. Schaudt. We are also grateful to the advisory committee of the conference, B. Bertotti, D. Brill, 1. Ehlers, and 1. Stachel, for really good advice, and to the editors of the Einstein Studies Series, D. Howard and J. Stachel, for their continuous support of the 'Mach project,' for the inclusion of this project in the Einstein Studies, and for help with the final editorial work. The camera copy for this volume was typed by Mrs. Kate Draper, to whom we are greatly indebted for her accuracy and unending patience. Our publishers, Birkhauser, have been supportive, helpful, and understanding. Finally, we should like to express our thanks to Dr Siegfried Wagner for the fascinating talk he gave about Einstein's ancestors and the life of Jews over several centuries in rural Southern Germany. This talk was on the occasion of an excursion from Tiibingen to the Jewish Cemetery at Bad Buchau, where many of Einstein's ancestors are buried. Dr Wagner's research is being prepared for publication in a book entitled Albert Einsteins Ahnen, ihre Herkunft und ihr Schicksal.
The Editors

1. Introduction and Historical
General Introduction
Hypothesen sind Netze, nur der wirdfangen der auswiift. Novalis.
I think it was Hermann Bondi who once said that physics is such a consis-
tent and connected logical structure that if one starts to investigate it at any point and if one pursues correctly every issue that branches away from
one's starting point, in the outcome one will be led to understand the whole ofphysics: With Mach's Principle it seems something like that.
Sir Fred Hoyle, this volume, p. 269.
This volume is based on the conference 'Mach's Principle: From Newton's Bucket to Quantum Gravity,' held July 26-30, 1993, at the Max-Planck-House in Tiibingen, Germany. As far as we know, this was the first conference exclusively devoted to Mach's Principle. (Sir Hermann Bondi in his closing remarks: "This conference was a splendid idea, and I am only surprised that nobody thought of having such a conference before.")
The so-called Mach's Principle is surely one of the most elusive concepts in physics: On one hand, Machian aspects have been present either explicitly or implicitly in theoretical astronomy, general physics, and dynamics from their Greek infancy up to the present day (Barbour 1989 and following article). On the other hand, most of practical physics is done, and successfully done, without ever thinking of the 'deep questions' connected with Mach's Principle. (The situation is similar in quantum theory, which functions extremely well using established prescriptions notwithstanding deep and unresolved questions about its interpretation, its measuring process, and its classical limit.)
In this volume, the notion 'Mach's Principle' is understood in as broad a sense as possible. Although it is certainly interesting (see Chap. 1) and may be important (see p. 215) to establish precisely what Mach
Einstein Studies, vol. 6: Mach's Principle: From Newton's Bucket to Quantum Gravity, pp. 1-5 © 1995 Birkhiiuser Boston, Inc. Printed in the United States.

2 General Introduction
said about absolute and relative elements in physics, and to see how Einstein (who coined the actual expression 'Mach's Principle' in 1918, p. 186) tried to incorporate Mach's ideas in general relativity, it would be ridiculous for a book published in 1995 to narrow these age-old questions about the foundations of physics to the pronouncements of just these two physicists, however eminent, and not to cover the contributions of their contemporaries. It is also very important to consider the development in thinking and the accumulation of experimental facts that have occurred in the intervening period.
The root of Mach's Principle, as understood in this volume, is deeply connected with the question of what constitutes the essence of the method of physics and the concept of a physical system: It is often not sufficiently appreciated how kind nature has been in supplying us with 'subsystems' of the universe which possess characteristic properties (literally in the sense 'proper to the system') that can be described and measured almost without recourse to the rest of the universe. The strategy of dividing the universe into ever smaller and 'simpler' parts has shaped physics, beginning with the investigations of the solar system, which resulted in the concept of a mass point for complicated objects such as planets, going on to atoms and elementary particles, and presumably coming to an end only at the level of the constituents (quarks, subquarks) of elementary particles. On the other hand, it is evident that basic concepts such as 'inertia' and 'centrifugal force' cannot be understood and explained within the context of the subsystems themselves, but at best by taking into account the rest of the universe.
As is well known, Newton 'solved' this conflict by the introduction of the extremely successful concepts of 'absolute space' and 'absolute time.' Newton recognized clearly that only relative quantities can be directly observed but, unlike his relationist contemporaries Huygens, Leibniz, and Berkeley, he was convinced that a scientifically useful notion of motion could not be based on relational quantities. Instead, he sought to demonstrate how absolute quantities could be deduced from relative observations. In this endeavor he was not entirely successful (Barbour 1989).
The most emphatic and most influential physicist to insist on a reformulation or extension of the foundations of physics in purely relational terms was Ernst Mach in the last quarter of the 19th century, though he made only tentative proposals for such a goal. Albert Einstein was very much influenced by Mach's writings, and his general relativity was at least partly conceived in the spirit of realizing Mach's dictum. Indeed, general relativity was the first theory to supply a dynamic

General Introduction 3
spacetime (dependent on the matter distribution) and to indicate at least possibilities of how inertia and centrifugal forces could result from interaction with the distant cosmic objects. On the other hand, general relativity in its present formulation does not, despite its name, fulfill the demand of using solely relational properties between physical objects composed of matter in the strict sense (as opposed to 'generalized matter' in the form of gravitational waves or spacetime curvature). One of the main debates at the conference concerned the question of how far general relativity realizes Mach's Principle: Does this principle make sense for the full theory with its huge manifold of (partly unphysical) solutions, or does it function as a selection principle for special classes of solutions (and if so which?), or has it meaning only in our unique universe? The cosmological context of Mach's Principle goes a long way towards explaining why this principle is so elusive: Cosmology lies somewhere at the edge of the physical method, which usually relies on the possibility of preparing physical systems and confirming results by repeated measurements on ensembles of similar systems. In this respect it is remarkable how much reliable information astronomy and astrophysics have already supplied about our cosmos. In the future we can expect information about still more distant, and therefore earlier, parts of the universe, and in this way information about the cosmos as a whole. This will surely have an impact on 'Machian questions.'
Investigations of the very early cosmos necessarily call for a unification of gravity and quantum theory, which is widely held to be the deepest open problem in contemporary theoretical physics. As it happens, many problems in so-called quantum gravity and quantum cosmology are intrinsically of a Machian character, for instance the goal to treat 'time' no longer as an absolute, external parameter, but to understand it as an intrinsic property of the considered system, i.e., the whole universe. It is clear that this volume cannot do full justice to these rather new and actively developing fields of quantum gravity and quantum cosmology. On the other hand, these may well be the fields in which most activity and progress in Machian questions can be expected from future research.
Although this volume is based on a conference, it is not a usual conference proceedings volume, to which all participants contribute only their latest, very specialized results without much interrelation between them. From the beginning, it was the intention that the conference and this volume should - very much in agreement with the general policy of the Einstein Studies Series - cover all aspects of Mach's Principle historical, philosophical, astronomical, theoretical and experimental- and

4 General Introduction
confront, where necessary, the different views on these aspects. Experts were invited to prepare general overviews, which were distributed to all participants already two months ahead of the conference. In some cases, these overviews were then supplemented by prepared reply talks. It was guaranteed that there was enough time for lively discussions after all talks. In addition, there were scheduled discussion sessions on selected, especially controversial topics. All these discussions were recorded on tape, and were edited by us and the contributors after the conference. Some talks and discussion contributions have been considerably improved and in part even rewritten after the conference. As organizers of the conference, we were very happy that it was possible to gather together in Tiibingen nearly all experts worldwide on the different views of Mach's Principle. Only a few prominent names are obviously missing, for example, Boris Al'tshuler, Bruno Bertotti, Jeffrey Cohen, Robert Dicke, Dennis Sciama, and John Wheeler. They had to decline their participation for different reasons, some of them at the last minute. Their influence can nevertheless be easily traced through this entire volume. For example, it turned out that one entire morning session, devoted to the initial-value problem in general relativity and based on Isenberg's paper (p. 188), was intimately related to the Machian ideas of John Wheeler and was exclusively presented by former collaborators of John. The session Chairman, Jayant Narlikar, introduced it as 'Wheeler without Wheeler. '
It should be mentioned that many important historic papers connected with Mach's Principle are scattered in hardly accessible journals or other sources; most of them are originally in German and have never been translated, and some of them have moreover been forgotten for decades. Indeed, one of the more important consequences of the conference was that it brought to light significant papers on Mach's Principle by Hofmann (1904), Reissner (1914, 1915), and SchrOdinger (1925) that were virtually unknown, even to experts in the field. Therefore we found it appropriate to collect such papers (partly in extracts) in English translation in this volume.
In summary, we hope that this volume represents a fairly complete status report and reference source on most aspects of Mach's Principle. In order to give greater unity to this collection of contributions, we have not hesitated to give cross references (indicated in square parentheses) to other places in the volume in which the same or related topics are discussed. In various places, especially following the translations and in the chapter introductions, we give commentaries. We have also prepared an index, in which we also attempt to draw the reader's attention to

General Introduction 5
common themes that run through the volume. Given the topic of the book, it is hardly to be expected that its two
editors will be in complete agreement on all aspects of Mach's Principle. Indeed, as will be evident from our own contributions, one of us (J.B.B.) believes Mach's Principle is in essence fully contained within general relativity whereas the other (H.P.) has reservations on this score. This divergence of opinion has not been any hindrance to productive and harmonious collaboration; indeed, we feel that the book gains from a certain friendly rivalry, each of us being keen to see the respective viewpoints properly represented. Somehow this seems very appropriate for Mach's Principle - see p. 630. Let the reader decide!
We should also mention a project to publish within the next year or two a book with the provisional title Relativity and Its Alternatives (J. Renn et al., eds.). This will cover much ground in common with the present volume; in particular it will include a long paper "The Third Way to General Relativity. Einstein and Mach in Context" by Jiirgen Renn. This paper is based on the talk he gave at Tiibingen but because of its length unfortunately could not be included in the present volume. The new book may also include translations of some papers with Machian context that also could not be included in this volume for lack of space. In particular, there may be a complete translation of Absolute oder relative Bewegung? by Benedict and Immanuel Friedlaender (1896) and also of Reissner's paper of 1915, partial extracts of which are included in this volume (p. 114, p. 309, p. 145ft).
The motto from Novalis - "Hypotheses are nets; only he that casts will catch" - has already been used: by Karl Popper at the head of his book The Logic of Scientific Discovery. We are grateful to Domenico Giulini for suggesting its appropriateness in connection with Mach's Principle. (It was also Giulini who drew our attention to the longforgotten papers of SchrOdinger and Reissner.) The idea to use the motto by Fred Hoyle came during the work of compiling the index! A glance at the index confirms the truth of Bondi's remark.
Julian B. Barbour, Herbert Pfister
REFERENCE
Barbour, Julian B. (1989). Absolute or Relative Motion?, vol. 1: The Discovery ofDynamics. Cambridge: Cambridge University Press.

Mach before Mach
Julian B. Barbour
The debate about motion - Is it absolute or relative? - extends back to antiquity, and 'Machian' attitudes can be readily identified in the writings of Aristotle, but I begin this brief survey at the dawn of the scientific age: with Copernicus and Kepler.
Not surprisingly - since astronomers cannot fail to be aware that observations are relational - both were 'Machians.' Copernicus defined his frame of reference thus: "The first and highest of all is the sphere of the fixed stars, which contains itself and everything, and is therefore immovable. It is unquestionably the place of the universe, to which the motion and position of all the other heavenly bodies are compared."
Kepler's standpoint is particularly interesting, since he was deeply impressed by Tycho Brahe's 'demolition' of the crystal spheres. Kepler posed the problem of astronomy in the famous words: "From henceforth the planets follow their paths through the aether like the birds in the air. We must therefore philosophize about these things differently." His response to the problem was very 'Machian' (Barbour 1989): The planets could not possibly follow such precise orbits by a mere inspection of empty space - they must be both guided and driven in their motion by the real masses of the universe, namely, the sun and the sphere of the fixed stars. This deeply held conviction was a decisive factor in Kepler's discovery of the laws of planetary motion - troly, a pre-Machian triumph of Mach's Principle.
Although Galileo retained many Aristotelian - and hence 'Machian' - concepts, he instinctively believed in motion relative to space. This comes out clearly in his theory of the tides, in the discussion of which he actually uses the expression absolute motion (Barbour 1989, p. 400).
The modern debate about motion had a most ironic origin. In 1632, Descartes was about to publish his Le Monde when he heard about Galileo's condemnation by the Inquisition. Since Copernicanism was central to his new mechanical philosophy, this put Descartes in a
Einstein Studies, vol. 6: Mach's Principle: From Newton's Bucket to Quantum Gravity, pp. 6-8 © 1995 Birkhiiuser Boston, Inc. Printed in the United States.

~ach before ~ach 7
quandary. He suppressed 1.£ Monde and only ventured to present his new physics in his Principia Philosophiae of 1645. To avoid censure, Descartes began by asserting, in a very Aristotelian manner, that both position and motion are relative. A convoluted argument enabled him to wriggle out of potential difficulties with the Inquisition. However, when he came to his laws of motion, he reverted, without explanation or warning, to the instinctively 'absolutist' position he had adopted in 1.£ Monde, in which he had advanced something almost identical to Newton's first law of motion as the foundation of his physics.
About 25 years later, Newton spotted the crass discrepancy between Descartes's espousal of relationalism and the use of the law of inertia as the foundation of mechanics. In De Gravitatione, which only came to light this century, Newton inveighed against Descartes. He saw that to set up a science of motion one must be able to define velocity as something definite. But if motion is relative and everything in the world is in motion - as it is in Cartesian philosophy - Descartes's own relationalism makes a mockery of the Cartesian law of inertia: "That the absurdity of this position may be disclosed in full measure, I say that thence it follows that a moving body has no determinate velocity and no definite line in which it moves." This is the nub - the fundamental problem ofmotion (Barbour 1989, Introduction): If all motion is relative and everything in the universe is in motion, how can one ever set up a determinate theory of motion?
Unlike his contemporaries Huygens and Leibniz, who both cheerfully used the law of inertia as the foundation of dynamics while stoutly maintaining the relativity of motion, Newton felt this problem so acutely that he could not conceive of any dynamics formulated without a rigid framework - absolute space. The Scholium in his Principia was simply a coded reworking of De Gravitatione in which Newton disdained to mention Descartes by name. Especially revealing is Newton's use of centrifugal force - in Cartesian philosophy the explicatory basis of both light and gravity - to exhibit the reality of absolute motion. Descartes is to be hoist with his own petard. The choice of a bucket was also at least in part mischievous in intent: By Descartes's philosophical concept of motion, the only 'true' motion of the water must be that relative to its immediate ambience (the bucket wall). This is why Newton said pointedly: "Therefore this endeavor does not depend upon any translation in respect of the ambient bodies, nor can true circular motion be defined by such translation."
Two centuries later, Mach (unaware of Newton's fixation with Cartesian absurdities) thought Newton naive to suppose the mere bucket

8 Julian B. Barbour
wall to have any relevance to centrifugal force and produced one of the great suggestive sayings in the history of physics: "Newton's experiment with the rotating vessel of water simply informs us, that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces .... No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass till they were ultimately several leagues thick." Given the effect of this remark - and the whole absolute-relative debate that Descartes initiated - on Einstein, it may not be too fanciful to suppose that if the Inquisition had condemned Galileo a few months later, and Descartes had published Le Monde, Newton might never have thought of the bucket nor Einstein of general relativity!
Let me conclude with a remark about Bishop Berkeley, who in De Motu (1721) comments that in empty space motion oftwo globes around a common center cannot be conceived by the imagination, but that if we "suppose that the sky of the fixed stars is created; suddenly from the conception of the approach of the globes to different parts of that sky the motion will be conceived." For this remark, Berkeley is often credited with having been a true precursor of Mach. Note, however, Berkeley's phrase 'fixed stars.' The stars were still very fixed in his mind, as we see from his earlier Principles ofHuman Knowledge (1710, §114):
Philosophers who have a greater extent of thought, and juster notions of the system of things, discover even the earth itself to be moved. In order therefore to fix their notions, they seem to conceive the corporeal world as finite, and the utmost unmoved walls or shell thereof to be the place, whereby they estimate true motions. If we sound our own conceptions, I believe we may fmd all the absolute motion we can frame an idea of, to be at bottom no other than relative motion thus defmed.
Thus, Berkeley looked backward to Kepler and Copernicus just as much as he looked forward to Mach. He never confronted the real problem of both Newton and Mach - the definition of determinate velocities if "the heavens began to move and the stars swarmed in confusion" (cf. p. 222).
But the exhortation to "sound our own conceptions" cannot be bettered at the start of our journey to the distant goal of quantum gravity - and perhaps even more remote consensus on Mach's Principle. The references are to my The Discovery ofDynamics, cited on p. 5.

Mach's Principle before Einstein
John D. Norton1
1. Introduction
The doctrine of the relativity of motion is attractive for its simplicity. According to it, the assertion that a body moves can mean nothing more than that it moves with respect to other bodies. Acceleration has long proved to be the stumbling block for the doctrine, for, in the case of acceleration, the simplest of observations seem to contradict the doctrine. When a test body rotates, for example, it is acted upon by centrifugal forces. The presence of these centrifugal forces seems to be completely independent of whether the test body rotates with respect to bodies immediately surrounding. Thus Newton observed in his famous bucket experiment that these centrifugal forces induced a concavity in the surface of a rotating body of water and did so independently of whether the water rotated with respect to the bucket containing the water. Therefore, using these inertial forces as a marker to indicate whether the body is accelerating, it seems possible to know that a body is accelerating without any concern for whether it accelerates with respect to the other bodies around it. This outcome contradicts the doctrine of the relativity of motion as applied to acceleration.
For about a century now, the most popular escape from this unwelcome refutation has been the following simple idea. Relativists point out that experiments such as Newton's reveal only that inertial forces are not noticeably related to motion with respect to nearby bodies. That, however, does not rule out the possibility that inertial forces are caused by acceleration with respect to more distant bodies. If this were the case, then inertial forces would not reveal an absolute acceleration but merely an acceleration relative to these distant masses. The core idea is that the inertial forces acting on an accelerating body arise from an interaction between that body and other bodies. The idea is not so much a proposal of a definite, new physical law; rather it is the prescription
Einstein Studies, vol. 6: Mach's Principle: From Newton's Bucket to Quantum Gravity, pp. 9-57 © 1995 Birkhiiuser Boston, Inc. Printed in the United States.

10 John D. Norton
that such a law should be found. The law recommended is only loosely circumscribed. It must be such that more distant masses play the decisive role in fixing the inertial forces on a given body, for example.
The proposal's most prominent sponsor was Albert Einstein. In the early years of his work on general relativity, he believed that his theory implemented the proposal, although he completely lost this belief in his later years. Nonetheless, the future of the proposal was guaranteed by the vigorous support of an Einstein who rapidly rose to celebrity status both inside and outside the scientific community. Einstein did not claim the proposal as his own invention. From the earliest moments, he attributed it to Ernst Mach and in 1918 gave a field theoretic formulation of the proposal its now standard name of 'Mach's Principle.' (Einstein 1918).2
The story of the role of the principle in Einstein's work, his enchantment with it, and his subsequent disenchantment, has been frequently told because of its enormous importance in the historical development of relativity theory and relativistic cosmology. My purpose in this paper is to explore another side of Mach's Principle, its earliest years prior to its adoption by Einstein, which so profoundly redirected and ruled its future. I will ask: What role did the principle play in Mach's own system? How was it received by Mach's contemporaries? In answering these questions, we shall find a story that is a little different from the one we might expect. With Mach now universally acclaimed as the patron of a growing literature on Mach's Principle and Machian theories, one expects to find in Mach's writings a penetrating voice of prescient clarity that easily transcends the generations that separate us from him. Instead we shall find:
• Mach's own writings that pertain to the principle were vague and ambiguous, bordering on the contradictory. The principle is never clearly stated, but at best obliquely suggested, and it remains unclear whether Mach endorsed the suggestion or condemned it as unscientific. • It was Mach's disciples and his contemporary and later readers who extracted an unequivocal proposal from his writings. Several even claimed the idea independently of Mach. • Mach's Principle proved to be an idea that fascinated Einstein so much that he sought to build his general theory of relativity around it. However he was in a minority in his fascination. • Prior to the advent of general relativity, the principle was a fringe idea, often opposed by those who would become Einstein's most ardent supporters. The philosophical community was largely uninterested in the proposal. As an empirical proposal, it had no foundations because of the failure of every experimental test actually tried. As a product of

Mach's Principle before Einstein 11
philosophical analysis, it smacked of a priori physics.
In Sec. 2 of this paper, I will pose the question of precisely what it is that Mach proposed concerning the origin of inertia. In Sec. 3, I will argue that cases can be mounted for each of two plausible answers. In Sec. 4, I will offer a reconciliation. In Sees. 5 and 6, I will assess the broader reaction to the proposal, considering both the favorable and unfavorable responses.
Although use of the term 'Mach's Principle' is anachronistic in much of the time period under consideration, I will use the term here for lack of anything better. Over the years it has come to label a proliferation of different ideas. Here I will understand it to refer to the proposal that the inertia of a body is caused entirely by an interaction with other bodies.
2. What Mach Actually Said
In his first published reference to the principle he attributed to Mach, Einstein (1912, p. 39) formulated it as " ...the entire inertia of a point mass is the effect of the presence of all other masses, deriving from a kind of interaction with the latter." A footnote appended to this sentence announced its origin:
This is exactly the point of view which E. Mach urged in his acute investigations on the subject. (E. Mach, The Development of the Principles of Dynamics. Second Chapter. Newton's Views of Time, Space and Motion.)
The attribution is deliberate and unequivocal. Einstein, who is notorious for the infrequency of citation in his writings, is carefully naming a section of the second chapter of Mach's celebrated The Science of Mechanics: A Critical and Historical Account of Its Development (Mach, 1960).
Readers who turn to the relevant section of The Science of Mechanics, a critique of Newton's notions of absolute time, space, and motion, will find many assertions reminiscent of the principle Einstein enunciated. But nowhere will they find it stated without distracting qualification or ambiguous hesitation. Indeed if the relevant section of Mach's text was intended to state clearly and advocate forcefully the principle Einstein enunciated, then it has failed. Rather, readers of the relevant section find Mach clearly devoting his expository energies to an attack on Newton's conceptions. The assault is based on two of Mach's

12 John D. Norton
favorite themes, which are enunciated clearly and repeatedly. These two themes, rather than some forerunner of Mach's Principle, are what readers find as the principal content of this section of The Science of Mechanics. The following remarks from this section are typical:
No one is competent to predicate things about absolute space and absolute motion; they are pure things of thought, pure mental constructs, that cannot be produced in experience. All our principles of mechanics are, as we have shown in detail, experimental knowledge concerning the relative positions and motions of bodies. Even in the provinces in which they are now recognized as valid, they could not, and were not, admitted without previously being subject to experimental tests. No one is warranted in extending these principles beyond the boundaries of experience. In fact, such an extension is meaningless, as no one possesses the requisite knowledge to make use of it. (Mach 1960, pp. 280) ...
When we say that a body K alters its direction and velocity solely through the influence of another body K', we have asserted a conception that it is impossible to come at unless other bodies A, B, C ... are present with reference to which the motion of the body K has been estimated. In reality, therefore, we are simply cognizant of a relation of the body K to A, B, C ... If now we suddenly neglect A, B, C ... and attempt to speak of the deportment of the body K in absolute space, we implicate ourselves in a twofold error. In the first place, we cannot know how K would act in the absence of A, B, C ... ; and in the second place, every means would be wanting of forming a judgment of the behavior of K and of putting to the test what we had predicated - which latter therefore would be bereft of all scientific significance. (Mach 1960, p. 281)
These passages recapitulate the two themes. First is the notion that physical science is or ought to aspire simply to provide economical descriptions of experience. Thus elsewhere Mach (1882) had pronounced "Physics is experience, arranged in economical order" (p. 197), and "The goal which it [physical science] has set itself is the simplest and most economical abstract expression of facts" (p. 207). The second theme is that Newton's absolute space, time, and motion are idle metaphysical excesses that are superfluous to this goal of economical description. Again elsewhere Mach (1872, 1911) had made the point very clearly. All our statements containing the terms 'space' and 'time' are really only statements of the relation of phenomena to phenomena, and the terms could be struck out without affecting the content of the statements. Mach (1872, 1911, pp. 60-61) even gave a prescription for how this striking out might be effected:3

