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July 27, 2001; Revised August 15, 2001
Einstein's Triumph over the Spacetime Coordinate System:
A Paper presented in Honor of Roberto Torretti
John D. Norton1
Department of History and Philosophy of Science
University of Pittsburgh
Pittsburgh PA 15260
1. Introduction
Each student of Einstein must eventually make his or her their peace with Einstein's
pronouncements on relativity and spacetime coordinate systems. Einstein saw the
development of relativity as the ultimately successful struggle to overcome certain spacetime
coordinate systems and thereby to implement a generalized principle of relativity. This
signal achievement of relativity is embodied in its general covariance. We now hold
spacetime coordinate systems merely to be convenient devices for smoothly labeling events.
The selection of a coordinate system amounts to little more than a conventional choice of
1 Dr. Torretti has inspired my generation: in scholarship, by setting the standard in his
researches in history and philosophy of space and time; and in humanity with his generosity
and kindness. I take this opportunity to thank him personally for the stimulating model of
scholarship in his Relativity and Geometry and related writings and for his encouragement,
patience and instruction when I first worked in history and philosophy of space and time,
especially during a year we shared at the Center for Philosophy of Science, University of
Pittsburgh, in 1983-1984. He helped make it one the most exciting years intellectually of my
life.


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numbers, much like the selection of definition. How can one proclaim victory over a
definition? If we are offended by a definition, the more appropriate attitude is just to decide
quietly not to use it.
Dr. Torretti's celebrated Relativity and Geometry and related writings represent a
landmark of scholarship. They provide our most detailed account of how Einstein's work in
relativity theory changed physical geometry. It is presented in a comprehensive historical
context with the uncompromised insistence that every geometric conception must be
explicated to the highest standards of mathematical rigor. So when Dr. Torretti makes his
peace with the problem of Einstein and spacetime coordinates in Section 5.5 "General
Covariance and the Einstein-Grossmann theory," this latter insistence ensures that the
peace will be uncomfortable—for Einstein. He takes Einstein's formulation of the postulate of
general covariance and rephrases in language that mimics Einstein's 1905 statement of the
principle of relativity of special relativity. Calling it the "principle of general relativity," Dr.
Torretti explains why the similarity of the two relativity principles is only superficial. Unlike
the case of the special principle, the general principle does not assert a physical equivalence
of states of motion. Dr. Torretti's analysis is careful, thorough and leaves no room to quibble.
So we are left with a puzzle. How could Einstein be so confused about the fundamentals of
his own theory?
My goal in this paper is small. I do not want to dispute Dr. Torretti's careful analysis.
Rather I offer an extended footnote to it. I want to try to explain what Einstein intended in
his remarks about coordinate systems. There is, I believe, a natural reading for Einstein's
claims that do make perfect sense. They require us to adopt a physical interpretation of
relativity theory that is now no longer popular, so the natural reading will no longer have
intrinsic interest. It will, however, allow us to make sense of Einstein's claims and his
program.


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2. "The Vanquishing of the Inertial System"
A Letter to Besso
When we face claims that are unintelligible in the writing of an Einstein, we are
often tempted to dismiss them as remarks made in haste in the frenzied first moments of
great discovery. Might they not be retracted or qualified in some essential way as time brings
sober distance from those heady moments? While time mellowed Einstein, we can be sure
this was not the case with his proclamations over coordinate systems. He brought the general
theory of relativity to a generally covariant formulation in November 1915. Nearly 40 years
later, after his theory had been much celebrated and its foundations subject to minute
scrutiny, Einstein wrote to his lifelong friend and confidant, Michele Besso.
His letter of August 10, 1954, lays out a brief account of the essence of the general theory of
relativity, explicitly intended to be free of entanglement with the history of the theory.
(Speziali, 1972, p.525)2
Your characterization of the general theory of relat.[ivity] characterizes the
genetic side quite well. It is also valuable afterwards, however, to analyze the
whole matter logically-formally. For as long as one cannot determine the
physical content of the theory on account of temporarily insurmountable
mathematical difficulties, logical simplicity is the only criterion of the value of
the theory, even if it is naturally an insufficient one.
The special th.[eory] of r.[elativity] is really nothing other than an adaptation
of the idea of the inertial system to the empirically confirmed conviction of the
constancy of the velocity of light with respect to each inertial system. It does not
vanquish the epistemologically untenable concept of the inertial system. (The
2 I thank Karola Stotz for help in this translation.


