zotero/storage/RARSYZU2/.zotero-ft-cache

Relational
Mechanics
Andre K.T. Assis
Apeiron Montreal

Published by Apeiron 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com
© Andre K.T. Assis
Dr. Andre Koch Torres Assis Institute of Physics State University of Campinas 13083-970 Campinas - SP, Brazil E-mail: assis@ifi.unicamp.br Homepage: www.ifi.unicamp.br/~assis
First Published 1999

Canadian Cataloguing in Publication Data

Assis, Andre Koch Torres, 1962Relational mechanics

Translation of Mecanica Relacional Includes bibliographical references and index. ISBN 0-9683689-2-1

1. Mechanics. 2. Gravitation. 3. Inertia (Mechanics). I. Title.

QC125.2.A88 1999

531

C99-900382-8

Front cover:
Newton’s bucket experiment. The water rises in the second case because of its rotation. But: rotation relative to what? This is the central theme of this book.
Back cover:
Galileo’s free fall experiment. Why do two bodies of different weight fall in vacuum with the same acceleration toward the surface of the earth? Relational Mechanics offers a simple explanation for this remarkable fact.

In memory of Isaac Newton
who paved the way for past, present and future generations

2

Contents

I Old World

13

1 Newtonian Mechanics

15

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Applications of Newtonian Mechanics

29

2.1 Uniform Rectilinear Motion . . . . . . . . . . . . . . . . . . . . . 30

2.2 Constant Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.2 Charge Moving Inside an Ideal Capacitor . . . . . . . . . 33

2.2.3 Accelerated Train . . . . . . . . . . . . . . . . . . . . . . 36

2.3 Oscillatory Motions . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1 Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.2 Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.3 Electrically Charged Pendulum . . . . . . . . . . . . . . . 42

2.4 Uniform Circular Motion . . . . . . . . . . . . . . . . . . . . . . 46

2.4.1 Circular Orbit of a Planet . . . . . . . . . . . . . . . . . . 47

2.4.2 Two Globes . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4.3 Newton’s Bucket Experiment . . . . . . . . . . . . . . . . 51

3 Non-inertial Frames of Reference

59

3.1 Constant Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.1 Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.2 Accelerated Train . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Uniform Circular Motion . . . . . . . . . . . . . . . . . . . . . . 62

3.2.1 Circular Orbit of a Planet . . . . . . . . . . . . . . . . . . 62

3.2.2 Two Globes . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2.3 Newton’s Bucket Experiment . . . . . . . . . . . . . . . . 66

3.3 Rotation of the Earth . . . . . . . . . . . . . . . . . . . . . . . . 67

3

4

CONTENTS

3.3.1 Kinematical Rotation of the Earth . . . . . . . . . . . . . 67 3.3.2 The Figure of the Earth . . . . . . . . . . . . . . . . . . . 71 3.3.3 Foucault’s Pendulum . . . . . . . . . . . . . . . . . . . . . 75 3.3.4 Comparison of the Kinematical and Dynamical Rotations 81 3.4 General ﬁctitious Force . . . . . . . . . . . . . . . . . . . . . . . . 82

4 Gravitational Paradox

85

4.1 Newton and the Inﬁnite Universe . . . . . . . . . . . . . . . . . . 85

4.2 The Force Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3 The Paradox based on Potential . . . . . . . . . . . . . . . . . . 88

4.4 Solutions of the Paradox . . . . . . . . . . . . . . . . . . . . . . . 90

4.5 Absorption of Gravity . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Leibniz and Berkeley

97

5.1 Leibniz and Relative Motion . . . . . . . . . . . . . . . . . . . . . 97

5.2 Berkeley and Relative Motion . . . . . . . . . . . . . . . . . . . . 101

6 Mach and Newton’s Mechanics

107

6.1 Inertial Frame of Reference . . . . . . . . . . . . . . . . . . . . . 107

6.2 Absolute Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.3 The Two Rotations of the Earth . . . . . . . . . . . . . . . . . . 110

6.4 Inertial Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.5 Mach’s Formulation of Mechanics . . . . . . . . . . . . . . . . . . 114

6.6 Relational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.7 Mach and the Bucket Experiment . . . . . . . . . . . . . . . . . . 116

6.8 Mach’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.9 What Mach did not Show . . . . . . . . . . . . . . . . . . . . . . 122

7 Einstein’s Theories of Relativity

125

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2 Einstein’s Special Theory of Relativity . . . . . . . . . . . . . . . 125

7.2.1 Asymmetry in Electromagnetic Induction . . . . . . . . . 127

7.2.2 Postulate of Relativity . . . . . . . . . . . . . . . . . . . . 131

7.2.3 Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.2.4 Constancy of the Velocity of Light . . . . . . . . . . . . . 133

7.2.5 Velocity in Lorentz’s Force . . . . . . . . . . . . . . . . . 140

7.2.6 Michelson-Morley Experiment . . . . . . . . . . . . . . . . 144

7.3 Einstein’s General Theory of Relativity . . . . . . . . . . . . . . 146

7.3.1 Relational Quantities . . . . . . . . . . . . . . . . . . . . . 146

7.3.2 Invariance in the Form of the Equations . . . . . . . . . . 148

7.3.3 Implementation of Mach’s Ideas . . . . . . . . . . . . . . 148

7.3.4 Newton’s Bucket Experiment . . . . . . . . . . . . . . . . 152

CONTENTS

5

7.4 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 157

II New World

161

8 Relational Mechanics

163

8.1 Basic Concepts and Postulates . . . . . . . . . . . . . . . . . . . 163

8.2 Electromagnetic and Gravitational Forces . . . . . . . . . . . . . 166

8.3 Spherical Shell Interacting with a Particle . . . . . . . . . . . . . 170

8.4 Implementation of Mach’s Principle . . . . . . . . . . . . . . . . 173

8.5 General Consequences . . . . . . . . . . . . . . . . . . . . . . . . 180

8.6 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.7 Ptolemaic and Copernican World Views . . . . . . . . . . . . . . 190

8.8 Implementation of Einstein’s Ideas . . . . . . . . . . . . . . . . . 192

9 Applications of Relational Mechanics

197

9.1 Uniform Rectilinear Motion . . . . . . . . . . . . . . . . . . . . . 197

9.2 Constant Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

9.2.1 Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

9.2.2 Charge Moving Inside an Ideal Capacitor . . . . . . . . . 205

9.2.3 Accelerated Train . . . . . . . . . . . . . . . . . . . . . . 206

9.3 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 208

9.3.1 Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

9.3.2 Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . 209

9.4 Uniform Circular Motion . . . . . . . . . . . . . . . . . . . . . . 210

9.4.1 Circular Orbit of a Planet . . . . . . . . . . . . . . . . . . 210

9.4.2 Two Globes . . . . . . . . . . . . . . . . . . . . . . . . . . 212

9.4.3 Newton’s Bucket Experiment . . . . . . . . . . . . . . . . 214

9.5 Rotation of the Earth . . . . . . . . . . . . . . . . . . . . . . . . 218

9.5.1 The Figure of the Earth . . . . . . . . . . . . . . . . . . . 218

9.5.2 Foucault’s Pendulum . . . . . . . . . . . . . . . . . . . . . 219

10 Beyond Newton

227

10.1 Precession of the Perihelion of the Planets . . . . . . . . . . . . . 227

10.2 Anisotropy of Inertial Mass . . . . . . . . . . . . . . . . . . . . . 231

10.3 High Velocity Particles . . . . . . . . . . . . . . . . . . . . . . . . 233

10.4 Experimental Tests of Relational Mechanics . . . . . . . . . . . . 236

11 History of Relational Mechanics

243

11.1 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

11.2 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

11.3 Weber’s Law Applied to Gravitation . . . . . . . . . . . . . . . . 249

6

CONTENTS

11.4 Relational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 250

12 Conclusion

259

Acknowledgments
To the undergraduate and graduate students who followed our courses on Relational Mechanics, Mach’s Principle, Weber’s Electrodynamics and Cosmology, for the many constructive remarks they presented. To our undergraduate and Ph.D. students who are developing researches in these areas. To our friends who are giving us support to continue this work. In special we wish to thank those to whom we gave a ﬁrst version of this book: C. Roy Keys, Thomas E. Phipps Jr., A. Ghosh, D. Roscoe, A. Martin, R. de A. Martins, M. A. Bueno, W. M. Vieira, M. A. de Faria Rosa, A. Zylbersztajn, D. S. L. Soares, J. I. Cisneros, H. C. Velho, D. Gardelli, J. A. Hernandez and J. E. Lamesa. The suggestions and ideas which we received contributed greatly to the improvement of the book.
To the following Publishers and individuals for granting permission to quote from their publications: University of California Press (I. Newton, Mathematical Principles of Natural Philosophy, 1934, Cajori edition); Open Court Publishing Company, a division of Carus Publishing Company, Peru, IL (E. Mach, The Science of Mechanics, 1960, translated by T. J. McCormack); Everyman’s Library, David Campbell Publications Ltd. (G. Berkeley, Philosophical Works including De Motu and A Treatise Concerning the Principles of Human Knowledge, 1992); Dover Publications (A. Einstein, H. A. Lorentz, H. Weyl and H. Minkowski, The Principle of Relativity, 1952); Cambridge University Press (J. B. Barbour, Absolute or Relative Motion?, 1989); Albert Einstein Archives, The Hebrew University of Jerusalem, Israel (A. Einstein, The Meaning of Relativity, 1980); Dr. Peter Gray Lucas, in name of Dr. H. G. Alexander (The Leibniz-Clarke Correspondence, edited by H. G. Alexander, 1984); Birkhauser Boston and Dr. J. B. Barbour for his translations of E. Schr¨odinger, H. Reissner, A. Einstein, B. and I. Friedlaender (Mach’s Principle - From Newton’s Bucket to Quantum Gravity, edited by J. B. Barbour and H. Pﬁster, 1995).
To the Center for Electromagnetics Research, Northeastern University (Boston, USA), which received us for one year in which we had the ﬁrst idea to write this book and discussed its contents with some friends. To the Institutes of Physics and Mathematics and to the Center of Logics, Epistemology and History of Sciences of the State University of Campinas - UNICAMP (Brazil),
7

8

CONTENTS

that gave the necessary support to undertake this work. Above all, to my parents, my wife and children for helping me grow as a
human being.

Preface
This book presents Relational Mechanics, a new mechanics which implements the ideas of Leibniz, Berkeley, Mach and many others. Relational mechanics is based only on relative quantities, such as the distance between material bodies, their relative radial velocity and relative radial acceleration. In this new mechanics the absolute concepts of space, time and motion do not appear. The same can be said of inertia, inertial mass and inertial frames of reference. When we compare relational mechanics with Newtonian mechanics, we will gain a new and clear understanding of these old concepts. Relational mechanics is a quantitative implementation of Mach’s ideas utilizing a Weber’s force law for gravitation. Many people have contributed to its development, including Erwin Schr¨odinger.
This is the ﬁrst time such a book has been written, bringing together all the features and characteristics of this new world view. This allows it to be seen in its proper light, and a comparison with old worldviews is easily accomplished.
Considerable emphasis is placed on Galileo’s free fall experiment and on Newton’s bucket experiment. These are some of the simplest experiments ever performed in physics. Despite this fact, no other experiment has had such far-reaching consequences for the foundations of classical mechanics. An explanation of these two experiments without utilizing the concepts of absolute space and inertia is one of the major accomplishments of relational mechanics.
In order to show all the power of relational mechanics and put it in perspective, we ﬁrst present Newtonian mechanics and Einstein’s theories of relativity. We address the criticisms of Newton’s theory made by Leibniz, Berkeley and Mach. Then we present relational mechanics and show how it solves all these problems quantitatively with a clarity and simplicity unsurpassed by any other model. We also discuss the history of relational mechanics in detail, emphasizing the achievements and limitations of all major works along these lines. In addition, we present several notions which are beyond the scope of Newtonian theory, such as the precession of the perihelion of the planets, the anisotropy of an eﬀective inertial mass, the adequate mechanics for high velocity particles, etc. Experimental tests of relational mechanics are also outlined.
9

10

CONTENTS

This book is intended for physicists, mathematicians, engineers, historians and philosophers of science. It is also addressed to teachers of physics at university or high school levels and to their students. After all, those who have taught and learned Newtonian mechanics know the diﬃculties and subtleties of its basic concepts (inertial frame of reference, ﬁctitious centrifugal force, inertial and gravitational masses, etc.) Above all, it is intended for young unprejudiced people who have an interest in the fundamental questions of mechanics: Is there an absolute motion of any body relative to space or only relative motion between material bodies? Can we prove experimentally that a body is accelerated relative to space or only relative to other bodies? What is the meaning of inertia? Why do two bodies of diﬀerent weight, form and chemical compositions fall with the same acceleration in vacuum on the earth’s surface? When Newton rotated the bucket and saw the water rising towards the sides of the bucket, what was responsible for this eﬀect? Was it due to the rotation of the water relative to some material body? What ﬂattens the earth at the poles in its diurnal rotation? Is it the rotation of the earth relative to something? Is the earth really rotating and translating? We show that the answer to these questions with relational mechanics is much simpler and more philosophically sound and appealing than in Einstein’s theories of relativity.
Nowadays the majority of physicists accept Einstein’s theories as correct. We show this is untenable and present an alternative theory which is much clearer and more reasonable than the previous ones. We know that these are strong statements, but we are sure that anyone with a basic understanding of physics will accept this fact after reading this book with impartiality and without prejudice. With an understanding of relational mechanics, we enter a new world, viewing the same phenomena with diﬀerent eyes and from a new perspective. It is a change of paradigm [1]. This new formulation will help put physics on new rational foundations, moving it away from the mystiﬁcations of this century.
We hope physicists, engineers, mathematicians and philosophers will adopt this book in their courses of mechanics, mathematical methods of physics and history of science, recommending it to their students. We believe the better way to create critical minds and to motivate the students is to present to them diﬀerent approaches for the solution of the same problems, how the concepts have been growing and changing throughout history and how great scientists viewed equivalent subjects from diﬀerent perspectives.
A Portuguese version of this book was published under the title Mecˆanica Relacional, [2].
In this book we utilize the International System of Units. When we deﬁne any physical concept we utilize “≡” as a symbol of deﬁnition. We utilize symbols with a double subscript with three diﬀerent meanings. Examples: Fji is the force exerted by particle j on particle i, a12 = a1 − a2 is the acceleration of

CONTENTS

11

particle 1 minus the acceleration of particle 2, and vmS is the velocity of particle m relative to the frame of reference S. In the text we clarify which meaning we are employing in each place.

12

CONTENTS

Part I
Old World
13

Chapter 1
Newtonian Mechanics
1.1 Introduction
The branch of knowledge which deals with the equilibrium and motion of masses is called mechanics. For the last three hundred years the mechanics taught in schools and universities has been based on the work of Isaac Newton (16421727). His main book is called Mathematical Principles of Natural Philosophy, usually known by its ﬁrst Latin name, Principia, [3]. Originally published in 1687, it is based on the concepts of space, time, velocity, acceleration, weight, mass, force, etc. In the next section we present Newton’s own formulation of mechanics.
Since long before Newton, there has always been a great debate between philosophers and scientists regarding the distinction between absolute and relative motion. In other words, motion of a body relative to empty space and relative to other bodies. For a clear discussion of this whole subject with many quotations from the original see the authoritative book by Julian Barbour, Absolute or Relative Motion? [4]. In our book we consider only Newton and others following him. The reasons for this are the impressive success of his mechanics and the new standard he introduced in this whole discussion with his dynamical arguments, as distinguished from kinematical arguments, in favour of absolute motion. In particular, we can cite his famous bucket experiment. This is one of the main subjects of this work.
1.2 Newtonian Mechanics
The Principia begins with eight deﬁnitions, [3]. The ﬁrst deﬁnition is “quantity of matter,” which nowadays we call the inertial mass of a body. Newton deﬁned
15

16

CHAPTER 1. NEWTONIAN MECHANICS

it as product of the density and volume occupied by the body:
Deﬁnition I: The quantity of matter is the measure of the same, arising from its density and bulk conjointly.

Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and ﬁne dust or powders, that are condensed by compression or liquefaction, and of all bodies that are by any causes whatever diﬀerently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body, for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shown hereafter.
Designating the quantity of matter (the inertial mass) of a body mi, its density ρ and its volume V we would have:

mi ≡ ρV .

(1.1)

Later on we will present Mach’s criticism of this deﬁnition. We will also discuss in detail the proportionality between the mass and weight of bodies, as well as Newton’s experiments on this matter. For the moment it is important to stress that with this proportionality Newton found a precise operational way of determining the mass of any body, as he needed only to weight it.
Then Newton deﬁnes the quantity of motion as the quantity of matter times the velocity of the body:

Deﬁnition II: The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly.

The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.
Denoting the vectorial velocity v and the quantity of motion p we have:
p ≡ miv .
Newton goes on to deﬁne the inertia of a body:

1.2. NEWTONIAN MECHANICS

17

Deﬁnition III: The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, continues in its present state, whether it be of rest, or of moving uniformly forwards in a right line.

This force is always proportional to the body whose force it is and diﬀers nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inert nature of matter, is not without diﬃculty put out of its state of rest or motion. Upon which account, this vis insita may, by a most signiﬁcant name, be called inertia (vis inertiae) or force of inactivity. (...)
His fourth deﬁnition is “impressed force,” namely: An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line.
Then follow deﬁnitions of centripetal force, of the absolute quantity of a centripetal force, of the accelerative quantity of a centripetal force and the motive quantity of a centripetal force.
After these 8 deﬁnitions there is a Scholium with the deﬁnitions of absolute time, absolute space and absolute motion. It is worthwhile quoting its main parts:
Hitherto I have laid down the deﬁnitions of such words as are less known, and explained the sense in which I would have them to be understood in the following discourse. I do not deﬁne time, space, place, and motion, as being well known to all. Only I must observe, that the common people conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.
I. Absolute, true, and mathematical time, of itself, and from its own nature, ﬂows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.
II. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is com-

18

CHAPTER 1. NEWTONIAN MECHANICS

monly taken for immovable space; such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position in respect of the earth. Absolute and relative space are the same in ﬁgure and magnitude; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be continually changed.
III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. (...)
IV. Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another. (...)