Mach's Principle before Einstein 13
We can eliminate time from every law of nature by putting in its place a phenomenon dependent on the earth's angle of rotation. The same holds of space. We know positions in space by the affections of our retina, of our optical or other measuring apparatus. And our x, y, Z in the equations of physics are, indeed, nothing else than convenient names for these affections. Spatial determinations are, therefore, again determinations of phenomena by means of phenomena.
These two themes comprise Mach's attack on Newton's conception. In his The Science of Mechanics, Mach now goes to some pains to emphasize the error that one may fall into if one forgets Mach's lesson and takes Newton's absolute space and time too seriously. Talk of motion of a body K in space is really only an abbreviated description of the change of relations between K and other bodies A, B, C .... If we forget that these abbreviated descriptions do depend essentially on these other bodies and try to anticipate the motion of K 'in absolute space,' that is, if these other bodies were not present, then we will illegitimately extend our science beyond its proper domain. The domain of science is experience. We have no experience of the motion of a body in a space devoid of other bodies. Our extension would cease to be science.4
These two themes would be the ones that every modern reader would find pursued by Mach with vigor and clarity in his critique of Newton, were it not for the modern obsession of recovering Mach's Principle from Mach's critique. As a result of this obsession, the modern reading of Mach focuses on passages that are certainly highly suggestive, but, in the last analysis, vague and ambiguous. Typical of them is the most quoted of all passages of Mach's critique (1960, p. 284), which I have broken up into three sentences, labeled s], S2' and S3' for discussion:
[s\] Newton's experiment with the rotating vessel of water simply informs us, that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotation with respect to the mass of the earth and the other celestial bodies. [sJ No one is competent to say how the experiment would tum out if the sides of the vessel increased in thickness and mass till they were ultimately several leagues thick. [S3] The one experiment only lies before us, and our business is, to bring it into accord with the other facts known to us, and not with the arbitrary fictions of our imagination.
The ambiguity of this famous passage lies in the admissibility of two

14 John D. Norton
readings that contradict one another: First is the reading that returns what we now call Mach's Principle.
Sentence Sl reminds us that, in our search for causes for the centrifugal forces within the bucket, we have overlooked one possibility, the rotation of the water with respect to other bodies. We cannot rule out such a cause, as long as it is a cause that only acts when very large masses are involved. Thus S2 agrees with Newton that rotation with respect to the walls of the bucket induce no noticeable centrifugal forces. But according to the new physical mechanism conjectured, this would not be so if the walls were substantially increased in mass and size. Sentence S3 closes by observing that we would never have been tempted with an explanation in terms of absolute space - the "arbitrary fictions of our imagination" - had we recalled that the real business of science is economical description of experience. In this case, the experience is of Newton's experiment and of the other bodies that surround it.
The second reading recalls the two themes of Mach's critique. Since the goal of physical science is economical description of experience, SI reminds us of what we should really infer from Newton's experiment. We should conclude merely that there is a correlation between two experiences, the presence of centrifugal forces and rotation with respect to the stars. There is no place for a metaphysical absolute space in such descriptions. Sentence S2 is a tease to shake the dogmatic belief of a Newtonian. It points out that the Newtonian has inferred far more than what is actually warranted by Newton's experiment. The experiment does not give us enough information to rule out the possibility of an alternative physical theory in which the centrifugal forces are caused by rotation with respect to other bodies. Sentence S3' however, reaffirms resoundingly that such speculation lies well beyond the compass of science as economical description of experience. This speculation requires us to think of cases in which we do not and cannot have experience: for example, the walls of the bucket enlarged to a thickness of several leagues - "an arbitrary fiction[s] of the imagination" if ever there was one. Therefore Mach will not entertain such speculation.
Thus we have two readings of Mach's famous analysis of Newton's bucket experiment:
-The first escapes Newton's conclusion by proposing a new physical mechanism for the generation of inertial forces that will later be associated with the label 'Mach's Principle.' -The second effects the escape essentially by insisting that Newton be restricted to describing the experiment only in terms of what is experienced

Mach's Principle before Einstein 15
and pointedly condemns as unscientific the proposal of Mach's Principle.
Our task now is to decide which if either is the correct reading.5 Our resources are Mach's other writings as well as the interpretations of his contemporaries. Unfortunately we shall see that quite strong cases can be mounted for both readings. My accusation of the broader ambiguity of Mach's analysis rests on this unhappy fact. I now proceed to develop the case for each reading of Mach's analysis.
3. Mach Escapes Absolute Space by Urging ...
3.1.... Mere Redescription. It is clear that a major component of Mach's analysis involved the simple recommendation to redescribe motion in space as experiences that do not invoke the term 'space.' Thus he wrote (1960, pp. 285-86; Mach's emphasis): "When..,we say that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe. "
How are we to decide if in addition to this project of simple redescription Mach is also proposing a new physical mechanism? I shall assume that a proposal for a new physical mechanism must make claims about counterfactual or hypothetical systems, that is, claims about systems which are known not to exist or are not known to exist. Certainly such a proposal cannot approach the proposal of Mach's Principle unless it is prepared to license inferences about such cases as the rotation of a hypothetical bucket with walls several leagues thick or perhaps about the inertial forces induced between two bodies in an otherwise (counterfactually) empty universe. 6
Under this criterion there would seem to be no possibility that Mach could be proposing a new physical mechanism. For the claim he repeats most in the entire analysis is that we have no business in science speculating about such systems that are beyond our experience. Merely in the passages already quoted above, Mach has made the point three times. And it appears elsewhere in his analysis. For example (1960, p. 285):
The comportment of terrestrial bodies with respect to the earth is reducible to the comportment of the earth with respect to the remote heavenly bodies. If we were to assert that we knew more of moving objects than this their last-mentioned, experimentally given comportment with respect to celestial bodies, we should render ourselves culpable of a falsity.

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Or again Mach considers a proposal by C. Neumann, who imagines that a rotating celestial body will still be deformed into oblateness by centrifugal forces even if the other heavenly bodies were absent. Mach (1960, pp. 340-41) insists that this latter assumption is meaningless and objects that one is simply not allowed to assume away these other masses as unimportant when experimenting in thought.7 But if Mach refuses to allow any consideration of such hypothetical or counterfactual systems, then it is hard to see how he could be proposing a principle that even vaguely resembles the later Mach's Principle. On the contrary he must condemn any such principle as unscientific.
In places Mach does seem to urge a reformulation of the principles of mechanics. He allows for examples: "The principles of mechanics can, presumably, be so conceived, that even for relative rotations centrifugal forces arise."
Is Mach suggesting a reconception of mechanics in which the principles are materially changed and a new physical mechanism introduced? Or is the reconception merely a restatement of the same laws in such a way that the idle metaphysical conceptions of space and time are no longer mentioned? We may answer by looking at what Mach proceeds to do. On the pages following, what Mach actually does corresponds to the latter alternative of simple redescription. He seeks ways of restating the law of inertia so that it does not use the term 'space.' This project of redescription proves quite simple for one case (p. 286)
Bodies very remote from each other, moving with constant direction and velocity with respect to other distant fixed bodies, change their mutual distances proportionately to the time. We may also say, all very remote bodies - all mutual or other forces neglected - alter their mutual distances proportionately to those distances.
Mach then shows (pp. 286-287) how this type of formulation of the law of inertia can be couched in the language of mathematical formulae. The usual form of the principle requires that the acceleration of a body remote from other masses be constant. That is, if the body has absolute spatial coordinates (x, y, z) and the time is t, then
-dx-2 _-d-y 2-_ -dz0 2 _ . dt2 dt 2 dt 2 Mach's goal is to rewrite the law without the absolute spatial coordinates (x, y, z). To achieve this, he considers the distances r, r', r", ... to the

Mach's Principle before Einstein 17

other distant masses m, m', m", ... from the test mass. In place of the

spatial coordinates, Mach uses the mass weighted sum of these distances

(Emr/Em) so that the principle becomes

:t ~:] 22 [

= O.

(l)

The project is clearly just one of redescription of existing laws and not the proposal of a new mechanism.9 Indeed Mach soon makes it very clear that his new expression for the principle of inertia is not intended
to be applied to cases remote from experience (p. 289):

It is impossible to say whether the new expression would still represent the true condition of things if the stars were to perform rapid movements among one another. The general experience cannot be constructed from the particular case given us. We must, on the contrary, wait until such an experience presents itself.

Thus it is possible to present a collection of Mach quotations that drives towards the conclusion that Mach is not advancing what we now know as Mach's Principle, but condemning it. Is this an example of selective and biased quotation? Apparently not in the view of several of Mach's contemporary readers. C. D. Broad (1916) reviewed the supplement that contained a compendium of Mach's additions to the third
English language edition of The Science of Mechanics. He reported that
he now understood more clearly Mach's ambiguous discussion surrounding Newton's bucket experiment. What he understood in that discussion was not a proposal for a new physical mechanism but merely Mach's strictures about redescription:

There is also a far clearer statement than before of Mach's much quoted remark (in connection with Newton's bucket) that "the universe is not given to us twice, but only once." It is now clear that Mach's meaning is that the Ptolemaic and the Copernican view are simply different ways of describing precisely the same set of facts, and that therefore there is no real difference between the bucket standing still with the fixed stars rotating and the bucket rotating with the fixed stars standing still. This is clearly a necessary result of the relative view, and it is one that is often overlooked.

Broad's remarks were those of a sympathetic reviewer. Far more significant was the evaluation of Paul Caruso Carus was born in Germany in 1852, received a doctorate from the University of Tiibingen in 1876, and emigrated to America. There he began working for the Open Court

18 John D. Norton
Publishing Company, editing its journals Open Court and the Monist. In particular, Carus became the medium, welcomed heartily by Mach, through which Mach's writings were made available in English to the American audience. Carus found a natural empathy with Mach's views lO and was able to engage Mach in a huge correspondence spanning almost three decades, one of Mach's largest correspondences. 11 What induced Carus to publish on precisely the question that concerns us was a talk given by Philipp Frank in 1909, "Is There Absolute Motion?," published the following year as Frank (1909). Frank clearly attributed to Mach the proposal of a new physical mechanism to explain inertial forces of the type of Mach's Principle. Carns's discussion (1913, pp. 23-40) contains extensive quotation from Frank's lecture and provides the foundation for his denunciation of the suggestion that Mach was proposing a new physical mechanism:
Another point where we feel justified in doubting Dr. Frank's exposition is the statement that Mach hypothetically assumes a new law of nature as to the efficacy of masses, besides the law of gravitation. The passage in Mach's writings to which Dr. Frank refers 12 does not (in my opinion) suggest the idea of an additional law of nature according to which the distant fixed stars should exercise a mysterious influence on the Foucault pendulum. We will later on let Mach speak for himself. In our opinion it seems that it would be sufficient to ascribe the rotation of the pendulum to its inertia while the earth revolves round itself, and this takes place in the space in which the earth has its motion, viz., the space of the Milky Way system. The pendulum remains in the plane of oscillation in which it started while the earth turns around underneath.... There seems to me no need of inventing a new force besides gravitation. The law of inertia seems to explain the Foucault pendulum experiment satisfactorily.
Carus supports his reading of Mach with his own selection of Mach quotes, similar to those discussed here, pointing out that Mach's endeavors are devoted to elimination of the terms 'space' and 'time.'
Carus's published argument is based on widely available published writing of Mach. However, because Carns also enjoyed the privileged view of an extensive correspondence with Mach, it is tempting to conjecture that Carus is also silently drawing on this correspondence or even on discussions with Mach during one of Carns's visits to Mach in 1893 or 1907. Whether such correspondence is still extant will have to be decided by a search of the relevant archives. However, the prospect that any such correspondence existed in 1913 seems slight. If it did exist, Carus would almost certainly have published it to buttress his case. As

Mach's Principle before Einstein 19
editor of the Monist, Carus had clearly been eager to publish a letter by Mach (Carus 1906a) on an earlier article by Carus (1906) on Mach's philosophy. He published it with obvious delight, embedding the letter in the pomp of an introduction and afterword by Carus, and retaining its original German "lest it lose many of the fine points in an English translation." (Carus (1913), however, showed no restraint in presenting extensive passages of Frank (1909) in English translation!)
Finally, whatever their differences over whether Mach did propose a new hypothetical law, both agreed that such a proposal is an anomaly in Mach's broader systematic proclamations in which such hypothesis is abhorred. Thus Frank notes (1909, p. 17; trans. Carus 1913, p. 32): "But Mach in this case stands in the opposite camp as in most other cases where his repugnance to all hypothesis has made him a pioneer in the phenomenological direction." And Carus (1913, p. 32) himself, speaking of Frank's broader reading of Mach, writes provocatively "Strange that Mach, with his reluctance to introduce anything hypothetical except what is absolutely indispensable, should range on the side of the theorists .... "
3.2.... a New Physical Mechanism. Or did Mach intend to recommend more than mere description? Did he intend to propose a new physical mechanism for the origin of inertial forces? Once again a case can be made for this possibility and it too rests on quotations from Mach's writings and on his interactions with colleagues and others. However, if the case for this possibility were to rest only on the first part, Mach's writings for publication, then the case would be considerably weaker than the corresponding case for his advocacy of mere redescription. For none of the writings unambiguously endorses a proposal for a new mechanism. Worse, it is not clear which of his writings even talks about such a proposal.
In Mach's critique of Newton's conceptions in The Science of Mechanics are several much cited remarks that could be taken as suggesting a new physical mechanism. However, precisely because they are rhetorical flourishes, they admit of many interpretations and do not provide a firm foundation for the case. He exclaims (p. 279): "Try to fix Newton's bucket and rotate the heaven of fixed stars and then prove the absence of centrifugal forces."
But could this not simply mean that Mach takes the case of bucket rotating/stars resting to be exactly the same as the case of bucket resting/stars rotating? Then to try to prove the absence of centrifugal forces, as Mach challenges, is obviously futile since the two cases are really just the one case described differently. Indeed the sentences

20 John D. Norton
immediately preceding the exclamation are devoted to arguing that the two cases are really one. Again, there is Mach's famous observation on Newton's bucket experiment (p. 284): "No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass till they were ultimately several leagues thick. "
As we saw above, a consistent continuation in Mach's voice would be "And I [Mach] certainly would not dare to speculate on such an unscientific thing!" - this being a plausible reading of what Mach actually says in the next sentence: "The one experiment only lies before us, and our business is, to bring it into accord with the other facts known to us, and not with the arbitrary fictions of our imagination."
Again, Mach concludes the paragraph preceding with an apparently unequivocal recommendation for a new physical mechanism for inertial forces: "The principles of mechanics can, presumably, be so conceived, that even for relative rotations centrifugal forces arise." 8
However, the appearance is deceptive, for, as we saw above, this reconception might just be referring to a simple redescription such as leads up to Mach's equation (1) above.
More promising are his later remarks that Barbour (1989, p. 692) identifies as "Mach's clearest statement of the ideal of a seamless dynamics" such as would arise were he proposing a new mechanism for inertia. Mach writes (p. 296, Mach's emphasis)
The natural investigator must feel the need of further insight - of knowledge of the immediate connections, say, of the masses of the universe. There will hover before him as an ideal an insight into the principles of the whole matter, from which accelerated and inertial motions result in the same way. The progress from Kepler's discovery to Newton's law of gravitation, and the impetus given by this to the fmding of a physical understanding of the attraction in the manner in which electrical actions at a distance have been treated, may here serve as a model. We must even give rein to the thought that the masses which we see, and by which we by chance orientate ourselves, are perhaps not those which are really decisive. On this account we must not underestimate even experimental ideas like those of Friedliinder [(1896)] and Foppl [(1904, 1904a)], even if we do not yet see any immediate result from them.
Once again I do not see that we can rule out the possibility that these remarks refer to Mach's project of redescription. The understanding of (1) is of immediate connection of the masses since the superfluous mediation of 'space' has been eliminated. And was not the progress from Kepler to Newton (in Machian terms) the discovery of a system of

Mach's Principle before Einstein 21
laws that yielded a far more economical summary of not just Kepler's astronomical discoveries but much else besides? It is also very possible, however, that Mach is referring to a new physical mechanism for the origin of inertia. For, as we shall see below, Friedlaender (1896) and Fappl (1904a) both clearly consider such a novel mechanism and conduct experiments to detect it.
If this passage does refer to such a novel mechanism, it still provides no evidence that the proposal of such a mechanism originated with Mach or that Mach endorsed it. The passage in question was added to the seventh German edition of 1912,13 presumably in response to Friedlaender (1896) and Fappl (1904, 1904a). Since these works already propose a new physical mechanism for inertia, one can hardly say that the proposal originated with Mach's remarks of 1912. Even Mach's vague suggestion of the use of the theory of electricity as a model had been anticipated and in more precise form. Friedlaender (1896, p. 17) had raised the possilibility of applying Weber's law of electrodynamics to gravitation in this context, as does Hafler (1900, p. 126), as we shall see below. Worse, Mach's language suggests that whatever he is introducing is novel and goes beyond what was already said in earlier editions. That is, after the Machian ideal of purification from meaninglessness has been achieved, there is a new goal, some "further insight," a speculative "ideal." In one sentence, we are invited "even [to] give rein to the thought [sagar dem Gedanken Raum geben] that the masses which we see, and by which we by chance orientate ourselves, are perhaps not those which are really decisive." This invitation would hardly be necessary if we had already made space for that thought in the earlier text of the earlier editions. The thought for which we are to make space might even be a distinctly non-Machian one. If the thought is that the decisive bodies are ones we cannot see, then it contradicts Mach's repeated and forceful pronouncements on the primacy of the observable. If the theoretical and experimental work of the Friedlaenders and Fappl is a part of such non-Machian speculation, then Mach can hardly be giving them unreserved endorsement. Indeed the passage quoted above closes with what seems to be a gentle rebuke: "Although the investigator gropes with joy after what he can immediately reach, a glance from time to time into the depths of what is uninvestigated cannot hurt him. "
One could read this as a very kind way for Mach to point out to the Friedlaenders and Fappl that he finds their work to have strayed well beyond science, the domain of economical descriptions of experience, into the murky depths of unscientific speculation. 14
Remarks published by Mach in 1872 [quoted here from Mach

22 John D. Norton
(1911)] support most strongly his advocacy of a new physical mechanism for the origin of inertia - although they are still subject to the same ambiguities. In the appendix, Mach stresses that in referring motions in the law of inertia to space we should never lose sight of the fact that this is really only an abbreviated reference to other bodies. He then begins to discuss how the motion of these other reference bodies might affect the law of inertia, arriving at the following puzzle (p. 78):
But what would become of the law of inertia if the whole of the heavens began to move and the stars swarmed in confusion? How would we apply it then? How would it have to be expressed then?
It seems clear enough that Mach's puzzle refers to the problem of stating - redescribing - the law of inertia in a form similar to (1), in the awkward case in which the heavenly bodies adopted chaotic motion. How can Mach be sure that an expression in terms of a simple mass weighted sum of distances (Emr/Em) will be adequate? This seems to be the same
problem that Mach discusses in The Science o/Mechanics (1960, p. 289)
(see above). Mach then proceeded to another example, a free body acted upon by an instantaneous couple so that it rotates. He continues (p. 79)
Here the body makes very strange motions with respect to the celestial bodies. Now do we think that these bodies, without which one cannot describe the motion imagined, are without influence on this motion? Does not that to which one must appeal explicitly or implicitly when one wishes to describe a phenomenon belong to the most essential conditions, to the causal nexus of the phenomenon? The distant heavenly bodies have, in our example, no influence on the acceleration, but they have on the velocity.
The ambiguity of these remarks resides in the unexplained terms 'influence' and 'causal nexus.' What do they mean? What sort of influence is suggested?15
Mach then makes the remarks that most strongly suggest that he is seeking a new physical mechanism. He seems to be conjecturing the form of the law that governs it:
Now, what share has every mass in the determination of direction and velocity in the law of inertia? No definite answer can be given to this by our experiences. We only know that the share of the nearest masses vanishes in comparison with that of the farthest. We could, then, be able completely to make out the facts known to us if, for example, we were to make the simple supposition that all bodies act in the way of determination

Mach's Principle before Einstein 23
proportionately to their masses and independently of the distance, or proportionately to the distance, and so on.
This talk of 'share' and 'masses' acting in proportion to their mass and distance might well be a conjecture of some new physical mechanism. However, it can also be read as a part of Mach's project of redescription. As we have seen, Mach gives such a redescription of the law of inertia in terms of the mass weighted sum of distances (Emr/Em) or its second time derivative d 2/dt 2(Emr/Em). The 'share' of each mass m, m', mil ... in the reformulated law would simply be the magnitude of the term each mass contributes to these sums. The functional dependence of these contributions are then exactly of the type Mach mentions. In the first sum, for example, each mass contributes a term proportional to its mass and to its distance from the test body. And the nearest masses certainly contribute vanishingly small terms in comparison with the remaining masses.
In my reading, one thing makes it clear that Mach intends in this passage to propose only a redescription of the law of inertia and not a new physical mechanism. That is the sentence immediately following the passage quoted above, which closes the paragraph and Mach's discussion: "Another expression would be: In so far as bodies are so distant from one another that they contribute no noticeable acceleration to one another, all distances vary proportionately to one another."
This expression is clearly offered as a variant or, possibly, a special form of the general laws discussed. Yet it is just a redescription of the inertial motion of a collection of noninteracting bodies that avoids mention of space. 16 There is no hint of some new physical mechanism that would enforce the proportional variation of distances.
This discussion is the best evidence in Mach's published writing for his advocacy of a new physical mechanism for the origin of inertial forces. But it does not make a good case. Even in the collective judgments of Mach's sympathetic contemporaries, its intent is unclear. As we have seen, Frank (1909) found it to advocate a new mechanism; Carns (1913) did not. My judgment is also that it is ambiguous, but I think its most natural reading is as a proposal for simple redescription. In my view, this same judgment must also hold of Mach's published corpus on Newton's bucket experiment and the law of inertia. The only unequivocal proposal Mach makes is for a simple redescription of the experiment and the law in a formulation that does not use the term 'space.' It remains unclear whether Mach intended to propose and endorse a new physical mechanism for the origin of inertial forces.

