Volume Two: Symposia and Invited Papers || Is General Relativity Generally Relativistic?

Is General Relativity Generally Relativistic?Author(s): Roger JonesSource: PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association,Vol. 1980, Volume Two: Symposia and Invited Papers (1980), pp. 363-381Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/192599 .

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Is General Relativity Generally Relativistic?

Roger Jones

The University of Tennessee-Knoxville

1. Introduction

Within the lore of nearly every scientific theory, among the facts, laws, theorems, and such that constitute the "truths" it expresses, are one or more principles. Physics textbooks, I know, usually mention them at the beginning, or during some interlude given over to general remarks or speculation. As examples, the uncertainty principle and correspondance principle of quantum mechanics come to mind (see, e.g., Messiah 1960). And associated with the general theory of relativity, which is particularly rich in principles, are a principle of general covariance, a principle of equivalence (or two), Mach's principle, and a general principle of relativity (see, e.g., Misner, Thorne, and Wheeler 1973).

Where do principles come from? Generally, as I said, they are supposed to be "truths" about the working of the world: observations and experiments are appealed to as their sources. But this doesn't distinguish them from facts and laws. What does, in the way principles are usually treated, is their place in a theory. Principles are usually said to "underlie" a theory, a role with at least four charac- teristics. First, of course, the principle is true in the theory. I will have more to say about this characteristic in a moment. Second, the principle is not true in competing theories; the theory is somehow distinguished as manifesting the principle. Often, in fact, the theory is considered to result from implementation of a principle. This is sometimes made as a historical claim (the third sense in which a principle underlies a theory), and it is usually meant to express a "logical" relationship as well: the theory which manifests the principle can be understood as arising uniquely, or at least with some kind of logical inevitability, as a modification of an earlier theory necessary to accommodate the principle.2

PSA 1980, Volume 2 , pp. 363-381 Copyright Q 1981 by the Philosophy of Science Association



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The principle I want to talk about here is the general principle of relativity. And I want to concentrate on the limited issue of whether or not it really is true in the general theory of relativity.3 Very roughly, investigating the truth of a principle within any theory involves formulating the principle as a claim within (or about) the mathematical structure of the theory, and searching for a proof of, or a counter-example to, the claim. This is usually a tricky kind of investigation. Arriving at a formal statement of a principle is usually not at all straightforward, and people will disagree about whether the fundamental intuitions of the principle have been cap- tured. They often disagree, for that matter, on the nature of the fundamental intuitions themselves.4 This is somewhat the case for the general principle of relativity, which must be formulated so as to be a natural extension, a generalization, of a special principle of relativity, and so as to be clearly distinguished from the other three principles of the general theory--Mach's principle, the equivalence principle, and the principle of general covariance. Einstein himself did not clearly distinguish these principles, as we shall see.

If a principle can be rendered as a precise claim though, one is generally in a favorable position to decide on its truth in the theory. After all, if the theory itself has been precisely described as a mathematical entity, the search for a proof, or a counter-example, really ought to be relatively straightforward. I will show a bit of how this goes for this case shortly.

2. Intuitions and Formal Framework

I have said that principles have their source in observations from the real world. What observations establish the content of relativity principles? One of the most graphic accounts is provided (though for a slightly different purpose) by Galileo, in the second day of his Dialog Concerning the Two Chief World Systems.

Shut yourself up with some friend in the main cabin below decks in some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drop falls into the vessel beneath. ...When you have observed all these things carefully, have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. ...The droplets will fall as before into the vessel beneath... .The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to



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bait placed anywhere around the edges of the bowl. Finally, the butterflies and flies will continue their flights indifferently toward every side... .(p. 186).