Then come his three “Axioms, or Laws of Motion” and six corollaries, namely:

Law I: Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.
Law II: The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
Law III: To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
Corollary I: A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately.
(...)
Corollary V: The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.
(...)

His ﬁrst law is usually called the law of inertia. His second law of motion might be written as:

dp d F = dt = dt (miv) .

(1.2)

1.2. NEWTONIAN MECHANICS

19

Here we have used F for the resultant force acting on the body. If the inertial mass mi is a constant, then this law can be cast in the simple and well-known form

F = mia ,

(1.3)

where a = dv/dt is the acceleration of the body. His third law is called the law of action and reaction. Denoting the force
exerted by a body A on another body B by FAB, and the force exerted by B on A by FBA, the third law states that:

FAB = −FBA .

Whenever Newton utilized the third law, the forces between two bodies were always directed along the straight line joining them, as in the law of gravitation.
His ﬁrst corollary is called the law of the parallelogram of forces. His ﬁfth corollary introduces the concept of inertial frames (frames which are at rest or which move with a constant velocity relative to absolute space). In Section XII of Book I of the Principia, Newton proved two extremely important theorems related to the force exerted by a spherical shell on internal and external points, supposing forces which fall oﬀ as the inverse square of the distance (as is the case with Newton’s gravitational law and Coulomb’s electrostatic force):

Section XII: The attractive forces of spherical bodies.

Proposition 70. Theorem 30: If to every point of a spherical surface there tend equal centripetal forces decreasing as the square of the distances from these points, I say, that a corpuscle placed within that surface will not be attracted by those forces any way.
If the body is anywhere inside the shell (not only on its center), it will not experience any resultant force from the shell as a whole.
Proposition 71. Theorem 31: The same things supposed as above, I say, that a corpuscle placed without the spherical surface is attracted towards the centre of the sphere with a force inversely proportional to the square of its distance from that centre.
This means that a body outside the shell is attracted as if the shell were concentrated at its center.
In the third book of the Principia Newton presented his law of gravitation. This can be stated as follows: every particle of matter attracts every other

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CHAPTER 1. NEWTONIAN MECHANICS

particle with a force varying directly as the product of their gravitational masses and inversely as the square of the distance between them.
Nowhere in the Principia Newton did express the gravitational law in this form. But we can ﬁnd statements similar to these in the following passages of the Principia: Book I, Props. 72 to 75 and Prop. 76, especially Corollaries I to IV; Book III, Props. 7 and 8; and in the General Scholium at the end of Book III. For instance, in Book I, Prop. 76, Cors. I to IV we read, referring to spheres with an isotropic distribution of matter, densities such as ρ1(r) and ρ2(r), in which every point attracts with a force which falls oﬀ as the square of the distance:
Cor. I. Hence if many spheres of this kind, similar in all respects, attract each other, the accelerative attractions of each to each, at any equal distances of the centres, will be as the attracting spheres.
Cor. II. And at any unequal distances, as the attracting spheres divided by the squares of the distances between the centres.
Cor. III. The motive attractions, or the weights of the spheres towards one another, will be at equal distances of the centres conjointly as the attracting and attracted spheres; that is, as the products arising from multiplying the spheres into each other.
Cor. IV. And at unequal distances directly as those products and inversely as the squares of the distances between the centres.
Proposition 7 of Book III states:
That there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they contain.
That all planets gravitate one towards another, we have proved before; as well as that the force of gravity towards every one of them, considered apart, is inversely as the square of the distance of places from the centre of the planet. And thence (by Prop. 69, Book I, and its Corollaries) it follows that the gravity tending towards all the planets is proportional to the matter which they contain.
Moreover, since all the parts of any planet A gravitate towards any other planet B; and the gravity of every part is to the gravity of the whole as the matter of the part to the matter of the whole; and (by Law III) to every action corresponds an equal reaction; therefore the planet B will, on the other hand, gravitate towards all the parts of the planet A; and its gravity towards any one part will be to the gravity towards the whole as the matter of the part to the matter of the whole. Q. E. D.

1.2. NEWTONIAN MECHANICS

21

This last paragraph is very important. It shows the key role played by Newton’s action and reaction law in the derivation of the fact that the force of gravity is proportional to the product of the masses of the two bodies (and not, for instance, proportional to the sum of the two masses, or to their product squared).
In the General Scholium at the end of the book we read:

Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but we have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, without suﬀering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes used to do), but according to the quantity of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always as the inverse square of the distances.

In the System of the World written by Newton we can also see the importance of the law of action and reaction for the derivation of the fact that the gravitational force is proportional to the product of the masses. Here we quote Section 20 of the Principia, [3, p. 568], just after the Section where Newton discussed his pendulum experiments which showed the proportionality between weight and inertial mass:

Since the action of the centripetal force upon the bodies attracted is, at equal distances, proportional to the quantities of matter in those bodies, reason requires that it should be also proportional to the quantity of matter in the body attracting.
For all action is mutual, and (by the third Law of Motion) makes the bodies approach one to the other, and therefore must be the same in both bodies. It is true that we may consider one body as attracting, another as attracted; but this distinction is more mathematical than natural. The attraction resides really in each body towards the other, and is therefore of the same kind in both.

Algebraically his law of gravitation might be written as:

F21

=

−G

mg1mg2 r2

rˆ

.

(1.4)

In this equation F21 is the force exerted by the material particle 2 on the material particle 1, G is a constant of proportionality, mg1 and mg2 are the

22

CHAPTER 1. NEWTONIAN MECHANICS

gravitational masses of particles 1 and 2, r is their distance and rˆ is the unit vector pointing from 2 to 1.
Here we are calling the masses which appear in Eq. (1.4) “gravitational masses,” to distinguish them from the “inertial masses” which appear in Newton’s second law of motion, Eqs. (1.2) and (1.3). They might also be called “gravitational charges,” by analogy with the electrical charges which appear in Coulomb’s force, to be discussed later on. The electrical charges generate and experience electrical forces, while gravitational masses generate and experience gravitational forces. In this respect and observing the form of the force laws of universal gravitation and of Coulomb, the gravitational masses have a greater resemblance to electrical charges than inertial masses. Later on we discuss this in greater detail.
Utilizing Newton’s law of gravitation, Eq. (1.4), and his theorems stated above, we ﬁnd that a spherically symmetrical body will attract an external body as if all the gravitational mass of the spherical body were concentrated at its center. In the case of the earth, neglecting the small eﬀects due to its form being not exactly spherical, this yields:

F

=

−G

Mgtmg r2

rˆ

,

where Mgt is the gravitational mass of the earth and rˆ points radially outwards. This force is usually called the weight of the body, and is represented by P :

P = mgg ,

(1.5)

where

g

=

−

GMgt r2

rˆ

.

Here g is called the gravitational ﬁeld of the earth. It is the downward acceler-
ation of freely falling bodies, as we will see.
If we are close to the surface of the earth, then r ≈ Rt, where Rt is the earth’s radius. Near the surface of the earth the measured value of this acceleration is found to be: g = |g| ≈ GMgt/Rt2 ≈ 9.8 m/s2.
By performing experiments with pendulums, Newton established that the
gravitational and inertial masses are proportional or equal to one another. He
expressed this as a proportionality between matter (mi) and weight (mg|g|) in Proposition 6 of Book III in the Principia:

That all bodies gravitate towards every planet; and that the weights of bodies towards any one planet, at equal distances from the centre of the planet, are proportional to the quantities of matter which they severally contain.

1.2. NEWTONIAN MECHANICS

23

Newton’s Propositions 70 and 71 given above are presented nowadays as
follows: We have a spherical shell of gravitational mass Mg and radius R centered on O, as in Figure 1.1. An element of mass dmg2 located at r2 in this spherical shell is given by dmg2 = σg2da2 = σg2R2dΩ2 = σg2R2 sin θ2dθ2dϕ2, where σg2 = Mg/4πR2 is the uniform surface mass density, dΩ2 is the element of spherical angle, θ2 and ϕ2 are the usual angles of spherical coordinates, θ2 ranging from 0 to π and ϕ2 from 0 to 2π.

Figure 1.1: Spherical shell of mass Mg interacting with a mass point of mass mg1.

The gravitational force exerted by this element of mass on the test particle mg1 located at r1 is given by Eq. (1.4), namely:

dF21

=

−G

mg1dmg2 r122

rˆ12

,

where r12 = r1 − r2, r12 = |r12| and rˆ12 = r12/r12. Integrating this equation yields the following results, with r1 = |r1|:

F=

−GMgmg1rˆ1/r12 , if r1 > R

0,

if r1 < R .

(1.6)

If the test particle is outside the spherical shell it will be attracted as if the

whole shell were concentrated at its center. If the test particle is anywhere

inside the shell it will not experience any net gravitational force.

Nowadays, these theorems are easily proved utilizing Gauss’s theorem. The

force exerted by several masses on mg located at r may be written as: F = mgg, where g is the gravitational ﬁeld at r due to the other masses. Gauss’s theorem

applied to the gravitational ﬁeld states that the ﬂux of g over a closed surface S is given by −4πGMignt, where Mignt is the gravitational mass internal to S:

24

CHAPTER 1. NEWTONIAN MECHANICS

g · da = −4πGMignt .
S

(1.7)

Gauss’s theorem is valid for any radial ﬁeld which falls as 1/r2, as is the case

for Newton’s law. Let us calculate the gravitational ﬁeld of a spherical shell of

gravitational mass Mg and radius R centered on the origin O of a coordinate system. For reasons of symmetry, the gravitational ﬁeld due to this spherical

shell can only be radial, namely: g = g(r)rˆ. We now consider a spherical surface

S centered on O and with a radius r > R. The element of area of this spherical surface is da = r2dΩrˆ, where dΩ = sin θdθdϕ is the element of spherical angle.

Utilizing Gauss’s theorem we obtain:

g4πr2 = −4πGMg ,
so that: g(r > R) = −GMgrˆ/r2. If we had integrated over a surface S such that r < R than Mignt = 0, so
that we would arrive at: g(r < R) = 0. With these results we recover Eq. (1.6). Newton was completely aware of the cosmological implications of his 70th
proposition, theorem 30 (the gravitational force on a test body anywhere inside a spherical shell is zero). The main implication is that we can essentially neglect the gravitational inﬂuence of the ﬁxed stars on planetary motions and in experiments conducted on the earth, as the stars are randomly scattered in all directions in the sky (neglecting the concentration of stars in the Milky Way). He expressed this clearly in the second corollary of Proposion 14, Theorem 14 (The aphelions and nodes of the orbits of the planets are ﬁxed), of Book III of the Principia:

Cor. I. The ﬁxed stars are immovable, seeing they keep the same position to the aphelion and nodes of the planets.
Cor. II. And since these stars are liable to no sensible parallax from the annual motion of the earth, they can have no force, because of their immense distance, to produce any sensible eﬀect in our system. Not to mention that the ﬁxed stars, everywhere promiscuously dispersed in the heavens, by their contrary attractions destroy their mutual actions, by Prop. 70, Book I.

Newton discussed the distance of the ﬁxed stars to the solar system at greater length in Section 57 of the System of the World, [3, pp. 596-7].
It is usually stated in textbooks that the gravitational constant G was measured by H. Cavendish (1731-1810) in 1798 with his torsion balance experiment. As a matter of fact, neither Newton nor Cavendish wrote the force law with G, as is given in Eq. (1.4), and they never mentioned the gravitational constant

1.2. NEWTONIAN MECHANICS

25

G. Cavendish’s paper is called “Experiments to determine the density of the earth,” [5]. What he found is that the mean density of the earth is (5.448±0.033) times greater than the density of water (Cavendish gave 5.48, due to an error in calculation corrected by A. S. Mackenzie, who reprinted Cavendish’s work in 1899. See [5], Gravitation, Heat and X-Rays, pp. 100-101 and 143). For a discussion of his work see [6].
The claim that he measured G deserves an explanation. Considering the earth to be exactly spherical, the force it exerts on a material particle of gravitational mass mg near its surface utilizing Eq. (1.4) is given by P = GMgtmg/Rt2 = mgg. The quantity of matter of the earth is given by its inertial mass Mit = ρt × Vt = ρt × 4πRt3/3, where ρt is its mean density, Vt its volume and Rt its radius. Newton found experimentally that the quantity of matter is proportional to the weight. Here we utilize this fact with a constant of proportionality equal to one, namely: Mit = Mgt = Mt. We then obtain G = 3g/4πRtρt. The gravitational ﬁeld of the earth g near its surface has the same value as the acceleration of free fall. In the MKSA system of units we have: g ≈ 9.8 m/s2 and Rt = 6.4 × 106 m. With Cavendish’s measurement we get ρt = 5.448 × 103 kg/m3 and Mt = 6 × 1024 kg, where we have used the fact that the density of water is given by ρwater = 1 g/cm3 = 103 kg/m3. This value of ρt in the previous expression for G yields: G = 6.7 × 10−8 cm3/gs2 = 6.7 × 10−11 m3/kgs2. The value given by modern tables is G = 6.67 × 10−11 m3/kg s2, which shows that Cavendish’s measurement of the mean density of the earth is quite accurate.
We can then see that the value of G depends not only on the system of units but also on the choice of the constant of proportionality between the inertial and gravitational masses. If we had chosen Mit = αMgt, where the constant α could even have dimensions, then the value and dimensions of G would need to change to: G = α2 × (6.67 × 10−11 m3/kgs2). But this would not aﬀect the results and predictions of any experiments. It is only a matter of convention to choose α = 1 and this yields the usual value of G.
It should be remarked that Newton had a very good idea of the mean density of the earth 100 years before Cavendish. For instance, in Proposition 10 of Book III of the Principia he wrote:
But that our globe of earth is of greater density than it would be if the whole consisted of water only, I thus make out. If the whole consisted of water only, whatever was of less density than water, because of its less speciﬁc gravity, would emerge and ﬂoat above. And upon this account, if a globe of terrestrial matter, covered on all sides with water, was less dense than water, it would emerge somewhere; and, the subsiding water falling back, would be gathered to the opposite side. And such is the condition of our earth, which

26

CHAPTER 1. NEWTONIAN MECHANICS

in a great measure is covered with seas. The earth, if it was not for its greater density, would emerge from the seas, and, according to its degree of levity, would be raised more or less above their surface, the water of the seas ﬂowing backwards to the opposite side. By the same argument, the spots of the sun, which ﬂoat upon the lucid matter thereof, are lighter than that matter; and, however the planets have been formed while they were yet in ﬂuid masses, all the heavier matter subsided to the centre. Since, therefore, the common matter of our earth on the surface thereof is about twice as heavy as water, and a little lower, in mines, is found about three, or four, or even ﬁve times heavier, it is probable that the quantity of the whole matter of the earth may be ﬁve or six times greater than if it consisted all of water; especially since I have before shown that the earth is about four times more dense than Jupiter. (...)
Newton estimated 5ρwater < ρt < 6ρwater and Cavendish found 100 years later ρt = 5.5ρwater!

1.3 Energy

Newton based his mechanics in the concepts of force and acceleration. There is another formulation based on the idea of energy. This formulation is due originally to Huygens and Leibniz, although it has been later on incorporated in newtonian mechanics. The basic concept is that of kinetic energy T . If we are in an inertial frame S and a particle of inertial mass mi moves in this frame with a velocity v then its kinetic energy is deﬁned by

T

≡

miv2 2

v·v = mi 2

.

This kinetic energy is an energy of pure motion in classical mechanics. It is not related to any kind of interaction (gravitational, electric, magnetic, elastic, etc.) As such, it depends on the frame of reference, because the same body at the same time may have diﬀerent velocities relative to diﬀerent inertial frames, so that its kinetic energy relative to each one of these frames may have a diﬀerent value.
The other kinds of energy are based on how the particle interacts with other bodies. For instance, the gravitational potential energy Ug between two gravitational masses mg1 and mg2 separated by a distance r is given by

Ug

= −G mg1mg2 r

.

1.3. ENERGY

27

If the body mg1 is outside the earth at a distance r1 from its center we can integrate this equation, replacing mg2 by dmg2 and assuming an isotropic matter distribution, to obtain

U = −G mg1Mg , r1
where Mg is the gravitational mass of the earth. If the body is near the earth of radius Rt, at a distance h from its surface,
r1 = Rt + h, with h Rt, this reduces to

U

= −G mg1Mg Rt + h

≈ mg1gh −

Gmg1Mg Rt

,

where g = GMg/Rt2 ≈ 9.8 m/s2 is the gravitational ﬁeld of the earth at its surface. Besides the constant term −Gmg1Mg/Rt this shows that the gravitational potential energy near the earth’s surface is given by mg1gh.
The analogous electrostatic potential energy Ue between two point charges q1 and q2 separated by a distance r is given by

Ue

=

1 q1q2 4πεo r

,

where εo = 8.85 × 10−12 C2 s2/kg m3 is the vacuum permittivity. The potential elastic energy Uk of a mass interacting with a spring of elastic
constant k is given by

kx2 Uk = 2 ,

where x is the displacement of the body from the equilibrium position (x = − o, with being the stretched length of the spring and o its relaxed length). We relate the concepts of force and energy by the equation

F = −∇U .

(1.8)

This is especially useful when the potential energy and the force depend only on the positions of the bodies.
When we utilize the formulation of mechanics based only on the concept of energy, we utilize the theorem for the conservation of energy instead of Newton’s three laws of motion. This law simply states that the total energy of the system (sum of the kinetic and potential energies) is a constant in time for conservative systems.
In this work we focus more on the Newtonian formulation based on forces.