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However it is dubious that this verdict represents Mach's real intentions. What speaks loudly against this verdict is that the majority of Mach's contemporaries and confidants understood him to be proposing a new physical mechanism. On this point Cams is in a clear minority. Indeed the view that Mach proposed a new mechanism is a commonplace of the literature from around 1900 and on to the year of Mach's death in 1916. It is mentioned17 by Friedlaender (1896, p. 9), Hofler (1900, pp. 122-26), Fappl (1904a, p. 383), Frank (1909), Cassirer (1910, p. 177),18 Petzoldt (1912, p. 1057), Schlick (1915, p. 166), and, of course, Einstein, whose repeated attributions, commencing with Einstein (1912), brought the viewpoint to the broadest audience. If this view were an outright misreading of Mach, then Mach had ample opportunity to correct it. But this correction never came. 19 He even mentioned the work of the Friedlaenders (1896) and Fappl (1904a) in later editions of his The Science of Mechanics (1960, p. 296). Surely that is the point at which Mach would issue a correction if both works were misrepresenting his position. Or are the somewhat indirect remarks quoted above (" ... a glance ... into the depths of what is uninvestigated... ") intended as a gentle rebuke?
It would seem that any corrections that Mach may have issued would have been so gentle as to escape later reporting, or, at least, any reporting of which I am aware. In particular, in a letter of June 25, 1913, Einstein reported to Mach that Einstein's new theory had yielded a new physical mechanism for the origin of inertia and Einstein attributed that idea directly to Mach20: " . .. inertia has its origins in a kind of interaction of bodies, quite in the sense of your reflections on Newton's bucket experiment."
Yet Einstein's later writings contain no trace of hesitation in continuing this attribution to Mach. Similarly, Frank (1957, p. 153) continues the attribution. Again, in a letter of January 11, 1910 (Blackmore and Hentschel 1985, pp. 66-67) to Mach, Fappl mentions his "treatment of the question of relative motion" - presumably Foppl (1904a). He commented with relief that Mach "at least had no fundamental misgiving [Bedenken] to raise against [it]." We might well wonder what Mach did say to evoke such a response!
Fortunately, within Mach's surviving correspondence there is a record of how Mach responded to such attributions. In a letter of September 3, 1904, which contained an enthusiastic response to the fifth German edition of Mach's The Science of Mechanics, Petzoldt put to Mach a series of questions and proposals about Mach's ideas on the law of inertia. In particular, he attributed to Mach the idea that the thickened

Mach's Principle before Einstein 25
walls of Newton's bucket could induce centrifugal forces and expressed his own doubts on this notion (Blackmore and Hentschel 1985, p. 36, Petzoldt's emphasis):
I still cannot reconcile myself to your observation (p. 247) on the possible variation of the experiment through the thickening of the bucket walls. With this you still make the appearance of centrifugal forces dependent on the magnitude of the surrounding bodies instead of the (relative) rotations of the bodies. The centrifugal forces are still aroused only through relative rotation against the locations of the masses of the earth and the other heavenly bodies. I am inclined, however, very much to the belief, which you also admit, that the heavenly bodies here play only a chance role like the axial rotation of the earth for the determination of temporal processes, and hope for future experiences on the deeper relations of things, without shutting my eyes to your doubt over whether such experiences will ever be accessible to us as people.
Mach's response in a letter of September 18, 1904, is lengthy and unfortunately never actually mentions the walls of Newton's bucket. He does make clear that he dislikes Petzoldt's idea that the locations of the masses may be the decisive thing. He objects (Blackmore and Hentschel 1985, p. 39, Mach's emphasis): "A bare, efficacious location has been observed by no one."
However, he does clearly leave the impression that Mach's own view of inertia is that it is a matter to be decided by experiment. After explaining that his original thoughts on inertia were formulated before the ascendancy of Faraday's conception of local action and of a medium or material intervening between bodies ("aether, space or whatever it is called"), he continued: (p. 38, Mach's emphasis)
As long as one attends to bodies alone, one conceives naturally of gravitational processes and inertial motions as determined by them alone or, correspondingly, through other masses. If one now also is not to expect a positive result from the Friedliinder fly-wheel experiment,[211since the mass and velocity of the wheel is too small, then a greatly refmed Foucault experiment could still show that a pendulum or gyroscope orients itself not only according to the fixed heavenly stars, but also in part is influenced by the earth, which is, after all, a powerful flywheel. However should such an experiment definitely come out negative, that would also be a great gain in insight. .,. If I conceive of gravitation as carried through a medium, then I can conceive of the state of this medium still as only determined by the masses of bodies, for the reaction accelerations depend on the masses of the bodies. But if one body that is very distant and unaccelerated with respect

26 John D. Norton
to the others is in motion, then its motion can only be described with reference to the latter. The idea that this motion is detennined by the latter bodies cannot be dismissed without further ado. In any case, the orientation of the motion through the distant bodies can be a merely apparent one. Perhaps the motion is a concern only of the moving body and the medium alone. Perhaps each body conducts itself in space like Dirichlet's bodies in a frictionless fluid. 22
The letter closes with a very brief sketch of an experiment designed to detect the Friedlaenders' effect arising from the earth's motion.
With a response such as this, it is no surprise that Petzoldt (1912, p. 1057) should proceed to attribute to Mach the conjecture that the relative rotation of masses induces centrifugal forces, the same effect sought experimentally by the Friedlaenders and Foppl. However the only definite point that Mach has made is to rule out Petzoldt's proposal with his disparagement of a "bare, efficacious location." His answer strongly suggests that he expects or would welcome a positive outcome of a Friedlaender style experiment. But he has still not positively asserted that he believes that a thickened bucket in Newton's experiment would induce inertial forces - his original passage in The Science of Mechanics insists that no one is competent to assert this! And for all our pursuit of Mach's writings, we still do not have a clear statement from Mach that he conjectures that the origin of inertia lies solely in an interaction of bodies through some new physical mechanism.
4. A Reconciliation?
This is the puzzle that Mach's writings on inertia pose for us. We must reconcile two facts. Mach's publications contain only a clear advocacy of the view that one ought to redescribe Newton's bucket experiment and the law of inertia in such a way that the term 'space' does not arise. If there is a suggestion of a new physical mechanism to explain the origin of inertial forces, then its discussion is vague, and the proposal of a new mechanism might even be condemned as unscientific. On the other hand, Mach must have been aware that the proposal of exactly such a new causal mechanism was routinely attributed to him, but, in spite of ample opportunity, there is no evidence that he ever moved to correct this misattribution - if it did in fact need correcting. In brute form, we are left wondering whether Mach did intend to propose a new mechanism but was simply incompetent in expressing his intention. Or, if he did not intend a new mechanism, we must ask why Mach allowed such

Mach's Principle before Einstein 27
widespread misinterpretation of his work. I can offer two reconciliations, although neither is attractive. The
first is that Mach was unwilling to see the need for a new physical mechanism in his system. That is, he was an adherent of the relativist doctrine with respect to motion, which leads to the need for a new mechanism to account for inertial forces. Mach, however, was simply unwilling to embrace this consequence, so willingly embraced by other relativists, and simply tried to avoid committing himself. There is some evidence for this view. It stems from Hugo Dingler, who had been sanctified by an extremely favorable mention from Mach in the
penultimate paragraph of his preface to the last edition of The Science of Mechanics. He reported in Dingler (1921, p. 157) that Mach's23
... only salvation [from the problem of centrifugal forces] was to bring the centrifugal appearances into relation with the fixed stars, and, in fact, Mach also accepts this in the last (7th) edition of his Mechanics (I cannot really decide how much this was already the case in earlier editions); he was forced to it, even though this also obviously contradicted his sensibilities.
To the last sentence, Dingler appended the crucial footnote
1 thank Herr Dr. Ludwig Mach for the fr[ien]dl[y] communication that this consequence was always "especially tormenting" [besonders qutilend] for his father, that he knew for a long time of the monstrous conclusions deducible from it, yet did not draw them, but rejected them.
Thus Mach's behavior could be explained by a horror and unwillingness to accept what his system had produced. In this account, his aversion would be so profound that he would be unable to address the horrific consequence squarely in both his writings and in his private correspondence and discussions.
There are two difficulties with this view. First, contrary to Dingler's suggestion, Mach's system offered a perfectly good reason for rejecting a new physical mechanism: It transcended the economical description of experience that was the proper domain of science. With perfect consistency and in clear conscience, Mach could denounce this new mechanism as unscientific, if he disliked it so much, and there would be no need to be tormented. Second, by 1921, Dingler had become an outspoken critic of relativity theory and, as a disciple of Mach, may well have been overeager to seek reasons to remove Mach's support from relativity theory.24

28 John D. Norton
A second more plausible reconciliation is the one I favor. It depends on Mach's somewhat idiosyncratic notion of the true nature of causation. In Sec. 3, when seeking to judge whether Mach's proposals advanced beyond mere redescription to a new physical mechanism, I used the criterion that such a mechanism must make claims about counterfactual or hypothetical systems, for that was clearly required if Mach's proposals were to approach what later became Mach's Principle. However Mach's view of physical science as merely economical description of experience rules out exactly such considerations. A causal connection for Mach is merely a functional dependence extracted from experience. He makes this very clear in (Mach 1911, p. 61; Mach's emphasis) when he writes
The present tendency of physics is to represent every phenomenon as a function of other phenomena and of certain spatial and temporal positions. If, now, we imagine the spatial and temporal positions replaced in the above manner [by phenomena], in the equations in question, we obtain simply every phenomenon as function of other phenomena.
Thus the law of causality is sufficiently characterized by saying that it is the presupposition of the mutual dependence ofphenomena. Certain idle questions, for example, whether the cause precedes or is simultaneous with the effect, then vanish by themselves.
The law of causality is identical with the supposition that between the natural phenomena a, {3, 'Y, 0, ... , w certain equations subsist.
One cannot overemphasize how different this view is from the common view of causation. Newton's inverse square law of gravity is commonly understood to legislate that the sun causes an acceleration of the earth that varies directly with the inverse square of the distance that separates them. And this is assumed to hold whether the two masses are the sun and earth of our actual universe or a sun and earth in some hypothetical universe devoid of all other matter. As Mach's frequent protestations above show, he does not allow, in general, this assuming away of the other masses of the universe. Now it is not clear whether Mach would want this prohibition to apply in this case. If it does apply, however, then the relevant causal law simply becomes the assertion of the functional relationship between the sun-earth distance and the
acceleration of the earth towards the sun that happens to obtain in our universe alone.
If we now apply precisely this same thinking to Newton's bucket experiment, we arrive almost verbatim at many of Mach's pronouncements on the experiment and the law of inertia. And we do so

Mach's Principle before Einstein 29
without Mach ever proposing the type of new physical mechanism soon to be suggested under the banner of Mach's Principle. If we seek the fundamental causal relation revealed by Newton's bucket experiment, we must recover the functional relation of the actual phenomena - and that is merely
... that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotation with respect to the mass of the earth and the other celestial bodies.
It now follows immediately that, using Mach's definition, the centrifugal forces in the bucket and the mass of the earth and other celestial bodies stand in a causal relation. Speaking loosely, in a way that risks 'idle questions,' we might identify these masses as the cause of the forces. Also, to identify the role that each of the masses play in the functional relation is just to identify their causal role. Mach might well describe this as their 'influence,' a term with obvious causal connotations. Or he might well ask: "What share has every mass in the determination of direction and velocity in the law of inertia?" And if the relevant functional relation is linear in mass, he might well describe the body as 'acting' in proportion to its mass. Further, a result such as (1) appears to non-Machian readers merely to describe a functional relation and nothing more. But to Mach, the very fact that it describes a functional relation between phenomena of our world makes it the statement of a causal relation. Finally, Mach can offer a functional relation such as (1) as the fundamental causal relation pertaining to inertia, that is, the law of inertia, without needing to suggest that this same relation would obtain were the motions of the masses of the universe to be very different. For the functional relation need only obtain for our actual experiences to qualify as a causal relation. 25
There is an unappealing aspect of this resolution. The resolution rests on the assumption that what Mach meant by causation is very different from what the same term meant for the many proponents of what came to be known as Mach's Principle. Thus, when Einstein wrote to Mach
that "inenia has its origins in a kind of interaction of bodies, quite in the
sense of your reflections on Newton's bucket experiment," Einstein's notion of causal interaction extended well beyond the simple functional relations of phenomena. It included relations on hypothetical and counterfactual systems of precisely the type denounced by Mach. What remains unexplained is how Mach could repeatedly allow such

30 John D. Norton
misattributions to pass without objection or correction by him.
5. Early Sponsors of Mach's Principle
Whatever may have been Mach's attitude to the principle that came to bear his name, his writing proved to be a continuing inspiration to the advocates of the principle and it prospered under their guidance. Prior to Einstein, the sponsors of the principle formed a scattered group, largely on the fringe of the physics community. Typically, the members of this group thought that the existence of the new physical mechanism was an issue to be settled by experiment.26 They devoted their energies to devising and executing such experiments - and to the writing of labored but generally inconsequential treatises.
Mach ensured remembrance of two such experiments, those of the
Friedlaenders and Foppl, by citing them in his The Science ofMechanics
(1960, p. 296). The work of the Friedlaenders is described in the short, two-part monograph, Friedlaender (1896). The first part, written by Immanuel Friedlaender, describes how Immanuel's pursuit of the relativity of motion and the problem of centrifugal forces lead him to what we would now call Mach's Principle (p. 14):
Without knowing that this had already been done by Mach, I have doubted the completeness of these foundations of mechanics for many years now. In particular I have come to the conviction that the appearance of centrifugal forces ought to be explicable also through regular mechanical knowledge [Erkenntnis] from the relative motions alone of the systems concerned, without resorting to absolute motion.
In just a few words, Immanuel is able to state clearly the call for a new physical mechanism which would supplement the existing laws of mechanics and explain centrifugal forces in terms of relative motions alone. Yet, ironically, he gives priority for this idea to Mach, even though I have been unable to find a similarly clear formulation of the idea in Mach's writings. Immanuel then proceeded to describe his efforts to detect this mechanism experimentally. He expected that the spinning of a fly wheel would produce forces directed away from its axis through this mechanism, just as the rotation of the heavens about the earth produces centrifugal forces. He proposed to detect these forces with a torsion balance, "the most sensitive of all physical instruments" (p. 15). However, when he sought to carry out these experiments in a rolling mill in Peine in November 1894, this necessary but extreme sensitivity of the

Mach's Principle before Einstein 31
balance proved to be his undoing. His results were inconclusive since he was unable to control disturbing influences. He lamented (p. 16): "A sensitive torsion balance is, however, a tricky instrument and a rolling mill certainly not the most comfortable or most favorable location for precision measurement. "
Upon the failure of these experiments, he turned to his brother, Dr. Benedict Friedlaender, who only then informed him of Mach's work. Jointly they developed their ideas, upon which Benedict reported in the second part of the monograph. Immanuel concluded by stating his expectation that the correct formulation of the law of inertia ought to lead to "a unified law" which combined both gravitation and inertia as an action of masses. The idea that this new mechanism be integrated with the law of gravitation is not usually attributed to Mach, but is considered Einstein's innovation. Of course, in the Friedlaenders' hands it was merely speculation, but at least we see that the unification Einstein effected was not so completely unanticipated.
Foppl (1904) described his attempt to perform an improved version of the Foucault pendulum experiment. The purpose of the experiment was to reveal the precise disposition of an inertial system, correcting for the acceleration of the laboratory on the surface of the earth. He explained that "Foucault's pendulum experiment is afflicted with such sources of error that its accuracy leaves much to be desired even with careful execution" (p. 5). Foppl described how his experiment employed a carefully suspended gyroscope. Its precessional motion would reveal the disposition of an inertial frame of reference. Foppl hoped his experiment might decide whether (p. 5): " ... the terrestrial phenomena of motion is itself influenced by the rotation of the earth in such as way that, for [these motions], the rotation of the earth does not coincide with that [rotation] with respect to the fixed star heaven."
In other words, Foppl is interested in comparing two reference systems. The first is the reference system of the fixed stars. The second is the inertial reference system in the neighborhood of the earth's surface revealed by the motions of bodies, such as the pendulum of Foucault's experiment. These systems are routinely assumed to coincide. Foppl conjectures that they may not because of "a possible, special influence of the rotation of the earth" (p. 5). In the event, Foppl reported that he could detect no deviation from coincidence within the accuracy of his experiment.
The report of this experiment was communicated to the Munich Academy on February 6. It was not until a further communication of November 5 (Foppl 1904a) that we find what led Foppl to conjecture

32 John D. Norton
such a special influence. His inspiration was the work of Mach on the relativity of motion. According to Mach, Foppl reported, an inertial system "obtains its orientation from the masses of the system of the universe in some kind of law governed manner." (p. 383). Foppllater (p. 386) considered the bodies of the universe divided into a large and a small group. An inertial system is determined by the combined group. Therefore, if the larger group is used to define a rest system of reference, the inertial frame will execute some motion in it, such as a rotation. This rotation would appear as Coriolis forces in the rest system of the larger group; they would not be regarded as merely artifacts of calculation but as "physically existing forces exerted by the smaller group on each test point." Foppl then explained that these were the considerations that led to the experiment described in his earlier communication. If one takes the fixed stars as the larger group of bodies and the earth as the smaller, then these forces would be the "special influence of the earth" sought.
If Mach ensured remembrance of the work of the Friedlaenders and Foppl, then Einstein similarly ensured remembrance of the work of Hofmann. In (Einstein 1913, §9), he discussed what he called the "hypothesis of the relativity of inertia," the hypothesis that inertial resistance is merely resistance to acceleration with respect to other bodies. As to the origin of the idea, Einstein wrote
It is well known that E. Mach, in his history of mechanics, first advanced this point of view with all sharpness and clarity, so that here I can simply refer to his exposition. I refer also to the ingenious pamphlet of the Viennese mathematician W. Hofmann, in which the same point of view is advanced independently.
The work referred to is (Hofmann 1904).27 The forty three page pamphlet is a wordy and labored defense of the relativity of motion. It seeks to escape the inference from centrifugal forces to absolute acceleration by urging that these forces arise from an interaction with the remaining masses of the universe. Unlike Foppl and the Friedlaenders, Hofmann (pp. 28-30) conjectured a new mechanical law that would lead to this interaction and perhaps this is what attracted the description of 'ingenious' from Einstein. Hofmann considered the standard result of traditional mechanics that the kinetic energy (die lebendige Kraft) of a body of mass m moving at velocity v is mv 2/2. He found this result unsatisfactory since, in the case of two masses m and M in relative
motion, the kinetic energy of m with respect to M is not the same as the

Mach's Principle before Einstein 33

kinetic energy of M with respect to m. Therefore Hofmann proposed a new, symmetric law for the kinetic energy L of two bodies of mass m and M in relative motion with speed v and at a distance r

L =kMmf(r)v2,

(2)

where k is a constant and f some function to be determined. For

consistency with known results in mechanics, Hofmann indicated that the

kinetic energy of a mass of actual experiment derives contributions from

all the masses of the universe according to (2), so that (2), upon

integration over all these masses, must yield the familiar mv2/2.

Hofmann's law contains a mechanism in which inertial resistance is

resistance to acceleration with respect to other bodies; for, in the case of

two masses m and M, an attempt to change the relative velocity v will

change the kinetic energy and thus require a force. In the case of a body

in relative rotation with respect to the bodies of the rest of the universe,

one would expect this same mechanism to yield centrifugal forces.

Hofmann did not develop the technical details and formal con-

sequences of his supposition (2) in any systematic or extensive manner.

This task was carried out by Reissner (1914, 1915). Reissner gave the

usual attribution to Mach. Curiously, however, he made no mention of

Hofmann, even though Hofmann's law (2) is the fundamental supposition

upon which Reissner's theory is built. Perhaps we should allow for the

possibility that Reissner independently arrived at the same supposition.

In any case, the years 1914 and 1915 were not the time to construct a

theory embodying the relativity of inertia, for such a theory would have

no chance of competing with Einstein's general theory of relativity,

whose brilliance came to outshine all competitors. By 1916, Reissner

(1916) had turned his attentions to work on the latter theory, developing

his celebrated solution of Einstein's field equations.

There is a small puzzle associated with the pamphlet. Einstein

attributes its positing of the relativity of inertia as independent of Mach.

Certainly the pamphlet itself makes no claim either way; no works by

other authors are mentioned, and Mach is never named. However there

is sufficient similarity between parts of Mach's and Hofmann's analysis

to raise suspicion of an unacknowledged debt by Hofmann to Mach.

Hofmann, for example, couches part of his discussion in terms of

Newton's bucket experiment. He even proposes consideration of what

would happen if the water-filled bucket were surrounded by a very heavy

ring which is set into as rapid a motion as possible (p. 32) - close indeed

to Mach's suggestion of the thickening of the walls of the bucket.

Perhaps Einstein's attribution of independence from Mach derives from

34 John D. Norton
the failure of the text of Hofmann (1904) to cite Mach. However, Einstein may also have the claim directly from a meeting with Hofmann, which might have happened during Einstein's visit to Vienna for the 85th Naturforscherversammlung in September 1913 - (Einstein 1913) is the text of a lecture he delivered at that meeting. Again, Einstein describes Hofmann as a Viennese mathematician. That information could not be gleaned from the pamphlet alone, which simply described Hofmann as a professor and gave no affiliation.
The work of the Friedlaenders, Foppl, and Hofmann enables us to start to assemble an image of the group working around 1900 on what is to become Mach's Principle. First, the group members are on the fringes of the physics community. Only Foppl has any status in this community.28 And they are an isolated and fragmented group. None of these authors cites any of the others. Indeed, the work of the Friedlaenders and of Hofmann were published in such obscure vehicles that we are now probably only aware of them because they happened to be cited by Mach and Einstein. In any case they are difficult works to procure. Thus we might well conjecture that the works discussed so far are but a random sample of other similar works which may be unknown because of their obscure vehicles of publication or a failure to publish at all.
This conjecture is confirmed by Hofler's (1900, pp. 122-26) report. He described experiments of which he was aware and which were designed to test the relativity of motion. Hofler knew of the Friedlaenders' experiment and described Mach's remark about the thickening of the walls of Newton's bucket as a thought experiment. In addition, he described an experiment due to Johannesson (1896). The experiment, only incompletely described by Hofler, involved rotation in connection with an oil droplet or sphere. Johannesson's results did not correspond at all with Johannesson's expectations. The design of the experiment seems flawed and Hofler devoted a page-long footnote to conjectures on where the deficiencies of the experiment may have been. He made clear that no positive result came from the experiment. Hofler also described another proposal for an experiment by Herr Dr. Karl Neisser.29 The proposal involved examining the behavior of a gyroscope in air and in atmospheres of reduced pressure. Neisser, a relativist about motion, somehow managed to infer from this doctrine that the behavior of a gyroscope must at least in part be dependent on the relative rotation of the wheel against the air. Therefore he expected that a spinning gyroscope would lose its stability if enclosed in a chamber from which the air is pumped and that it would fall down like a gyroscope that is not

Mach's Principle before Einstein 35
spinning. Hofler added a remark to the proofs of his volume that Neisser had informed him that he had been able to perform the experiment, but the expected effect had not occurred. Hofler's report confirms that there was more interest around 1900 in what became Mach's Principle. But it would also seem that these further investigations were not as competently executed.
6. Early Critics of Mach's Principle
When Einstein incorporated Mach's Principle into the foundations of his general theory of relativity, he drew it in from these fringes into a new mainstream. In fact, Einstein's work defined what the new mainstream was to be in the physics of space, time, and gravitation and also, as it happened, a new mainstream in philosophy of space and time. Thus the principle enjoyed an enviable prominence. Einstein incorporated the principle or its precursors into most of his accounts of general relativity in the 1910s and 1920s. And, in his hands, the principle acted as midwife at the birth of modern relativistic cosmology. Einstein's efforts to ensure the place of the principle in his theory in 1917 led to his modification of his gravitational field equations and the introduction of the Einstein universe - not to mention the Einstein-de Sitter controversy.3D The principle also rapidly entered into a popular and semi-popular literature on relativity, written for a wider, popular audience eager to come to grips with Einstein's great revelations. [See, for example, Born (1924, Ch. VII).] Finally the principle came to enjoy the sponsorship of leading philosophers and became a paradigm of the fruitfulness of the interplay of physics and philosophy. Prominent among these sponsors was Hans Reichenbach, leading figure in the logical empiricist movement, whose works in philosophy of space and time would dominate the discipline for several generations. [See (Reichenbach 1928, Sec. 34).]
6.1. Among the Physicists. The rapidity of the principle's rise and its lasting prominence tend to obscure the fact that it ascended only over a considerable if somewhat quiet opposition that persisted throughout this period as a tenacious skepticism towards the principle. That opposition can be located clearly in two areas: among physicists both before and after the advent of general relativity, and among philosophers, both of the neo-Kantian old guard and ofthe new generation that spawned logical positivism.
Prior to Einstein's championing of the principle, it is difficult to find