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untenability of this concept was brought to light especially clearly by Mach and
was, however, already recognized with lesser clarity by Huygens and Leibniz.)
The core of this objection against Newton's fundamentals is best explained
through the analogy with the "center point of the world" of Aristotelian physics:
there is a center point of the world, towards which heavy bodies strive. This
explains, f[or] e[xample], the spherical shape of the earth. The ugliness in it is
that this center point of the world acts on all others, but that all these others
(i.e. bodies) do not act back on the center point of the earth. (One-sided causal
nexus.)
It is just like this with inertial systems. They determine the inertial relations
of things everywhere, without being influenced by them. (Really one ought better
to speak of the aggregate of all inertial systems; however this is inessential.) The
essence of the gen.[eral] th.[eory] of rel.[ativity] (G. R.) lies in the vanquishing
[Ueberwindung] of inertial systems. (This was still not so clear at the time of the
setting up of G. R., but was subsequently recognized principally through Levi
Civita.) In the setting up of the theory I had chosen the symmetric tensor gik as
the starting concept. It provided the possibility of defining the "displacement
field" Γlik...
Einstein briefly explained the notion of the displacement field and its independence from the
metric gik. He continued:
But how is it that the displacement field really led to the vanquishing of
inertial systems? If one has vectors with the same components at two arbitrarily
distant points P and Q in an inertial system, then this is an objective (invariant)
relation: they are equal and parallel. On this rests the circumstance that one
obtains tensors again through differentiation of tensors with respect to the
coordinates in an inertial system and that e.g. the wave equation represents an
objective expression in inertial systems. The displacement field now allows such


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tensor formation by differentiation in relation to an arbitrary coordinate system.
Therefore it is the invariant substitute of inertial systems and thereby--as it
appears--the foundation of every relativistic field theory.
Einstein then continued to explain how the metric and displacement field are used to
formulate general relativity and his unified field theory.
Its Unusual Treatment of Coordinate Systems
Einstein finds the essence of the general theory to lie in the vanquishing of inertial
systems, that is, inertial coordinate systems. Part of his account is that these systems have
the objectionable feature of acting without being acted upon. That aspect has been subject to
much discussion and analysis. It is usually explicated by the notion of "absolute object,"
geometric objects that act but are not acted upon. In special relativity, the pertinent absolute
object is the Minkowski metric.3 Here I pass over the problem of explicating the absoluteness
Einstein raises; I am interested in just one other aspect. Einstein's notion of the absolute
inertial [coordinate] system has been transmogrified into an absolute geometric object, the
Minkowski metric.
It is so tempting to say that this transformation is what Einstein really intended. But
then we must be amazed at his tenacity in avoiding the assertion. His remarks to Besso
mention the metric field and the displacement field, both geometric objects, but condemns
the inertial system for its absolute character—and this forty years after his achievement of
general covariance.4
3 For discussion see Norton (1993, Section 8).
4 Similar remarks on inertial systems span Einstein's life. They appear, for example, as early
as Einstein (1913, pp. 1260-61) and as late as a letter to George Jaffé of January 19, 1954
(Einstein Archive, document with duplicate archive control number 13 405).


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2. Einstein's use of Coordinate Systems
Their Physical Content
There is a simple way to understand Einstein's remarks.5 He did not regard
coordinate systems as we now do, as essentially arbitrary systems of numerical labels of
events. In his theorizing, they initially carried significant physical content. The journey to
the completion of general relativity required the systematic elimination of this content.
That coordinate systems can be used to represent significant physical content is not
the modern view and it is tempting to think that no other view is possible. But that
narrowmindedness is quite incorrect. Our physical theories use mathematical structures to
represent aspects of interest of the physical world. We routinely use a manifold that is
topologically R4 to represent the set of physical events in special relativity. Nothing prevents
us using the structurally richer number manifold of quadruples of reals as this manifold. If
we do use a number manifold in this way, then we are assigning quadruples of reals to
events in spacetime. That is just what a coordinate system does.
A number manifold has considerably more structure than we use in standard
theories of spacetime. It has a preferred origin (0,0,0,0), for example. How are we to interpret
that? Does this preferred origin correspond to a real physical center point of the world?
Whether it does or not cannot be decided purely by the mathematics of the theory. The
mathematics can only affirm that (0,0,0,0) is indeed different from all other points in R4, but
not that the differences amount to nothing physically. This last judgment must be made by
the physical interpretation we supply for the mathematical structures. The modern view is to
discount it as physically insignificant. Einstein's default was the opposite. The various
5 I have developed the approach to Einstein's use of coordinate systems sketched below in
greater detail in Norton (1989, 1992).