28

CHAPTER 1. NEWTONIAN MECHANICS

Chapter 2
Applications of Newtonian Mechanics

Here we discuss several well-known applications of Newtonian mechanics. Later on we present Mach’s criticisms of classical mechanics utilizing these examples. Lastly we present these examples from the point of view of relational mechanics to illustrate the diﬀerent approach it makes possible.
In Newton’s second law of motion, Eqs. (1.2) and (1.3), there appear a velocity and an acceleration (assuming a constant inertial mass). These velocities and accelerations are to be understood as referred to absolute space, and measured by absolute time. According to the ﬁfth corollary we may also refer motion to any frame of reference which moves relative to absolute space with a constant velocity. Nowadays we call these frames of reference “inertial frames.” In what follows, we assume that we are describing the motion of bodies in one inertial frame. Later on we will discuss this concept in more detail.
Here we consider only situations in which the inertial mass is a constant. In these cases Newton’s second law of motion takes the form

F = mia .

(2.1)

Bodies with negligible dimensions compared with the distances involved in the problems are called particles. Usually we can neglect its internal properties and represent them by material points. A particle will be characterized by its mass, and for its localization we will utilize three coordinates describing its position: x, y, z. We are interested here in the motion of particles in paradigmatic situations.

29

30

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

2.1 Uniform Rectilinear Motion
If we have a particle which is free from external forces, or if the resultant force acting on this particle is zero, then the particle will move with a constant velocity v according to the ﬁrst law of motion:
dr v = = constant ,
dt

r = ro + vt .
Here r(t) is the position vector of the body relative to an inertial system of reference, ro the initial position of the particle and t the time. The velocity v is the velocity of the test body relative to an inertial frame of reference, or relative to Newton’s absolute space.
The direction and magnitute of the velocity will be constant in time. This can only make sense if we know how to say when a particle is free from external forces (if we know in which conditions this happens). We also need to ﬁnd an inertial system of reference without utilizing Newton’s ﬁrst law of motion (to avoid vicious circles). None of this is in any way simple or trivial.

2.2 Constant Force
We can easily integrate Eq. (2.1) when the force is a contant, yielding: dv F
a = = = constant , dt mi

v = vo + at ,
at2 r = ro + vot + 2 . Here vo is the initial velocity.
2.2.1 Free Fall
As the ﬁrst example of a constant applied force we have the free fall of a body near the surface of the earth, neglecting air resistance, Figure 2.1.
The only force acting on the test body is the gravitational attraction of the earth, namely, its weight P = mgg. With Eq. (2.1) we get:

2.2. CONSTANT FORCE

31

Figure 2.1: Free fall of a body of mass m.

a = mg g . mi

The value of g depends only on the earth and on the location of the test body, but does not depend of mi or mg. The gravitational ﬁeld g does not depend on the inertial or gravitational mass of the test body.
It is a fact of experience that all bodies fall in vacuum with the same acceleration near the surface of the earth. This fact cannot be derived from any of Newton’s laws or mathematical theorems. This result is valid no matter what the weight, form or chemical composition of the bodies. If we have bodies 1 and 2 falling in vacuum at the same location near the earth’s surface, we know from experience that a1 = a2 = g, Figure 2.2. This means that mg2/mi2 = mg1/mi1 = constant.
The ﬁrst to arrive at this conclusion was Galileo (1564-1642) when working with bodies falling on inclined planes. Some of the main results obtained by Galileo in mechanics date from the period 1600 to 1610. From these experiments (bodies falling on inclined planes with negligible resistance, or falling freely in vacuum) we obtain for all bodies:

mi = constant = α . mg

(2.2)

When we say that these two masses (inertial and gravitational) are equal (α = 1), we are specifying G = 6.67 × 10−11 m3/kg s2. If we said that these
two masses were proportional to one another, mi = αmg (where α might be a constant diﬀerent from one and could even have dimensions), all the results would remain valid provided we had put G = α2 × (6.67 × 10−11 m3/kgs2) instead of G = 6.67 × 10−11 m3/kgs2 in Newton’s law of gravitation. To see
this we need only write the acceleration of free fall as:

32

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

Figure 2.2: Two diﬀerent bodies fall to the ground with the same acceleration in vacuum.

a=

P mi

=

GMgtmg Rt2mi

.

With Eq. (2.2) and Mit = ρtVt = ρt4πRt3/3 (from Newton’s ﬁrst deﬁnition) we get:

G

3a

α2 = 4πRtρt .

Putting the observed values of a = 9.8 m/s2, Rt = 6.4×106 m and ρt = 5.5×103 kg/m3 (from Cavendish’s experiment) yields:

G = α2 × (6.7 × 10−11 m3/kgs2) .
The experiments of free fall only say that these two masses are proportional to one another and not that they are equal. As the choice of α has no inﬂuence in the predictions of experiments, it is simpler to say that they are equal to one another, choosing by convention α = 1. From now on we will take this choice of α:

mi = mg .

(2.3)

The fact that in vacuum all bodies fall with the same acceleration was expressed as follows by Newton in the Principia: “It has been, now for a long time, observed by others, that all sorts of heavy bodies (allowance being made

2.2. CONSTANT FORCE

33

for the inequality of retardation which they suﬀer from a small power of resistance in the air) descend to the earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums” (Book III, Proposition 6). In the Opticks he expressed it as follows: “(...) The open air in which we breathe is eight or nine hundred times lighter than water, and by consequence eight or nine hundred times rarer, and accordingly its resistance is less than that of water in the same proportion, or thereabouts, as I have found by experiments made with pendulums. And in thinner air the resistance is still less, and at length, by rarefying the air, becomes insensible. For small feathers falling in the open air meet with great resistance, but in a tall glass well emptied of air, they fall as fast as lead or gold, as I have seen tried several times” [7] (Book III, Query 28, p. 366).

2.2.2 Charge Moving Inside an Ideal Capacitor

We now present another example of a constant force. In 1784-5 Augustin Coulomb (1738-1806) obtained the law of force between two point charges q1 and q2. In modern vectorial notation and in the International System of Units the force exerted by q2 on q1 is given by:

F21

=

q1q2 4πεo

rˆ r2

.

(2.4)

In this equation εo = 8.85 × 10−12 C2 s2/kg m3 is the vacuum permittivity, r is the distance between the charges and rˆ is the unit vector pointing from q2 to q1.
This force is very similar to Newton’s law of gravitation, as it is directed along the straight line connecting the bodies, follows the law of action and reaction and falls as the inverse square of the distance. Moreover, it depends on the product of two charges, as in Newton’s law it depends on the product of two masses. It would appear that Coulomb was led to this expression more by analogy with Newton’s law of gravitation than by the results of his doubtful experiments [8]. The similarity between Coulomb’s force (2.4) and Newton’s law of gravity, Eq. (1.4), shows that the gravitational masses have the same role as the electrical charges: both generate and experience some kind of interaction with equivalent bodies, whether electrical or gravitational. The form of the interaction is essentially the same.
An ideal capacitor is represented in Figure 2.3. Two large square plates are separated by a distance d , where is the length of any plate. The plates situated at z = zo and z = −zo are uniformly charged with charges Q and −Q, respectively. In each plate we have a constant charge density given by σ = Q/ 2 and −σ, respectively. If we integrate the force exerted by the

34

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

capacitor on an internal test charged particle utilizing Coulomb’s force and neglecting edge eﬀects we obtain the well known result:

σzˆ F = −q = qE .
εo

(2.5)

Here zˆ is the unit vector pointing from the negative to the positive plate and E = −σzˆ/εo is the electric ﬁeld generated by the capacitor in the region between the plates. Outside the capacitor there are no electric or magnetic ﬁelds.

Figure 2.3: Ideal capacitor generating a uniform electric ﬁeld between its plates.
In Weber’s electrodynamics there will be a component of the force exerted by the capacitor on the test charge q moving inside it which depends on the velocity of q relative to the plates ([9], [10], [11, Section 5.6], [12, Sections 6.7 and 7.2], [13, Section 5.5], [14] and [15]). But supposing v2/c2 1, as we can consider in this experiment, Eq. (2.5) will also be valid in Weber’s electrodynamics.
In classical electrodynamics (Maxwell’s equations plus Lorentz’s force) this is the total force exerted by the capacitor on the internal test charge, regardless of the velocity or acceleration of q relative to the plates, assuming ﬁxed charges over the plates of the capacitor. This can be obtained by assuming a capacitor made of dielectric charged plates (with a vacuum between the plates) which do not allow a free motion of charges over its surface. Accordingly, the capacitor generates a constant electric ﬁeld only between the plates, and no magnetic ﬁeld.
Equating Eq. (2.5) with (2.1) yields:

2.2. CONSTANT FORCE

35

q a= E .
mi
The electric ﬁeld depends only on the surface density of charge over the plates of the capacitor, and is independent of q or mi. It is analogous to the gravitational ﬁeld near the surface of the earth in our previous example. The diﬀerence now is that in the same electric ﬁeld we can have bodies experiencing diﬀerent accelerations. For instance, a proton (p) undergoes double the acceleration of an alpha (α) particle (nucleus of the helium atom, with two protons and two neutrons) if both are accelerated by the same capacitor: ap = 2aα, as in Figure 2.4. This is due to the fact that the charge of an alpha particle is twice that of a proton, while its mass is four times that of the proton due to the two neutrons and two protons it contains. This does not happen in free fall, as all bodies, regardless of their weight, chemical composition, etc., fall with the same acceleration in vacuum near the surface of the earth.

Figure 2.4: A proton and an alpha particle being accelerated inside a capacitor.
This is an extremely important fact. Comparing these two examples (see Figures 2.2 and 2.4) we can see that the inertial mass of a body is proportional to its gravitational mass, but not to its electrical charge. This fact suggests

36

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

that the inertia of a body is related to its weight or gravitational property, but not to its electrical properties. Later on we will come back to this point.

2.2.3 Accelerated Train
The third example discussed here is an accelerated train moving along a straight line. From top of one of the wagons there is a small body suspended by a string, as in Figure 2.5.

Figure 2.5: Accelerated train with a small body suspended by a string.

Here we analyse the equilibrium situation in which the body is at rest relative to the accelerated train. In other words, we analyse the situation when both of them have the same constant acceleration relative to the earth or to an inertial frame of reference. There are two forces acting on the body: the gravitational force of the earth (the weight P = mgg), and the force exerted by the string due to its tension, T . The equation of motion is

P + T = mia . Utilizing the angle θ of Figure 2.5:

P = T cos θ ,

(2.6)

T sin θ = mia . From these expressions and from P = mgg we obtain:
tan θ = mi a . mg g

(2.7) (2.8)

2.3. OSCILLATORY MOTIONS

37

From the experimental fact that θ is the same for all bodies independent of their weight, chemical composition etc. we obtain once more that mi = mg or that the inertia of the body is proportional to its weight.

2.3 Oscillatory Motions
In this section we deal with forces which depend on position and which generate oscillatory motion.
2.3.1 Spring
The ﬁrst example to be discussed here is that of a mass fastened to a spring which is connected to the earth, Figure 2.6. The weight of the test body is balanced by the normal force exerted by a frictionless table. The only remaining force is the horizontal force exerted by the spring.

Figure 2.6: Spring on a frictionless table.

The force exerted by the spring on the body of inertial mass mi is given by

F = −kxxˆ ,

(2.9)

where k is the elastic constant, x is the displacement of the body from the
equilibrium position (x = − o, with being the extended length of the spring and o its relaxed length) and xˆ the unit vector along the length of the spring.
Equating this with Eq. (2.1) with a = (d2x/dt2)xˆ = x¨xˆ yields the one-
dimensional equation of motion:

38

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

mix¨ + kx = 0 . This equation can be easily solved:

(2.10)

where

x(t) = A sin(ωt + θo) ,

k

ω=

.

mi

(2.11)

The constant A is the amplitude of oscillation, θo is the initial phase and ω the frequency of oscillation. The constants A and θo may be related to the constant total energy E of the body and the initial position xo by:

E = T + U = mix˙ 2 + kx2 = kA2 ,

2

2

2

xo = A sin θo .
2.3.2 Simple Pendulum
The second and most important example to be discussed here is a simple pendulum, Figure 2.7. A small body of typical dimension d oscillates in a vertical plane fastened to a string of constant length such that d .

Figure 2.7: Simple pendulum of length .
Neglecting air resistance, there are two forces acting on the pendulum, its weight P = mgg = −mggzˆ and the tension in the string, T . The equation of

2.3. OSCILLATORY MOTIONS

39

motion is simply P + T = mia. Utilizing the angle represented in this Figure,
the fact that the length of the string is a constant and a polar coordinate system (with s = θ, vθ = θ˙ and aθ = θ¨ instead of x, x˙ and x¨) yields

T

−P

cos θ

v2 =m

=

m

θ˙2

,

−P sin θ = miaθ = mi θ¨ .
If we consider only small oscillations of the pendulum (θ θ, and this last equation reduces to:
miθ¨ + mg g θ = 0 .

π/2) then sin θ ≈

This equation has the same form as Eq. (2.10). Its solution is

θ = A cos(ωt + B) , with

(2.12)

ω = mg g . mi

(2.13)

The constant A is the amplitude of oscillation for θ, B is the initial phase and ω the frequency of oscillation.
We now compare the frequencies of oscillation ω for the spring and for the simple pendulum, Eqs. (2.11) and (2.13). The periods of oscillation are given simply by T = 2π/ω. The most striking diﬀerence is that while in the spring the frequency of oscillation depends only on mi but not on mg, in the pendulum the frequency of oscillation depends on the ratio mg/mi. Now suppose we have a test body of inertial mass mi and gravitational mass mg. If it is oscillating horizontally fastened to a spring of elastic constant k, its frequency of oscillation is given by ω1 = k/mi. If we connect two of these bodies to the sam√e spring, the new frequency of oscillation is given by ω2 = k/2mi = ω1/ 2, as in Figure 2.8.
On the other hand, if the ﬁrst body were connected to a string of constant length and oscillating like a pendulum, its frequency of oscillation would be given by: ω1 = mgg/mi . Connecting two of these bodies to the same string, the new frequency of oscillation is given by ω2 = 2mgg/2mi = ω1, as in Figure 2.9.
The same happens whatever the chemical composition of the test particle. In simple pendulums of the same length and at the same location on the

40

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

Figure 2.8: Two diﬀerent masses attached to the same spring.

Figure 2.9: Two diﬀerent masses attached to the same pendulum.
earth (same g), all bodies oscillate with the same frequency, regardless of their weight or chemical constitution, when air resistance is neglected. This is an experimental fact which cannot be derived from Newton’s laws of motion (from Newton’s laws we cannot derive that mi = mg nor that mi/mg = constant). Only experience can tell us that the frequency of oscillation of a simple pendulum in vacuum does not depend on the weight or chemical constitution of the bodies, while the frequency of oscillation on an horizontal spring is inversely proportional to the square root of the mass of the body.
This experimental fact shows that we can cancel the masses in Eq. (2.13), writing the frequency of oscillation ω of the pendulum and its period T as:
g 2π ω= = .
T
In section 2.2.2 we saw that the inertial mass of a body is proportional to the gravitational mass or weight of the body, but is not proportional to its charge or electrical properties. Here we see that the inertial mass of a body is not

2.3. OSCILLATORY MOTIONS

41

proportional to any elastic property of the body or of the surrounding medium (the spring in this case). Analogously, it can be shown that the inertial mass (or inertia) of a body is not related to the magnetic, nuclear, or any other property of the body or of the surrounding medium. Newton expressed this in Corollary V, Proposition 6 of Book III of the Principia, our words in square brackets: “The power of gravity is of a diﬀerent nature from the power of magnetism; for the magnetic attraction is not as the matter attracted [the magnetic force is not proportional to the inertial mass of the attracted body]. Some bodies are attracted more by the magnet; others less; most bodies not at all. The power of magnetism in one and the same body may be increased and diminished; and is sometimes far stronger, for the quantity of matter, than the power of gravity; and in receding from the magnet decreases not as the square but almost as the cube of the distance, as nearly as I could judge from some rude observations.”
The inertial mass is only proportional to the weight or gravitational mass of the body. Why does nature behave like this? There is no answer in Newtonian mechanics. We might imagine that a piece of gold could fall in vacuum with a larger acceleration than a piece of iron or silver of the same weight, but this is not the case. We might further imagine that a heavier lump of gold could fall in vacuum with a larger acceleration than a lighter lump of gold, or than another piece of gold with a diﬀerent shape. Once more, this is not what happens. If any of these things did happen, all results of Newtonian mechanics might be kept, provided we did not cancel mi with mg. We would then conclude that mi would depend on the chemical composition of the body, or on its form, or that it is not linearly proportional to mg, or ..., depending on what were found experimentally.
Although this striking proportionality between inertia and weight does not prove anything, it is highly suggestive. It indicates that the inertia of a body (its resistance to acceleration) may have a gravitational origin. Later on, we show that this is indeed the case.
For the moment we present here Newton’s own careful experiments with pendulums performed in order to arrive at this proportionality of inertia and weight (or proportionality between the quantity of matter mi and mg, as we would say today). In the ﬁrst deﬁnition of the Principia, quantity of matter, Eq. (1.1), Newton wrote: “It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body, for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shown hereafter.” These experiments are contained in the previously mentioned Proposition 6, Theorem 6 of Book III of the Principia:
That all bodies gravitate towards every planet; and that the weights of bodies towards any one planet, at equal distances from the centre

42

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

of the planet, are proportional to the quantities of matter which they severally contain.
It has been, now for a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suﬀer from a small power of resistance in the air) descend to the earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums. I tried experiments with gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal: I ﬁlled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes, hanging by equal threads of 11 feet, made a couple of pendulums perfectly equal in weight and ﬁgure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forwards and backwards, for a long time, with equal vibrations. And therefore the quantity of matter in the gold (by Cor. I and VI, Prop. XXIV, Book II) was to the quantity of matter in the wood as the action of the motive force (or vis motrix) upon all the gold to the action of the same upon all the wood; that is, as the weight of the one to the weight of the other: and the like happened in the other bodies. By these experiments, in bodies of the same weight, I could manifestly have discovered a diﬀerence of matter less than the thousandth part of the whole, had any such been. (...)
From this experiment Newton found that mi = mg within one part in a thousand:
mi − mg = ±10−3 . mi
With E¨otvos’s experiments at the turn of the century the precision of this relation improved to one part in 108. Nowadays it is known as one part in 1012. For references, see [16].