36 John D. Norton
broad measures of the overall feeling of the physics community concerning it. Little was said in opposition to it. But it was not a strong position which could expect or demand response from its critics, since, as we have seen, support for the principle lay scattered and disorganized in the fringes of the community. Of course, this fact itself indicates a broader lack of support. However, we have two fairly clear expressions of opposition. Toward the end of the first decade of this century, Ernst Mach and Max Planck engaged in a fairly bitter, polemical exchange (planck 1909, 1910; Mach 1910). At issue was the reality of atoms, defended resolutely by Planck against Mach's skepticism, and the viability of Mach's notions of economy of thought in science and the elimination of metaphysics. As Planck's assault become more bitter, he decided to mention another area of disagreement with Mach, the relativity of motion, even though this was not the focus of their dispute. He wrote (planck 1910; taken from Blackmore's translation 1992, p. 145 with minor corrections)
Where Mach attempts to move forward by relying on his theory of knowledge quite often he runs into error.
Here belongs Mach's strenuously fought for but physically entirely useless thought that the relativity of all translational movements also corresponds to a relativity of all rotary movement, that therefore, one cannot decide at all in principle whether the fixed stars rotate around the earth at rest or the Earth rotates around the fixed stars. The equally general and simple principle that in Nature the angular velocity of an infinitely distant body circling a finite, rotating axis cannot possibly possess finite value is therefore for Mach either false or not applicable. According to Mach's mechanics, one is just as bad as the other.
The conceptual errors about physical matters which this unallowable transfer of the principle of the relativity of rotary movements from kinematics into mechanics has already caused, if they were depicted more closely at this point, would lead us too far astray. It therefore naturally follows that Mach's theory cannot possibly account for the immense progress which is intimately associated with the introduction of the Copernican theory - a circumstance which should suffice by itself to put Mach's theory of knowledge into considerable doubt.
The target of Planck's skeptical ridicule is the relativity of all motion. Since this relativity is the motivation for what soon becomes known as Mach's Principle, Planck's scorn would presumably extend to that principle. It might well be the "conceptual errors about physical matters" to which Planck alludes. Frank (1957, p. 153) did report Planck's

Mach's Principle before Einstein 37
remark as aimed directly at this principle. It is tempting to dismiss Planck's intemperate remarks as a petulant
outburst. Even if it was, there is no reason to dismiss its basic sentiments as insincere. Whatever its origin, opposition from Max Planck was very serious. Perhaps it reflected a broader consensus. If not, Planck was sufficiently influential that his views could foster such a consensus. Worse, while we now principally remember Planck for his contribution to quantum theory, he was also one of the earliest well-placed supporters of Einstein's special theory of relativity. He energetically threw in his lot and his prestige with Einstein's theory at a time when the theory's author was still a little-known patent clerk with a proclivity for incorporating bizarre philosophy into his physics.31 Clearly Planck's opposition to a full relativity of motion did not derive from any ill-considered antipathy to the general idea of the relativity of motion.
Philipp Frank was both physicist and philosopher. As physicist, like Planck, he was one of the early group that took up active research in special relativity. With Hermann Rothe, he first discovered one of the most frequently rediscovered results in special relativity - that the group property and requirement of linearity already powerfully constrain the possible transformation laws between inertial coordinate systems: The only viable options remaining are the Galilei transformation or the Lorentz transformations, with r? an undetermined factor (Frank and Rothe 1911). This publication, which was not Frank's first on special relativity, introduced the term 'Galilei transformation.' Frank also had very favorable relations with Einstein: Einstein recommended Frank as Einstein's own successor at the German University in Prague and Frank later published a biography of Einstein (Frank 1947). Thus we might well expect that Frank would have been sympathetic to the view that played such a prominent role in Einstein's thinking. Yet the final
outcome of Frank's 1909 lecture, discussed above, is a decision against
the Machian view, which, in Frank's hands, contains Mach's Principle. Frank (1909) attributed to Mach the view that inertia arises through "a formal, new law of nature about the action of masses" (p. 17). This view allows Mach to retain his relativist position and to answer affirmatively to the question of whether the future behavior of a system of bodies is determined solely by their relative motions and not any absolute motion of the entire system. Frank prefers a view intermediate between the relativism of Mach and antirelativism or absolutism. He considers the absolute motion of mechanics merely a special case of relative motion, that is, it is motion relative to 'fundamental bodies' or 'inertial bodies,' such as the fixed heavenly stars. This somewhat

38 John D. Norton
tortured, hybrid position enables him to claim establishment of his conclusion, stated in emphasized text (p. 18): "Physical phenomena do not depend only on the relative motion of bodies; this however still does not admit the possibility of the concept of an absolute motion in the philosophical sense."
Whatever may have been the broader feeling about Mach's Principle in the physics community in this early period, one would expect that, after its endorsement by Einstein, the principle would enjoy the broader support of the physics community, at least through the late 1910s and 1920s, the period of the euphoria over Einstein's discovery of general relativity. Of course, it is widely known that at least one member of the astrophysical community dissented. Willem de Sitter was clearly an enthusiastic supporter of Einstein's general theory of relativity. For example, in 1916 and 1917, when relations between the English and German physics communities were stretched by the bitterness of the Great War, de Sitter took upon himself the task of informing his English colleagues of Einstein's new theory by means of a series of communications to the Royal Astronomical Society. At the same time, however, he found himself disputing sharply Einstein's view that his general theory of relativity satisfied the relativity of inertia or what came to be called Mach's Principle. (See Kerszberg 1989 for a recent account of the controversy.)
There is some evidence that a majority in the physics community at this time did not agree with Einstein's view that Mach's Principle, in some suitable form, was one of the fundamental postulates of general relativity. (Einstein (1918) had listed Mach's Principle along with the principle of [general] relativity and of equivalence as the fundamental postulates of general relativity, when he published a carefully worded defense of his view of the foundations of the theory.) This is an outcome of a survey of expositions of general relativity which I recently completed (see Norton 1993, especially Sec. 4.2). Emphasis on Mach's Principle as a fundamental postulate of general relativity tended to be concentrated in popular and semi-popular expositions. Otherwise, most typically for serious textbook expositions, the principle found no place in the accounts of the foundations of the theory, with Einstein's own expositions comprising the major exception. Or the principle may appear later in discussion devoted to the cosmological problem, as in (Pauli 1921). It is difficult to know what to read into this treatment - or lack of treatment - of the principle. It certainly does not rule out the possibility that many of these authors regarded the principle as an uncontroversial consequence of the theory that they simply did not

Mach's Principle before Einstein 39
choose to discuss. Laue (1921), at least, makes clear that his omISSIon of Mach's
Principle was based on reservations concerning the place of the principle in the theory. The goal of Einstein's (1917) famous cosmological paper was to eliminate the need to posit Minkowskian boundary conditions for the metric tensor in general relativity, for Einstein held that such boundary conditions violated the Machian requirement that the inertia of a body be fully determined by other masses alone. His ingenious solution was to abolish spatial infinity by means of the Einstein universe, which became an admissible solution of this field equations after the introduction of the cosmological term. Laue (1921, p. 180) discussed Einstein's proposal in the context of Laue's treatment of Minkowskian boundary conditions:
According to the fundamental idea of the general theory of relativity, the inertia of a single body should vanish if it is at a sufficient distance from all other masses. For inertia can only be a relational concept, which can be applied only to two or more bodies. ... With the boundary conditions mentioned, however, the inertia [of a single body] continues to exist. Such considerations have led Einstein to the hypothesis of a space which runs back on itself like the surface of a sphere.32 To us the whole question seems clarified too little physically for us to want to go into the matter. In the following we understand 'infinity' to be regions inside our fixed star system for which the mentioned boundary conditions hold, but which are sufficiently far distant from the bodies of the gravitational field under consideration.
6.2. Among the Philosophers. When it comes to the philosophical community in the period prior to the mid 191Os, it is more difficult to assess the broader view towards what will become Mach's Principle. The principle seems not to have been a major focus of philosophical debate and, for this reason, not to have many supporters or detractors. In 1912, Joseph Petzoldt wrote an article on special relativity and its epistemological connection to relativistic positivism. Because of Petzoldt's close connection and sympathy with Mach and his positivist views and because they had corresponded on precisely this question, we might expect the principle to figure in his article. It is mentioned only briefly in a footnote (p. 1057), and Petzoldt takes no position on it, beyond merely suggesting that further experiments like those of the Friedlaenders and Fappl may settle the question. Perhaps his correspondence with Mach in 1904 had not assuaged the doubts he initially expressed to Mach as quoted above in Sec. 3.2. Frank (1909),

40 John D. Norton
in mapping out 'relativist' and 'antirelativist' posItions, wrote of the work of Hofler (1900) and more recently Hamel (1909a) as opposing Mach, characterizing their disagreement as a controversy (p. 12) and Hofler as writing a "polemic against Mach's thesis."
However, a reading of the sources Frank cites does not give one the impression of a polemical dispute over the specific question of whether inertia arises from some interaction of accelerated bodies mediated by a new physical mechanism. Hamel [(1909a), and the closely related (1909)] was devoted to developing Hamel's own axiomatic development of mechanics, with the discussion of Mach's views in preliminary surveys of the alternatives. Hamel does not directly address the question of a new physical mechanism for inertia. The closest is a critique of Mach's strictures against absolute space (for example Hamel 1909a, pp. 363-64). Hofler does rehearse lengthy debates over the relativist and absolutist positions. Yet his specific attitude to the possibility of a such a new physical mechanism is very sober and undogmatic. He seems fully prepared to let actual experiment decide. For this reason, presumably, he gave the careful review (discussed above) of experiments designed to detect the mechanism. He then stated his view (or rather buried it in grammar of bewildering complexity!) (pp. 125-26):
From my point of view I must admit in any case that, in so far as it is allowed at all, or even is ones duty, before an experiment, to form ideas over what can reasonably emerge from it, I expect nothing from all such experiments that could become somehow in the future a direct experimental proof for the relativity of rotational motion. I hold this negative expectation not without expressed experiential, even in part experimental foundations. Rather I believe that, flor] i[nstance], according to the total experiences of mechanics so far, in an axially symmetric· system, such as would be a bucket with miles thick walls in rotation about its geometric axis, no force couples would arise on the water mass inside and therefore according to these mechanical experience so far, it cannot be set into rotation. More precisely: It cannot be set into rotation any more than [a water mass] in a bucket with walls of ordinary thickness, of which we know of course (or at least for the present believe to know), that only the innermost layer is acted upon by friction.
He continued to quote Hertz and Mach to stress the dependence of the question on experiment and the possibility of new experiments overturning the outcomes of old experiments, concluding:- "But do I have to give up our current law of inertia, the foundations of our whole present mechanics, for such a 'possibility?'"

Mach's Principle before Einstein 41
It is difficult to fault the good sense of this unadventurous assessment. Let experiment decide, Hofler says. But he notes his skepticism about a positive outcome, since the mechanism sought would have to be quite unlike anything encountered so far in mechanics. The footnote to the word 'symmetric' sought to drive this last point home. Yet, ironically, in the attempt to dismiss them, the footnote ended up anticipating a Machian class of mechanical theories modeled after electrodynamicallaws!
One must at least say that a geometrically axially symmetric system is not also phoronomically [kinematically] and dynamically axially symmetric, even only because it rotates about its axis. But in this case force effects would be ascribed to mass particles propagating in different directions, f[or] i[nstance], antiparallel, and [those effects] should be functions of the direction (and speed?); and also this assumption (an analogy to Weber's electrodynamic hypothesis) is certainly at least suggested by nothing in the current experiences of mechanics and would hardly allow explanation of the current experiences, upon which, after all, the thesis of the relativity of motion depends.
Hofler's work lies in the neo-Kantian mainstream. It is actually an afterword to an edition of Kant's Metaphysische AnfangsgrUnde der Naturwissenschaft, and the two are bound as one volume. Thus it would seem that the neo-Kantians, a dominant force in German language philosophy at this time, had no objection of Kantian principle to the possibility of inertia arising from some new physical mechanism. But that did not guarantee assent from the neo-Kantians. The leading neo-Kantian, Ernst Cassirer (1910, pp. 176-77) attributed to Mach the notion that the fixed stars are "one of the causative factors on which the law of inertia is dependent." He felt the view untenable since it amounted to robbing the law of inertia of its status as a law:
If the truth of the law of inertia depended on the fixed stars as these definite individuals, then it would be logically unintelligible that we could ever think of dropping this connection and going over to another system of reference. The principle of inertia would in this case not be so much a universal principle of the phenomena of motion in general, as rather an assertion concerning definite properties and 'reactions' of a given empirical system of objects; - and how could we expect that the physical properties found in a concrete individual thing could be separated from their real 'subject' and transferred to another? ... [the meaning of principle in this view] corresponds in no way to the meaning and function it has actually fulfilled

42 John D. Norton
in scientific mechanics from the beginning.
While we would not expect unqualified support from neo-Kantians for ideas attributed to Ernst Mach, we would expect such ideas to receive a more sympathetic hearing from members of the Vienna Circle, a discussion group which met in Vienna in the 1920s and out of which the logical positivist movement sprang. Ernst Mach was the spiritual inspiration for the group - Frank (1949, p. 79) called Mach "the real master of the Vienna Circle." Frank himself was one of the longest standing members of the Circle; his early discussion meetings with the mathematician Hans Hahn and economist Otto Neurath starting in 1907 had laid the foundations for the group of the 1920s. Yet as we have seen, Frank: (1909) did not endorse the proposal for a new physical mechanism for inertia that he read in Mach's works. This opposition was no longer voiced in Frank's later writings, however. (See, for example, Frank 1947, Ch. 2, §8; 1957, p. 153.)33
In 1922, Moritz Schlick was appointed to Ernst Mach's old chair at the University of Vienna, and it was around Schlick that the Vienna Circle organized itself. Thus it is somewhat surprising to discover that the principal burden of Schlick (1915) was to drive a wedge between Mach's analysis of inertia and the treatment given by Einstein in the context of his general theory of relativity.34 Einstein's approach is praised and Mach's is condemned. Schlick states Mach's escape from Newton's argument in his bucket experiment as follows (p. 166): "Experience does not show us that centrifugal forces do not also arise if the entire fixed heavenly stars were to rotate around it."
Against Mach's view, Schlick levels two objections. The first is aimed at Mach's often stated view that there is no distinction between the cases of the bucket rotating and stars at rest and the case of the bucket at rest and stars rotating, so that (Schlick quoted Mach as saying)
The experiment [of testing whether rotating stars induce centrifugal forces] cannot be carried out, the idea is completely meaningless, since the two cases do not sensibly differ from one another. I hold the two cases to be the same case and the Newtonian distinction an illusion.
Schlick responds that Mach's proposal is not at all beyond test. He refers to Einstein's work on the relativity of rotation that has led to experimentally testable conclusions. Presumably Schlick means that if rotation relative to the distant stars induces inertial forces, then one would also expect rotation relative to other bodies to induce forces, such

Mach's Principle before Einstein 43
as Einstein (1912, 1913) found in his developing general theory of relativity. For example, a rotating shell of matter induces Coriolis forces within it. Schlick's second objection is (p. 166): " ...the assertion that the two cases do not differ sensibly, a petitio principii, is evoked by ignoring the difference between kinematic and dynamic ways of consideration. "
That is, he objects that one can define motion purely kinematically if one wishes; but that does not ensure that all the physical facts associated with motion are reducible to kinematics. Newton's theory supposes otherwise. It posits the possibility of kinematically identical systems which differ dynamically - for example a rotating and nonrotating body. And the difference between the two is a fact of sense experience (p. 168): "We can also ascertain the absolute rotation of a body, according to the Newtonian view, through muscular sensation, for we will find with its help that centripetal forces are needed for the body to keep its shape and to hold together its parts."
Mach's analysis ironically had turned into an exercise in a priori physics (p. 167): "It is curious to observe how sometimes exactly the attempt always to stick with just sensible experience leads to clever, a priori postulates, since one forgets that experiences can only be isolated from one another in abstraction."
Schlick proceeded to compare Mach's view with that of Einstein in his general theory of relativity. He asked if Einstein's theory amounted to "a great triumph of Mach's philosophy, since it had asserted the relativity of all motion as necessary." Schlick felt it did not represent such a triumph for three reasons:
The first reason, which is already completely decisive, is one we have already presented, in that we have showed the arguments that led Mach to his conclusion are completely untenable. If, nevertheless, it turns out to be correct, it would result more from an accidental coincidence than a proper verification. With Mach the conclusion arises as a necessity of thought; with Einstein it is posited as a fundamental assumption of a theory and the decision of how far it may be considered valid is fmally still left to expenence.
The second objection referred to the lack of general covariance of the then current version of general relativity and to Einstein's belief that a generally covariant theory would be physically uninteresting. Thus Einstein's theory contradicted Mach's view, which required the equivalence of all reference systems. This objection could not stand for long, since, in November 1915, Einstein advanced the final generally

44 John D. Norton
covariant version of his theory and retracted his objections to general covariance. (See Norton 1984 for an account of this episode.) The third objection repeated parts of the first: Mach had just had a clever idea; but Einstein had built a theory on it. Schlick however was anxious - if not over anxious - to deprive Mach even of much credit for having a clever idea. He called the idea "very obvious" and explained in a footnote (p. 171):
In order to show just how obvious the idea is, I might perhaps mention that I had already thought of it as a 6th form boy [Primaner] and in conversations stubbornly defended the assertion following from it that the cause of inertia must be assumed to be an interaction of masses. I was delighted, but not at all surprised, to come across the idea again shortly, when I got to know Mach's Mechanics.
It is difficult to overlook the unpleasant, dismissive tone of Schlick's remarks. It is somewhat reminiscent of Planck's tone, as is Schlick's general argument. For Planck was clearly happy to endorse a relativity of inertial motion, which formed the foundation of Einstein's special theory of relativity. He was unable to find kind words for Mach's proposal that this relativity be extended to all motion. Thus we may wonder if it is mere coincidence that Schlick studied physics under Max Planck at the University of Berlin, taking his doctorate in 1904. Is there some kind of unhealthy conspiracy against Mach plotted by the students of Mach's opponents? If one wants to, one can always find fragments of evidence for conspiracies. Laue, too, was an assistant of Planck in Berlin, and Frank was a student of Mach's arch rival, Boltzmann! However, I think there is no weight of evidence for such a conspiracy theory. The opposition of Frank and Laue is mild and mildly stated. It is more compatible with seriously considered disagreement. Schlick, however, was more intemperate. He was not prepared to concede anything to Mach. He closed his paper by noting that particular relativistic assertions made by positivists such as Mach were more likely to be refuted than confirmed by advances in physical science. Moreover, the investigations of Mach or other positivists on the concept of time did not pave the way for Einstein's special theory of relativity. "No one anticipated, f[or] i[nstance], the relativization of simultaneity."35

Mach's Principle before Einstein 45
7. Conclusion
Mach presents us with a perplexing puzzle in his analysis of Newton's bucket experiment and the law of inertia. On the one hand, in his publications, the only unequivocal proposal is that we eliminate the odious notion of space by redescribing the relevant experiment and law in a way that does not use the term 'space.' If there is a suggestion of a new physical mechanism that would reach from the distant stars to cause the inertial forces in Newton's bucket, then the proposal is made vaguely and we are left to wonder whether Mach endorses it or condemns it as unscientific. On the other hand, if Mach did not wish to propose a new physical mechanism for the origin of inertia, then, in the course of the final two decades of his life, he passed over numerous opportunities to correct many who publicly attributed such a proposal to him.
I favor the view that Mach's published pronouncements cease to be ambiguous when we recognize that Mach held an extremely restrictive view of causation. Specifically, Mach held a causal relation to be nothing more than a functional relation between actual phenomena and prohibited speculation on hypothetical or counterfactual systems as unscientific. All we are allowed to infer from Newton's bucket experiment is that centrifugal forces arise when there is relative rotation between the water in the bucket and the other bodies of the universe. That alone is the causal relation. We have no license to infer to an absolute motion or even what would happen if (counterfactually) the walls of the bucket were made several leagues thick. This reading exonerates Mach of equivocation, ambiguity and inconsistency in his publications. However, it requires that the proposal of a new physical mechanism, as commonly attributed to Mach, is incorrect, and it leaves unexplained why he failed to correct this frequent misattribution in the final decades of his life.
If Mach did not propose such a mechanism, then at least the proposal was widely attributed to him in the 1890s and 1900s. It was then the focus of work of a scattered and disconnected group of investigators, largely on the fringes of the physics community. August Fappl was perhaps the only member of this group with any standing in the physics community. There is some indication of the proposal arising independently of Mach. Immanuel Friedlaender claimed his own version of the proposal came prior to knowing of Mach's work. Einstein attributed independent introduction of the proposal to Hofmann. Because of their lack of cohesion and because they tended to publish in obscure vehicles, it is likely that the full extent of their work is not now appreciated. The known work tends towards actual experimental test of

46 John D. Norton
the mechanism (unlike Mach) and labored but rather inconsequential treatises.
It is difficult to gauge the broader view of the proposal for a physical mechanism to explain inertia, prior to its sponsorship by Einstein. The difficulty is that the proponents of the view were largely on the fringes of the physics community and could not expect or demand a response from the mainstream. At least Max Planck, in 1910, spoke out strongly against Mach's insistence on the relativity of motion, while at the same time energetically supporting the relativity of inertial motion in Einstein's special theory of relativity. In 1909, Philipp Frank also weighed the possibility of a new physical mechanism to explain inertia and decided against it. After Einstein's sponsorship of what became Mach's Principle, the notion was widely celebrated by both physicist and philosopher. It seemed to provide a paradigm of fruitful interaction between the two disciplines. However, in the physics community its celebration tended to be focused in the popular and semi-popular expositions of general relativity. In general, as a review of expositions of general relativity from the 1910s and 1920s shows, the broader physics community did not wish to present Mach's Principle as one of the fundamental postulates of general relativity.
Prior to Einstein's sponsorship, the philosophy community devoted little attention to the proposal. If it was noticed at all, it accrued a mention in passing in the more traditional debates over absolute and relative motion. Criticism offered was sober and, in my view, largely justified. For example, if Mach's proposal was to be construed narrowly as urging the replacing of the Newtonian law of inertia by the observation that free bodies move uniformly with respect to the fixed stars, then Cassirer objected that this was a retrograde step for science, for it replaced a general law by an extremely restrictive description of one case. If Mach's proposal was for a new law, then Hofler felt that its merit was to be settled by experiment. But all experiments so far had yielded no positive results. This was hardly an encouraging foundation for overturning the Newtonian principle of inertia, then one of the most successful of scientific laws. In a similar vein, Schlick complained that Mach was engaged in a priori physics, an ironical twist given Mach's emphasis on the supremacy of experience. One could, Schlick noted, define motion purely kinematically. But this by no means guaranteed that the complete physics of moving bodies ought to be determined solely by kinematic relations. Newton had certainly supposed otherwise - and the dynamic effects to which he appealed, inertial forces, were matters of direct experience. The centrifugal forces that distinguish rotation are

Mach's Principle before Einstein 47
directly sensed by the muscles. Thus Schlick was at pains to distinguish Mach's view, which legislated a priori and seemed uninterested in experimental test, from Einstein's view, in which one constructed a definite theory with definite predictions that could be subject to experimental test.
If there is a moral in the early history of Mach's Principle, it lies exactly in Schlick's last point. As long as the relativity of all motion is posited dogmatically and Mach's Principle derived from it as a priori physics, then it is moribund. Its promise lies in the realm of empirical science, in the attempt to draw the doctrine of relativity and Mach's Principle into a physical theory that can be subject to experimental test, where one allows that experience may speak against it. 36 It was Einstein's recognition of this point that enabled him to breathe life into Mach's Principle.
NOTES
11 am grateful to John Earman, Peter Galison, and Ulrich Maier for assistance in procuring sources for this paper and to Julian Barbour for helpful discussion.
2While Einstein is usually credited with christening the principle, Schlick (1915) had already used the term ["das Machsche Prinzip" (p. 170) and" ... des Machsches Postulats" (p. 171)]. It is a little unclear precisely to what Schlick's terms referred. They most likely referred to Mach's general proposal for a relativity of all motion, from which, Schlick noted (p. 171), it follows that "the cause of inertia must be assumed to be an interaction of masses."
3In his synopsis of his critique of Newton's ideas, Mach (1960, pp. 303-304) gives another example of how the terms 'space' and 'time' can be eliminated in this case from the fundamental propositions of Newton's mechanics.
4Mach (1960, pp. 272-73) gives a similar analysis of time. When we say that some process takes time, this is simply an abbreviated way of saying that the process has a dependence on another thing such as the changing position of the earth as it rotates. If we forget that this is all we may mean, we can fall into the error of thinking of time as an independent entity. In fact time has "neither a practical nor a scientific value" and "It is an idle metaphysical conception. "
sIn recent literature, it has been urged that Mach did not intend to propose a new physical law and merely intended a redescription of Newton's theory that preserved its true empirical content. See, in particular, Strauss (1968). Wahsner and von Borzeszkowski (1988, pp. 602-603) also found Mach's goal merely to be redescription of existing Newtonian theory.
6This criterion may contradict Mach's own highly restrictive