The central intuition of a relativity principle, as it emerges from these kinds of observations, can be expressed in three ways. First, of course, is relativity: certain states of motion are only defined relative to some fixed framework; they are not "absolute". Second is indistinguishability: certain states of motion are indistinguishable; they cannot be differentiated by appeal to effects "internal" to the system, by appeal to the (internal) "physics of the system", or, in the customary terminology, by appeal to "the laws of physics" (Einstein 1916, p. 113). The states of motion see the same set of physical laws. Third is invariance: since the laws of physics are identical in certain states of motion, they are invariant with respect to transformations from one such state to another; they are state- independent.

The second of these concepts seems to lie at the heart of the other two: certain observers (characterized by their states of motion) are indistinguishable in terms of the "lawlike behavior" of their physical circumstances. Or, to be perfectly true to Galileo's "observations", an observer in one state of uniform motion who examines the "lawlike behavior" of his physical circumstances, will obtain precisely the same impression if he and his whole physical situation are "boosted" to any other state of uniform motion. Since the observer himself and his laboratory, his whole physical situation, are to be boosted, this notion of indistinguishability is customarily termed "active" indistinguishability.

So it is this notion of the "active indistinguishability of observers" that must be formalized if a principle of relativity is to be rendered as a precise claim.

The formal framework in which these issues are now customarily discussed is that of space-time theories.5 To specify a space-time theory is to specify a certain set of space-time models. A model for a theory in general is a dynamically possible history for the physical system described by the theory. A space-time model consists of a 4- dimensional differentiable manifold (representing the collection of all space-time point locations) on which is defined various geometrical object fields (characterizing whatever structure the theory takes to be present on space-time). Generally, metric structure is provided, a derivative operator, and various "physical object" or "matter" fields characterizing whatever physical system is described by the theory. All these geometrical object fields are required to satisfy various field equations.

Now the "laws" of a theory in general are what determine the set of dynamically possible physical histories of the system described by the theory. So the "laws" of a space-time theory certainly include the field equations of the theory. These do, in a sense, pick out the



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model set of the theory. One might even consider them to be implic- itly provided by the model set of the theory.

Again, the root notion of the relativity principle as I have characterized it is that of the "active indistinguishability" of observers with respect to "the laws of physics". And two observers are indistinguishable with respect to some theory only if they "construe the laws of the theory identically". In terms of the formal framework of space-time theories, then, we will have to make certain choices of explication.

First, what structure on the space-time manifold will we take to represent an observer? We might take a global reference frame, as defined by a smooth, non-vanishing, unit, timelike vector field; or a local frame--such a vector field defined only on some region of the manifold; or even a single timelike curve--a world-line. Second, in what regions are observers to be compared? Shall we compare them globally, locally, or perhaps even at a single point? Third, in terms of what are they to be compared? What is it for two observers to "construe the laws of a theory identically"? I have indicated briefly what I mean by 'the laws of a theory'; but what is it for an observer to "construe the laws of a theory"? And under what conditions are two such construals to be taken to be "identical"?

One thing is quite clear here already, though: it is not the coordinate form of the field equations of a theory that is at issue. It is not as Einstein presented it in his famous paper of 1916 when he said, "The general laws of nature are to be expressed by equations which hold good for all systems of coordinates, that is, are co-variant with respect to any substitution whatever (general co-variant)," and then immediately followed with,"It is clear that a physical theory which satisfies this postulate will also be suitable for a general principle of relativity." (p. 117).

All space-time theories cast in the form I have described satisfy the "principle of general covariance" Einstein describes. This is very easy to see, in two ways. First, taking the laws of a theory simply to be the field equations, one notices that they can be, and usually are these days, expressed in completely coordinate-free no- tation: they are coordinate-independent laws. So they trivially "hold good for all systems of coordinates." Or, if we take the laws of a theory to be exemplified by the model set of the theory, it must be the case, if the laws of the theory are to "hold good for all systems of coordinates," as Einstein says, that the model set be "pre- served" under arbitrary coordinate transformations. Since the characterization of a space-time model I have provided is itself completely coordinate-independent, this requirement is met automatically as well. So the principle of general covariance serves not at all to distinguish space-time theories. Newtonian space-time theories can be formulated to be just as generally covariant as are general relativistic theories. Neither does it imply suitability for a general principle of relativity. For though all space-time theories



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can be considered to satisfy the principle of general covariance, certainly not every theory satisfies the general principle of relativity. Far from it, as we shall see.