2.3.3 Electrically Charged Pendulum
We now discuss the motion of a simple pendulum of length and inertial mass mi performing small oscillations due to the gravitational attraction of the earth. Once more we suppose the earth to be a good inertial frame for this problem. The diﬀerence as regards subsection 2.3.2 is the following: Beyond its gravitational mass mg, we suppose the pendulum to have an electrical charge q and to be in the presence of a permanent magnet, as in Figure 2.10.

2.3. OSCILLATORY MOTIONS

43

Figure 2.10: Charged simple pendulum oscillating near a magnet.

In this case the forces acting on the simple pendulum will be the gravitational force of the earth (the weight P ), the tension T in the string and the magnetic force due to the magnet. In classical electromagnetism this force is represented by Fm = qv × B, where v is the velocity of the charge q relative to an inertial frame of reference (the earth or the laboratory in this case) and
B is the magnetic ﬁeld generated by the magnet. In Weber’s electrodynamics the force exerted by the magnet on the charge has essentially the same value, although we don’t necessarily need to speak of the magnetic ﬁeld, and the velocity v will be the velocity of the charge relative to the magnet ([9], [17], [18], [19] and [12, Sections 6.7, 7.3 and 7.4]). As we are supposing the magnet to be at rest relative to the earth (assumed to be a good inertial frame in this experiment), there will not be any fundamental diﬀerence between the expressions for the magnetic force according to Weber’s electrodynamics and classical electromagnetism. We then have a uniform gravitational ﬁeld g = −gzˆ pointing downwards and a magnetic ﬁeld B = Bzˆ pointing vertically upwards. To simplify the analysis we will assume a uniform magnetic ﬁeld (constant magnitude in space and time). The equation of motion takes the form

P + T + qv × B = mia .

(2.14)

We now suppose small oscillations (θ π/2) and that the pendulum is released from rest (vo = 0) from the initial position so = −|θo| ≈ xo with initial motion along the xz plane. With these conditions we ﬁnd from (2.12) that in the absence of a magnetic ﬁeld the velocity of the pendulum along the x axis is given approximately by

vx ≈ vθ = θ˙ = |θo|ω sin ωt .

(2.15)

44

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

Here we put the initial phase equal to zero due to the initial conditions that the pendulum was released at rest from the initial angle θo. Moreover, we utilized vx ≈ vθ because for small oscillations the motion is practically along the horizontal x axis. If there were no magnetic ﬁeld, the pendulum would remain oscillating along the xz plane of our inertial frame of reference.
Supposing now the presence of the magnetic ﬁeld, the motion of the pendulum will no longer remain along the same plane. With an initial velocity along the x axis, the vertical magnetic ﬁeld will exert a force along the y axis given by

qv × B = qvxxˆ × Bzˆ = −qvxByˆ .

(2.16)

This force will modify the motion of the pendulum as indicated in Figure 2.11. In this Figure we are observing the projection of the motion of the pendulum in the yz plane as if we were on top of the pendulum. Assuming an initial motion along the positive x direction, vx > 0, and a positive charge, q > 0, the magnet will deviate the motion of the pendulum in the direction y < 0. On the other hand, when the same pendulum is returning (vx < 0) the magnet deviates it in the direction y > 0. This creates a clockwise rotation of the plane of oscillation of the pendulum with an angular velocity Ω (looking at the pendulum from above, supposing q > 0 and a magnetic ﬁeld pointing vertically upwards).

Figure 2.11: Rotation of the plane of oscillation of a charged pendulum due to a magnet.
We now calculate Ω assuming a weak magnetic ﬁeld, namely, qB/miω 1. This is analogous to having the greatest velocity in the x direction much larger

2.3. OSCILLATORY MOTIONS

45

than the greatest velocity in the y direction, or to saying that the velocity in the x direction is essentially unaﬀected by the magnet. From Eqs. (2.14), (2.15) and (2.16) the equation of motion in the y direction is given by (observing that P = −mggzˆ and that the tension T is in the xz plane):

−qvxB = −q|θo| ω sin ωt = miay .

(2.17)

This equation can be easily integrated twice utilizing that vy(t = 0) = 0 and y(t = 0) = 0, yielding:

y = qB|θo|

sin ωt −t .

mi

ω

The value of Ω can be obtained from Figure 2.12.

(2.18)

Figure 2.12: Geometry for calculating the precession of the plane of oscillation of a charged pendulum.

In half a period, t = π/ω, the pendulum has moved from xo = −|θo| to x = |θo| , such that x = 2|θo| . On the other hand it has moved from yo = 0 to y(π/ω) = −qB|θo| π/miω, such that y = −qB|θo| π/miω. The value of Ω is then given by

y/ x qB

Ω=

=− .

t

2mi

(2.19)

The negative value of Ω indicates a rotation in the clockwise direction when the pendulum is seen from above. To arrive at this result we neglected friction, and assumed uniform gravitational and magnetic ﬁelds, and that qB/miω 1.

46

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

We conclude that the magnet causes a precession of the plane of oscillation of the charged pendulum oscillating in an inertial frame due to the action of a uniform gravitational force.

2.4 Uniform Circular Motion
In this section we discuss three situations of uniform circular motion which were analysed by Newton: A planet orbiting around the sun, two globes connected by a string and the spinning bucket.
We ﬁrst consider a single body under the inﬂuence of a central force F , shown in Figure 2.13.

Figure 2.13: Uniform circular motion under a central force.
We consider a central force always directed toward the origin O of an inertial coordinate system S: F = −F ρˆ, where F = |F | and ρˆ is a unit vector pointing radially from O. With a polar coordinate system to describe the position, velocity and acceleration of a particle we have, respectively:
r = ρρˆ ,

dr v = = ρ˙ρˆ + ρϕ˙ ϕˆ ,
dt

where ρ = |r|.

a

=

dv dt

=

d2r dt2

=

(ρ¨ − ρϕ˙ 2)ρˆ + (ρϕ¨ + 2ρ˙ϕ˙ )ϕˆ

,

2.4. UNIFORM CIRCULAR MOTION

47

With Eq. (2.1), and a constant ρ (uniform circular motion) we obtain:

F = miac = miρϕ˙ 2 ,

(2.20)

ϕ˙ = constant .

We represent the centripetal acceleration ac arising from the motion in the ϕ direction by:

ac = ρϕ˙ 2

=

vt2 ρ

,

where

vt = ρϕ˙ .
Huygens (1629-1695) and Newton were the ﬁrst to obtain this value for the acceleration of a body orbiting with a constant velocity around a center. Huygens calculated the centrifugal force (a name created by him, meaning a tendency to depart from the center), arriving at his result in 1659. His manuscript on this topic was only published posthumously in 1703. However, in his book Horologium Oscillatorum, of 1673, he presented the main properties of the centrifugal force, but did not supply the proofs of how he arrived at these results. In any event he was the ﬁrst to publish the correct value of this acceleration. Newton calculated the centripetal force (a name he framed later on in order to oppose to Huygens centrifugal force) between 1664 and 1666, without knowing Huygens’s results. In the Principia, of 1687, he made great use of this result. See [4, Sections 9.7-9.8 and 10.5-10.6].
It should be observed that this central force changes only the direction of motion, leaving the magnitude of the tangential velocity constant: |vt| = constant.

2.4.1 Circular Orbit of a Planet
The ﬁrst situation analysed here is a planet orbiting around the sun due to their mutual gravitational attraction. We consider the gravitational mass of the planet, mgp, much smaller than the gravitational mass of the sun, mgs, so that we can neglect the motion of the sun, see Figure 2.14. Although the orbit of the planets is in general elliptical, we consider here only the particular case of circular orbits in which the distances of the planets to the sun are constants in time.
From Eqs. (2.20) and (1.4) we obtain:

48

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

Figure 2.14: Planet orbiting around the sun.

F

=

G

mgsmgp r2

=

mipacp

=

mip

vt2p r

.

From Eq. (2.3):

acp =

vt2p r

=

Gmgs r2

.

The centripetal acceleration and the orbital velocity do not depend on the mass of the planet, but only on the mass of the sun.
How does the planet maintain a constant distance to the sun (or the moon to the earth, for instance) despite the gravitational attraction between them? According to Newton it is because the planet has an acceleration relative to absolute space (we might say relative to an inertial frame of reference). If the planet and the sun were initially both at rest relative to an inertial frame, they would attract and approach one another due to this attraction. What keeps the planet at a constant distance from the sun despite their gravitational attraction is the centripetal acceleration of the planet in absolute space (its tangential motion relative to the sun).

2.4. UNIFORM CIRCULAR MOTION

49

2.4.2 Two Globes
We now consider two globes connected to one another by a string and spinning relative to an inertial system with a constant angular velocity ω = ϕ˙ = vt/ρ around the center of mass O, Figure 2.15.

Figure 2.15: Two spinning globes connected by a string.

The only force exerted on each globe is due to the tension in the string. We call this tension T . By applying Eq. (2.20) to globe 1 we obtain:

T

= mi1ac1

=

mi1

vt21 ρ1

= mi1w2ρ1

,

For the second body we have, analogously:

(2.21)

T

= mi2ac2

=

mi2

vt22 ρ2

= mi2w2ρ2

.

(2.22)

The faster the rotation of the globes (i.e. the larger ω) the larger will be

the tension in the string supporting the rotation. If instead of a string we had

a spring of elastic constant k, the tension might be measured by T = k( − o), where is the stretched length of the spring ( = ρ1 +ρ2, see Figure 2.15) and o its relaxed length. By measuring , given k and o, we could know the tension. Knowing mi1, ω and ρ1 we can also obtain the tension T applying Eq. (2.21).
Newton discussed this problem of the two globes as a possible way of dis-

tinguishing the relative from absolute motion (or, more speciﬁcally, the relative

50

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

rotation from the absolute rotation). By this experiment we could know if the globes were really rotating or not rotating relative to absolute space (or relative to an inertial frame). His discussion appears in the Scholium in the beginning of Book I of the Principia, following the ﬁrst 8 deﬁnitions and before the three axioms or laws of motion. Here we present the entire discussion, with our emphasis:
It is indeed a matter of great diﬃculty to discover, and eﬀectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate; for we have some arguments to guide us, partly from the apparent motions, which are the diﬀerences of the true motions; partly from the forces, which are the causes and eﬀects of the true motions. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavor of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord, we might infer the increment or decrement of their motions; and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented; that is, we might discover their hindmost faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might ﬁnd both the quantity and determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position to one another, as the ﬁxed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest; and then, lastly, from the translation of the globes among the bodies, we should ﬁnd the determination of their motions. But

2.4. UNIFORM CIRCULAR MOTION

51

how we are to obtain the true motions from their causes, eﬀects, and apparent diﬀerences, and the converse, shall be explained more at large in the following treatise. For to this end was that I composed it.
Suppose the ﬁxed stars to be at rest relative to absolute space. Spinning the globes with an angular velocity ω relative to absolute space (or relative to the ﬁxed stars in this case) would, according to Newton, generate a tension in the string. This might be visualized by an increase in the length of a spring replacing the cord. Now suppose the same kinematical situation as above, namely, the globes rotating relative to the ﬁxed stars with a constant angular velocity ω. But if in this second case the globes were at rest relative to absolute space and the ﬁxed stars were revolving as a whole with an angular velocity −ω relative to absolute space, then, according to Newton in this passage, there would be no tension in the cord (or the spring would not be streched or under tension). In this way we might distinguish the true or absolute rotation of the globes (relative to absolute space) from the apparent or relative rotation of the globes (relative to the ﬁxed stars). Observing if there is or not a tension in the cord we might know if the globes were spinning or not relative to absolute space (or relative to an inertial frame of reference), although in both cases there would be the same relative or kinematical rotation of the globes relative to all other matter (the ﬁxed stars here). Later on we will discuss this experiment further.

2.4.3 Newton’s Bucket Experiment
We now analyse Newton’s bucket experiment. This is one of the simplest and most important of all experiments performed by Newton. It is described just before the two-globes experiment presented above, in the Scholium following the eight deﬁnitions in the beginning of Book I of the Principia, before the presentation of the axioms or laws of motion (our emphasis):
The eﬀects which distinguish absolute from relative motion are, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion, they are greater or less, according to the quantity of motion. If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then ﬁlled with water, and held at rest together with the water; thereupon, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion; the surface of the water will at ﬁrst be plain, as before

52

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

the vessel began to move; but after that, the vessel, by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave ﬁgure (as I have experienced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavor to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, becomes known, and may be measured by this endeavor. At ﬁrst, when the relative motion of the water in the vessel was greatest, it produced no endeavor to recede from the axis; the water showed no tendency to the circunference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavor to recede from the axis; and this endeavor showed the real circular motion of the water continually increasing, till it had acquired its greatest quantity, when the water rested relatively to the vessel. And therefore this endeavor does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be deﬁned by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavoring to recede from its axis of motion, as its proper and adequate eﬀect; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and, like other relations, are altogether destitute of any real eﬀect, any otherwise than they may perhaps partake of that one only true motion. (...)
Let us obtain the form of the surface and the pressure anywhere within the spinning bucket. We consider the water to be an ideal homogeneous incompressible ﬂuid of density ρ = 1 g/cm3.
In the ﬁrst situation the bucket and water are at rest relative to an inertial system, shown in Figure 2.16.
Then the surface of the water is ﬂat and the pressure within the liquid increases as a function of the depth h according to p(h) = po + ρgh, where po = 1 atm = 760 mm Hg = 1 × 105 N/m2 is the normal atmospheric pressure and g ≈ 9.8 m/s2 is the gravitational ﬁeld of the earth. From this expression we may obtain Archimedes’ (287-212 b. C.) rule: The upward force exerted by the water on any immersed body of volume V is given by the weight of the

2.4. UNIFORM CIRCULAR MOTION

53

ﬂuid displaced (in modern terms this is given by ρgV ). See On Floating Bodies in [20] and [21, pp. 538-560, especially Propositions 6 and 7]. This force does not depend on the mass of the body, but only on its immersed volume and the density of the surrounding liquid (or on the weight of the displaced ﬂuid).
Now consider the bucket and water spinning together at a constant angular velocity ω relative to an inertial frame of reference (we may consider the earth as a good inertial system in this case). The water forms a concave ﬁgure, as represented in Figure 2.17.

Figure 2.16: Bucket and water at rest relative to the earth.

Figure 2.17: Bucket and water spinning together relative to the earth (Newton’s bucket experiment).

The simplest way to obtain the form of the surface is to consider a frame of reference centered on the lower part of the spinning liquid with the z axis pointing vertically upwards, as in Figure 2.18.
Let us consider a small volume of liquid dmi = ρdV just below the surface. It is acted upon by the downward force of gravity, dP = dmgg, and by a force normal to the surface of the liquid due to the gradient of pressure, dE. As this portion of liquid moves in a circle centered on the z axis, there is no net vertical force. There is only a centripetal force pointing towards the z axis changing its direction of motion, but not the magnitude of the tangential velocity. From Figure 2.18 we obtain in this case (x being the distance of dmi to the z axis):

dE cos α = dP = dmgg ,

(2.23)

54

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

Figure 2.18: Geometry to calculate the form of the water surface when spinning.

dE sin α

=

dmiac

=

dmi

vt2 x

=

dmiω2x

.

(2.24)

From these two equations and utilizing the fact that dmi = dmg we have

ω2 tan α = x .
g

(2.25)

Utilizing tan α = dz/dx, where dz/dx is the inclination of the curve at each point, and the fact that we want the equation of the curve which contains the origin x = z = 0 yields

z = ω2 x2 . 2g

(2.26)

The curve is a paraboloid of revolution. The greater the value of ω, the larger the concavity of the surface.
We can also calculate the pressure anywhere within the liquid by similar reasoning, with Figure 2.19. The equation of motion of a small quantity of water dmi is dP + dE = dmia, where dE is the force due to the gradient of pressure.
For the element of mass represented in Figure 2.19:

∂p ∂p dE = −(∇p)dV = − xˆ + zˆ dV .
∂x ∂z

2.4. UNIFORM CIRCULAR MOTION

55

Figure 2.19: Forces in a volume element of water within the spinning bucket.

Utilizing the fact that there is only a centripetal acceleration yields a = −(v2/x)xˆ = −ω2xxˆ. With dP = −dmggzˆ and dmi = dmg we obtain
∂p = −ρg ,
∂z

∂p = ρω2x . ∂x

Integration of the ﬁrst of these equations yields: p = −ρgz + f1(x), where f1(x) is an arbitrary function of x. Integration of the second equation yields: p = ρω2x2/2 + f2(z), where f2(z) is another arbitrary function of z. Equating these two solutions and utilizing the fact that p(x = 0, z = 0) = po yields the ﬁnal solution (valid within the water):

p(x,

z) =

ρω2 2

x2

−

ρgz

+

po

.

All over the surface of the liquid we have p(x, z) = po. Substituting this in the previous result once more yields the equation of the concave surface, namely, z = ω2x2/2g. This completes the solution of this problem in classical
mechanics.