48 John D. Norton
pronouncements on causality which do not seem to admit such hypothetical or counterfactual claims. However if we rule out the possibility that Mach did allow such claims in his analysis - possibly in contradiction with his own general view of causality - then it seems to me that we have settled the question in advance of whether Mach actually proposed what we now know as Mach's Principle. (Added in proof) Julian Barbour (this volume, p. 218) has identified a remark found only in early editions of Mach's, Mechanics, as employing counterfactuals in the way I require. The remark is suggestive but still contains no positive proposal for a new mechanism. Rather it casts doubt on whether a particle in the set up described would move according to Newtonian prescriptions if the fixed stars were absent or not unvarying. The remark makes no positive claim about how the particle would move in this counterfactual circumstance. It does not even deny outright that the motion will not be Newtonian. It is merely "very questionable." These sentiments fit well with Mach's repeated exhortation that we have no business proclaiming what would happen in situations beyond our experience, such as if there were no fixed stars. All such attempts are dubious.
7As almost everywhere, Mach's precise point remains clouded by ambiguity. We cannot assume away these bodies, he says, since we cannot assume that "the universe is without influence on the phenomenon here in question." Is Mach assuming there is some influence? If so, he does not say so. What does Mach mean by 'influence' in any case?
8The text is a slightly corrected version of the standard English translation (1960, p. 284) "The principles of mechanics can, indeed, be so conceived, that even for relative rotations centrifugal forces arise." I am grateful to Herbert Pfister for pointing out an error in this standard translation of the wohl of Mach's original "Die mechanischen Grundsiitze konnen also wohl so gefasst werden, dass auch flir Relativdrehungen Zentrifugalkriifte sich ergeben. "
9Mach continues in a similar vein, using mass weighted sums of distances, to treat the motion of two bodies which do exert a force upon each other.
lOCarus (1906, p. 332) calls Mach a "kindred spirit" and is "proud to count him among my dearest personal friends," although "there are no doubt differences between Mach's views and mine."
llFor more details see (Holton 1992, pp. 30-33) and (Thiele 1971). 12He refers to a passage from Mach (1911) which will be discussed below. 13See Mach (1915, p. 44). 14These passages do not exhaust the relevant passages from The Science of Mechanics, although I have found none that provide brighter illumination. I leave to readers the task of deciding what Mach intended when he asked of the bodies A, B, C, ... " ... whether the part they play is fundamental or collateral" and that "it will be found expedient provisionally to regard all motions as determined by these bodies" (p. 283). 151 cannot resist observing that if this consideration is intended to show a Newtonian that the distant masses engage in some causal interaction with the

Mach's Principle before Einstein 49
body in question, then it is an extremely odd argument. Under the Newtonian viewpoint, the reason that distant celestial bodies are so valuable for describing the motion of the rotating body is precisely that there are no causal interactions between them and the body.
16(Added in proof) Julian Barbour (this volume, p. 216) doubts this claim. He points out that distances between inertially moving bodies do not in general vary in direct proportion to one another. In support he cites Mach's own equation for the distance between two inertially moving bodies [Barbour's equation (1)]. I do not fmd the situation so unequivocal. Since a is constant and Idr/dtl :::;;a, the equation does give the stated linear dependence in the limit in which r becomes sufficiently large. Mach's words are all too few, but he is considering bodies separated by great distances. Are these distances great enough to bring us towards this limit? (If not, so that the bodies are close but just not interacting, how can Mach escape this equation, whose derivation requires little more than simple geometry?)
17Many of these authors were sufficiently close to Mach to meet or enter into correspondence with him, including Petzoldt, Frank, Fappl, and Einstein.
18Cassirer is sufficiently unsure of the attribution to indicate that he infers it from Mach's writing by introducing it as "Mach himself must, according to his whole assumption, regard the fixed stars ... as one of the causalfactors" and felt the need to support the attribution with a lengthy quotation from Mach (1911).
19Mach's celebrated July 1913 renunciation of the role of 'forerunner of relativity' in the preface to his Optics (1921, pp. vii-viii) is far too vague to be such a correction, since it clearly refers to Einstein's relativity theory in general. Wolters (1987) also urges that this famous renunciation was forged by Ernst Mach's son Ludwig.
2oBlackmore and Hentschel 1985, p. 121. Otto Neurath also wrote in about 1915 along similar but vaguer lines to Mach (Blackmore and Hentschel 1985, pp. 150-152), although one might no longer reasonably expect a response from an ill Mach who would die in 1916.
21The experiment sought inertial dragging effects in the vicinity of a spinning fly-wheel.
22Mach here cites a passage in his The Science ofMechanics (1960, p. 283, Mach's emphasis) where he reports the result that" ... a rigid body experiences resistance in a frictionless fluid only when its velocity changes." He conjectures about the possibility of this result as a "primitive fact," introduced prior to the notion of inertia, were our world filled with some hypothetical, frictionless medium, which would be an alternative to "the forlorn idea of absolute space. "
23Dingler here raises the possibility that Mach's position on this matter may have altered considerably through the years 1883-1912 of the various editions of The Science of Mechanics. I have been unable to check this possibility thoroughly. However the task of comparison has been eased considerably by a remarkable and unusual volume (Mach 1915). This volume contains, in

50 John D. Norton
English translation, a compendium of the extensive additions and alterations made in preparation of the 7th German edition of the work. It is interesting to speculate why such a compendium, useless without the earlier volume, should be published at all, rather than simply publishing a complete, updated text. In any case, in examining the volume, I could see no evidence of a significant shift in Mach's viewpoint with respect to the matters at issue here.
24Recall also that, on Wolters's (1987) account, Ludwig Mach was hardly a reliable source for his father's views pertaining to relativity theory, since Wolters accuses Ludwig of forging his father's famous renunciation of his role as 'forerunner of relativity theory. '
25It is helpful to compare the analysis of Newton's bucket experiment under Mach's view of causation and under a view that leads to some version of Mach's Principle. In both, we arrive at the result that the centrifugal forces in the bucket arise from the rotation of the water relative to the distant masses A, B, C, ... Mach requires that we halt analysis at this point. The other view makes the assumption, decried by Mach, that this one relation can be decomposed into parts. It regards the interaction between the water and the masses A, B, C, ... as the compounding of many smaller interactions between the water and mass A, between the water and mass B, ... These smaller interactions are understood to obtain in the circumstance in which we have a universe devoid of all matter excepting the water and mass A, etc.
26This emphasis is quite different from Mach's. He seems less interested in experimental tests. His The Science of Mechanics only mentions the possibility of real experimental test in later editions in response to the experiments of the Friedlaenders and Foppl and does so in an equivocal way. He did however propose an experiment to Petzoldt in their correspondence of 1904, as we have seen.
271 am grateful to the editors of The Collected Papers of Albert Einstein (Draft of 1992) for determining that this was the work referred to by Einstein.
28For example, he was the author of (Foppl 1894), one of the most important German language introductions to Maxwell's electrodynamics, and first of the famous series. Many German physicists learned vector analysis from its self-contained exposition of vector analysis.
29Neisser is identified as one of the 'Teilnehmer des Kant-Maxwell-Collegs' and the only reference given is to a conference in 1893 on the question "Is absolute motion, if not discernible, at least conceivable?" at the philosophical society of the University of Vienna.
30For discussion of the role of the relativity of inertia and Mach's Principle in Einstein's accounts of the foundations of general relativity and of Einstein's later disenchantment with the principle, see (Norton 1993, Sec. 3).
31Planck had entered into an encouraging correspondence with Einstein by 1906. He had given a colloquium on the theory in Berlin in the fall of 1905 and encouraged work on the theory, supervising von Mosengeil's doctoral thesis on the theory. Planck also is believed to be the one that approved Einstein's 1905

Mach's Principle before Einstein 51
special relativity paper for publication in Annalen der Physik (Miller 1981, p. 2). His immensely important paper (Planck 1908) on relativistic dynamics is credited by Pais (1982, p. 150) as the first paper on relativity authored by someone other than Einstein. See (StacheI1989, pp. 266-67). Planck's support for Einstein did not wane. He was instrumental in engineering Einstein's move to Planck's own Berlin in 1914.
32Laue's footnote merely cites Einstein (1917). 33The latter discussion does, however, recapitulate Planck's (1910) objections, but proceeds to allow that Einstein's work eventually vindicated Mach's VIew. 34The same viewpoint is advanced far more briefly and in far more muted voice in (Schlick 1920, pp. 37-40). 35Perhaps Schlick might have agreed with Abraham's (1914, p. 520) gibe that Einstein's new theory scarcely fitted Mach's requirement of economy, for it replaces the then standard single gravitational potential with the complication of ten potentials, the components of the metric tensor. 361. Friedlaender's and Foppl's experiments fell short of this goal. While the experiments could in principle reveal a positive effect, a null outcome could not provide a decisive refutation. Since they had no definite theory that fixed the magnitude of the effect, they could not rule out the possibility that a small positive effect lay hidden behind the random error that shrouds all null results.
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published as first book of William Curtis Swabey and Marie Collins Swabey trans., Substance and Function and Einstein's Theory ofRelativity. Chicago: Open Court (1923); reprinted New York: Dover (1953). Dingler, Hugo (1921). Physik und Hypothese. Berlin & Leipzig: Vereinigung wissenschaftlicher Verleger, Walter de Gruyter. Einstein, Albert (1912). "Gibt es eine Gravitationswirkung, die der elektrodynamischen Induktionswirkung analog ist?" Vierteljahrsschrift jar gerichtliche Medizin und offentliches Sanitiitswesen. 44: 37-40. Einstein, Albert (1913). "Zum gegenwiirtigen Stande des Gravitationsproblems. " Physikalische Zeitschrift 14: 1249-1262. Einstein, Albert (1917). "Kosmologische Betrachtungen zur allgemeinen Relativitiitstheorie." Preussische Akademie der Wissenschaften, Sitzungsberichte (1917), 142-152; trans. as "Cosmological Considerations on the General Theory of Relativity." In H. A. Lorentz et al., The Principle of Relativity. Dover (1952), pp. 175-188. Einstein, Albert (1918). "Prinzipielles zur allgemeinen Relativitiitstheorie." Annalen der Physik 55: 240-44. Foppl, August (1894). Einjahrung in die Maxwell'sche Theorie der Elektricitiit. Leipzig: B. G. Teubner. Foppl, August (1904). "Uber einen Kreiselversuch zur Messung der Umdrehungsgeschwindigkeit der Erde." Akademie der Wissenschaften, Miinchen, Mathematisch-WissenschaftlicheKlasse, Sitzungsberichte, Sitzung vom 6. Februar, 1904, pp. 5-28. [Partial translation in this volume, p. 312.] Foppl, August (1904a). "Uber absolute und relative Bewegung." Akademie der Wissenschaften ,Miinchen,Mathematisch-WissenschaftlicheKlasse,Sitzungsberichte, Sitzungvom 5. November, 1904, pp. 383-395. [Partial translation in this volume, p. 120.] Frank, Philipp (1909). "Gibt es eine absolute Bewegung?" Vortrag, gehalten am 4. Dezember, 1909. Wissenschaftliche Beilage, 1910, 1-10. Frank, Philipp (1947). Einstein: His Life and Times. trans. George Rosen. rev. 1953. New York: A. A. Knopf; reprinted New York: Da Capo, n.d. An abridged translation of Einstein: Sein Leben und seine Zeit. Munich: Paul List (1949); reprinted Braunschweig: Vieweg (1979). Frank, Philipp (1949). Modern Science and its Philosophy. Cambridge: Harvard University Press. Frank, Philipp (1957). Philosophy of Science: The Link Between Science and Philosophy. Englewood Cliffs, N.J.: Prentice-Hall. Frank, Philipp, and Rothe, Hermann (1911). "Uber die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme." Annalen der Physik 34: 825-855. Friedlaender, Benedict and Friedlaender, Immanuel (1896). Absolute oder

Mach's Principle before Einstein 53
Relative Bewegung? Berlin: Leonhard Simion. [Partial translations in this volume, p. 114 and p. 309.] Hamel, Georg (1909). "Uber die Grundlagen der Mechanik." Mathematische Annalen 66: 350-97. Hamel, Georg (1909a). "Uber Raum, Zeit und Kraft als apriorische Formen der Mechanik." Jahresbericht der Deutschen Mathematiker-Vereinigung 80: 357-85. Hofler, Alois (1900). Studien zur gegenwiirtigen Philosophie der Mechanik. Als Nachwort zu Kant's [sic] Metaphysische Anfangsgriinde der Naturwissenschaft. Veroffentichung der Philosophischen Gesellschaft an der Universitiit zu Wien, Band /lIb. Leipzig: Verlag von C. E. M. Pfeffer. Hofmann, W. (1904). Kritische Beleuchtung der beiden Grundbegriffe der Mechanik: Bewegung und Triigheit und daraus gezogene Folgerungen betreffs der Achsendrehung der Erde und des Foucault'schen Pendelversuchs. Wien und Leipzig: M. Kuppitsch. [Partial translation in this volume, p. 128.] Holton, Gerald (1992). "Ernst Mach and the Fortunes of Positivism in America." Isis 83: 27-60. Johannesson, Paul (1896). "Das Beharrungsgesetz." Jahresbericht des SophienRealgymnasiums in Berlin, Ostern. Kerszberg, Pierre (1989). "The Einstein-de Sitter Controversy of 1916-1917 and the Rise of Relativistic Cosmology." in Einstein and the History of General Relativity, Einstein Studies, Vol. 1. Don Howard and John Stachel, eds. Boston: Birkhauser, pp. 325-366. Laue, Max (1921). Die Relativitiitstheorie. Vol. 2: Die allgemeine Relativitiitstheorie und Einsteins Lehre von der Schwerkraft. Braunschweig: Friedrich Vieweg und Sohn. Mach, Ernst (1872). Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit. Prag: Calve'sche Buchhandlung. Mach, Ernst (1882). "The Economical Nature of Physical Inquiry." in Popular Scientific Lectures, 5th ed. T. J. McCormach, trans. LaSalle, Illinois: Open Court, 1943, pp.186-213. Mach, Ernst (1910). "Die Leitgedanken meiner naturwissenschaftlichen Erkenntnislehre und ihre Aufnahme durch die Zeitgenossen." Physikalische Zeitschrift 11: 599-606; abbr. trans. as Blackmore (1992), pp. 133-37. Mach, Ernst (1911). History and Root of the Principle of the Conservation of Energy, P. Jourdain, trans. Chicago: Open Court. [Translation of Mach (1872)]. Mach, Ernst (1915). The Science of Mechanics: A Critical and Historical Account ofits Development. Supplement to the Third Edition, Containing the Author's Additions to the Seventh German Edition. P. E. B. Jourdain, trans. and annot. Chicago and London: Open Court. Mach, Ernst (1921). Die Prinzipien der physikalischen Optick. Leipzig, 1921; trans. as The Principles of Physical Optics. 1926; reprinted New York:

54 John D. Norton
Dover. Mach, Ernst (1960). The Science of Mechanics: A Critical and Historical
Account of Its Development, 6th ed. T. J. McCormach, trans. LaSalle, Illinois: Open Court. Miller, Arthur (1981). Albert Einstein's Special Theory ofRelativity: Emergence (1905) and Early Interpretation (1905-1911). Reading, MA: AddisonWesley. Norton, John D. (1984). "How Einstein found his Field Equations: 1912-1915." Historical Studies in the Physical Sciences, 14: 253-316; reprinted in Einstein and the History of General Relativity, Einstein Studies, Vol. 1. Don Howard and John Stachel, eds. Boston: Birkhiiuser (1989) pp.101-159. Norton, John D. (1993). "General Covariance and the Foundations of General Relativity." Reports on Progress in Physics 56: 791-858. Pais, Abraham (1982). 'Subtle is the Lord... ': The Science and the Life ofAlbert Einstein. Oxford: Clarendon. Pauli, Wolfgang (1921). "Relativitiitstheorie." In Encyklopadie der mathematischen Wissenschaften, mit Einschluss ihrer Anwendung. Vol.5, Physik, Part 2. Arnold Sommerfeld, ed., Leipzig: E.G. Teubner, 1904-1922, pp.539-775. [Issued November 15, 1921] English translation Theory of Relativity. With supplementary notes by the author. G. Field, trans. London: Pergamon, 1958; reprint New York: Dover. Petzoldt, Joseph (1912). "Die Relativitiitstheorie im erkenntnistheoretischen Zusammenhange des relativistischen Positivismus. " Deutsche Physikalische Gesellschaft. Verhandlungen 14: 1055-1064. Planck, Max (1908). "Zur Dynamik bewegter Systeme." Annalen der Physik 26: pp. 1-34. Planck, Max (1909). "Die Einheit des physikalischen Weltbildes." Physikalische Zeitschrift 10: 62-75; abbr. trans. as Blackmore (1992), pp. 127-32. Planck, Max (1910). "Zur Machschen Theorie der physikalischen Erkenntnis." Physikalische Zeitschrift 11: 1186-90; abr. trans. as Blackmore (1992), pp. 141-46. Reichenbach, Hans (1928). Philosophie der Raum-Zeit-Lehre. Berlin: W. de Gruyter; Maria Reichenbach and John Freund, trans., Philosophy of Space and Time. New York: Dover, 1957. Reissner, Hans (1914). "Uber die Relativitiit der Beschleunigungen in der Mechanik." Physikalische Zeitschrift 15: 371-75. Reissner, Hans (1915). "Uber eine M6glichkeit die Gravitation als unmittelbare Folge der Relativitiit der Triigheit abzuleiten." Physikalische Zeitschrift 16: 179-85. Reissner, Hans (1916). "Uber die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie." Annalen der Physik 50: 106-120. Schlick, Moritz (1915). "Die phi1osophische Bedeutung des Relativitiitsprinzips. " Zeitschrift flir Philosophie und Philosophische Kritik 159: 164-171. Schlick, Moritz (1920). Space and Time in Contemporary Physics. New York:

Mach's Principle before Einstein 55
Oxford University Press. Stachel, John, ed. (1989). Collected Papers ofAlbert Einstein. Volume 2. The
Swiss Years: Writings, 1900-1909. Princeton: Princeton University Press. Strauss, M. (1968) "Einstein's Theories and the Critics of Newton." Synthese
18: 251-284. Thiele, Joachim (1971). "Paul Carus und Ernst Mach: Wechselbeziehungen
zwischen deutscher und amerikanischer Philosophie um 1900." Isis 62: 208-219. Wahsner, Renate and von Borzeszkowski, Horst-Heino (1988). "Nachwort" to Mach, Ernst, Die Mechanik in ihrer Entwicklung dargestellt: Historischkritisch dargestellt. Berlin: Akademie-Verlag, pp. 563-647. Wolters, Gereon (1987). Mach I, Mach II, Einstein und die Relativitiitstheorie: Eine Fiilschung und ihre Folge. Berlin & New York: Walter de Gruyter.
Discussion
Nordtvedt: Did you consider all the experiments a failure in the sense that they saw nothing, or did they have problems? Null experiments are good experiments even when they see no effects, and perhaps you were being a bit hard on the experimentalists, particularly FappI. Norton: I did not mean to say that the experiments were bungled. Rather what I meant was that the results had to be inconclusive since the experimenters had no idea of the magnitude of the effect sought. Therefore a null result could not eliminate the possibility of a positive effect smaller than their experimental error. Of course the experiments were not uninformative, since they did place an upper bound on the size the effect could have. Ehlers: I'd like to have your reaction to the following: If one takes the redescription interpretation, then it seems to me that although Mach hinted at possibilities of redescription, he would not have been able to reconstruct the whole body of Newton's theory. Newton was, I think, much more of a mathematical physicist than Mach. Mach was perhaps more an intuitive and empirically oriented physicist. The Newtonian system needs a basis for concepts such as velocities, accelerations, and so on, some framework, relatively to which these concepts are well defined. Even people like Euler struggled with the question: How can you give a meaning to the concept of velocity if you don't have some space to which you refer it? It requires a considerable amount of abstraction to consider velocity as meaningful without absolute space, namely, only have a certain class of inertial frames. So my question is: Could a redescription be given which does not lose an essential part of

56 John D. Norton
the Newtonian system as a quantitative mathematical theory? The second remark is a comment only. I think if one, as a physicist,
compares these two interpretations, namely, the redescription interpretation and the interpretation that one would like to have a new mechanics, then if one cannot decide, as a historian, which of the two interpretations is truer to the text, then I think it matters that physicists are interested in history, not so much because they want to know what has been said by such and such a person but which useful suggestions are contained in earlier works. The second view, namely, looking for a new mechanics, is fruitful and interesting for bringing physics further, whereas the redescription point of view, in that sense, is not of interest. Therefore I feel, even if one cannot decide, that for a physicist the other point of view is more fruitful and interesting. Norton: Briefly, on the second point, as a historian, I'm fairly constrained by what happens [laughter], at least, I try to be. As a physicist you try to be constrained by the world. If the two can coincide, and I can find useful things happening, all the better, but I have to stick with what was there.
On the first point, I think you can redescribe everything that Newton had in his science without talk of absolute space and time. It's simply a matter of doing what Mach prescribed. You work through Newton's texts replacing every metaphysical claim by a statement of the observational content of the claim. Whether the resulting description will be economical is the real question. And this, I think, is what has always troubled Mach's system. There was a tension between the need for the descriptions to be restricted to observation and for them to be economical. We see this clearly in the case of Mach's skepticism over atoms. We like them since they do provide a very economical systematization of many physical phenomena. But the price of the economy is talk of entities that transcend observation. So it is with spacetime structures; they are unobserved, but, as you point out, they do enable just the systematization we want. In the end, I think this problem was a major part of the transition from the simple positivism of Mach to the logical positivism of those who followed Mach. It was the realization that one cannot be so narrow and restrict all talk to experience. You also need theoretical terms. Then follows the long debate over what to do with these theoretical terms. Are spacetime structures real entities or merely convenient aids to prediction? Rindler: Did you say Fappl and Friedlaender had no idea of the magnitude of the effect they were looking for? Why didn't they have an idea? In those days there were a number of people who had played with

Mach's Principle before Einstein 57
Maxwellian theories of gravitation. Dennis Sciama later pointed out that the Maxwellian type of gravitational theory has various Machian features. My question is, surely somebody before Dennis Sciama must have thought of that. Why isn't it that people used some kind of a Maxwellian estimate for the magnitude of the Machian effects they were looking for before they did those experiments? Of course, this would have totally discouraged them from even trying. Norton: You're referring, I take it, to the literature in gravitation theory towards the end of the nineteenth century. They were trying to start modeling extra terms for Newton's theory on the basis of electrodynamics. I believe that a Weber-like law was one of them; there are many different variants. However, I did not find any cases of experimentalists using such laws to estimate the magnitude of the effect sought. As you point out, that is odd. Renn: I think the answer to the question as to why the scientists who were looking around the turn of the century for Machian effects did not come up with precise ideas on the magnitude of these effects can be found in the split of two conceptual traditions, that of mechanics and that of electrodynamics, which I discuss at some length in my contribution [see p. 5]. Without much exaggeration one can say that those interested in electrodynamic theories of gravitation did not link this interest with a critique of mechanics along the lines of Mach and vice versa. A short footnote in the paper of the Friedlaender brothers, referring to a Weberian theory of gravitation, and the work of Einstein are exceptional in establishing the link between these two traditions. Editorial Note (J.B.B.): The reader may be puzzled by the limited discussion recorded above of the issue raised by Norton of whether Mach truly intended a physically new theory of inertia or merely a redescription of Newtonian theory in relational terms. In fact, there was a fairly extended discussion at Tiibingen around the passage by Mach reproduced in its entirety on p. 110 (beginning line 5) and discussed by Norton on pp. 16-17. However, examination of the discussion transcript showed that quite a large proportion of the comments, which were made without benefit of the complete exact text for examination, were either irrelevant or misleading, though Kuchar did make the important point that, irrespective of the physical significance Mach may have read into Eq. (1) on p. 17, the equation itself is mathematically incomplete, since it is a single scalar equation and therefore insufficient to describe either absolute or relative motion (cf. my comments on p. 217). Since the issue of whether Mach merely intended a redescription is discussed in some length in my own contribution (pp. 215-218) and Notes 1 and 2 on p. 230) and both Norton (in his Notes 6 and 16) and von Borzeszkowski and Wahsner (pp. 65-66) have responded to my comments, there seems little point in reproducing here the Tiibingen discussion.