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features of coordinate systems represent physical features of the world. Most crudely, the
origin (0,0,0,0) is a physical center point. In Einstein's program, we must find a way of
depriving coordinate systems of this default physical content.
...and How it is Systematically Denied
Einstein used a single technique that was not his own invention. He used a strategy
codified by Felix Klein in the nineteenth century.6 Each geometric theory would be
associated with a class of admissible coordinate systems and a group of transformations that
would carry us between them. The cardinal rule was that physical significance can be
assigned just to those features that were invariants of this group. In special relativity, that
group is the Poincaré group. The origin (0,0,0,0) is not an invariant; under translations
within the group, the origin is not mapped back to itself. Thus it has no physical significance.
But the light cone structure –the complete catalog of the pairs of events that are lightlike
separated7—is invariant and thus has physical significance.
3. The Development of Relativity Theory
The Default Interpretation of Spacetime Coordinate Systems...
Einstein's natural starting point is to assign physical significance to the natural
features of a coordinate system. Using the familiar (t, x, y, z) as the spacetime coordinates,
we can list some of them:
6 For a more detailed account of the connection to nineteenth century geometry, see Norton
(1999).
7 In coordinate terms, the pair satisfies the condition ∆t2 - ∆x2 - ∆y2 - ∆z2 = 0, where (t, x, y, z)
are the usual spacetime coordinates, ∆ represents the coordinate differentials and the speed
of light is set to unity.


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(a) The origin (0,0,0,0) corresponds to a central point; the distinction between the x, y and z
coordinates makes space anisotropic.
(b) The curves picked out by constant values of x, y and z are a state of rest.
(c) In a Lorentz or Galilean covariant theory, the set of all curves picked out by (b) for all
coordinate systems are the inertial states of motion.
(d) Coordinate differences have metrical significance; they represent the possible results of
clock and rod measurements by observers in the state of rest picked out by (b).
...and Their Loss of Physical Significance
The development of relativity theory brings the systematic elimination of these
default physical interpretations. As our starting point, we might imagine a one-coordinate
system theory. It would have all the physical structures of the list above (a)-(e). The first step
had already been taken in the nineteenth century. The spatial sections of the spacetime are
covered by coordinates x, y, z. The Euclidean character of space entails that we can use many
coordinate systems related by translations, rotations and reflections. None of the structures
of (a) are invariants of these transformations. They lose physical significance.
The Relativity of Motion
The theory would retain an absolute state of rest (b), however. That is eliminated by
the transition to a Newtonian spacetime, with the characteristic group the Galilean group, or
to special relativity with the characteristic group the Poincaré group. The states of rest (b)
are no longer invariant.


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The next step marks the starting point of Einstein's 1907 quest for his general theory
of relativity.8 Einstein sought to expand the covariance of his theory further so as to deprive
the inertial states of motion (c) of physical significance. This, he believed, was achieved with
his postulation of the principle of equivalence which now allowed him to extend the Poincaré
group with transformations that represented uniform acceleration, although in only limited
circumstances. Einstein immediately interpreted the expansion as representing an extension
of the principle of relativity to acceleration. In this account, we see why: the inertial motions
of (c) are no longer invariants of the admissible transformations.
Metrical Significance
Presumably this much was all Einstein expected in 1907. In 1912, Einstein realized
that the development of his theory required him to take another step in depriving
coordinates of physical significance. He saw an analogy between the problem of gravitation
and relativity and the theory of curved surfaces of Gauss. The latter has led to a new
mathematics in which one could use arbitrary coordinate systems and in which coordinate
differences cease to have the direct metrical significance of (d).9
8 There has been very considerable investigation in recent decades of Einstein's passage to
the general theory of relativity. They span from early work including Torretti (1983), Norton
(1984) and Stachel (1980) to Renn (in preparation).
9 Here I will report Dr. Torretti's repeated lament that the group structure—or lack of it—of
Einstein's expanded coordinate systems brought many unintended problems apparently
ignored by Einstein. For example (Torretti, 1983, p. 153) observes that the ranges of two
coordinate charts may not overlap, so that the point transformation by induced the
corresponding coordinate transformation may have degenerate properties. Einstein largely
maintained a physicist's silence on these mathematical niceties.