56

CHAPTER 2. APPLICATIONS OF NEWTONIAN MECHANICS

The importance of this experiment to Newton lies in the distinction it allows between absolute and relative rotation. According to Newton the surface will be concave only when the water is spinning relative to absolute space. This means that to him the ω which appears in Eq. (2.26) is the angular rotation of the water relative to absolute space and not the angular rotation of the water relative to any “ambient bodies.” It is not the rotation of the water relative to the bucket, nor relative to the earth, nor even its rotation relative to the distant universe, such as the ﬁxed stars. Remember that to Newton the absolute space has no relation to anything external, so that it is not related to the earth nor to the ﬁxed stars.
We will now show that Newton had no other alternative at that time than to arrive at this conclusion. As the angular rotation of the bucket in Newton’s experiment is much larger than the diurnal rotation of the earth or the annual rotation of the solar system, we may consider the earth to be without acceleration relative to the frame of ﬁxed stars, and as a good inertial system during this experiment. In the ﬁrst situation the bucket and the water are essentially at rest relative to the earth, so that they have at most a constant velocity relative to the ﬁxed stars. The surface of the water is ﬂat and there are no problems in deriving this conclusion. We now consider the second situation in which the bucket and the water are spinning together relative to the earth (and so relative to the ﬁxed stars) with a constant angular velocity ωbe = ωwe ≡ ω = ωzˆ. Here the z axis points vertically upwards from that location (zˆ = rˆ, where rˆ points radially outwards from the earth’s center), ωbe is the angular velocity of the bucket relative to the earth and ωwe is the angular velocity of the water relative to the earth. In this case the surface of the water is concave, rising towards the sides of the bucket. The key questions which need to be answered are: Why is the surface of water ﬂat in the ﬁrst situation and concave in the second? What is responsible for this diﬀerent behaviour? The rotation of the water relative to what?
Let us analyse this from the Newtonian point of view. There are three main natural suspects for this concavity of the water: The rotation of the water relative to the bucket, relative to the earth, or relative to the ﬁxed stars. That the bucket is not responsible for the diﬀerent behaviour of the water can be immediately grasped by observing that there is no relative motion between the water and the bucket in both situations emphasized above. This means that whatever the force exerted by the bucket on each molecule of water in the ﬁrst situation, it will remain the same in the second situation, as the bucket remains at rest relative to the water.
The second suspect is the rotation of the water relative to the earth. After all, in the ﬁrst situation the water was at rest relative to the earth and its surface was ﬂat, but when it was rotating relative to the earth in the second situation, its surface became concave. Thus, this relative rotation between

2.4. UNIFORM CIRCULAR MOTION

57

the water and the earth might be responsible for the concavity of the water. Newton maintained that this is not the case (“And therefore this endeavor [to recede from the axis of circular motion] does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be deﬁned by such translation”). We show here that Newton was consistent and correct in this conclusion when using his own law of gravitation. In the ﬁrst situation, the only relevant force exerted by the earth on each molecule of water is of gravitational origin. As we saw in chapter 1, utilizing Eq. (1.4) and Newton’s Theorem 31 we obtain that the earth attracts any molecule of water as if the earth were concentrated at its center, Eqs. (1.6) and (1.5):
P = mgg = −mggzˆ. In the second situation, the water is rotating relative to the earth, but the force exerted by the earth on each molecule of water is still given simply by P = −mggzˆ pointing vertically downwards. This is due to the fact that Newton’s law of gravitation (1.4) does not depend on the velocity or acceleration between the interacting bodies. This means that in Newtonian mechanics, the earth cannot be responsible for the concavity of the surface of the water. Whether the water is at rest or spinning relative to the earth, it will experience the same gravitational force due to the earth, the weight P pointing downwards, without any tangential component of the force perpendicular to the z direction and depending on the velocity or acceleration of the water.
The third suspect is the set of ﬁxed stars. In the ﬁrst situation the water is essentially at rest or moving with a constant linear velocity relative to them and its surface is ﬂat. In the second situation it is spinning relative to them and its surface is concave. This relative rotation might be responsible for the concavity of the water. But in Newtonian mechanics this is not the case either. The only relevant interaction of the water with the ﬁxed stars is of gravitational origin. Let us analyse the inﬂuence of the stars in the ﬁrst situation. As we saw in chapter 1, utilizing Eq. (1.4) and Newton’s Theorem 30, we ﬁnd that the net force exerted by all the ﬁxed stars on any molecule of water is essentially zero, assuming that the ﬁxed stars are distributed more or less at random in the sky and neglecting the small anisotropies in their distribution. This is the reason why the ﬁxed stars are seldom mentioned in Newtonian mechanics. This will remain valid not only when the water is at rest relative to the ﬁxed stars, but also when it is rotating relative to them. Once more, this is due to the fact that Newton’s law of gravitation (1.4) does not depend on the velocity or acceleration between the bodies. Thus, his result (1.6) will remain valid no matter what the velocity or acceleration of body 1 relative to the spherical shell.
As we have seen, Newton was aware that we can neglect the gravitational inﬂuence of the set of all ﬁxed stars in most situations. Recall what he wrote in the Principia: “(...) the ﬁxed stars, everywhere promiscuously dispersed in the heavens, by their contrary attractions destroy their mutual actions, by Prop.

58

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70, Book I.” The conclusion is then that the relative rotation between the water and the ﬁxed stars is not responsible for the concavity of the water either. Even introducing the external galaxies (which were not known by Newton) does not help, as they are known to be distributed more or less uniformly in the sky. So the same conclusion Newton reached for the ﬁxed stars (that they exert no net force on other bodies) applies to the distant galaxies.
An important consequence of this fact is that even if the ﬁxed stars and distant galaxies disappeared (were literally annihilated from the universe) or doubled in number and mass, the concavity of the water would not change in this experiment (according to Newtonian mechanics).
Since the eﬀect of the concavity of the water is real and can be measured (the water can even pour out of the bucket), Newton had no other choice than to point out another cause for it, namely, the rotation of the water relative to absolute space. This was his only alternative, assuming the validity of his universal law of gravitation, which he proposed in the same book where he presented the bucket experiment. Moreover, this Newtonian absolute space cannot have any relation with the mass or quantitity of matter of the water, of the bucket, of the earth, of the ﬁxed stars, the distant galaxies nor any other material body, as all these other possible inﬂuences have been eliminated.
A quantitative explanation of this key experiment without introducing absolute space is one of the main accomplishments of relational mechanics as developed in this book.

Chapter 3
Non-inertial Frames of Reference
We now discuss some of the examples of the previous chapter in non-inertial frames of reference S’. As we have seen, Newton’s second law of motion is valid only in absolute space or in frames of reference which move with a constant translational velocity relative to it, by his ﬁfth corollary. These are called inertial frames of reference which we represented by S. When the frame of reference is accelerated relative to an inertial frame, diﬃculties with the application of Newton’s laws of motion arise. To overcome these diﬃculties it is necessary to introduce so-called ﬁctitious forces. We analyse these situations here.
3.1 Constant Force
3.1.1 Free Fall
The ﬁrst situation is the case of free fall. Suppose we are falling to the earth in vacuum. We will consider the earth an inertial frame of reference S. This means that our frame of reference S’ is falling freely towards the earth with a constant acceleration, so that it is non-inertial. If we try to apply Newton’s laws to study our own motion, we would write F = mia in order to ﬁnd out our own acceleration a in our frame of reference S’. The only force acting on us is the gravitational attraction of the earth, so that, F = mgg. We would conclude that a = g ≈ 9.8 m/s2 - which is wrong. After all we are at rest relative to ourselves, and the correct result we should have arrived at was a = 0, Figure 3.1. In other words, we see ourselves at rest while the earth approaches us with an acceleration ae = gzˆ, where zˆ points vertically upwards, from the earth to
59

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CHAPTER 3. NON-INERTIAL FRAMES OF REFERENCE

us.

Figure 3.1: A person studying his own fall to the earth. The person is at rest
relative to himself, while the earth moves towards him with an acceleration aE = g = 9.8 m/s2.

To arrive at the correct result we need to apply Newton’s second law in the form

F − miao = mia ,

(3.1)

where ao is our acceleration relative to absolute space or relative to an inertial frame of reference, and a is the acceleration of mi relative to S’. In this case ao = g = −gzˆ (supposing the earth to be an inertial frame of reference, neglecting thus the small eﬀects due to the non-inertial character of the earth’s rotation).
Utilizing this and the fact that F = mgg, we would ﬁnd that our acceleration relative to ourselves is given by (as always with mi = mg): a = 0. This is the correct answer in our own frame of reference.
The force −miao is called a ﬁctitious force. The reason for this name is that all other forces which appear in F of Eq. (3.1) have a physical origin due to the interaction of the test body with other bodies, such as a gravitational interaction between the test body and the earth or the sun, an elastic interaction with a spring, an electric or magnetic interaction with another charge or magnet, a force of friction due to its interaction with a resistive medium, etc. On the other hand, the force −miao in classical mechanics has no physical origin. It is not due to an interaction of the test body with any other body. It only appears

3.1. CONSTANT FORCE

61

in non-inertial frames of reference which are accelerated relative to absolute space. At least this is the usual interpretation in Newtonian mechanics. Later on we will see that this need not be the case.
Despite its ﬁctitious character, this force −miao is essential in non-inertial frames of reference in order to arrive at the correct results applying Newton’s laws of motion.

3.1.2 Accelerated Train
The second example analysed here is an accelerated train. Once more we suppose the earth to be an inertial frame of reference as a good approximation here. In the previous chapter we analysed the motion and inclination of a pendulum in the frame of reference ﬁxed to the earth. We now analyse the same problem in a frame of reference ﬁxed in the accelerated wagon, as for instance, for a passenger who is inside the train, Figure 3.2. In this case the bob of mass m is at rest relative to the train and passenger, while the earth is accelerated to the left with an acceleration ae = −aexˆ.

Figure 3.2: Passenger in an accelerated train. The mass m is at rest relative to him, while the earth moves to the left with an acceleration ae = −|ae|xˆ.
If the passenger applied Newton’s second law to the bob of mass m in the form of Eq. (1.3) in the situation which he were observing, he would arrive at the same conclusion as Eqs. (2.6), (2.7) and (2.8), namely:
a = g mg tan θ = 0 . mi
But obviously this is the wrong answer in the frame of reference of the train. After all the pendulum is not moving relative to the train or to the passenger in the equilibrium situation being analysed here, so that the passenger should arrive at a = 0. He can only obtain this with Eq. (3.1). He needs to introduce the ﬁctitious force −miao in order to arrive at the correct result. In this case ao = aexˆ. This ﬁctitious force balances the gravitational force exerted by the

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earth and the force exerted by the string, in order to cancel the motion of the pendulum relative to the passenger and keep it in an inclined position relative to the vertical, as represented in Figure 3.2. Then he would arrive at the situation described in Figure 3.3:

so that a = 0.

P + T − miao = mia ,

Figure 3.3: Forces in the passenger’s frame of reference.
In this frame of reference the vertical component of the tension in the string balances the weight of the body, while −miao balances the horizontal component of the tension in the string, yielding zero motion of the bob relative to the train.
Once more in Newtonian mechanics there is no physical origin for this force −miao, but it is essential to utilize it in the train’s frame of reference to arrive at correct results.
3.2 Uniform Circular Motion
We now analyse some problems of the previous chapter in the frame of reference of the rotating bodies.
3.2.1 Circular Orbit of a Planet
We begin with the planet orbiting around the sun. Once more we consider here only the particular case of circular orbits. In the inertial frame of reference S considered previously, with the sun much more massive than the planet, the sun was considered essentially at rest and the planet was orbiting around the sun. Application of Eq. (1.3) yielded a centripetal acceleration given by acp = Gms/r2.
We now analyse this problem in the non-inertial frame of reference S’ in which the sun and the planet are at rest. In other words, in a frame of reference

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63

S’ centered on the sun but that rotates together with the planet relative to S. In this frame we should ﬁnd that the planet would not be accelerated, namely: a = 0. But this is not the case if we apply Newton’s second law of motion in the form of Eq. (1.3). In this new frame of reference, how can we explain the fact that the planet is at rest while subject to the gravitational attraction of the sun? How can the planet keep an essentially constant distance to the sun? To arrive at the correct result, i.e. that there is no acceleration of the planet in this frame of reference, and to explain why the distance between the planet and the sun is essentially constant, we need to introduce another ﬁctitious force. In this case this ﬁctitious force has a special name, centrifugal force, and is given by:

Fc = −miω × (ω × r) ,

(3.2)

where r is the position vector of the test body relative to the origin of the non-inertial system of reference and ω is the vector angular velocity of the noninertial system of reference relative to absolute space, or relative to any inertial frame of reference. In the previous chapter we considered the inertial frame of reference S centered on the sun. In this frame S the planet orbitated around the sun with an angular frequency ω. The non-inertial frame of reference S’ considered here is also centered on the sun, but it rotates relative to S with the same angular frequency as the planet’s orbit, shown in Figure 3.4.

Figure 3.4: Frame S’ rotating together with the planet relative to S.
If the planet were the earth, the period of rotation of S’ relative to S would be T = 2π/ω = 365 days. If the non-inertial frame of reference S’ is rotating relative to the inertial frame S around the vertical z axis, ω = ωzˆ. Then ω × (ω × r) = −ω2ρρˆ, where ρ is the distance of the test body to the axis of

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rotation and ρˆ is the unit vector pointing from the axis of rotation to the body, in a plane orthogonal to the axis of rotation (in polar coordinates: r = ρρˆ+zzˆ). In this case the centrifugal force is given simply by Fc = miω2ρρˆ. This shows that this ﬁctitious force, which appears only in the non-inertial frame of reference S’ but not in S, is directed away from the center. This is the reason for the name “centrifugal,” the case represented by Figure 3.5.

Figure 3.5: centrifugal force in the frame S’.

In this rotating non-inertial frame of reference Newton’s second law of motion should be written as (in order to predict correct results):

F + Fc = mia ,

(3.3)

where F is the resultant force due to all other bodies acting on mi, Fc is given by Eq. (3.2) and a is the acceleration of mi relative to this non-inertial frame of reference.
In the problem of the planet we have the situation of Figure 3.6. Utilizing the fact that a = 0 in this frame of reference gives the centrifugal force, namely

Fc

=

G

mgsmgp r2

rˆ

=

−mipω

×

(ω

×

r)

.

3.2. UNIFORM CIRCULAR MOTION

65

Figure 3.6: Planet “orbiting” around the sun, as seen in S’.
This yields: ω = Gmgs/r3. Alternatively we might use the fact that ω = Gmgs/r3 to ﬁnd that a = 0 in the frame of reference S’ in which the planet and the sun are at rest.
Once more there is no physical origin for this centrifugal force, while the gravitational force in this case is due to the attraction between the sun and the planet.
3.2.2 Two Globes
We now brieﬂy discuss the experiment with two globes described by Newton. In a frame of reference S’ which rotates with the globes and centered on the center of mass of the system, we have the situation depicted in Figure 3.7.

Figure 3.7: Two globes in the frame S’.
In this frame there is no motion of the globes despite the tension T in the string. The centripetal force due to this tension is balanced by a centrifugal force given by miω2ρ, Figure 3.8: Fc1 = m1ω2ρ1 = T and Fc2 = m2ω2ρ2 = T .
There are two interpretations of this equilibrium: Either we say that the tension is balanced by the centrifugal force, which does not let the globes ap-

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Figure 3.8: Tension balanced by centrifugal force in the frame S’.
proach one another, or we say that the centrifugal force generates the tension in the string.
We might easily apply the same analysis to the previous problem of the sun and the planet, generalizing it to take into account the motion of the sun, and replacing the tension T in this example by the gravitational attraction Gmg1mg2/r2. In this more real situation, the sun and the planet would be orbiting around the center of mass relative to an inertial frame of reference. In a non-inertial frame of reference centered on the center of mass and in which both bodies are at rest, the gravitational attraction would be balanced by the centrifugal force.
3.2.3 Newton’s Bucket Experiment
We now consider the bucket experiment. We will concentrate on the situation in which the bucket and water rotate together with a constant ω relative to an inertial frame of reference S. In a frame of reference S’ which rotates with the bucket there is no motion of the water, so that Eq. (3.3) reduces to (with Eq. (3.2) and ω = ωzˆ):
−(∇p)dV − dmggzˆ + dmiω2ρρˆ = dmia = 0 .
This yields the same result obtained previously, remembering that dmi = dmg and that here we are utilizing ρρˆ instead of xxˆ to represent the distance to the axis of rotation.
What is important to stress here and in the previous examples of the circular orbit of the planets and of the two globes, is that this centrifugal force has no physical origin in Newtonian mechanics. It appears in non-inertial frames of reference, and in this sense we might say that they are real (balancing the gravitational attraction of the sun, creating a tension in the string, pushing the water towards the sides of the bucket, etc.) On the other hand, unlike real forces such as the gravitational attraction exerted by the sun or by the earth, the electric force exerted by charges, the magnetic force exerted by magnets or current-carrying wires, or the elastic force exerted by a streched spring or tensioned cord, we cannot locate the material body responsible for the centrifugal force or for the ﬁcitious forces in general. Let us analyse this in the case of the

3.3. ROTATION OF THE EARTH

67

bucket experiment (a similar analysis can be carried out for all other examples discussed here).
We consider the situation in which the bucket and the water are rotating together relative to the earth and the ﬁxed stars with a constant angular velocity around the vertical axis. We analyse the problem in the non-inertial frame of reference of the bucket, so that in this frame the surface of the water is concave, although the water is at rest. Is the bucket responsible for this concavity? No, after all the bucket is at rest relative to the water. Is the earth responsible for this concavity? In other words, is the rotation of the earth relative to the water, to the bucket and to this frame responsible for the centrifugal force? Once more the answer in Newtonian mechanics is no. As we saw in chapter 1, the gravitational force exerted by a spherical shell on material particles outside it points towards the center of the shell. As Newton’s law of gravitation does not depend on the velocity or acceleration between the bodies, this will remain valid when the spherical shell is spinning. This means that according to Newtonian theory even when the earth is spinning relative to a material point or to a frame of reference, it will exert only the usual downward force of gravity, without any tangential force orthogonal to the radial direction. Are the ﬁxed stars and external galaxies responsible for this concavity? In other words, is the rotation of the ﬁxed stars (or of the external galaxies) relative to the water, to the bucket and to this non-inertial frame of reference responsible for the centrifugal force? The answer is no once more, due to Newton’s 30th Theorem stated above, and to result (1.6). In other words, spherically symmetric distributions of matter do not exert any net gravitational forces on any internal point particles, regardless of the rotation or motion of these spherical distributions relative to the internal particle or to any frame of reference. This means that in Newtonian mechanics the ﬁxed stars and distant galaxies might disappear without having any inﬂuence on the concavity of the water.
As we will see, relational mechanics will give a diﬀerent answer here.