Mach's Criticism of Newton and Einstein's Reading of Mach: The Stimulating Role of Two Misunderstandings
Horst-Heino v. Borzeszkowski and Renate Wahsner
In the present paper we will give some arguments in favor of the thesis that the so-called Mach's Principle owes its existence to two misunderstandings, namely first to Mach's misunderstanding of fundamentals of Newtonian mechanics, mainly of the Newtonian notion of space, and second to a misreading of Mach by Einstein. The latter was admittedly a reading of genius, but nevertheless a misreading.
To start with, it should be mentioned that in order to discuss this matter it is not sufficient to study appropriate passages of Mach's Mechanik. Rather, one has also to analyze the other critical-historical treatises and, first of all, the philosophical work of Mach. Since this, however, is not the place for discussing Mach's philosophy in detail, here we shall make only a few remarks summarizing some aspects of Mach's philosophy that are of interest in the context of this topic. (For a detailed consideration, see Wahsner and v. Borzeszkowski 1988.)
Mach's main intention was to free mechanics, optics, and other physical branches from metaphysics, so that the real nature of physics becomes visible. In the search for a way toward his aim, he arrived at the conclusion that one has to study the history of physics. In his 1872 pamphlet Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit, which was programmatic for his life's work, Mach said:
Denn metaphysisch pflegen wir diejenigen Begriffe zu nennen, von welchen wir vergessen haben, wie wir dazu gelangt sind. Man kann jedoch nie den thatsachlichen Boden unter den Fussen verlieren oder gar mit den Thatsachen in Collision gerathen, wenn man stets auf den Weg
Einstein Studies, vol. 6: Mach's Principle: From Newton's Bucket to Quantum Gravity, pp. 58-66 © 1995 Birkhiiuser Boston, Inc. Printed in the United States.

Mach and Einstein: Two Misunderstandings 59
zuriickblickt, den man gegangen. [For the notions that we usually call metaphysical are the ones for which we have forgotten how we arrived at them. However, we can never lose the real ground from under our feet or, worse, come into collision with the facts if we always look back over the way that we have gone.] (Mach 1872, p. 2)
As far as mechanics is concerned, Mach considered it correct but, for historical reasons, represented by Newton in a manner containing a lot of metaphysical elements. Therefore, he intended to remove these elements by reformulating mechanics, and the result was his criticalhistorical account of mechanics: Die Mechanik in ihrer Entwicklung. Historisch-kritisch dargestellt (Mach 1883; later editions published in 1889, 1897, 1901, 1904, 1908, 1912).
As is well known, Mach was dissatisfied in particular with Newton's definition of mass and his representation of the axioms of mechanics. Therefore, he started by reformulating them.
First he replaced Newton's definition of masses, saying that the ratio of the masses m1 and m2 of two bodies is equal to the ratio of their weights, m/m2 =G1/G2 , by the definition that states: The ratio of the
masses m1 and m2 of two bodies is equal to the negative and inverse ratio
of the accelerations b1 and b2 caused by their mutual interaction, m/m2= -b2/b 1•
From this point of view, Mach considered Newton's second law as a convention and the third law as a consequence of his definition of mass. So for him there remained only the task to answer the question as to the first Newtonian law. Mach's answer was: This law is a fact first perceived by Galileo, but it has only a definite meaning when one can answer the question as to the reference system one needs in order to determine the motion.
He argued as follows. When one says "a system or a body on which no forces act is either at rest or in uniform motion," one has to ask "uniform motion, relative to what?" Newton's answer was "to absolute space." But this is a metaphysical element that can be replaced by the totality of the cosmic masses, more precisely, by the fixed stars realizing a rigid reference system.
To demonstrate this, Mach showed that the center of mass of an Nbody system on which no external masses act realizes a system to which the motion described by mechanics can be referred. Assuming then that this N-body system is the system of cosmic masses (on which per dejinitionem no external forces act), he has in this way determined a cosmic reference system. For Mach this was a proof that one can free

60 Horst-Heino v. Borzeszkowski and Renate Wahsner
mechanics of the metaphysical element 'absolute space' by replacing it with something that is nearer to experience. In this way, inertia seemed to him caused by cosmic masses.
Mach was encouraged to use this formulation by his analysis of Newton's arguments concerning the behavior of water in a rotating bucket. According to Newton, the curved surface of water arising when the water was rotating with respect to the heavens showed that inertial forces are caused by the motion with respect to the absolute space. In contrast to Newton, Mach believed that this experiment is no proof of the existence of an absolute space, since one can ask what would happen if the whole of the heavens rotated around the bucket. He believed that it should lead to the same result, namely to a curved surface of water as a consequence of the rotating cosmic masses. So one cannot - he argued - distinguish between relative and absolute motions by experience. One can, however, talk about the real (relative) motion with respect to the cosmic masses.
Now it is not intended to discuss here the problem of the extent to which Mach did really provide a formulation of mechanics which could be used in physical work (for a detailed discussion of this, see, for instance, Bunge 1966). The point we want to stress is rather that the starting point of Mach's considerations was a misunderstanding of mechanics. When Mach started he believed that the space of Newtonian mechanics is a rigid background given once and for all like a stage in front of which physical processes unfold. He did not see that the socalled absolute space is the totality of all inertial systems and thus is not
a metaphysical ghost but a constructive element like the quantity mass
and other notions that are determined by the entire system of classical mechanics.
To be fair, it should be mentioned that in Mach's day classical mechanics was taught in a version which indeed was loaded with metaphysical ballast. Furthermore, when the first edition of Mach's Mechanik was published, the clarifying papers of Carl Neumann (Neumann 1870) and Ludwig Lange (Lange 1886) were not well known or even not yet published. Finally, the meaning of inertial systems was only understood when the role of Galileo's principle of relativity was cleared up, and this was only done in the context of the discussion around Einstein's special theory of relativity.
Mentioning these objective reasons for Mach's misreading of classical mechanics, however, one has also to state that it was too a consequence of his philosophical standpoint, i.e., of his empiropragmatic philosophy. This philosophy replaced the system (the physical

Mach and Einstein: Two Misunderstandings 61
theory) by a catalog of experimental data and their mutual relations. Mach wrote:
Wenn man eine vollstiindige Theorie als das Endziel der Forschung bezeichnen wollte, ... mussten [wir] unter diesem Namen vielmehr erne vollstiindige systematische Darstellung der Thatsachen begreifen ... Das Ideal aber, dem jede wissenschaftliche Darstellung wenn auch sozusagen asymptotisch zustrebt, ... ist ein vollstiindiges ubersichtliches Inventar der Thatsachen eines Gebietes. (Mach 1896, p. 461) [If we wish to say that a complete theory is the [mal aim of research ... we [must] understand by this word a complete and systematic representation of the facts ... But the ideal to which every scientific representation tends (even though only so to speak asymptotically) ... is a complete and clear inventory of the facts of a domain (quoted with slight alteration from Mach 1986, p. 415).]
Therefore, Mach did not and could not realize in what manner a physical theory determines its notions. He overemphasized the role of that what he called the real (das Tatslichliche), so that his expurgation of metaphysics from physics degenerated into a liquidation of basic epistemological prerequisites of physics (Wahsner and v. Borzeszkowski 1988, pp. 595-597).
Because of his missing insight into the inevitability of transcendental assumptions of physics, it was difficult for Mach to incorporate results of authors like Lange clarifying the notion inertial system into later editions of his Mechanik. In the second edition of 1889, one finds, for instance, an Appendix with remarks on Lange's 1886 paper, but no change of the main text of his book. Subsequent editions then incorporate the supplements and other insertions into the main text. Here one feels that he has a lot of problems accepting Lange's definition of inertial systems without changing his own criticism of Newtonian mechanics. His way out of this dilemma is to say that Lange's answer to the question as to the reference system of mechanics and thus to the notion of space is purely mathematical, while his own is physical.
Let us now turn to Einstein and his attitude to Mach's ideas. As is well known, Einstein did not refer to Mach during the first period of the foundation of the theory of general relativity. Only when he arrived at the conclusion that the principle of relativity should be extended to arbitrarily moving reference systems and that gravitation is to be described by the metric tensor of a curved spacetime did he begin to talk of the relativity of inertia (this was about in 1912). In a 1912 paper he writes:

62 Horst-Heino v. Borzeszkowski and Renate Wahsner
Es legt dies die Vermutung nahe, dass die ganze Triigheit eines Massenpunktes eine Wirkung des Vorhandenseins aller iibrigen Massen sei, auf einer Art Wechselwirkung mit den anderen beruhend. [This makes it plausible that the entire inertia of a point mass is the effect of the presence of all other masses, deriving from a kind of interaction with the latter.] (Einstein 1912, p. 39).
And in 1918 he even used the expression Mach's Principle (Einstein 1918).
Although Einstein's attitude to Mach's ideas changed in his later years (see, for example, Pais 1982; Wahsner and v. Borzeszkowski 1988), this principle played a stimulating and constructive role in physical discussions in the course of years. While initially the question as to validity of the principle in the theory of general relativity was in the center of interest, later this principle became the point of departure for the construction of alternative gravitational theories. In this connection, different authors were working with different formulations of this principle. In an analysis of this situation, it was stated (Goenner 1981) that this is due to the fact that Mach did not propose a definite ansatz for an induction of inertia by cosmic masses, so that Mach's principle says more about Einstein's and other authors' reading of Mach than about Mach's intention.
The thesis in favor of which we will give arguments here goes a step further. It says that Mach did not only not create a cosmic principle of the type gathered by Einstein from Mach's Mechanik but such a principle is even in conflict with Mach's ideas.
To this end, let us return to Mach's philosophy. As mentioned above, as a consequence of his empiro-pragmatic standpoint, Mach could not understand the status of a physical theory. The analysis of the discussion between Mach and Boltzmann, Planck, Hertz, and Einstein (Wahsner and v. Borzeszkowski 1988, pp. 604-642) makes this especially clear. Thus Mach could not grasp in what a manner a physical theory determines its notions and, in particular, not understand that Newton's axioms determine simultaneously the physical dynamics and the systems of reference to which this dynamics refers or has to be referred. Since he could only conceive of a catalog of single statements and facts but not of a theory, he could only ask whether a statement under consideration is a fact or not. In this scheme there is no room left for the space notion of classical mechanics. Therefore, he did not think of another physical theory as an answer to his criticism of Newtonian mechanics. He did not think at all in terms of theories, and Einstein's

Mach and Einstein: Two Misunderstandings 63
theory of general relativity had to seem to him further from experience than Newton's theory. To repeat, the aim he had was to reformulate classical mechanics so that its notions and statements were nearer to experience.
This is one line of argument showing that Mach did not think of a new cosmic principle that would lay the foundation of a new theory. But there are also more explicitly formulated arguments that one can find in Mach and which show the same.
When Mach had shown that the law of inertia can also be referred to the cosmic masses, he added that this reading implies the same difficulties as Newton's. For, in the Newtonian version one has to refer
to absolute space, on which one cannot get a hold. In the other case,
only a limited number of masses is accessible to our knowledge but not the totality of cosmic masses. In Mach's words:
In dem einen Fall konnen wir des absoluten Raumes nicht habhaft werden, in dem anderen Fall ist nur eine beschriinkte Zahl von Massen unserer Kenntnis zugiinglich, und die angedeutete Summation ist also nicht zu vollenden (Mach 1912, p. 230). [In the one case we are unable to come at an absolute space, in the other a limited number of masses only is within the reach of our knowledge, and the summation indicated can consequently not be fully carried out (Mach 1960, p. 289).]
For Mach, these obstacles were of a fundamental nature, so that he did not believe that one could overcome them by modifying the physical theory. According to him, the universe as a whole is not tractable as a
physical system. Notions like energy of the universe or entropy of the universe have no tangible sense because they imply applications of
measurement notions to an object that is not accessible to measurement (Mach 1896, p. 338). The only thing he wanted to do was to bring to our attention the fact that the law of inertia (and other physical laws) is based on experience, on experience that is never complete and, even more, that can never be completed.
To conclude, Einstein introduced a cosmic principle into physics, and the irony of the story is that this was initiated by Mach and called by his name, although the possibility of cosmology as a physical discipline was the very thing that Mach himself denied.

64 Horst-Heino v. Borzeszkowski and Renate Wahsner
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Neumann, Carl (1870). Ober die Principien der Galilei-Newtonschen Theorien. Leipzig: Teubner.
Pais, Abraham (1982). 'Subtle is the Lord. ' The Science and the Life ofAlbert Einstein. Oxford/New York: Oxford University Press.
Wahsner, Renate and v. Borzeszkowski, Horst-Heino (1988). "Nachwort der Herausgeber." In Ernst Mach, Die Mechanik in ihrer Entwicklung. Historisch-kritisch dargestellt. Renate Wahsner und Horst-Heino v.Borzeszkowski, eds. Berlin: Akademie-Verlag. pp. 563-647.
Discussion
Norton: I wanted to see if you have had any more luck with the historical puzzle than I have had. We seem to agree that Einstein is misreading Mach. I have tried to get some idea of where Einstein got the reading from. Is it possible that he just read Mach by himself, we know he read it in his early years, and produced this reading or is it

Mach and Einstein: Two Misunderstandings 65
possible that he had some help? Was there some intermediate source? Did he read about Mach elsewhere? For a while I conjectured that Philip Frank had played some intermediate role on the basis of Frank's 1909 paper that I cited and the fact that Frank was, I believe, Einstein's successor at Prague, but I couldn't come up with anything. I don't know if you might have come across something. Borzeszkowski: I don't know. Maybe there was such an intermediate stage, but I think the main reason is that Einstein read only Mach's Mechanics, and then he did what a physicist should do. He tried to win from it a constructive idea.
One finds it also in other connections that when authors discussed epistemological questions and they were talking about philosophy, Einstein read it in a physicalizing manner, and, to repeat, this can be useful for purely physical considerations. One encounters, however, another situation when one wants to discuss such matters as we did this morning, namely the relation to philosophy and historical context. Then one has, of course, to analyze the whole edifice of thoughts of the author one refers to. Bondi: You have not mentioned the very interesting statement of Mach's "the universe is only given once," which I think influenced Einstein. It certainly influenced me. To me, it means all our physics is learned in the presence of just the universe we've got and of no other. Borzeszkowski: Yes, I wanted to mention it, but I didn't due to the shortage of time. Because it is a further hint that Mach did not really mean that one should construct a physics that starts from a cosmological principle. In his Wlirmelehre, for instance, he says that one can't use the notions which we know from physics like energy and so on which one applies to several different finite systems to the universe as a whole, because it is only once given. More precisely, Mach says (Wlirmelehre 1900, p. 338) that sentences on the energy, entropy, and so on of the universe have no conceivable sense since they contain applications of measuring notions to an object which is not accessible to measurement. I completely agree.
Von Borzeszkowski and Wahsner: Two comments on Julian Barbour's comments (pp. 215-218). (i) With Eq. (1) describing the change of the relative distance between two bodies moving purely inertially, Mach presents a further simple implication of Newton's laws - here, in particular, of the first law. This passage shows once more, first, that he considered Newtonian mechanics to be true and, second, that he believed that the laws and statements of this physics can be reformulated so that, instead of absolute, relative distances occur. That

66 Horst-Heino v. Borzeszkowski and Renate Wahsner
this, to some extent, is possible, C. Neumann (Uber die Principien der Galilei-Newtonschen Theorie, Leipzig 1870) had already shown by demonstrating that, choosing an arbitrary body alpha, mechanics can be written in Jacobian coordinates. Roughly speaking, Mach intended to rewrite Newtonian mechanics by replacing the body alpha by something one can call 'the totality of cosmic masses,' maybe, the center of cosmic masses.
(ii) Mach's criticism of Lange that one finds in some editions of his Mechanik shows that he did not mention that, accepting - as he did - Lange's construction of an inertial system, for reasons of logical self-consistence, a fourth force-free material point must follow with respect to one of Lange's inertial systems a straight line (uniformly). The passage here under consideration shows again Mach's initial misunderstanding, not only of Newton but also of Lange. This led him to a dim formulation. One should not, however, forget that Mach himself dropped this passage later. In later editions, in particular in the last edition supervised by the author, Mach agrees with Lange. There his point then was to state that Lange's point of view need not be the last word. Mach could imagine a physics describing Friedlaender-F6ppl effects. Anyone looking for a passage in Mach that can be read as something like Einstein's version of Mach's Principle should take this one (cf. Chap. 2, Sec. 6, Subsec. 11 in the 7th edition).

Einstein's Formulations of Mach's Principle
Carl Hoefer
It is well known that Einstein first used the term 'Mach's Principle' in his 1918 paper on the general theory, "Prinzipielles zur allgemeinen Relativitatstheorie." In that paper Einstein expresses his current understanding of the requirements of Mach's ideas on inertia:
Mach's Principle: The G-fie1d is without remainder determined by the masses of bodies. Since mass and energy are, according to results of the special theory of relativity, the same, and since energy is formally described by the symmetric energy tensor (TI'.)' this therefore entails that the G-fie1d be conditioned and determined by the energy tensor. 1
What is less well known is that Einstein struggled with other ways of understanding Mach's ideas on inertia in the context of the general theory, only arriving at his 1918 conception after failing adequately to cash out Mach's ideas in other ways in the years from 1912 to 1917; and that Einstein had to abandon this 1918 formulation of Mach's Principle by the middle of that year.
In this paper I will discuss some of the history of Einstein's work on Mach's Principle, by identifying several distinct ways that Einstein adopted, at various times, of formulating the general idea that we now call 'Mach's Principle. '2 Many important details of Einstein's work in 1916 on Mach's Principle are not widely known and deserve greater attention from those interested in Machian ideas on inertia.. In addition to highlighting some very puzzling aspects of Einstein's work on Mach's Principle, I will also contend that, compared with the 1918 formulation, Einstein's 1917 formulation in his cosmological paper (Einstein 1917) was correct in some crucial respects, even though it conflicts with much current usage of the term 'Mach's Principle' in the physics community.
Einstein Studies, vol. 6: Mach's Principle: From Newton's Bucket to Quantum Gravity, pp. 67-90 © 1995 Birkhiiuser Boston, Inc. Printed in the United States.

68 Carl Hoefer
1. Pre-1916 Formulations and Puzzles
I want to begin with a quote from 1912, which I believe is the earliest expression of Einstein's understanding of Mach's Principle. This is from the paper "Gibt es eine Gravitationswirkung, die der elektrodynamischen Induktionswirkung analog ist?":
This suggests the hypothesis that the whole inertia of any material point is an effect of the presence of all other masses, depending on a kind of interaction with them.3
A footnote citing Mach's The Science of Mechanics follows this sentence in the text. This is only the first of many such passages to appear in Einstein's writings between 1912 and 1918. In 1912, Einstein had not yet made the move to working on gravitation through field equations linking a metric tensor to material tensors, using what was then called the absolute differential calculus. When he did make this move, it affected his expressions of Mach's Principle in two main ways. First, it immediately suggested to Einstein that the metric tensor should be 'determined' by the tensor describing matter and energy (the core of the 1918 formulation). Second, it led Einstein to equate the achievement of generally covariant field equations for the metric with a complete relativization of motion - and therefore, presumptive satisfaction of Mach's Principle. These connections will be discussed further below.
The most important differences between Einstein's understanding of Mach's Principle in the 1913-1915 period, and the 1918 formulation, are two: First, in the early period there is (apparently) no recognition of the fact that an empty spacetime with Minkowski structure is incompatible with Mach's Principle; and second, Einstein substantially equated general covariance, Mach's Principle, and the equivalence principle. This equation was responsible for several conceptual problems that plagued Einstein prior to November, 1915, and also helped shape the next stage of Einstein's thinking on Mach's Principle, in 1916. An examination of Einstein's understanding of covariance, the equivalence principle, and their relations to Mach's Principle, is therefore necessary for understanding Einstein's later formulations.
The next passage of interest is from Einstein's 1913 exposition of the Entwuif theory that appeared in the Vierteljahrsschrijt der Natuiforschenden Gesellschajt Zurich. In the paper Einstein describes the highest goal of a theory of gravitation as being the task of determining the (components of the) metric, when the 'field-creating' material contents

Einstein's Formulations of Mach's Principle 69
of the world are given. Here we see the core of the 1918 formulation expressed. Later, Einstein takes up Mach's ideas explicitly and claims that the Entwuif theory overcomes the 'epistemological defect' of absolute acceleration that Mach criticized:
The theory sketched eliminates an epistemological defect, emphasized particularly by E. Mach, that affects not only the original relativity theory but also Galilean mechanics. It is plausible to suppose that the concept of the acceleration of a material particle can no more have an absolute meaning ascribed to it than the concept of velocity.... One must demand that the occurrence of an inertial resistance be tied to the acceleration of the body under consideration relative to other bodies....4
Einstein appears to claim that Mach's Principle is fully implemented in the Entwuif theory, since the epistemological defect of earlier mechanics has been overcome. In other passages from this period, Einstein's claims are more modest, describing the Machianization of inertia as still not yet complete in the Entwuiftheory. Still, this claim of having overcome the epistemological defect, which is repeated in other writings of 1913-1914 (and in the 1916 review paper on the final GTR), is remarkable to modern readers. The reason is that it is clear that Minkowski spacetime is a solution of the Entwuif equations and that Einstein realized this. But Minkowski spacetime is the most clearly anti-Machian spacetime possible: It has a well-defined inertial and metrical structure, without any matter being present that could be said to 'determine' or explain that structure. We will come back to this point below; first, some further intriguing passages from this period.
In a long exposition of the whole of his current general relativity theory of October 1914, Einstein invokes the Machian conception of inertia in a way that suggests strongly that he viewed it as the same as the principle of equivalence. After mentioning the Newtonian argument from centrifugal effects to the existence of absolute motion, Einstein expresses the Machian response particularly clearly:
We need not necessarily trace the existence of these centrifugal forces back to a[n absolute] movement of K'; we can instead just as well trace them back to the rotational movement of the distant ponderable masses in relation to K', whereby we treat K' as 'at rest. ' ... On the other hand, the following important argument speaks for the relativistic perspective. The centrifugal force that works on a body under given conditions is determined by precisely the same natural constants as the action of a gravitational field on the same body [i.e., its mass], in such a way, that we have no means to

70 Carl Hoefer
differentiate a 'centrifugal field' from a gravitational field .... This quite substantiates the view that we may regard the rotating system K' as at rest and the centrifugal field as a gravitational field. 5
The reader will recognize the spirit (if not the letter) of the equivalence principle in Einstein's reference to the fact that one natural constant is involved in the definition of both gravitational and centrifugal forces, and in the idea of our being able to regard the system K' (here rotating, rather than uniformly accelerated) as being at rest in a gravitational field. To regard a centrifugal force field as being a gravitational field is, on the one hand, a natural Machian move: Inertial forces of all kinds are produced by interaction with other masses, and hence are just gravitational forces. On the other hand, it is an extension of the equivalence principle, extending what Einstein had claimed about an inertial system K and a uniformly accelerated system K', to systems K' accelerated in different ways.
The connection between general covariance and the equivalence principle can be seen as follows. The equivalence principle shows that we can extend the validity of the equations of a theory of motion to reference frames that are uniformly accelerated, so long as we regard them as in the presence of a uniform gravitational field. But the
extension of the validity of the equations of a theory to all reference
frames, including uniformly accelerated ones, is just what is achieved by general covariance. Therefore, as Einstein wrote in 1916, "The
requirement of general covariance of equations embraces the principle of equivalence as a quite special case.,,6 To modern readers, it seems
clear that this reasoning confuses reference frames with coordinate systems, and that the purely formal requirement of general covariance is in fact unrelated to the equivalence principle. But this is somewhat unfair. Einstein's understanding of general covariance was more robust
than the modern view, and this is clear from his 1918 Prinzipielles
paper. In particular, he viewed the absence of prior absolute spatiotemporal structure in GTR (a feature not shared by generally covariant formulations of other theories) as crucially part of what he understood by 'general covariance.' Therefore, GTR did implement
general covariance in a way that does not necessarily make it misleading
to say that the equations of the theory apply in accelerated reference frames just as they do in unaccelerated frames (in so far as such frames can be meaningfully defined and related to coordinate systems) - and so such frames may be considered 'at rest' in a kind of gravitational field.
Having now seen the links between Mach's Principle and the