Back now to formalizing intuitions about the notion of active indistinguishability of observers with respect to a space-time theory. Let's return for a moment to Galileo's ship's cabin and see how the active indistinguishability of observers with respect, say, to the theory of butterflight would be described in a more formal way. One can consider a ship's-cabin-filling set of butterfly trajectories over some long time to define a model of the theory. With the ship standing still, the observer in the cabin records the set of trajectories in a particular way and thus obtains an impression of the model. The ship is then boosted to some fixed speed and the observer records again in the same way the new set of trajectories, obtaining an impression of another model. Galileo's claim, for my purposes, is that the two model-impressions are the same.

Take an observer in a space-time theory to determine a reference frame, where a reference frame, for now, will be defined on a model of the theory by a non-vanishing, smooth, unit, timelike vector field on the space-time manifold of the model, In general, given a vector at some point on a space-time manifold, and given any geometrical object at that point, one can "resolve" the geometrical object into its components orthogonal and parallel to the vector. Since reference frames are defined by non-vanishing vector fields, let me say that a reference frame "sees" a model of a theory by resolving each of its physical object fields at each point of the space-time manifold. Two reference frames' impressions of respective models will then be identical when, given any point p in the manifold of one model, there is a unique point p' in the manifold of the other such that the components of the physical object fields of the first model as resolved by the first frame at p are identical to those of the second model as resolved by the second frame at p'.

Now a boost from one observer-state to another is formally represented as a diffeomorphism--a structure-preserving mapping--of the underlying manifold of (a model of) the theory onto itself, a diffeomorphism that maps the vector field defining one frame to that defining the second. But active indistinguishability of frames involves boosting more than just the frames. Galileo would have us consider the whole physical situation to be boosted--ship (i.e., ship's cabin, observer, butterflies) and all. The physical situation is characterized by the physical object fields defining the models of the theory, so if we are to compare impressions of reference frames in unboosted and boosted physical situations, we must compare the impression obtained by the unboosted frame of some model of the theory, and the impression obtained by the boosted frame of the physical situation which is characterized by physical object fields boosted in the same way. There's no doubt that these two impressions will be identical; that's guaranteed by the action of the boosting diffeomorphism on the physical object fields. The question is, is the boosted physical



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situation itself a model of the theory? That is, will the boosted observer experience a physical situation that satisfies the laws--the field equations--of the theory?

If he does, then the two frames are indeed actively indistinguishable. Given any model of the theory, there is another (unique) model--that boosted with respect to the first--such that the unboosted observer sees the unboosted model in the same way as the boosted observer sees the boosted model. And since the boosting is quite arbitrary, this result holds for observers in any relative states of uniform motion. That is, the laws of the theory, manifested in its model set, provide no basis for distinguishing, in an active sense, among observers in uniform relative motion: all such observers per- ceive all physical situations consistent with the field equations of the theory in the same way.

Let me say this same thing in a slightly different way. Any two actively indistinguishable reference frames determine an isomorphism of the model set of a space-time theory: given any model of the theory, there is another model such that the first model is seen by the first frame in the same way as the second model is seen by the second frame. The "boosting" operation on the model set defines this isomorphism. So, in this sense, the two frames "see the model set in the same way". Since the laws of the theory are implicit in the model set, the two frames "see the same laws", they "construe the laws of the theory in the same way". They are, again, indistinguishable with respect to the laws of the theory.