3.3 Rotation of the Earth
There are two main ways of determining the rotation of the earth. The ﬁrst is kinematical or visual and the second, dynamical. We discuss the problem of the earth’s rotation in this section.
3.3.1 Kinematical Rotation of the Earth
The simplest way to know that the earth rotates relative to something is by the observation of the astronomical bodies. Standing on the ground we do not observe the rotation of the earth directly; after all, we are essentially at rest

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relative to it. But looking at the sun we see that it moves around the earth with a period of one day. There are two obvious interpretations for this fact: The earth is at rest (as in the Ptolemaic system) and the sun translates around the earth; or the sun is at rest and it is the earth that spins around its axis (as in the Copernican system). Both interpretations are represented in Figure 3.9, considering the reference represented by the paper to be the rest frame.

Figure 3.9: Relative rotation between the earth and the sun in the Ptolemaic (P) and Copernican (C) world views.
We can add to the motions of the earth and sun a common motion (a translation or rotation relative to absolute space, for instance) without altering their relative motion. What is important to realise here is that from the observed relative rotation between earth and sun we cannot determine which one of them is really moving relative to absolute space. The only thing observed and measured in this case is their relative motion. In this regard the Ptolemaic and Copernican systems are equally reasonable and compatible with the observations. It is a matter of taste to choose one or the other, considering only the relative rotation between the earth and the sun.
Another kinematical rotation of the earth is rotation relative to the ﬁxed stars (relative to the stars which belong to our galaxy, the Milky Way). Although the moon, the sun, the planets and comets move relative to the background of stars, there is essentially no motion of one star relative to the others. The sky seen today with its constellations is essentially the same sky seen by the ancient Greeks or Egyptians. Although the set of stars rotate relative to the earth, they essentially do not move relative to one another and for this reason they are usually called ﬁxed stars. Although the stellar parallax had been predicted by Aristarchus of Samos around 200 B.C., the ﬁrst observation of this parallax (motion or change of position of one star relative to the others) was only obtained without doubt by F. W. Bessel in 1838. If we take a picture of the night sky with a long exposure we observe in the Northern hemisphere

3.3. ROTATION OF THE EARTH

69

that all the stars rotate approximately around the north pole star, typically with a period of one day.
Once more, we may say that the real rotation belongs to the stars or to the spinning earth. We cannot decide between these two interpretations based only on these observations. It may be simpler to describe motions and planetary orbits in the frame of reference ﬁxed with the stars than in the earth’s frame of reference, but both of them are equally reasonable.
With a period of rotation of one day we get ωk = 2π/T = 7 × 10−5 rad/s, where ωk is the kinematical rotation of the earth. The direction of this kinematical rotation is approximately the direction of the north pole star. In this way we have a complete description of the kinematical rotation, i.e., the rotation of the earth relative to the ﬁxed stars.
To simplify the analysis we are not distinguishing here the solar day to the sidereal day (time for the ﬁxed stars to give a complete turn around the earth). We are putting both of them as 24 hours. As a matter of fact, while the sidereal day is essentially constant (when compared, for instance, with a mechanical or atomic clock), the solar day varies according to the month of the year. This was known to the Ancients, and Ptolemy (100-170 A.D.), for instance, presented the so called “equation of time” describing the variation of the solar day compared with the sidereal one. The mean solar day (obtained by an average taken over the year of the duration of the solar days) has by deﬁnition 24 hours, while the measured sidereal day has 23 hours, 56 minutes and 4 seconds. In a year the sun turns essentially 365 times around the earth, while the ﬁxed stars turn 366 times. Another diﬀerence between these two motions is that while the stars rise at the same place relative to the earth’s horizon all year long, the same does not happen with the sun, which rises at diﬀerent locations at diﬀerent epochs of the year. For a discussion of these points and further references, see [22, pp. 9-10 and 266-268] and [4, Sections 3.15 and 11.6]. Newton mentions the equation of time in the Scholium at the end of his deﬁnitions:
Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the apparent time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time; astronomers correct this inequality that they may measure the celestial motions by a more accurate time. It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded, but the ﬂowing of absolute time is not liable to any change. The duration of perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all: and therefore this duration ought to be distinguished from what are only sensible measures thereof; and from which we

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deduce it, by means of the astronomical equation. The necessity of this equation, for determining the times of a phenomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter.
In Proposition XVII, Theorem XV, Book III of the Principia he says: “That the diurnal motions of the planets are uniform, and that the libration of the moon arises from its diurnal motion. The Proposition is proved from the ﬁrst Law of Motion, and Cor. XXII, Prop. LXVI, Book I. Jupiter, with respect to the ﬁxed stars, revolves in 9h 56m; Mars in 24h 39m; Venus in about 23h; the earth in 23h 56m; the sun in 25 1/2d, and the moon in 27d 7h 43m. These things appear by the Phenomena. (...)” In Section [35] of his System of the World he says: “The planets rotate around their own axes uniformly with respect to the stars; these motions are well adapted for the measurement of time.” The importance of this statement is that Newton is presenting an operational way of measuring absolute time. The curious fact is that this measurement has to do with the diurnal rotation of the planets relative to the ﬁxed stars, while in his deﬁnition of absolute time there should be no relation with anything external. For a general discussion of the time concept in physics see [23], [24] and [25].
Nowadays we have two other kinematical rotations of the earth. The ﬁrst is the rotation of the earth relative to the background of distant galaxies. The reality of external galaxies was established by Hubble in 1924 when he determined that the nebulae are stellar systems outside the Milky Way (after ﬁnding Cepheid variables in the nebulae). We can then determine kinematically our translational and rotational velocities relative to the isotropic frame of the galaxies. This is the frame relative to which the galaxies have no translational or rotational velocity as a whole, in which the galaxies are essentially at rest relative to one another and to this frame, apart from peculiar velocities. Our angular rotational velocity relative to this frame is essentially the same as that relative to the ﬁxed stars.
The second modern kinematical rotation of the earth is its rotation relative to the cosmic background radiation (CBR) discovered by Penzias and Wilson in 1965, [26]. This radiation has a blackbody spectrum with a characteristic temperature of 2.7 K. Although it is highly isotropic, there is a dipole anisotropy due to our velocity relative to this radiation. This motion generates Doppler shifts which are detected and measured. In this way we can, at least in principle, determine our translational and rotational velocities relative to the frame in which this radiation is isotropic.
We have indicated four diﬀerent kinematical rotations of the earth. They have to do with a relative motion between the earth and external bodies or external radiation. We cannot determine by any of these means which body is really rotating, the earth or the external ones. Up to now we can adopt

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71

any point of view without problems, namely: the earth is at rest (relative to Newton’s absolute space, for instance) and these bodies rotate around the earth, or these bodies are at rest and the earth spins around its axis (relative to Newton’s absolute space, for instance).
In the next sections we will see how to distinguish these two points of view dynamically.

3.3.2 The Figure of the Earth
The simplest way to know that the earth is a non-inertial frame of reference is to observe its ellipsoidal form: the earth is ﬂattened at the poles. Newton discussed this in Props. XVIII and XIX of Book III of the Principia:
Proposition XVIII. Theorem XVI
That the axes of the planets are less than the diameters drawn perpendicular to the axes.
The equal gravitation of the parts on all sides would give a spherical ﬁgure to the planets, if it was not for their diurnal revolution in a circle. By that circular motion it comes to pass that the parts receding from the axis endeavor to ascend about the equator; and therefore if the matter is in a ﬂuid state, by its ascent towards the equator it will enlarge the diameters there, and by its descent towards the poles it will shorten the axis. So the diameter of Jupiter (by the concurring observations of astronomers) is found shorter between pole and pole than from east to west. And, by the same argument, if our earth was not higher about the equator than at the poles, the seas would subside about the poles, and, rising towards the equator, would lay all things there under water.

Proposition XIX. Problem III
To ﬁnd the proportion of the axis of a planet to the diameters perpendicular thereto. (...); and therefore the diameter of the earth at the equator is to its diameter from pole to pole as 230 to 229. (...)
This theoretical prediction of Newton (until that time there was no measurement of this quantity) is quite accurate compared with modern experimental determinations [3, pp. 427 and 664, note 41].

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The reason for this ﬂattening of the earth at the poles is explained in Newtonian mechanics due to the rotation of the earth relative to absolute space or to an inertial frame of reference. The earth and all frames of reference which are at rest relative to it are non-inertial. For this reason we need to introduce in the earth’s frame a centrifugal force −ωd2ρρˆ in order to apply Newton’s laws of motion and get correct results. Here ωd is the dynamical rotation of the earth relative to absolute space or to any inertial frame of reference. In principle it has no relation to ωk discussed previously. In the earth’s frame of reference it is this centrifugal force responsible for the ﬂattening of the earth. In an inertial frame of reference the ﬂattening of the earth is explained by its dynamical rotation relative to this inertial frame of reference. According to Newtonian mechanics even if the stars and distant galaxies disappeared or did not exist, the earth would still be ﬂattened at the poles due to its rotation relative to absolute space. As we will see, relational mechanics will give a diﬀerent prediction in this case.
We present here some quantitative results for this case. We will assume the earth to be composed only of water with a constant density α at any point in its interior. We will assume that the earth spins with a constant angular velocity ωd = ωdzˆ relative to an inertial frame of reference, where we have chosen the z axis along the axis of rotation to simplify the analysis. The equation of motion in this inertial frame for an element of mass dm of the water occupying an inﬁnitesimal volume dV (dm = αdV ) is given by

dmg − (∇p)dV = dma .

(3.4)

In this equation g is the gravitational ﬁeld at the point where dm is located and −(∇p)dV is the force on dm due to the gradient of pressure p. These are the only forces acting on dm. We will solve this equation utilizing spherical coordinates (r, θ, ϕ) with an origin at the center of the earth. As the only motion of dm is a circular orbit around the z axis, its acceleration is only centripetal, given by a = ωd ×(ωd ×r) = −ωd2ρρˆ = −ωd2ρ(rˆ sin θ +θˆcos θ), where ρ is the distance of dm to the axis z (and not to the origin, as this distance is given by r, being the relation between the two ρ = r sin θ). Moreover, rˆ, θˆ and ρˆ are the unit vectors along the directions r and θ (spherical coordinates), and ρ (cylindrical coordinate), respectively. The pressure gradient can be expressed in cylindrical or spherical coordinates, without diﬃculty. To solve this equation we need an expression for the gravitational potential. As a ﬁrst approximation we utilize the gravitational ﬁeld of a spherically symmetrical, homogeneous distribution of matter: a total mass M = 4πR3α/3 distributed uniformly over a sphere of radius R (later on we will improve on this approximation). Utilizing Eqs. (1.6) or (1.7) it is easy to show that the gravitational ﬁeld at a point r in the interior of this sphere is given by g = −4παGr/3 = −GM rrˆ/R3. Solving

3.3. ROTATION OF THE EARTH

73

the equation above, as was done in the case of a spinning bucket, yields the pressure anywhere in the ﬂuid as given by

p

=

GM αr2 − 2R3

+

αωd2r2 sin2 2

θ

+

GM α 2R

+

Po

,

(3.5)

where Po is the atmospheric pressure at the North pole (r = R and θ = 0). The
surfaces of constant pressure are ellipsoids of revolution. Taking p = Po at the
equator (θ = π/2) yields the largest distance of the water to the origin (R>). That is (supposing ωd2R3/2GM 1 as is the case for the diurnal rotation of the earth):

R> ≈ 1 + ωd2R3 ≈ 1.0017 .

R

2GM

(3.6)

To arrive at this number we put ωd = 7.3 × 10−5 s−1 (one day period), R = 6.36 × 106 m, G = 6.67 × 10−11 Nm2/kg2 and M = 6 × 1024 kg.
This value is approximately half of what is observed by performing measurements over the earth. The problem with this calculation is that due to its rotation, a ﬂuid earth modiﬁes its form. In other words, it does not remain spherical but becomes approximately ellipsoidal. Therefore, the gravitational ﬁeld inside and outside the rotating earth is that given by an ellipsoid. This ﬁeld can be obtained utilizing the results of exercises 6.17 and 6.21 of [27]. We will not present here all calculations but only the results we obtained following this procedure.
Consider then an ellipsoid centered on the origin of the coordinate system with semi-axes a, b and c along the axes x, y and z of the coordinate system, respectively, such that a = b = R> and c = R< = R>(1 − η), with η 1. We will suppose once more a constant density of matter α at all points of the ellipsoid. If M is the total mass of the ellipsoid and R its average radius we have: M = 4πR3α/3 = 4πR>2 R<α/3.
The gravitational potential energy U between two point masses m1 and m2 separated by a distance r is given by U = −Gm1m2/r = miΦ(r1), where Φ(r1) is the gravitational ﬁeld at the point where m1 is located, r1, due to m2 located at r2. Analogously, we can calculate the gravitational potential at any point in space due to the mass of the ellipsoid. The gravitational potential Φ which we found inside the ellipsoid is given by (up to ﬁrst order in η):

Φ

=

GM − 2R3

(3R2

−

r2)

−

GM R3

ηr2 5

(1

−

3

cos2

θ)

.

(3.7)

The potential outside the ellipsoid is given by (once more up to ﬁrst order in η):

74

CHAPTER 3. NON-INERTIAL FRAMES OF REFERENCE

GM Φ=−
r

1

+

ηR2 5r2

(1

−

3

cos2

θ)

.

(3.8)

The potential energy dU of a mass element dm interacting with this ellipsoid is given by dU = dmΦ. The force exerted by the ellipsoid on dm is given by dF = −∇(dU ) = −dm∇Φ = dmg. Applying this in the results above yields the gravitational ﬁeld inside the ellipsoid as given by (once more up to ﬁrst order in η):

GM r g = − R3

1

−

2 η(1

−

3

cos2

θ)

rˆ −

6 η

sin

θ

cos θθˆ

.

5

5

(3.9)

From this equation we see that the force inside the ellipsoid, for each ﬁxed θ, grows linearly with the distance. This fact was known by Newton (Principia, Book I, Prop. 91, Prob. 45, Cor. III).
Outside the ellipsoid we have:

GM g = − r2

1

+

3 5

R2 r2

η(1

−

3 cos2

θ)

rˆ

−

6 5

R2 r2

η

sin

θ

cos

θθˆ

.

(3.10)

The gravitational ﬁeld at the surface of the ellipsoid is given by

GM

3

cos2 θ

g = − R2

1+ η+η

5

5

.

(3.11)

From this relation we ﬁnd that the force on a point on the surface of the ellipsoid at the pole (r = R<, θ = 0) to the force on a point at the surface of the ellipsoid at the equator (r = R>, θ = π/2) is given by

Fpole

η ≈1+ .

Fequator

5

(3.12)

Up to now we have supposed the ellipsoid to be at rest relative to an inertial frame of reference.
At this point we will return to the problem of a spinning earth. We can then apply Eq. (3.9) to Eq. (3.4). In this case η still needs to be determined. But from the analysis of the previous case of an spinning spherical shell, we expect η to be of the order of ωd2R3/GM . With (3.9) in (3.4) we obtain the following expression of the pressure p at any point inside the ﬂuid ellipsoidal spinning earth:

GαM r2 p = − 2R3

4 1+ η
5

+

ωd2 2

+

3 GM

η 5

R3

αr2 sin2 θ + C ,

(3.13)

3.3. ROTATION OF THE EARTH

75

where C is a constant.
Equating the pressure at r = R<, θ = 0 with the pressure at r = R>, θ = π/2, utilizing η 1, ωd2R3/GM 1 and the fact that η is of the same order of magnitude as ωd2R3/GM yields η = 5ωd2R3/4GM and:

R> ≈ 1 + η ≈ 1 + 5ωd2R3 ≈ 1.0043 .

R<

4GM

(3.14)

This is essentially the value given by Newton, R>/R< ≈ 230/229 ≈ 1.0044. There are two important things to observe here. The ﬁrst is that to obtain

this result we utilized together the rotation of the earth and the gravitational

ﬁeld of an ellipsoid (the previous result (3.6) did not yield a precise value, since

we assumed the gravitational ﬁeld of a sphere). The second point is that the

ωd which appears in Eq. (3.14) is the dynamical rotation of the earth relative to absolute space or to an inertial frame of reference. In principle this ωd has nothing to do with the kinematical rotation of the earth relative to the ﬁxed

stars discussed above, ωk. But to arrive at the correct value for the ﬂattening
of the earth as observed by the measurements (R>/R< ≈ 1.004) it is necessary to have ωd ≈ 7.3 × 10−5 s−1. In other words, ωd needs to be equal to ωk, or the
dynamical rotation of the earth needs to have the same value as its kinematical

rotation relative to the ﬁxed stars! This should not be a coincidence; the

problem is to ﬁnd the connection between these two facts.