Einstein's Formulations of Mach's Principle 71
equivalence principle, and between the equivalence principle and general covariance, there remains the question of the link between general covariance and Mach's Principle. This latter link proceeds through the idea of an extended principle of relativity: General covariance is sufficient to ensure that no reference systems are privileged, this ensures the extension of the principle of relativity to arbitrary motions, and this is what is demanded by the Machian criticism of absolute motion. In the Entwuif period, this line is never pursued to completion by Einstein because of his uneasy conviction that, despite lacking general covariance, the Entwuif equations do implement Mach's Principle and a general relativity of motion. A discussion from the 1914 paper "Die formale Grundlage der allgemeinen Relativitatstheorie" (Einstein 1914a) illustrates Einstein's dilemma. After arguing for the special principle of relativity from the fact that from a kinematical standpoint all coordinate systems should be considered equal, Einstein continues:
This argument, however, immediately provokes a counter-argument. The kinematic equivalence of two coordinate systems, namely, is not restricted to the case in which the two systems, K and K', are in uniform relative translational motion. The equivalence exists just as well, from the kinematical standpoint, when for example the two systems rotate relative to one another. One feels therefore forced to the assumption that the previous relativity theory is to be generalized in a far-reaching way, so that the apparently incorrect privileging of uniform translations as opposed to other sorts of relative motions disappears. 7
Einstein follows this passage immediately with a discussion of the Newtonian argument against such a widened relativity: the argument from inertial effects of rotation for the absoluteness of such motion. Einstein goes on in the usual way to give the Machian response, thus showing the identification of Machianization of inertia and a general relativity of motion in his thinking. But what about general covariance? Because of the non-general covariance of the Entwuifequations, Einstein postpones the question until after he has had a chance to explain the reasons for this apparent failure.
The question now naturally arises, what kinds of reference systems and transformations we should regard as 'justifiable.' This question will however first be answered much later (section D). In the meantime we shall take up the standpoint that all coordinate systems and transformations are to be allowed, so long as they are compatible with the always-presupposed conditions of continuity.s

72 Carl Hoefer
At least three puzzles arise in the above passages on Mach, relativity, and covariance:
(l) Einstein throughout this period viewed the extension of the covariance of a theory to be crucial to an extension of the relativity of motion in the theory. But the Entwurf theory had covariance properties that Einstein recognized to be far short of what would be required for an extension of relativity to acceleration and, in particular, rotation. For a brief period in 1914, Einstein thought that covariance over a wide class of transformations, including uniform rotations, pertained to the theory's equations; but this proved to be an error.
(2) Einstein equated the extension of covariance to cover acceleration transformations with the principle of equivalence, on the one hand; and on the other hand, he equated an extended equivalence principle with the implementation of Mach's ideas on the origin of inertial forces (as seen in the quote just above). But Einstein maintained that Mach's ideas were implemented in the Entwurftheory, without clearly explaining how this could be the case without a corresponding general (or at least wide-ranging) covariance of all the theory's equations.
(3) Einstein articulated Mach's ideas as the demand that the metric field gp.. should be fully determined by the material distribution Tp.•. And Einstein was aware that in the Minkowski spacetime of the Special Theory (with Tp..=O), Mach's ideas are violated and spacetime has a structure of its own. But Einstein repeatedly referred to the Minkowski metric as the proper case of no gravitational field being present, and never discussed the obvious problem that this metric is a valid solution of the Entwurf equations.
I believe that some of these puzzles can be resolved through the following story on Einstein's thinking in this period.
The Entwurftheory is not generally covariant, and even at the time Einstein was prone to view this as a serious defect of the theory given its goal of generalizing the relativity of motion. A letter to Lorentz from August 1913 makes this clear, as Einstein writes:
But the gravitation equations themselves unfortunately do not have this property of general covariance. Only their covariance under linear transformations is established. But now, the entire trust in the theory rests on the conviction that acceleration of a reference system is equivalent to a gravitational field. 9
And in a letter just two days after this one, Einstein refers to the lack of general covariance as an "ugly dark spot" on the theory.

Einstein's Formulations of Mach's Principle 73

But Einstein had two arguments that he believed justified and

explained the lack of general covariance. 10 One was the now-notorious

'hole argument.' The other was Einstein's temporary belief that this

failure was justified by the necessity of restricting coordinate systems to

those in which the conservation law below holds11 :

E•

ax. aCT o.

+t ) o.

=0.

(1)

As an ordinary partial differential equation, this equation is not generally covariant if the quantities Ta> and ta> are tensors; given that it holds in a coordinate system, it will then hold also only in coordinate systems related to the first by a linear transformation. By laying the blame for failure of general covariance at the feet of the conservation law, Einstein was able to persuade himself that this failure of the equations to hold in arbitrary systems did not bring with it a failure of the Machian idea that the metrical structure of spacetime should be determined by the material distribution. For, as Einstein (19l4b) pointed out in the Scientia article of 1914, there is no prior selection of certain coordinate systems or frames as privileged; instead, the specification of the material distribution appears subsequently to pick out certain coordinate systems, naniely those in which (1) hold. Einstein held - and this was his belief in the Machian character of the Entwurf theory - that the material distribution determines the metric field. So, if one imagines that the material distribution had instead been laid out differently on the spacetime manifold, the metric (and, hence, the class of privileged coordinate systems, since the metric restricts the systems in which the conservation law can hold) would 'follow' the material distribution; and this shows that they are only frames privileged by the actual material distribution, not frames privileged in an absolute way as in Newtonian mechanics. 11
There is still some tension left, since Einstein never describes how the covariance limitations imposed by the hole argument affect the question of relativity of motion and the existence of privileged coordinate systems. Einstein never fully dropped the linkage of covariance to the relativity of motion in the back of his mind (even after 1918); rather, he was greatly relieved in November 1915 when he finally achieved the generally covariant equations of GTR. Nevertheless, this interpretation clears up some puzzles about Einstein's thinking on Mach's Principle in the Entwurfperiod.
But the problem remains that the material distribution does not fully determine the class of inertial systems in the Entwurf theory, despite

74 Carl Hoefer
Einstein's claims. This is shown by the compatibility of Minkowski spacetime with the field equations. The Entwurftheory faces the same problem of models with absolute or quasi-absolute inertial structure that the final GTR faces.
How did Einstein reconcile the Minkowski spacetime solution with the allegedly Machian character of the theory? This is a problem that carries over into the early period after the discovery of the final GTR field equations, since at that time too Einstein claimed that his theory overcame the problem of the Machian epistemological argument against privileged frames. 12 I believe that insofar as there is a solution to this puzzle, it is the same for both periods: Einstein was aware of the difficulty, even though he did not mention it in his published writings of
the time; and he intended to overcome it by finding suitable boundary conditions to impose on physically realistic solutions, conditions that
would rule out the empty Minkowski spacetime. At the latest, Einstein was already working on such conditions by May 1916, as they are mentioned in a letter to Besso of that month. Einstein may have been working on them much further back, in late 1915 or early 1916, and he may well have had the idea in the Entwurfperiod.
2. 1916: Mach's Principle as Boundary Conditions
In the next stage of Einstein's work on Mach's Principle, then, there are two kinds of formulations of Mach's Principle to be found. First, continuing expressions of the type "The metric g,... should be completely determined by the material distribution T",v'" And second, mathematical expressions of Mach's Principle through the idea of Machian boundary conditions that would supplement the field equations and eliminate non-Machian solutions. In the face of Einstein's recognition in 1916 that the field equations alone do not satisfy Machian demands, the Machian boundary conditions sought would have amounted to the implementation of Mach's Principle in GTR.13
Einstein thought that Mach's Principle should be implemented in GTR through boundary conditions rather than by some more general mathematical constraint, because of the way in which he saw the problematic models as violating Mach's Principle. Minkowski spacetime and Schwarzschild's solution both violate Mach's Principle because they display metrical/inertial structure that cannot be attributed to a material distribution. In the case of the Schwarzschild solution, this structure is evident at large T, where spacetime is essentially Minkowskian and the

Einstein's Formulations of Mach's Principle 75

central mass evidently is not responsible for that structure. Instead, on Einstein's way of thinking, the Minkowski boundary conditions imposed in deriving the solution are to blame for that absoluteness. The same perspective can be used in thinking of empty Minkowski spacetime. The metrical structure at any point is a result of the 'absolutist' boundary conditions, plus the local (lack ot) material distribution.
Given this perspective, the way to avoid violations of Mach's Principle is to come up with boundary conditions that do not impart any absolute structure to spacetime, so that the structure of spacetime at finite distances from the center is attributable only to the global matter distribution, not to the boundary conditions as well. Einstein expresses this idea in a letter to Willem de Sitter, from June 1916:

I am sorry to have plagued you with too much emphasis on the question of boundary conditions.... But I must add that I have never thought about a temporally finite extension of the world; and even spatially, the finite extension is not what matters. Rather, my need to generalize drove me to the following view: It is possible to give a spatial envelope (massless geometrical surface) (in four dimensions, a tube) outside of which a gram weight has as little inertia as I choose to specify. Then I can say that inside the envelope, inertia is determined by the masses present there; and to be sure, only by these masses. 14

Very little survives about Einstein's work on such conditions. All we have to go on are the clues from the above verbal expression, the 1917 Betrachtungen paper discussion (Einstein 1917), and brief reports by de Sitter in two 1916 articles. The boundary conditions that de Sitter provides, as well as others that he himself proposes as cashing out Mach's Principle, turn out to be in themselves meaningless, for reasons I will discuss below. But a mathematical reconstruction of what Einstein's calculations may have involved in this period might be able to shed more light on Einstein's temporary belief in a boundary conditions approach to Mach's Principle.
De Sitter reported in September 1916 that Einstein 'found' that the following set of boundary conditions at infinity satisfies the demand for the complete relativization of inertia15:

000 00

o 0 0 00

(2)

o 0 0 00

00 00 00 00 2

76 Carl Hoefer
De Sitter gives no explanation of the exponent on g44; he does remark that the invariance of these values is restricted to transformations in which x4' is a function of x4 alone.
This set of boundary values, as well as two others proposed by de Sitter in 1917, were intended to cash out an interesting idea that Einstein seems to have held for a while in 1916: The General Theory needed supplementation by generally covariant boundary conditions in order to secure the complete relativization of inertia. 16
The idea is that the boundary values of giL> at infinity should be such that they are left unchanged by a wide group of coordinate transformations (at least those corresponding to rigid motions - a requirement well short of general covariance). Choosing boundary values of either 0 or 00 for the relevant metric components seems superficially to be a good way to approach this idea, but this alone falls well short of guaranteeing anything about what will happen under a coordinate transformation. With a given metric expressed in some coordinate system, for example, a component that approaches zero or infinity in some limit may well fail to do so after a transformation such as a linear acceleration or rotation. Whether this is so or not depends on the given metric. In fact, it is not too strong to say that boundary conditions such as (2) are completely meaningless, until they are linked to one or more concrete metrics.
Unfortunately, this set of boundary conditions is not accompanied by any discussion by de Sitter of how they arise, i.e., what sort of actual functions might be compatible with the field equations, and also take these limiting values. Without a concrete example, of course, there is no way to verify that they do in fact represent a boundary region in which inertia 'disappears,' in some intrinsic sense. Further, much general work delimiting the class of metrics that can take such boundary values (with a given definition of 'at infinity') would be necessary in order to establish that the boundary conditions correctly capture Machian demands.
Einstein may have subscribed to these boundary conditions for perhaps as long as four or five months, from before September 1916, to December 1916, at which point Einstein had already turned to the cosmological constant and his closed universe, abandoning the idea of Machian boundary conditions.
A letter to Besso from December 1916 shows Einstein giving, in brief, the argument against boundary conditions and in favor of a closed world, that he would repeat in more detail in the Kosmologische

Einstein's Formulations of Mach's Principle 77
Betrachtungen paper. In this letter Einstein formulates what he sees as the dilemma facing him:

It's certain that inftnitely large differences of potential would have to give rise to stellar velocities of very signiftcant magnitude, and these would surely have already have manifested themselves long ago. Small potential differences in combination with an inftnite [spatial] extent of the world demand the emptiness of the world at inftnity (constancy of the gil-vat inftnity given appropriate choice of coordinates [Minkowski conditions]), in contradiction with a meaningfully understood relativity. Only the closure of the world frees us from this dilemma. 17

The technical part of the argument against the boundary conditions (equation) based on stellar velocities is made only somewhat more clear in Einstein's Kosmologische Betrachtungen paper (Einstein 1917). I present the entire relevant excerpt below (the English translation only, due to its length):

The opinion wllich I entertained until recently, as to the limiting conditions to be laid down in spatial inftnity, took its stand on the following

considerations. In a consistent theory of relativity there can be no inertia

relatively to "space," but only an inertia of masses relatively to one

another. If, therefore, I have a mass at a sufftcient distance from all other masses in the universe, its inertia must fall to zero. We will try to formulate this condition mathematically.

According to the general theory of relativity the negative momentum is

given by the ftrst three components, the energy by the last component of the covariant tensor multiplied by (_g)1/2

mFigil-a ~a ,

(3)

where, as always, we set

ds 2 =g dxdx.

(4)

ILl! p. '"

In the particularly perspicuous case of the possibility of choosing the system

of coordinates so that the gravitational fteld at every point is spatially

isotropic, we have more simply

ds 2 = -

A(dx

2 1

+

d

x

i

+dx

;)

+Bdx;.

(5)

If, moreover, at the same time

Fi =1=,jA3B

we obtain from (4), to a ftrst approximation for small velocities,

m_A _dx1 , mA-d-x2, mA-d_x3 VB dx4 VB dx4 VB dx4

78 Carl Hoefer

for the components of momentum, and for the energy (in the static case)

m.;B.

From the expressions for the momentum, it follows that m(A/-JB) plays the part of the rest mass. As m is a constant peculiar to the point of mass, independently of its position, this expression, if we retain the condition (-gy/2= I at spatial infmity, can vanish only when A diminishes to zero, while B increases to infmity. It seems, therefore, that such a

degeneration of the coefficients gl'> is required by the postulate of relativity of all inertia. This requirement implies that the potential energy m./B becomes infinitely great at infmity. Thus a point of mass can never leave the system; and a more detailed investigation shows that the same thing applies to light-rays. A system of the universe with such behavior of the

gravitational potentials at infmity would not therefore run the risk of wasting away which was mooted just now in connexion with the Newtonian theory ....
At this stage, with the kind assistance of the mathematician J. Grommer, I investigated centrally symmetrical, static gravitational fields, degenerating at infmity in the way mentioned. The gravitational potentials

gl'> were applied [angesetzt], and from them the energy-tensor TI'> of matter was calculated on the basis of the field equations of gravitation.1 8 But here it proved that for the system of the fixed stars no boundary conditions of the kind can come into question at all, as was also rightly emphasized by the astronomer de Sitter recently.
For the contravariant energy-tensor P> of ponderable matter is given

by

TI'>

=p

dxdx I' >

didi'

where p is the density of matter in natural measure. With an appropriate choice of the system of coordinates the stellar velocities are very small in comparison with that of light. We may, therefore, substitute (g44)1/2dx4 for ds. This shows us that all components of P> must be very small in comparison with the last component T 44 . But it was quite impossible to reconcile this condition with the chosen boundary conditions. In the
retrospect this result does not appear astonishing. The fact of the small

velocities of the stars allows the conclusion that wherever there are fixed

stars, the gravitational potential (in our case-JB) can never be much greater than here on earth. This follows from statistical reasoning, exactly as in the case of the Newtonian theory. At any rate, our calculations have convinced me that such conditions of degeneration for the gl'> in spatial infmity may not be postulated. 19

This discussion contains all the evidence there really is, about why

Einstein's Formulations of Mach's Principle 79
Einstein abandoned his boundary-conditions formulation of Mach's Principle. When one considers that this transition was Einstein's motivation for introducing the :\-term, and thus beginning modern cosmology with his 1917 closed model of the universe, the absence of discussion of this passage in the literature is quite remarkable.20 A reconstruction of Einstein's calculations on the question of boundary conditions would significantly enhance our understanding of the early history of GTR.
Without yet having such a reconstruction in hand, it is still possible to note some doubtful aspects of Einstein's reasoning in the 1917 paper. Because Einstein at this time (with everyone else) did not fully grasp the difference between coordinate effects (for example, singularities) and intrinsic effects, his results concerning large potentials and large stellar velocities have to be clearly reconstructed before they can be accepted as sound. Even if large stellar velocities are derived in some intrinsically meaningful sense, it has further to be shown whether these velocities would be observable from the earth, and whether they would be velocities in a static space, or 'velocities' like the velocities of recession of distant galaxies, which are a function of the expansion of spacetime. There are ample reasons to doubt that Einstein's arguments against the boundary-conditions approach are sound - though there are also ample reasons to doubt that the approach itself makes sense to begin with.
3. 1917: The Closed-Universe Formulation
The boundary-conditions expression of Mach's Principle was replaced by the demand of a closed universe, and not by any explicitly mathematical reformulation of the principle. Instead, Einstein's 1917 - early 1918 understanding of Mach's Principle can be cashed out only verbally, as comprised of two demands:
(I) The universe should be finite and closed, Le., have no boundary region; then, the local metric cannot be thought of as determined in part by boundary conditions of space (but rather only by the global matter distribution). This, Einstein thought, would assure that the metric is fully determined by the matter distribution in spacetime.
(II) The modified field equations should allow no matter-free, singularity-free solutions. This is necessary to ensure that the theory as a whole (not just some subset of models) is Machian in character.
Demand (I) emerges clearly from the Kosmologische Betrachtungen

80 Carl Hoefer
paper and the December 1916 letter to Besso. Both (I) and (II) are present in the 1918 discussion in the Prinzipielles paper but are stated even more explicitly by Einstein in a letter to de Sitter of March 24, 1917:
In my opinion it would be dissatisfying, if there were a conceivable world without matter. The ~v-field should rather be determined by the matter, and not be able to exist without it. This is the heart of what I understand by the demand for the relativity of inertia. One could just as well speak of the "material conditionedness of geometry." As long as this demand was not fulfilled, for me the goal of general relativity was not yet completely achieved. This was first achieved through the introduction of the A term. 21
In demand (I) we see the residue of Einstein's conviction that the non-Machian character of certain models is a product of their absolutist boundary conditions: if there is no boundary region, Einstein assumes, there is no room for a non-Machian determination of the metric. This reasoning can only hold, of course, if spacetime is nonempty; this explains the importance of demand (II) for Einstein.
As is now well known, the introduction of the A-term failed to achieve condition (II). In early 1917, de Sitter found a T,.v=O solution to the new field equations. Einstein struggled for over a year to show either that the solution was physically unacceptable due to a singularity, or not really matter-free after all; he gave up the struggle in June 1918, and in an important sense this marks the end of Einstein's advocacy of Mach's Principle. 22
4. Post-1918 Formulations
After accepting the failure of the modified field equations to meet demand (II), Einstein's attempts to implement Mach's Principle in GTR ended, and his enthusiasm for Mach's Principle began a steady decline that culminated, near the end of his life, in complete repudiation of the principle.23 The decline can be explained in part as due to the evident failure to make GTR perfectly Machian, and in part as due to Einstein's growing interest in unified field theories, in which a realistic (as opposed to reductionistic) attitude towards the metric field is presupposed. But Einstein did not cease to discuss Mach's ideas on inertia, in a positive manner, for many years after 1918. Instead, he tended to emphasize the respects in which it seems that Machian ideas are fulfilled in the general theory, and to advocate his closed-universe cosmology as fulfilling the

Einstein's Formulations of Mach's Principle 81
Machian demands due to its lack of a boundary region. The discussion in the textbook The Meaning ofRelativity is representative:
The theory of relativity makes it appear probable that Mach was on the right road in his thought that inertia depends upon a mutual action of matter. ... What is to be expected along the line of Mach's thought? 1. The inertia of a body must increase when ponderable masses are piled up in its neighborhood. 24 2. A body must experience an accelerating force when neighboring masses are accelerated, and, in fact, the force must be in the same direction as the acceleration. 3. A rotating hollow body must generate inside of itself a 'Coriolis field,' which deflects moving bodies in the sense of the rotation, and a radial centrifugal field as well.
Two pages later Einstein continues, discussing his cosmological model
Although all of these effects are inaccessible to experiment, because K is so small, nevertheless they certainly exist according to the general theory of relativity. We must see in them a strong support for Mach's ideas as to the relativity of all inertial actions. If we think these ideas consistently through to the end we must expect the whole inertia, that is, the whole gl'p-field, to be determined by the matter of the universe, and not mainly by the boundary conditions at infinity. 25
After this passage, Einstein discusses his closed universe model, citing with particular favor its lack of boundary conditions.
The differences between Einstein's expression of Mach's Principle here and his 1917 expression described above are subtle but important. Here the emphasis is on Machian-seeming effects that are present in GTR, and on one model that seems both to satisfy the core demand that the metric be determined by the energy tensor and to be physically reasonable. There is no demand that the theory exclude empty (and hence anti-Machian) models in general. And with the absence of this demand, there is no trace of any precise mathematical expression of Mach's Principle.
Einstein seems content with the possibility that the actual world is well described by a model that appears to be Machian by the light of intuition. This attitude toward Mach's Principle is quite common among current relativists who, like Einstein, are sympathetic to Mach's ideas on inertia. Aside from a few, such as Wheeler and Raine, who do try to formulate an explicit mathematical version of Mach's Principle, most

82 Carl Hoefer
physicists are content to rely on their intuitions about Machian effects in the absence of mathematical criteria and to follow Einstein in regarding a closed matter-filled universe as automatically Machian.26
5. Conclusion: The Correct Formulation
I will end with some critical remarks about Einstein's formulations of Mach's Principle and current assumptions among working physicists.
The widespread assumption that a closed, matter-filled cosmology such as Einstein's spherical cosmology must satisfy Mach's Principle is questionable. It is based on the reasoning discussed above, that since in anti-Machian models the trouble seems to come from the boundary conditions, if one eliminates the boundary region one eliminates the problem. But this reasoning is clearly fallacious. There is a missing premise: The only way a model can fail to be Machian is to have an empty boundary region in which an absolute spatiotemporal structure is posited. This premise is by no means intuitively obvious, and it could
only be established if we had a general, mathematical explication of
Mach's Principle and could show that this premise follows from it. Such a mathematical version of Mach's Principle would itself have to be supported by arguments showing that it correctly captures the core of Mach's ideas on the origin of inertia. It would entail a restriction of the class of models of GTR and delimit exactly those models in which inertia is fully determined by matter-energy. I do not know if such a mathematical expression is possible for GTR; but its attainment is necessary before we can claim that any given model does satisfy Mach's Principle.
In the meantime, it seems more clear that we can (as did Einstein) rule out some models of GTR as definitely anti-Machian and use these judgments as constraints on any explication of Mach's Principle. The clearest case is that of empty spacetimes. Since they do not contain any matter-energy and have a definite spatial (and hence inertial) structure, they clearly run contrary to the core of Mach's ideas on the origin of inertia. Therefore, I believe that Einstein was absolutely right to demand (II) as a necessary condition for the relativization of inertia, or satisfaction of Mach's principle by a gravitation theory. Demand (II) should remain our most secure touchstone in theorizing about how to create a Machian gravitation theory.
I stress this point because apparently it has become a minority view among physicists working on Mach's Principle.27 The reasons have to do, I believe, with work done by Wheeler and others on initial-value