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Independent Existence
With this development, Einstein's quest for depriving coordinate systems of their
default physical significance has taken an unanticipated turn. It proved to be a trifle in
comparison to the final hurdle that Einstein needed to overcome in arriving at a generally
covariant formulation of his general theory of relativity. Having failed to find what he
thought were admissible generally covariant gravitational field equations in 1912 and 1913,
Einstein eventually found a way to discount the failure. He developed arguments that
purported to show that general covariance would be physically uninteresting, were it to be
achieved. The best known and most important of these was the "hole argument."
The error of Einstein's argumentation is now well known. He had generated two
intertransformable metric fields gik(xm) and g'ik(xm) in the same coordinate system, xm. He
had assumed that the two fields represented two distinct physical possibilities. That proved
to be the elusive error that took several years to find. Einstein presumed that it made sense
to say that the two fields were in the same coordinate system. That tacitly accorded an
existence to the coordinate system independent of the metric field defined on it. Figuratively,
it meant that it makes sense to say that we can remove the first field from the coordinate
system, leave a bare coordinate system behind and then deposit the second fieldin the very
same coordinate system.
One of the final stages of Einstein's development of a generally covariant theory was
to recognize that coordinate systems have no such independent existence. He described his
error to Besso in a letter of January 3, 1916:10
10 Schulmann et al. (1998) Papers, Vol. 8A, Doc. 178; Einstein's emphasis. I have argued
elsewhere that Einstein's according independent reality to coordinate systems may have had
catastrophic effects at an earlier stage of his quest for general covariance. See "What Was
Einstein's Fatal Prejudice?" in Renn et al. (in preparation).


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There is no physical content in two different solutions G(x) [gik(xm)] and G'(x)
[g'ik(xm)] existing with respect to the same coordinate system K. To imagine two
solutions simultaneously in the same manifold has no meaning and the system
K has no physical reality.
4. Conclusion
These considerations, however, have little force with modern readers. We now
proceed from a quite different starting point. We do not accord default physical significance
to coordinate systems. If we wish to endow a spacetime with inertial structures, absolute or
otherwise, we start where Einstein ended. We start by endowing a manifold with an affine
connection (displacement field) whose natural straights are the inertial motions. In all this,
coordinate systems are little more than convenient labels for spacetime events.
For Einstein, however, matters looked quite different. His default was to load
physical content into the coordinate systems. The conceptual development through special to
general relativity is characterized by depriving coordinate systems of their default physical
significance in progressively greater measure. He had initially intended to end up just
depriving coordinate systems of absolute inertial motions. Once Einstein had started the
process, it could not be stopped. The natural development of the theory ended up forcing
much more. The coordinate systems lost their metrical significance and, after much
suffering, he finally recognized the need to dispense with a notion of independent existence
he had tacitly accorded them.
References
Einstein, Albert (1913a) "Zum gegenwärtigen Stande des Gravitationsproblems,"
Physikalische Zeitschrift, 14, pp.1249-1262.
Norton, John D. (1984) "How Einstein found his Field Equations: 1912-1915," Historical
Studies in the Physical Sciences, 14, 253-316; reprinted in Don Howard and John


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Stachel (eds.) Einstein and the History of General Relativity: Einstein Studies, Vol. 1
Boston: Birkhäuser, 1989, pp.101-159.
Norton, John D. (1989) "Coordinates and Covariance: Einstein's view of spacetime and the
modern view," Foundations of Physics, 19, 1215-1263.
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A. Kox eds., Studies in the History of General Relativity: Einstein Studies, Vol.3,
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