3.3.3 Foucault’s Pendulum
The most striking demonstration of the rotation of the earth was obtained in 1851 by Foucault (1819-1868). The original French paper can be found in [28], while the English translation can be found in [29]. The importance of this experiment is that it can be performed in a closed room, so that we obtain the rotation of the earth without looking at the sky.
It is simply a long pendulum which oscillates to and fro many times with a long period. The pendulum is not charged and the only forces acting on it are the gravitational attraction of the earth and the tension in the string. Foucault initially utilized a pendulum with a length of 2 meters and a sphere of 5 kg oscillating harmonically. Later on he utilized another pendulum with a suspension cord of 11 meters. Although he does not mention it in the paper, soon afterward he performed his experiment at the dome of the Pantheon, with a cord of 65 meters ([29, see the footnote on page 352 by E. Fr. Jr.]). The period of an oscillating simple pendulum of length is given by T = 2π /g, where g ≈ 9.8 m/s2. Suppose the pendulum initially at rest relative to the earth, and released from an initial angle θo. Neglecting the eﬀects of wind we might expect the pendulum to always oscillate in the same plane formed by the vertical direction of the weight and the direction of tension along the string. But this

76

CHAPTER 3. NON-INERTIAL FRAMES OF REFERENCE

is not what happens. The plane of oscillation changes slowly with time relative to the earth’s surface, with an angular velocity Ω. In Newtonian mechanics this is explained by means of another ﬁctitious force, the Coriolis force given by −2miωd × v, where ωd is the angular rotation of the earth relative to an inertial system of reference (the centrifugal force does not change the plane of oscillation, so that we do not consider it here to simplify the analysis). Coriolis (1792-1843) discovered this force while doing his doctoral work under Poisson, as related in [30].
The simplest way to understand this behaviour is to consider a pendulum oscillating at the North pole. The pendulum will keep its plane of oscillation ﬁxed relative to an inertial frame of reference (or relative to space, as it is usually termed). As the earth turns beneath it, the plane of oscillation relative to the earth changes with an angular velocity Ω = −ωd = −ωdzˆ, because the earth is rotating relative to the inertial system with an angular velocity ωd = ωdzˆ. At the equator, Foucault’s pendulum does not precess because here ωd × v is either zero (when v = ±vzˆ) or points vertically along the length of the string (when there is a velocity component perpendicular to zˆ and rˆ). In general the precession of the pendulum relative to the earth is given by Ω = −ωd cos θ, where θ is the angle between the radial direction rˆ (the direction in which the pendulum hangs at rest without oscillation) and the earth’s axis of rotation ωd/ωd = zˆ, shown in Figure 3.10.
We derive this result here utilizing some approximations which are valid for the problem. Accordingly, we neglect air resistance and the centrifugal force. The equation of motion in the earth’s frame of reference is then given by:
T + mgg − 2miωd × v = mia .
Here T is the tension in the string. The novelty compared with the equation of motion of a simple pendulum in an inertial frame of reference is the introduction of the Coriolis force −2miωd × v, where ωd is the dynamical angular rotation of the earth relative to absolute space or to an inertial frame of reference.
We choose a new coordinate system (x , y , z ) with its origin O’ directly below the point of support, at the point of equilibrium of the pendulum bob, with the z -axis pointing vertically upwards: zˆ = rˆ. The x -axis is chosen such that the pendulum would oscillate completely in the x z plane if it were not the Coriolis force, shown in Figure 3.11.
In this frame of reference we have ωd = ωd sin θxˆ + ωd cos θzˆ . The angle of oscillation of the pendulum with the vertical from the point of support is called β. For β π/2 we can utilize the approximation of small amplitude of oscillation so that the equation of motion yields the approximate solution (not taking into account for the moment the Coriolis force): β = βo cos ωot, where ωo = g/ is the natural frequency of oscillation of the pendulum and βo

3.3. ROTATION OF THE EARTH

77

Figure 3.10: Foucault’s pendulum.

the angle of release of the pendulum from rest. As we have small amplitudes of oscillation, the motion of the pendulum is essentially horizontal with x = β, so that v ≈ −x˙ xˆ = βoωo sin ωotxˆ . The only force component in the y direction is given by the Coriolis force −2miωd × v. With the previous values for ωd and v we ﬁnd that the equation of motion in the y direction takes the form:

y¨ = −2(ωd cos θ) βoωo sin ωot .

Integrating this equation twice and utiling the fact that y˙ (t = 0) = 0 and y (t = 0) = 0 yields:

y = 2ωd cos θ βo

sin ωot − t ωo

.

Between half a period (t = 0 and t = π/ωo) the bob moved in the y direction an amount of y = −2ωd cos θ βoπ/ωo. During this time the bob moved in the x direction an amount of x = 2 βo, Figure 3.12. This means that the plane of oscillation of the pendulum moved by an angle of y/ x = −ωd cos θπ/ωo. The angular rotation Ω of the plane of oscillation is this amount divided by the
time interval of t = π/ωo − 0 = π/ωo, so that:

Ω = −ωd cos θ .

(3.15)

78

CHAPTER 3. NON-INERTIAL FRAMES OF REFERENCE

Figure 3.11: Forces in Foucault’s pendulum.

Foucault did not present the calculations, but arrived at this result, stating that the angular rotation of the plane of oscillation is equal to the angular rotation of the earth multiplied by the sine of the latitude, [28] and [29]. At Paris, where Foucault performed his experiments, we have a latitude given by α = 48o51 . As the angle of latitude is given by α = π/2 − θ, we get Foucault’s result:

Ω = −ωd cos(π/2 − α) = ωd sin α .

(3.16)

It is curious to note Foucault’s description of his experiment. Sometimes he speaks of the rotation of the earth relative to space and other times relative to the ﬁxed stars (heavenly sphere). He does not distinguish these two rotations or these two concepts (dynamical rotation relative to absolute space and kinematical rotation relative to the celestial bodies). For instance, he begins by stating that his experiment showing the rotation of the plane of oscillation “gives a sensible proof of the diurnal motion of the terrestrial globe.” To justify this interpretation of the experimental result he imagines a pendulum placed exactly at the North pole oscillating to and fro in a ﬁxed plane, while the earth rotates below the pendulum. He then says (our emphasis), [29]:

Thus a movement of oscillation is excited in an arc of a circle whose plane is clearly determined, to which the inertia of the mass gives an invariable position in space. If then these oscillations continue for a certain time, the motion of the earth, which does not cease turning from west to east, will become sensible by contrast with the

3.3. ROTATION OF THE EARTH

79

Figure 3.12: Rotation of the plane of oscillation of Foucault’s pendulum.
immobility of the plane of oscillation, whose trace upon the ground will appear to have a motion conformable to the apparent motion of the heavenly spheres; and if the oscillations could be continued for twenty-four hours, the trace of their plane would have executed in that time a complete revolution about the vertical projection of the point of suspension.
When describing his real experiments, he states: “In less than a minute, the exact coincidence of the two points ceases to be reproduced, the oscillating point being displaced constantly towards the left of the observer; which indicates that the direction of the plane of oscillation takes place in the same direction as the horizontal component of the apparent motion of the celestial sphere.”
It must be stressed that the ωd which appears in Eqs. (3.15) and (3.16) is the dynamical rotation of the earth relative to absolute space or to any inertial frame of reference. Experimentally it is found that this ωd has the same value (in direction and order of magnitude) as the kinematical rotation of the earth relative to the ﬁxed stars, ωd = ωk. But there is no explanation of this fact in Newtonian mechanics.
The mathematical analysis leading to Eq. (2.19), Ω = −qB/2mi, was anal-

80

CHAPTER 3. NON-INERTIAL FRAMES OF REFERENCE

ogous to the mathematical derivation leading to Eq. (3.15), Ω = −ωd cos θ. The diﬀerence is that in the ﬁrst case we were in an inertial frame of reference and the precession of the plane of oscillation of the charged pendulum was due to its interaction with the magnet. On the other hand, in Foucault’s pendulum we have an electrically neutral pendulum and we do not ﬁnd the material agent (analogous to the magnet in the ﬁrst case) responsible for the precession of the plane of oscillation. The Coriolis force −2miωd × v is called a “ﬁctitious” force because it only appears in non-inertial frames of reference which are rotating relative to absolute space. Conversely, the magnetic force qv ×B is due to a real interaction between the charge q and the source of B (a magnet, a solenoid, a spinning charged spherical shell, etc.) In the earth’s frame of reference we see the set of stars and galaxies rotating around us with a period of one day relative to the North-South direction (around the North pole star). In Newtonian mechanics this set of spherical shells (stars and galaxies) rotating around the earth does not generate any net force on the pendulum, whether it is at rest or moving relative to the earth. It might be thought that this set of spherical shells composed of stars and galaxies would, when rotating around the earth, generate some kind of “gravitational magnetic” ﬁeld Bg which might explain the Coriolis force by a gravitational interaction analogous to the magnetic force. In other words, by an expression like mgv × Bg. However, even if this is the case, it cannot be due to Newton’s law of universal gravitation. As we have seen, a spherical shell does not exert any force inside itself, whether the shell is at rest or rotating, no matter what the postion, velocity and acceleration of the internal test body. We will see that there is something analogous to mgv × Bg in Einstein’s general theory of relativity, but that it does not have exactly the same value as the Coriolis force. On the other hand, in relational mechanics this term will appear (due to the rotation of the distant matter) with the precise value of the Coriolis force. This will allow us to show that the Coriolis force is a real force due to an interaction between the test body and the rotating universe around it, contrary to what happens in Newtonian mechanics.
Max Born discussed several examples of bodies in rotation and the dynamical eﬀects which appear. He presented the fundamental conclusion of Newtonian mechanics in simple and clear terms [31, p. 84]. In particular he emphasized that the centrifugal force of classical mechanics is universal and cannot be due to interactions between bodies, since it is due to rotation relative to absolute space.
3.3.4 Comparison of the Kinematical and Dynamical Rotations
Here we analyse these two rotations of the earth. The kinematical rotation is a relative rotation between the earth and surrounding bodies, such as the sun, the

3.3. ROTATION OF THE EARTH

81

ﬁxed stars, the distant galaxies or the CBR. The period of rotation is essentially one day (ωk ≈ 7 × 10−5 rad/s) and the direction is north-south (pointing towards the north pole star in the northern hemisphere). This kinematical rotation may be equally well attributed in classical mechanics to two opposite causes: the rotation of the external world while the earth remains at rest; or a spinning of the earth around its axis while the external world does not rotate. Kinematically, we cannot distinguish between these two situations.
A completely diﬀerent rotation of the earth is obtained by its ﬂattened ﬁgure or by Foucault’s pendulum. The rotation obtained dynamically by these means is a rotation of the earth relative to an inertial frame of reference. According to Newtonian mechanics, these dynamical eﬀects (deformation of the spherical form of the earth or rotation of the plane of oscillation of the pendulum) can only be explained by a rotation of the earth relative to absolute space or to an inertial frame of reference. These eﬀects would not appear if the earth were at rest relative to absolute space or to an inertial frame of reference, while the surrounding bodies (ﬁxed stars and distant galaxies) were rotating in the opposite direction relative to this inertial frame. The kinematical rotation would be the same in this case, but the dynamical eﬀects would not appear. As we will see, Mach had a diﬀerent point of view, namely, that if the kinematical situation is the same, the dynamical eﬀects must also be the same. Relational mechanics implements this quantitatively.
In classical mechanics it is a great coincidence that these two rotations happen to be the same. In other words, the rotation determined by looking at the stars kinematically is the same as the dynamical rotation determined in a closed room by Foucault’s pendulum. There is no explanation for this remarkable fact in Newtonian mechanics. In the same way, there is no explanation for the equality mi = mg. Classically, we can only say that nature happens to behave this way, but a closer understanding is not supplied. The inertial mass of a body did not need to be connected to its gravitational mass. It could have been a completely independent property of the body without any relation to mg, or it could depend on a chemical or nuclear property of the body without being in conﬂict with any law of classical mechanics. It only happens that experimentally the inertia is found to be proportional to the weight. A similar situation happens with the equality between the kinematical and dynamical rotations of the earth. The fact that the kinematical and dynamical rotations of the earth are the same indicates that the universe as a whole does not rotate relative to absolute space or relative to any inertial system of reference. The earth spins around its axis with a period T of one day (T = 8.640 × 104 s), or with an angular frequency of ω = 2π/T = 7 × 10−5 rad/s. The earth orbits around the sun with a period of one year (T = 3.156 × 107 s), or an angular frequency of ω = 2 × 10−7 rad/s. The planetary system orbits around our galaxy with a period of 2.5 × 108 years, (T = 8 × 1015 s), or an angular frequency of

82

CHAPTER 3. NON-INERTIAL FRAMES OF REFERENCE

ω ≈ 8 × 10−16 rad/s. Most astronomical bodies in the universe rotate, except the universe as a whole. Why does the universe as a whole not rotate relative to absolute space? There is no explanation for this fact in classical mechanics. This is a fact of observation, but nothing in classical mechanics obliges nature to behave like this. The laws of mechanics would remain the same if the universe as a whole were rotating relative to absolute space. We would only need to take this into account when performing calculations (this would cause a ﬂattening in the distribution of galaxies, similar to the essentially plane form of the solar system or of our galaxy due to their rotation).
These two coincidences of classical mechanics (mi = mg and ωk = ωd) form the main empirical foundations for Mach’s principle.

3.4 General ﬁctitious Force

In an inertial frame S we can write Newton’s second law of motion as
d2r m dt2 = F ,
where r is the position vector of the particle m relative to the center O of S. Suppose now we have a non-inertial frame of reference S’ which is located
by a vector h with respect to S (r = r + h, where r is the position vector of m relative to the origin O’ of S’), moving relative to it with translational velocity dh/dt and translational acceleration d2h/dt2. Suppose, moreover, that the axes x , y , z rotate relative to the axes x, y, z of S with an angular velocity ω. In this frame S’ taking into account the complete “ﬁctitious forces” Newton’s second law of motion should be written as ([27], Chapter 7):
d2r m dt2 = F − mω × (ω × r )

dr

dω

d2h

− 2mω ×

dt

−m ×r dt

− m dt2 .

(3.17)

The second term on the right is called the centrifugal force, the third term

is called the Coriolis force, the fourth and ﬁfth terms have no special names.

They are all “ﬁctitious forces” in Newtonian mechanics, and appear only in

non-inertial frames of reference. Although their eﬀects are real in these non-

inertial frames (ﬂattening of the earth, concave form of the water in Newton’s

bucket experiment, Foucault’s pendulum, ...), we cannot ﬁnd a physical origin

for these forces. That is, we cannot ﬁnd the body responsible for them and the

possible nature of this interaction (if gravitational, electric, magnetic, elastic,

nuclear, etc.) Certainly in classical mechanics these ﬁctitious forces are not

3.4. GENERAL FICTITIOUS FORCE

83

caused by the ﬁxed stars or by the external galaxies. The reason is that even if the stars or galaxies disappeared or doubled in number and mass, the ﬁctitious forces would still be there with the same values in any non-inertial frame of reference. Hence the name “ﬁctitious” forces.

84

CHAPTER 3. NON-INERTIAL FRAMES OF REFERENCE

Chapter 4
Gravitational Paradox
In this chapter we discuss the gravitational paradox. References can be found at: [32], Chapter 2 (Cosmological Diﬃculties with the Newtonian Theory of Gravitation), pp. 16-23; [33], pp. 194-195; [34], Chapter 8 (The Gravitational Paradox of an Inﬁnite Universe), pp. 189-212; [35]; [36]; [12], Chapter 7, Sections 203-222.
4.1 Newton and the Inﬁnite Universe
The cosmological conceptions of Isaac Newton have been clearly analysed by E. Harrison in an interesting paper [37]. Harrison’s work shows that during his early years (1660’s), Newton believed that space extended inﬁnitely in all directions and was eternal in duration. The material world, on the other hand, was of a ﬁnite extent. It occupied a ﬁnite volume of space and was surrounded by an inﬁnite space devoid of matter.
After his complete formulation of universal gravitation, Newton became aware that the ﬁxed stars might attract one another due to their gravitational interaction. In the General Scholium at the end of the Principia Newton wrote: “This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the ﬁxed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One; especially since the light of the ﬁxed stars is of the same nature with the light of the sun, and from every system light passes into all the other systems: and lest the systems of the ﬁxed stars should, by their gravity, fall on each other, he hath placed those systems at immense distances from one another.” However, putting the ﬁxed stars very far away from one another does not avoid another problem: if the
85

86

CHAPTER 4. GRAVITATIONAL PARADOX

universe existed for an inﬁnite amount of time, then a ﬁnite amount of matter occupying a ﬁnite volume would eventually collapse to its center due to the gravitational attraction of the inner matter.
In correspondence exchanged with the theologian Richard Bentley in 169293, Newton perceived this fact and changed his cosmological views. He abandoned the idea of a ﬁnite material universe surrounded by an inﬁnte void, and defended the idea of an inﬁnite material world spread out in inﬁnite space. This can be seen in his ﬁrst letter to Bentley [38, p. 281].
With an inﬁnite amount of matter distributed more or less homogeneously over the whole of an inﬁnite space, there would be approximately the same amount of matter in all directions. In this way there would be no center of the world to where the matter would collapse. Two hundred years later, however, a paradoxical situation was identiﬁed with this cosmological system. This is the subject of the next sections.

4.2 The Force Paradox
There is a simple but profound paradox which appears with Newton’s law of gravitation in an inﬁnite universe which contains an inﬁnite amount of matter. The simplest way to present the paradox is the following: Suppose a boundless universe with an homogeneous distribution of matter. We represent its constant density of gravitational mass by ρ. To simplify the analysis we deal here with a continuous mass distribution extending uniformly to inﬁnity in all directions. We now calculate the gravitational force exerted by this inﬁnite universe on a test particle with gravitational mass m located at a point P , as in Figure 4.1.

Figure 4.1: Inﬁnite and homogeneous universe with a constant mass density ρ.
If we calculate the force with our coordinate system centered on P, all the universe will be equivalent to a series of spherical shells centered on P. From

4.2. THE FORCE PARADOX

87

Eq. (1.6) we learn that there will be no net force acting on m. This might be expected by symmetry.
Now let us calculate the force on m utilizing a coordinate system centered on another point Q, as in Figure 4.2.