Einstein's Formulations of Mach's Principle 83
formulations of Mach's Principle (which do not rule out empty solutions), and also with the widespread view that gravitational field energy [to> of Eq. (1)] should count as part of the energy that helps to determine the metrical structure of spacetime. If gravitational stress-energy were a tensor quantity (and hence well-defined and localizable), this attitude would clearly be appropriate; the Machian demand would then be, roughly, that the combination of material and gravitational stress-energy uniquely determines the whole metric field gp.,' But since this is not the case, the status of gravitational stress-energy as a second kind of matter-energy in the universe is dubious. Furthermore, it is one thing to suppose that gravitational stress-energy present on a hypersurface or thin-sandwich in a matter-containing world should be included as part of the material distribution that determines future inertial structure, but it is quite another to suppose that a matter-free universe could count as Machian in virtue of some conditions satisfied by the gravitational waves present. At any rate, it seems to me that much further work clarifying the status of gravitational stress-energy is needed if we are to abandon the initially compelling view that a matter-free (Tp.,=O) universe is automatically anti-Machian.
Acknowledgments. Citations from Einstein's personal correspondence are reproduced with the kind permission of the Albert Einstein Archives at the Hebrew University of Jerusalem.
NOTES
1Einstein (l918), pp. 241-242. "Machsches Prinzip: Das G-Feldistrestlos durch die Massen der Korper bestimmt. Da Mass und Energie nach den Ergebnissen der speziellen Relativitiitstheorie das Gleiche sind und die Energie formal durch den symmetrischen Energietensor (TI") beschrieben wird, so besagt dies, daB das G-Feld durch den Energietensor der Materie bedingt und bestimmt sei." Throughout, all translations are my own unless otherwise noted.
2A discussion of some of the history of Einstein's work on Mach's ideas can be found in Kerszberg (1989a, b). Some of the material in this paper is also covered in (Hoefer 1993).
3Einstein (1912), p. 39. "Es legt dies die Vermutung nahe, da13 die ganze Tragheit eines Massenpunktes eine Wirkung des Vorhandenseins aller iibrigen Massen sei, auf einer Art Wechselwirkung mit den letzteren beruhend."
4Einstein (1913a), p. 290. "Durch die skizzierte Theorie wird ein erkenntnistheoretischer Mangel beseitigt, der nicht nur der urspriinglichen Relativitiitstheorie, sondem auch der Galilei'schen Mechanik anhaftet und insbesondere von E. Mach betont worden ist. Es ist einleuchtend, dafi dem

84 Carl Hoefer
Begriff der BescWeunigung eines materiellen Punktes ebensowenig ein absolute Bedeutung zugeschrieben werden kann wie demjenigen der Geschwindigkeit .... [Es wird] gefordert werden miissen, daB das Auftreten eines Triigheitswiderstandes an die Relativbeschleunigung des betrachteten Korpers gegeniiber andem Korpem gekniipft sei.... "
5Einstein (1914), pp. 1031-2. "Die Existenz jener Zentrifugalkriifte brauchen wir niimlich nicht notwendig auf eine Bewegung von K' zuriickzufiihren; wir konnen sie vielmehr ebensogut zuriickfiihren auf die durchschnittliche Rotationsbewegung der ponderabeln femen Massen der Umgebung in bezug auf K', wobei wir K' als 'ruhend' behandeln.... Fiir die relativistische Auffassung spricht anderseits folgendes wichtige Argument. Die Zentrifugalkraft, welche unter gegebenen Verhiiltnissen auf einen Korper wirkt, wird genau durch die gleiche Naturkonstante desselben bestimmt wie die Wirkung eines Schwerefeldes auf denselben, derart, daB wir gar kein Mittel haben, ein 'Zentrifugalfeld' von einem Schwerefeld zu unterscheiden.... Dadurch gewinnt die Auffassung durchaus an Berechtigung, daB wir das rotierende System K' als ruhend und das Zentrifugalfeld als ein Gravitationsfeld auffassen diirfen."
6Einstein (1916), p. 641. Here I use John Norton's translation (Norton 1989b, p. 26). I am greatly indebted to this paper of Norton's for the above points on the equivalence principle. Needless to say, Norton might not agree with my interpretation on all points.
7Einstein(l914), p. 1031. "Dies Argument fordert aber ein Gegenargument heraus. Die kinematische Gleichberechtigung zweier Koordinatensysteme ist niimlich durchaus nicht auf den Fall beschriinkt, daB die beiden ins Auge gefassten Koordinatensysteme K und K' sich in gleichformiger Translationsbewegung gegeneinander befmden. Diese Gleichberechtigung vom kinematischen Standpunkt aus besteht z.B. ebensogut, wenn die Systeme relativ zueinander gleichformig rotieren. Man fiiWt sich daher zu der Annahme gedriingt, daB die bisherige Relativitiitstheorie in weitgehendem Mass zu verallgemeinem sei, derart, daB die ungerecht scheinende Bevorzugung der gleichformigen Translation gegeniiber Relativbewegungen anderer Art aus der Theorie verschwindet. "
8Einstein (1914), p. 1032. "Es erhebt sich nun naturgemiiB die Frage, was fiir Bezugssysteme und Transformationen wir in einer verallgemeinerten Relativitiitstheorie als "berechtigte" anzusehen haben. Diese Frage wird sich jedoch erst viel spiiter beantworten lassen (Abschnitt D). Einstweilen stellen wir uns auf den Standpunkt, daB alle Koordinatensysteme und Transformationen zuzulassen seien, die mit den bei physikalischen Theorien stets vorausgesetzten Bedingungen der Stetigkeit vereinbar sind. "
9EA 16-434. "Aber die Gravitationsgleichungen selbst haben die Eigenschaft der allgemeinen Kovarianz Leider nicht. Nur deren Kovarianz linearen Transformationen gegeniiber ist gesichert. Nun beruht aber das ganze Vertrauen auf die Theorie auf der Uberzeugung, daB BescWeunigung des

Einstein's Formulations of Mach's Principle 85
Bezugsystems einem Schwerefeld iiquivalent sei. " 10See Norton's (1989a) pp. 126-132, for an enlightening discussion of
Einstein's arguments against general covariance in 1913 and 1914. llEinstein (1913b), p. 1258. 12Einstein (1923b), p. 113. BEarly 1916 is the latest date that can be placed on Einstein's full
recognition of the non-Machian character of the field equations: The Schwarzschild solution showed that even solutions containing matter might be radically non-Machian. But as I indicated earlier, Einstein's recognition of the problem was probably complete at an earlier stage.
14EA 20-539. "Es thut mir leid, Ihnen gegeniiber zu viel Nachdruck auf die Frage der Grenzbedingungen gelegt zu haben. '" Aber ich muss doch sogleich hinzufiigen, dass ich an eine zeitlich endliche Ausdehnung der Welt niemals gedacht habe; auch bei dem Riiumlichen kommt es auf eine endliche Ausdehnung nicht an. Sondem es trieb mich mein Verallgemeinerungsbediirfnis mir zu folgendes Auffassung:
Es sei moglich eine riiumliche Riille (masselose geometrische Fliiche) (in vierdimensionalen einen Schlauch) anzugeben, ausserhalb welcher ein Grammgewicht eine so geringe Triigheit hat, als ich nur immer will. Dann kann ich sagen, dass innerhalb der Riille die Triigheit durch die dort vorhandenen Massen bedingt sei; und zwar nur durch diese."
15De Sitter (1916), p. 531. 16It is impossible to be sure that Einstein's search for boundary conditions is accurately described in this way; only de Sitter uses the term 'generally covariant boundary conditions,' in texts that survive. But since de Sitter and Einstein were in intensive correspondence in the period from June-December 1916 (only some of which correspondence survives), it is likely that de Sitter's reports on Einstein's thinking are accurate. 17Speziali, p. 97. Speziali dates this letter as probably mid-December, 1916, but the dating is not certain. "Sicher ist, dass unendlich grosse Potentialdifferenzen zu Stemgeschwindigkeiten von sehr bedeutender Grosse Anlass geben miissten, die sich wohl schon lange eingestellt hiitten. Kleine Potentialdifferenzen im Verein mit unendlicher Ausdehnung der Welt verlangen Leersein der Welt im Unendlichen (Konstanz der gIL" im Unendlichen bei passender Koordinatenwahl), im Widerspruch mit einer sinnvoll aufgefassten Relativitiit. Nur Geschlossenheit der Welt befreit aus dem Dilemma. 18This sentence tempts one to suppose that Einstein had constructed a complete solution to the field equations that embodied the boundary conditions (5), thus constructing a (possibly) Machian cosmological model before the famous Einstein universe of the Betrachtungen paper. Of course, Einstein's calculations may well have fallen far short of that. 19Einstein (1923a), pp. 180-182. 2°1 have searched many of the most important current and older textbooks on GTR for any discussion of this passage - in vain. One reason for the lack

86 Carl Hoefer
of attention to this episode may be the fact that, like de Sitter, most relativists were and are hostile to the goal of Machianizing GTR. This was especially true in the late teens and '20s (long before the work of Wheeler, Brans, Dicke, and others revived interest in the '50s and '60s). By the time later physicists returned to Mach's Principle in GTR, this episode had been completely written out of the textbook history of general relativity.
21EA 20-548. "Es ware nach meiner Meinung unbefriedigend, wenn es eine denkbare Welt ohne Materie gabe. Das .(v-Feld solI vielmehr durch die Materie bedingt sein, ohne dieselbe nicht bestehen kOnnen. Das ist der Kern dessen, was ich under der Forderung von der Relativitiit der Tragheit verstehe. Man kann auch ebensogut von der 'materiellen Bedingtheit der Geometrie' sprechen. Solange diese Forderung nicht erfiillt war, war fiir mich das Ziel der Aligemeinen Relativitiit noch nicht ganz erreicht. Dies wurde durch das A-Glied erst herbeigefiihrt. "
22Einstein conceded the singularity-free nature of de Sitter's metric in a postcard to Felix Klein, EA 14-449.
23Einstein to Felix Pirani, EA 17-448. 24Julian Barbour (1992) has recently argued, persuasively I believe, that this effect is not really a consequence of Mach's ideas. 25Einstein (1922), pp. 100, 103. 26At the 1993 Tiibingen conference, Julian Barbour organized several strawpolls of the attendees concerning their views on GTR and Mach's Principle. My claims here are based partly on these straw polls [po 106], as well as the evidence of current literature. 27Again, this fact emerged clearly from straw-polls and discussions at the 1993 Tiibingen conference.
REFERENCES
Barbour, Julian B. (1992). "Einstein and Mach's Principle." In Studies in the History of General Relativity, Vol. 3 of Einstein Studies. Jean Eisenstaedt and A.J. Kox, eds. Boston: Birkhauser, pp. 125-153.
de Sitter, Willem (1916). "On the Relativity of Inertia in Einstein's Theory." Proc. ofthe Section of Sciences, Koninklijke Akademie van Wetenschappen 19: 527-532, September 30.
Einstein, Albert (1912). "Gibt es eine Gravitationswirkung die der elektrodynamischen Induktionswirkung analog is?" Vierteljahrsschrift fur gerichtliche Medizin und offentliches Sanitiitswesen 44: 37-40.
Einstein, Albert (1913a). "Physikalische Grundlagen einer Gravitationstheorie. " Vierteljahrsschrift der Natuiforschenden GeseUschaft Zurich 58: 284-290.
Einstein, Albert (1913b). "Zum gegenwartigen Stande des Gravitationsproblems." Physikalische Zeitschrift 14: 1249-1266.
Einstein, Albert (1914a). "Die formale Grundlage der allgemeinen Relativitiitstheorie." Sitzungsberichte der Preussischen Akademie der Wissenschaften,

Einstein's Formulations of Mach's Principle 87
pp. 1030-1085, Part 2. Einstein, Albert (l914b). "Zum Relativitiits-Problem." Scientia (Bologna) 15:
337-348. Einstein, Albert (1916). "Uber Friedrich Kottlers Abhandlung 'Uber Einsteins
Aquivalezhypothese und die Gravitation'." Annalen der Physik 51: 639642. Einstein, Albert (1917). "Kosmologische Betrachtungen zur allgemeinen Relativitiitstheorie." Sitzungsberichte der Preussischen Akademie der Wissenschaften, pp. 142-152. Translation: Einstein (1923a). Einstein, Albert (1918). "Prinzipielles zur allgemeinen Relativitiitstheorie." Annalen der Physik 55: 241-244. Einstein, Albert (1922). The Meaning of Relativity. Princeton: Princeton University Press. Einstein, Albert (1923a). "Cosmological Considerations on the General Theory of Relativity." In The Principle of Relativity. Papers by Albert Einstein et al. New York: Dover, pp. 177-188. Einstein, Albert (1923b). "Foundations of the General Theory of Relativity." ibid. Hoefer, Carl (1993). "Einstein's Struggle for a Machian Gravitation Theory." Studies in History and Philosophy of Modern Physics 25: 287-335. Kerszberg, Pierre (1989a). "The Einstein-de Sitter Controversy of 1916-1917 and the Rise of Relativistic Cosmology." In Einstein and the History of General Relativity, Vol. 1 of Einstein Studies. Don Howard and John Stachel, eds. Boston: Birkhiiuser, pp. 325-366. Kerszberg, Pierre (1989b). The Invented Universe: The Einstein-De Sitter Controversy (1916-1917) and the Rise of Relativistic Cosmology. Oxford: Oxford University Press. Norton, John (1989a). "How Einstein Found his Field Equations." In Einstein and the History of General Relativity, Vol. 1 of Einstein Studies. Don Howard and John Stachel, eds. Boston: Birkhiiuser, pp. 101-159. Norton, John (1989b). "What was Einstein's Principle of Equivalence?" In Einstein and the History of General Relativity, Vol. 1 of Einstein Studies. Don Howard and John Stachel, eds. Boston: Birkhiiuser, pp. 5-47. Speziali, Pierre, ed. (1972). Albert Einstein, Michele Besso: Correspondence 1903-1955. Paris: Hermann.
Discussion
Earman: People sometimes talk as if there is a dichotomy between universes that are spatially open and universes that are spatially closed. Now, of course, there are universes you can slice up with open sections and there's another way of slicing with closed sections, so is it enough for Mach's Principle that there exists a way of slicing it with spatially

88 Carl Hoefer
closed sections? Hoefer: My way of thinking about Mach's Principle, and I can only speak for my own understanding, is that closed vs open and infinite vs noninfinite has nothing to do with Mach's Principle. I have never seen any reason to connect Mach's Principle with any kind of demand on the topology. Barbour [comment after conference]: I believe the argument for closure is rather obvious. Mach said that motion is with respect to the universe as a whole. Now motion of one body relative to a finite universe is easy to defme but to defme motion, as a definite quantity, relative to an infmite universe is not at all easy. Virtually all actual implementations of Machian ideas have assumed the universe is fmite. Also in Wheeler's geometrodynamic approach, in which the three-geometry is the basic concept, two slightly different closed threegeometries in principle determine a complete spacetime by themselves (thinsandwich principle). However, if space is infinite, boundary conditions have to be imposed arbitrarily. The dynamics of the universe is no longer selfcontained. It was this sort of arbitrariness Einstein sought to avoid. Hoefer [response to above comment]: Dr. Barbour's comment illustrates exactly the widespread conceptions of the relation of fmitude/closure to Mach's ideas that I believe to be misconceptions. Motion is not more difficult to defme relative to an infmite universe than to a fmite universe, if by 'defming' we simply mean a nonmetrized description via a coordinate system. If we mean something stronger - specifying relative velocities or accelerations for pairs of bodies, for example - then problems do arise, in relativity in general, but especially difficult problems arise for the Machian. Spatiotemporal structure is needed to characterize these motions, yet the structure of spacetime is, for the Machian, supposed to arise out of those very relative motions. This problem is not resolved, conceptually speaking, just by assuming closure of space.
Wheeler's approach to Mach's ideas illustrates what I mean. As Barbour points out, the idea is that two different closed three-geometries determine the whole structure of spacetime. Is it an acceptable Machian strategy for the relativist to help herself to the whole geometry of these slices, which is clearly more than just a summary of relative motions of bodies at a time (for example, they may contain gravity waves)? I don't know whether it is or not - but I do claim that their being closed three-geometries does not automatically validate them for Machian use. Nor, if one used open, infinite spatial slices, do I see that this would automatically violate any Machian ideas. Boundary conditions would have to be imposed, but would they have to be arbitrary? Einstein thought that there might be naturally Machian boundary conditions, and while his attempt to work out this idea was a failure, I have argued that this doesn't show that the idea itself is necessarily mistaken. Bondi [response to same comment]: I disagree with Barbour. As I see it, any radius of curvature significantly greater than the Hubble distance is of little

Einstein's Formulations of Mach's Principle 89
relevance, whether it is positive, negative or infInite. Barbour: In response to both Bondi and Hoefer, I still maintain it is far easier to defme a defInite relative motion of either mass particles or fIelds that can be used in an action function in the case of a fmite or closed universe. As regards the role of gravitational degrees of freedom, when Mach fIrst criticized Newton, the 'ontology' of the world was mass points in Euclidean space. Einstein changed the ontology and worked with fIelds and dynamic geometry, but he never seems to have asked himself seriously this question: What precisely is the Machian problem in the new context of fIelds and dynamic geometry? The Poincare-type criteria of Machianity that I develop in my paper (p. 214, Sec. 3) translate immediately into the new context, but frankly it seems to me anachronistic in a world of fIelds and dynamic geometry to say only matter, and not other degrees of freedom, can determine the inertial frames.
Gravitational waves are just as observable as matter fIelds. The fact that there is no proper energy-momentum tensor of the gravitational fIeld presents no problem in the formulation of the thin-sandwich problem, which operates exclusively with 3-metrics (and fIelds if they are also present). See gravitational degrees of freedom, role in Mach's Principle in the Index and also the immediately following comments of Ciufolini made at Tiibingen. Ciufolini: This discussion is related to the view of John Wheeler. He thinks that a model universe is in agreement with Mach's Principle if it has a Cauchy surface that is a closed manifold, that is compact and without boundary; that is a model that admits a closed Cauchy surface. I think Jim [Isenberg] has done some work on that. Isenberg: Yes that is essentially Wheeler's view: That if a spacetime admits a closed Cauchy surface, and if it also satisfies Einstein's equations (with the constraints thus imposed on the initial data on any Cauchy surface) then the spacetime should be Machian. Oh, he also includes that topological restriction on the Cauchy surface. Ciufolini: So according to Wheeler the corresponding initial-value formulation clarifies the origin of inertia... Isenberg: There is no ambiguity about whether a Cauchy surface is open or closed. Ciufolini: Yes, but in the spatially nonclosed case you have to admit some kind of prior geometry such as asymptotic Minkowskian geometry. Hoefer: Well, I have always been puzzled about this, exactly why that demand expresses anything Machian. Narlikar: From the last transparency, it was not clear to me that Einstein wanted singularities or no singularity. Hoefer: Singularity free. The confusion arises because Einstein desperately wanted the de Sitter solution to have some kind of singularity, because it was a matter-free solution and his demand for a

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physically reasonable solution was that it be singularity free. Renn: You mentioned the problem of the Entwurftheory not being fully covariant. That was actually only one of its 'Machian problems,' so to say. Another problem is related to Einstein's claim that his theory would also do justice to the requirement that inertial mass is created by the presence of other masses in the universe. Max Abraham, who wrote a critical review of the Entwurf theory in 1914, actually calculated the effect that other masses have on the mass of a given body according to this theory. He found this effect to be so small that he concluded that Einstein's claim can only be maintained if the existence of invisible matter is assumed, an assumption he considered absurd. Hoefer: I was curious when you mentioned that earlier. I am not clear about that, why the smallness of the effect should be a stumbling block. Any effect at all would seem to fulfill your Machian expectations. Lynden-Bell: No, no. All of it has to be. Hoefer: You mean removing all the rest of the mass from the universe only subtracts a negligible amount of inertia? Ehlers: It seems to me that the term inertia was used in a somewhat unclear fashion even in the quotations which you showed. One could either think that by saying that the inertia of a particle should be determined by the cosmic masses it is to be interpreted as saying a local piece of the inertial trajectory of the particle, or one could interpret the term as meaning the value of the inertial mass, and these are rather different requirements. I am not sure which requirement was considered as a Machian requirement at that period by Einstein. Hoefer: Well Einstein thought both that the inertial mass should be a product of the presence of other masses and also that the local piece of the inertial trajectory should be determined by the distribution of masses. I believe Professor Barbour has argued that the first requirement shouldn't be thought as a true Machian requirement. Barbour: That is certainly my view. I am delighted with Jiirgen's question. That's one of the things I'm hoping we will discuss in the session this afternoon (p. 91). Norton: I have a brief remark on why Einstein thought the theory was Machian. As early as 1912 and 1913, he could derive the weak field effects associated with the dragging of inertial frames by accelerating masses. Even though his theory was not generally covariant at this early stage, he did believe (erroneously) that it was covariant under transformations to rotating frames of reference. That problem was fixed in November 1915, when he found the generally covariant version of his theory.

General Discussion: What is the Machian Program?
Because Mach's Principle is surrounded by so much controversy, the final session of the first day of the Tiibingen conference was devoted to a general discussion, led by Barbour, on the theme What is the Machian Program? The edited transcript of the discussion, to which a few comments made at other times during the conference have been added, follows. The editors feel that the discussion session did achieve its purpose - to identify all the main issues associated with Mach's Principle. At the end of the discussion, a straw poll on certain issues was held. The questions and results of the poll are given at the end of the discussion transcript [p. 106]. At the end of the final day of the conference, the straw poll was repeated to see if any significant changes of opinion had occurred. The results of that poll too are given.
Barbour: There are at least four questions that I feel we should discuss, the first of which has already been raised by John Norton [po 9].
Question 1: What was Mach actually advocating? Was he advocating a mere redescription of Newtonian theory without any change in its physical content, or was he advocating a genuinely new theory?
This next question has already been precisely formulated by Jiirgen Ehlers [p. 90]:
Question 2: Should the Machian principle be something to do with a cosmic derivation of the inertial mass, some sort of formula where m, the inertial mass, is equal to some integral stretched over the entire universe, something like that, or is it just to do with a cosmic derivation of the local inertial frames of reference?
Einstein Studies, vol. 6: Mach's Principle: From Newton's Bucket to Quantum Gravity, pp. 91-106 © 1995 Birkhiiuser Boston, Inc. Printed in the United States.

92 General Discussion
My own view is that it's the latter. I think Einstein brought in a red herring by requiring a cosmic derivation of the inertial mass. This is an important issue because it determines the sort of theory we want to find. Indeed, for me the main purpose of this meeting, and especially this session, is: Can we establish, if it's possible, what are the true criteria of 'Machianity'? When can we say that a theory is truly Machian? In fact, as I argue in my contribution [po 214, Sec. 3], I believe that Poincare has given us a very useful and precise criterion.
The third issue has not yet been mentioned today and has seldom been raised in the literature. When you read Mach's Mechanics, the first five or six pages of his critique of Newtonian mechanics are not about motion; they are about time. He starts by making a big issue about time.
In fact, Mittelstaedt (1976) even wondered whether one should not formulate a Second Mach's Principle, which is to do with relativity of time. This would then match the First Mach's Principle, or the first Machian requirement if you like, to do with the relativity of motion. When we get into quantum gravity, I think we shall see it's extremely important, and that it is a Machian issue. Therefore:
Question 3: Is there a Second Mach's Principle to do with the relativity of time?
Finally, if we do accept that the Machian requirement is to show how the local inertial frames of reference are determined by the universe at large, then: What agents do that determining?
Question 4: In the context of general relativity, must the local inertial frames of reference be determined completely by the energy-momentum tensor of matter in the narrow sense, or can the gravitational degrees of freedom themselves contribute to the determination of the local frames of reference?
Hoyle: Specified under what mathematical conditions? On a Cauchy surface? Barbour: That is fair enough; a Cauchy surface. I think this is an important issue, because quite a lot of interpretations of Mach's Principle, which stem from Einstein himself [po 180], suggest that giLP' which determines the local frames of reference, should be determined by the matter alone. Therefore, if you are going to give a Machian interpretation of general relativity, is Einstein right about the agents that can determine giLP? There's quite a large body of opinion that thinks it must