Let me be quite concrete about the central idea in all this discussion, and offer the following (quasi-)formal condition:

Rl Let two frames be defined on the underlying manifold of (some model of) a space-time theory. The two frames are actively indistinguishable with respect to the (laws of) the theory if: 1) there exists a diffeomorphism of the manifold that

takes one frame to the other; and 2) given any model of the theory, the physical situation

described by boosting the physical object fields defining the model by the frame-diffeomorphism is also a model of the theory.

A theory satisfies the special principle of relativity, then, only if its inertial frames (those frames defined by vector fields with vanishing acceleration) are Rl-indistinguishable. And a "dynamical symmetry group", or "invariance group", or "indistinguishability group" is associated with a space-time theory by the diffeomorphisms in this condition in a straightforward way. Applying the condition to a theory, one obtains an equivalence class of Rl-indistinguishable frames and a set of diffeomorphisms of the underlying manifold of the theory onto itself such that some member of the equivalence class is equivalent to some other member by virtue of each of these diffeo-



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morphisms. This set generally has the structure of a group, and given any frame in the equivalence class and any diffeomorphism in the group, the frame obtained by the action of that diffeomorphism on the vector field defining the first frame is also a member of the equivalence class.

My main concern, now, is with the suitability of this condition Rl as a foundation for a general principle of relativity applied in general relativistic theories. But before I talk about this, let me review the results of its application in "earlier" space-time frameworks.

Hand-tailored as Rl is for theories satisfying the special principle of relativity, one would certainly hope that it provides the "right" results for such theories. And it does. Inertial frames are actively indistinguishable in Newtonian theories without a preferred state of rest, and the Galilean group is the invariance group of such theories, in the sense just described. This is true in some Newtonian theories formulated with an absolute rest state as well. It's not true for the Newtonian theory of the electromagnetic field, however: given some model of the theory, the physical situation described by boosting the electromagnetic field tensor by a Galilean diffeomorphism is not a model. The boosted tensor field does not satisfy the field equations of the theory. (And boosted inertial observers see light travelling at different speeds from one absol- utely "at rest".) This theory does not satisfy the special principle of relativity.

All special relativistic theories do, of course. That's just the point of the special relativistic space-time framework. And the Poincare group is the invariance, or indistinguishability group of special relativistic theories.

So the active indistinguishability of reference frames, as formalized by condition Rl, seems to be just the right notion on which to ground the special principle of relativity. The intuitions involved have a firm basis in "observation"' and the results of the application of the condition to Newtonian and special relativistic theories are just what is expected. But what about the general principle of relativity? As the general principle is to be a generalization of the special principle, shall we simply require the active indistinguishability of all observers, all reference frames? That is, can we take our intuitions and formal condition Rl straight into general relativistic theories and expect to find universal indistinguishability?

No--to all three questions. Rl itself has technical problems. Everything about it is global: globally-defined reference frames, global diffeomorphisms from one to another and from one physical object field to another. One might try to fix this. Things could be treated locally, even infinitesimally. The main problem however is that Rl is just not the right condition. And that goes back to



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intuitions.

Galileo's basic idea, recall, was that an observer in uniform motion be boosted, along with his physical surroundings, to another state of uniform motion. He was then to compare impressions of his physical situation in the two states of motion. Formally, the physical situation was characterized by physical object fields on the space-time manifolds of the models of the theory; thus a boosted physical situation was characterized by boosting these physical object fields.

Throughout the formalization of Galileo's intuitions, we were able essentially to ignore the impression of the observer of his own dynamical state, as determined, say, by accelerometer readings. Inertial observers emerge as indistinguishable in Newtonian and special relativistic theories because they "see the same set of physical situations", butterfly trajectories, for instance, or electromagnetic fields. Their inertiality per se has nothing to do with it. And an inertial observer and an accelerated observer are distinguishable precisely because the physical object fields the accelerated observer sees do not satisfy the field equations of the theory. His distinguishability from the inertial observer has nothing to do with forces he may feel (or measure with his accelerometer) that the inertial observer does not. That is, there is really nowhere in these intuitions any consideration of the relation of the observer to the "background geometry" of space-time.