Figure 4.2: Force on m calculated from Q.
In order to calculate the net force we divide the universe into two parts centered on Q. The ﬁrst one is the sphere of radius RQP centered on Q and passing through P. The mass of this sphere is M = ρ4πRQ3 P /3. It attracts the material point m with a force given by GM m/RQ2 P = 4πGρmRQP /3 pointing from P to Q. The second part is the remainder of the universe. This remainder is composed of a series of external shells centered on Q containing the internal test particle m. By Eq. (1.6) this second part exerts no force on m. This means that the net force exerted on m by the whole universe calculated in this way is proportional to the distance RQP and points from P to Q.
Following a similar procedure but utilizing a coordinate system centered on another point R, as in Figure 4.3, we would ﬁnd that the net force exerted by the whole universe on m is proportional to the distance between P and R, pointing from P to R: F = 4πGρmRRP /3.
This means that depending on how we perform the calculation we obtain a diﬀerent result. This is certainly unsatisfactory.
However, the problem is not with the mathematics. For instance, if we were calculating the force exerted by a ﬁnite distribution of mass on a test particle m utilizing Newton’s law of gravitation, the result would be the same no matter how we calculated the result or where we centered the coordinate system. We assume, for instance, the ﬁnite body with constant density ρ of Figure 4.4. It is surrounded by an inﬁnite void space. If we calculate the net gravitational

88

CHAPTER 4. GRAVITATIONAL PARADOX

Figure 4.3: Force on m calculated from R.
force exerted by this body on one of its particles of mass m (or ρdV , where dV is the inﬁnitesimal volume of the particle) located at T, we always obtain the same result pointing from T to S. We can perform the calculations placing the coordinate system centered on S, on T, on U, on V or on any other point, and the ﬁnal result will always be the same: a force of the same magnitude pointing from T to S.
Another way of presenting the paradox is to consider the force on mg located at P calculated from an origin at Q, shown in Figure 4.2. As we have seen, the net force on mg points from P to Q and is proportional to the distance P Q. This means that the net force on a material particle located on P becomes inﬁnite if it is located at an inﬁnite distance from Q.
This is called the gravitational paradox. It was discovered by Seeliger and Carl Neumann at the end of last century (1894-96).
4.3 The Paradox based on Potential
Instead of calculating the force, we could just as well calculate the gravitational potential or the gravitational potential energy.

4.3. THE PARADOX BASED ON POTENTIAL

89

Figure 4.4: Finite body attracting one of its particles.

The gravitational potential at a point ro due to N gravitational masses mgj located at rj is given by:

Φ(ro)

=

−

N j=1

G

mgj roj

,

where roj ≡ |ro − rj|. The gravitational energy of a material particle mgo located at ro interacting with these N masses is given by U = mgoΦ.
We now calculate the gravitational potential at a point ro = rozˆ due to a spherical distribution of mass of radius R > ro, thickness dR and mass dMg = 4πR2dRρg (ρg being the uniform volumetric density of mass of the shell) with the previous expression. Substituting the sum by a surface integral over the shell and mgj by d3Mg = ρgR2dRdϕ sin θdθ yields the well-known result

2π π
dΦ(ro < R) = −GρgR2dR
ϕ=0 θ=0

sin θdθdϕ R2 + ro2 − 2Rro cos θ

=

−

GdMg R

= −4πGρgRdR

.

The contribution of the shell is proportional to the radius of the shell. This means that if we integrate this from R = 0 to inﬁnity we obtain an inﬁnite result. This was obtained by Seeliger and Neumann. The same can be said of the gravitational energy of a point particle interacting with this inﬁnite homogeneous universe, i.e., it becomes inﬁnite. The force on the test particle should be obtained by minus the gradient of this potential energy, but this becomes indeﬁnite.
There is another way to present the paradox. The equation satisﬁed by the gravitational potential in the presence of matter is known as Poisson’s equation:

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CHAPTER 4. GRAVITATIONAL PARADOX

∇2Φ = 4πGρg .
This is easily obtained observing that ∇2(1/r) = −4πδ(r), where δ(r) is Dirac’s delta function. Utilizing Φ = −Gmg/r and the fact that mgδ(r) = ρg(r) yields this equation.
If we have a homogeneous universe with a constant density of mass we should expect a constant Φ. But supposing Φ to be a ﬁnite constant yields ρg = 0 from this equation, which is against the initial supposition of a constant and ﬁnite density diﬀerent from zero. There is no solution of Poisson’s equation with a constant Φ and a constant ρg diﬀerent from zero.
4.4 Solutions of the Paradox
There are three main ways of solving the paradox: (I) The universe has a ﬁnite amount of mass. (II) Newton’s law of gravitation should be modiﬁed. (III) There are two kinds of mass in the universe, positive and negative.
(I) In the ﬁrst solution we maintain Newton’s law of gravitation and the constituents of the universe as usually known. We only require a ﬁnite amount of matter in order to avoid the paradox. For instance, if the universe has a total ﬁnite mass M uniformly distributed around a center P of radius R, with a constant mass density ρ = M/(4πR3/3) the net gravitational force exerted on a test particle m located at r < R is given simply by Gm(4ρπr3/3)/r2 = 4πGρmr/3 pointing from m to P, no matter how we perform the calculation. We can center the coordinate system on P, on m or at any other point, and the ﬁnal result will be the same.
However, this solution creates other problems. As we have seen, Newton abandoned this cosmological model of the universe because it leads to a collapsing situation. The external matter tends to concentrate on the center due to the gravitational attraction of the inner matter. To avoid this problem we would need to suppose the universe to be rotating relative to absolute space (the planetary system does not collapse into the sun due to its rotation, so that the centripetal gravitational force of the sun is balanced by mia, or by a centrifugal force in a frame of reference rotating with the planets). But we saw previously that the universe as a whole does not rotate relative to absolute space (the best inertial frame we have is the frame in which the distant galaxies are seen without rotation). This means that this proposed solution would be refuted by observations. We would then need to postulate some kind of repulsive force as yet unknown to avoid the gravitational collapse of the ﬁnite universe.
(II) The second solution was proposed by Seeliger and C. Neumann in 189596. Essentially, they proposed that the gravitational potential Φ = −Gm/r

4.4. SOLUTIONS OF THE PARADOX

91

should be replaced by −Gme−αr/r, where α has dimensions of length−1 and gives the typical range of interaction (the order of magnitude up to where gravitation is really eﬀective). It should be stressed that Seeliger and Neumann proposed this potential 50 years before Yukawa suggested a similar law describing nuclear interactions. If we have two interacting bodies mg1 and mg2 separated by a distance r12 their gravitational potential energy would be given by

U = −G mg1mg2 e−αr12 . r12

From this point on, we present our own calculations. Utilizing the fact that F = −∇U we can obtain the force exerted by mg2 on mg1, assuming α to be a constant:

F

=

−G

mg1mg2 r122

rˆ12(1

+

αr12)e−αr12

.

(4.1)

We now integrate this equation, assuming a universe with constant mass
density ρ2. The test particle of gravitational mass mg1 is located on the z-axis at a distance d1 from the origin of the coordinate system at O, r1 = d1zˆ. We consider an element of mass dmg2 located at r2 = r2rˆ2. Once more we divide the universe in two parts centered at O: The ﬁrst part is at r2 > d1 while the second is at r2 < d1, shown in Figure 4.5.
We now integrate the gravitational force exerted by a spherical shell of radius
r2 on mg1 utilizing spherical coordinates, with ϕ2 going from zero to 2π and θ2 going from zero to π. With r2 > d1 we get:

dF

=

2πGmg1ρ2r2e−αr2 dr2zˆ d21α

(1 + αd1)e−αd1 − (1 − αd1)eαd1

.

This is diﬀerent from zero if d1 = 0. This means that a spherical shell will exert a net force on an internal test particle according to Seeliger-Neumann’s potential if it is not at the center. There is a striking diﬀerence between this result and the Newtonian case, which yields zero no matter the position of the internal test particle.
In the limit in which αd1 1 we recover the Newtonian result where a spherical shell exerts no force on a test particle localized anywhere inside the shell.
Integrating this result from r2 = d1 to inﬁnity yields:

F

=

−

2πGmg1ρ2(1 + αd1)zˆ d21α3

(1 − αd1) − (1 + αd1)e−2αd1

.

(4.2)

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CHAPTER 4. GRAVITATIONAL PARADOX

Figure 4.5: Coordinate system to calculate the force on mg1.

We now calculate the force on mg1 due to the second part r2 < d1. We ﬁrst calculate the force of a spherical shell attracting an external particle. Integrating Eq. (4.1) in ϕ2 going from zero to 2π and in θ2 going from zero to π yields, with r2 < d1:

dF

=

2πGmg1ρ2(1

+ αd1)e−αd1 r2dr2zˆ d21α

e−αr2 − eαr2

.

In the limit in which αr2 1 and αd1 1, we recover the Newtonian result that a spherical shell attracts an external point as if it were concentrated at its center, namely:

dF

=

−

4πr22dr2ρ2Gmg1 d21

zˆ

.

Integrating the previous result in r2 going from zero to d1 yields:

F

=

2πGmg1ρ2(1 + d21α3

αd1)zˆ

(1 − αd1) − (1 + αd1)e−2αd1

.

(4.3)

This result is valid for α = 0 and cannot be applied for α = 0.

Adding Eqs. (4.2) and (4.3) yields zero as the resultant force acting on mg1 due to the whole universe. The same result is obtained choosing any other

4.4. SOLUTIONS OF THE PARADOX

93

point as the origin of the coordinate system. This shows that the paradox is solved with the Seeliger-Neumann potential energy, even keeping an inﬁnite and homogeneous universe.
We now analyse this solution of the paradox as regards the potential. The equation satisﬁed by a potential Φ = −Gmge−αr/r due to a point mass mg is given by:

∇2Φ − α2Φ = 4πGρg .

There is now a solution for this equation with a constant and ﬁnite ρg yielding a constant and ﬁnite Φ = −4πGρg/α2. With the known values of G,
ρg and utilizing α = Ho/c (Ho is Hubble’s constant) yields a potential close to −c2.
Another way of obtaining this result is directly integrating the potential due to a spherical shell of radius r. To this end we replace mg by d3mg = ρ sin θdθr2drdϕ and calculate the potential at the origin. Integrating:

Φ = −Gρg

∞ r=0

π θ=0

2π (r2 sin θdrdθdϕ) e−αr

ϕ=0

r

=

−

4πGρg α2

.

The same result is obtained calculating the potential at any other point ro diﬀerent from the origin.
This shows the solution of the gravitational paradox based on the potential. (III) The third way of solving the gravitational paradox is to suppose the existence of negative gravitational masses. The ﬁrst to propose this idea of a negative gravitational mass seems to have been F¨oppl in 1897 [39, p. 234]. He proposed this based on electromagnetic analogies, and was not concerned with the gravitational paradox. Calling the ordinary mass positive, we would have the following rule: positive mass attracts positive mass but repels negative mass, while negative mass attracts negative mass and repels positive mass. This would be the opposite of what happens with electrical charges. This being the case, we could have a universe with an equal amount of positive and negative masses, in which Newton’s law of gravitation would be obeyed and in which the gravitational paradox would not appear, even with an inﬁnite amount of positive mass. Now there is a solution of the equations in which both masses are equally distributed everywhere, so that the net gravitational force on any body is zero on the average. The gravitational potential energy would also be zero everywhere on the average. There exists a solution of Poisson’s equation with a constant Φ and a zero ρg. We can understand this third solution more easily observing that there is no electrical paradox analogous to the gravitational paradox. The reason is that usually we consider the universe as a whole to be electrically neutral. In

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CHAPTER 4. GRAVITATIONAL PARADOX

other words, apart from local anisotropies, the negative charge in one region is compensated by a corresponding positive charge somewhere else. This means that on average there is no electrostatic force on any charge due to all the charges in the universe. The same would be true for gravitation, provided there is negative gravitational mass.
The gravitational paradox is very simple to state and understand. It is amazing that with a situation so simple we can arrive at such a far-reaching conclusion, namely: We cannot have a universe with an inﬁnite amount of ordinary matter in which Newton’s law of gravitation is obeyed. At least one of these components must be modiﬁed: The inﬁnite amount of matter in the universe, Newton’s law of gravitation, or the nature of the constituents in the cosmos (if we have negative masses).
Our own preferred cosmological model is a universe that is boundless and inﬁnite in space, which has always existed without any creation, and with an inﬁnite amount of matter in all directions. In this model the universe extends in all directions without end, with an inﬁnite amount of matter on the whole, but with a ﬁnite matter density on average. The simplest universe model along these lines is an homogeneous distribution of mass in the large scale with a ﬁnite matter density. This means that it has no preferred center, so that any point can be arbitrarily chosen as its center. We could also perform the calculations beginning from any point. For this reason we do not adopt the ﬁrst solution of the gravitational paradox. We prefer the second and third solutions. In this book we explore quantitatively only the second solution.

4.5 Absorption of Gravity
There have been other reasons for people to propose an exponential decay in the gravitational potential of a point mass, in the gravitational potential energy between two point masses, or in the gravitational force. These ideas are not directly related to the gravitational paradox, but sometimes the proposed modiﬁcation is along the same lines. We have reviewed this situation elsewhere ([35], [36] and [12, Sections 7.5 to 7.7]); all the references and further discussion can be found in these studies. In each paragraph below we discuss a diﬀerent kind of idea leading to an exponential decay for gravitation.
Light ﬂowing from a source is absorbed by an intervening medium, so that its power falls as e−λr/r2. Those who suppose that gravitation propagates from a source like light (in the form of gravitational waves or in the form of particles like gravitons) are led to propose an exponential decay of gravitation.
When we interpose a dielectric between two point charges, this medium becomes aﬀected by the charges and becomes electrically polarized. The eﬀect of this polarization is to change the net force on each of the charges, as compared

4.5. ABSORPTION OF GRAVITY

95

with the situation in which there was no medium interposed between them. In this case we don’t need to speak of a propagation of the electric force, and the situation can be described by a simultaneous many-body interaction. Nevertheless, if we assume an analogy between electromagnetism and gravitation, we might expect some inﬂuence of the intervening medium for the net gravitational force on any material body. We may once more need to introduce an exponential decay for gravitation, although in this case there is nothing propagating at a ﬁnite speed. The only thing which happens here is that an action at a distance between many bodies may have this behaviour.
Astronomical observations, such as the ﬂat rotation curves of spiral galaxies, also led people to propose modiﬁcations in Newton’s law of gravitation or to postulate the existence of dark matter. Discussions of these topics can be found elsewhere ([40] and [41]). The problem of the ﬂat rotation curves of galaxies can be understood in a simple way. Let us suppose a gravitational interaction between a large body of mass M and a small body of mass m M describing a circular orbit around M in an inertial frame of reference. With Newton’s law of universal gravitation, his second law of motion and the expression for the centripetal acceleration we get: GM m/r2 = ma = mv2/r = mω2r. Here r is the distance between the bodies, v is the tangential velocity of m, a is its centripetal acceleration and ω = v/r is the angular velocity of m around M . From this expression we obtain v = GM/r and ω = GM/r3. With a larger r we have a smaller velocity v. This prediction is perfectly corroborated in the case of the planetary system, with M the sun and m any one of the planets. These relations of v and ω as a function of r are another form of Kepler’s third law in the case of a circular orbit (the square of the period of revolution is proportional to the cube of the radius): T 2 = (2π/ω)2 = 4π2r3/GM = Kr3. On the other hand, this relation is not valid for galaxies. Let m be a star belonging to a galaxy and far from its center, and M the mass of the nucleus of the galaxy (determined from its visible or bright part). Observations indicate that in most galaxies the tangential vel√ocity v becomes approximately constant as r increases, instead of falling as 1/ r (as would be expected according to Newtonian mechanics). To solve this problem there are two main approaches. One is to suppose the existence of dark matter not yet observed in any wavelength, that could interact gravitationaly with the stars. From the observed ﬂat rotation curves the distribution of this supposed dark matter can be estimated, assuming the validity of Newton’s laws. Another approach is to suppose that all existing matter has already been detected and then try to ﬁnd a modiﬁcation in Newtonian mechanics in order to explain the ﬂat rotation curves. Usually the modiﬁcation should be relevant for distances of the order of 1020 m, which is the typical size of a galaxy. We can try to modify either Newton’s second law of motion mia or the gravitational force GM m/r2. In this last case some trials have been made with relative success based on an exponential decay for

96

CHAPTER 4. GRAVITATIONAL PARADOX

gravitation [42], [43] and [44]. The main problem with this approach is how to derive simultaneously the ﬂat rotation curves of galaxies and Tully-Fisher’s law (luminosity proportional to the square of the tangential velocity of a galaxy). An alternative model has been developed elsewhere ([45] and [46]). Although it does not deal with absorption of gravity nor with its exponential decay, it leads to eﬀects involving an exponential decay which has some mathematical analogies with what is being discussed here. Further research is necessary before we can draw a ﬁnal explanation for the ﬂat rotation curves of galaxies.
Some terrestrial experiments have been made to detect modiﬁcations in Newton’s law of gravitation. Some of these have met with positive results, as those of Majorana in the beginning of this century. For this reason they should be repeated (see [47], [48], [49], [50] and [51]).
We have already discussed this problem in the references cited above, and will not analyse the subject further here. What should be stressed is that Newton’s law of gravitation or any other expression may be approximately valid in some conditions, although it may be necessary to modify it due to observations of astronomical bodies or terrestrial laboratory experiments. It is important to be open-minded in this regard.
For further discussions and references on all these topics can be found elsewhere ([16] and [52]).

Chapter 5
Leibniz and Berkeley
Before we present Mach’s criticisms of Newtonian mechanics we discuss the points of view of G. W. Leibniz and of the Bishop G. Berkeley as regards absolute and relative motion. These philosophers anticipated many points of view later advocated by Mach.
5.1 Leibniz and Relative Motion
Leibniz (1646-1716) was introduced to the modern sciences of his time by C. Huygens (1629-1695). They were in close contact during Leibniz’s stay in Paris during 1672-1676. Huygens may have inﬂuenced him as regards the concepts of space and time, and the signiﬁcance of centrifugal force. A detailed study of Huygens reactions to Newtonian mechanics can be found elsewhere ([33, pp. 119-126] and [53]).
Leibniz never accepted Newton’s concepts of absolute space and time. He maintained that space and time depend on things, with space being the order of coexistent phenomena and time the order of successive phenomena. There is a very interesting correspondence (in the years 1715-1716) between Leibniz and S. Clarke (1675-1729), a disciple of Newton. Leibniz wrote in French and Clarke in English. This correspondence illuminates this whole issue and can be found in English [54].
In the fourth paragraph of his third letter to Clarke, Leibniz wrote:
4. As for my opinion, I have said more than once, that I hold space to be something merely relative, as time is; that I hold it to be an order of coexistences, as time is an order of successions. For space denotes, in terms of possibility, an order of things which exist at the same time, considered as existing together; without enquiring into
97