This is perfectly appropriate in Newtonian and special relativistic theories, because the background geometry in such theories is "fixed"; it is the same model to model. There is in such theories a basic dichotomy between the structure of space-time and the physical situations characterized in space-time. This is what enables us to consider separately the boosting of an observer from one state of uniform motion to another and the boosting of his physical situation in the same way. The boosting of a physical situation in itself has no effect on the dynamical state of an observer, because it is not linked in any way to the background geometry of space-time.

This is, of course, not true in the general relativistic framework. For it is a distinctive feature of general relativistic theories that the physical content of space-time is intimately linked to itsgeometry, through Einstein's equation, one of the field equations of all such theories (see, e.g., Misner, Thorne, and Wheeler 1973, Ch. 1). Thus the boosting of a physical system--a ship's cabin full of butterflies, for instance--would certainly be expected in general to alter the background geometry of space-time, and thus the dynamical state of an observer boosted along with it. We can no longer take for granted that there exists any pair of observers, boosted with respect to each other, whose impressions of their own dynamical states will agree, that is, who will measure the same forces, or accelerations, on themselves.



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Such a circumstance could happen only for general relativistic theories whose derivative operator (and metric tensor) were left invariant, at least locally, by local diffeomorphisms of the model manifold onto itself. Observers related by these diffeomorphisms would have the same relationship to background geometry, and thus would agree on their own dynamical states, at least locally. But, at best, this means that aspects of our "classical" intuitions might apply in general relativistic theories involving metric tensors with local symmetries. In general, of course, there will be no such symmetries. So by no means do these intuitions lead to a condition which, applied to all local reference frames say, in all general relativistic theories, characterizes them as indistinguishable. Rather, if we consider indistinguishability of reference frames to require identical impressions, (at least locally) of physical object fields and identical inertial properties in pairs of models of some theory, then it seems appropriate to say--as a characterization of local relationships among reference frames in arbitrary general relativistic theories--that no two frames are indistinguishable. Far from leading to an extension of the special principle of relativity true in the general theory, the classical intuitions about relativity principles lead to a characterization of general relativity as satisfying no relativity principle at all.

3. Other Formulations

Well, if we reject the active indistinguishability of observers as a foundation for principles of relativity, where do we turn? The main difficulty with the active indistinguishability idea lies in estab- lishing some kind of "inter-model" relationship of frames, in particular, in preserving the dynamical state of a frame in a boost "from one model to another". One might think of avoiding this difficulty by looking for an "intra-model" criterion for observer indistinguishability. Since it was the boosting of the physical system that led to problems before, it might be more promising to consider only the observer to be boosted.

I want to take seriously the idea that observers can, in principle, determine their dynamical states to all orders--the totality of their inertial properties. They can certainly measure them at second order by means of an accelerometer. Perhaps a speedometer could be attached to the accelerometer bob to measure to third order. But details are unimportant with in-principle measurements. It certainly seems reason- able to think of indistinguishability in terms of at least local inertial properties of two reference frames as necessary for any sort of more elaborate equivalence in terms of "seeing the same laws of physics" generally. So take a reference frame again to be defined by a smooth, unit, timelike vector field, non-vanishing this time on a region of a space-time manifold. Roughly, two frames are to be taken as locally indistinguishable only if they are "locally identical" to arbitrary orders on their regions of definition. This relationship would define local diffeomorphisms from one region to the other. In somewhat more detail,



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R2 Given a space-time theory, a model of the theory, and two reference frames defined on regions of the manifold of the model, the two frames are locally indistinguishable to order n only if one can map one region patch by patch onto the other by means of diffeomorphisms that take the field defining the first frame and its first n-l (covariant) derivatives on the one patch to the field defining the second frame and its corresponding first n-l (covariant) derivatives on the second patch.

Now what frames are thus indistinguishable to what orders in which general relativistic theories? Any two frames in any model of any general relativistic theory are locally indistinguishable to first order, since any two vector fields on any manifold can be mapped to each other by local diffeomorphisms. Two frames will be locally indistinguishable to second order, however, only if the diffeomorphisms that relate them are local symmetries of the derivative operator (and metric) of the model. But, as mentioned before, there are general relativistic theories with models in which the metric tensor has no (non-trivial) symmetries. In such a model no frames are locally indistinquishable. So again, if this condition is used as a foundation for relativity principles, the general theory satisfies no relativity principle at all.

Well, how about an even more restricted condition? Take an "observer" to traverse a single, smooth, timelike curve on the space- time manifold, and consider some other similarly represented observer who crosses the path of the first at some point. Take them to be distinguishable at their point of intersection only if they are "trans- formationally identical" to all orders at the point. That is,

R3 Given a theory, a model of the theory, and two timelike curves defined on the model manifold intersecting at some point p, the two curves are indistinguishable to order n at p only if there is a diffeomorphism of the manifold onto itself that, at p, takes the 4- velocity vector of one curve and its first n-l derivatives to the 4-velocity vector and its corresponding first n-i derivatives of the other curve.

Again, what curves are equivalent at their intersection points to what orders under what conditions? Well, any two curves through a point of any model of any theory are equivalent to first order: one can always find a diffeomorphism of the manifold that at the intersection point maps one 4-velocity to the other. And all curves which are inertial at a point will be indistinguishable to all orders: their higher derivatives all vanish. In general, however, no arbi- trarily chosen curves through a point will be equivalent at second or higher orders. (Certainly no inertial curve could be indistinguishable at a point of intersection to one with non-vanishing acceleration.)



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And what of the principles of relativity? Obviously, all general relativistic theories satisfy the special principle under this con- strual: all trajectories through any point of any model that are inertial at the point are unqualifiedly indistinguishable at the point. But again, the general principle of relativity is not satisfied for any theory.

Finally, one might araue that this whole notion of "transforma- tional identity to all orders", whether applied to reference frames locally, or to individual worid lines at a point, is again just the wrong kind of notion to mandate as necessary for "dynamical indistinguishability of observers". Consider again two observers represented by smooth, timelike curves defined on some region of the manifold of a model of a theory, who meet at a point. Let each be equipped with what is, in-principle measurements aside, the favored relativistic instrument--an accelerometer. What they will do to determine their relative dynamical relationship is simply to compare accelerometer readings. An appropriate comparison will surely lead to agreement that they are "relatively non-accelerating" at their common point. Let's try this relationship as a necessary condition for their dynamical indistinguishability in the theory.

As it turns out, the spelling out of just what constitutes an appropriate comparison of accelerometer readings in general is a little tricky. It's not, of course, in the special case of Newtonian theories. Because of the preferred slicing of Newtonian space-time conferred by the universal time function, two observers at a point would agree on spatial relationships, and they could simply compare their accelerometer bob displacements. If they were identical--in magnitude and direction--the two observers would surely be entitled to declare themselves relatively non-accelerating. Obviously, all inertial observers at a point are relatively non-accelerating in this way, so all Newtonian theories satisfy the special principle of relativity.

The problem for relativistic theories, of course, is that observers whose world lines intersect, but whose 4-velocities are different at the point of intersection,will not agree on spatial orientations at the point. Thus, although the magnitude of their acceleration vectors might be the same, and they might experience the same "accelerative effects", their (spatial) acceleration vectors will not be identical. What is needed is a broadening of the Newtonian account of relative non-acceleration of trajectories suitable in the relativistic context.

Two considerations point strongly to the proper account. First, of course, the magnitudes of acceleration vectors (their Lorentz norms) must be identical. Second, surely the only candidates for relatively non-accelerating trajectories at a point are those whose velocities differ in magnitude there, but not in direction, those, that is, whose velocity vectors are oriented within a single (timelike) plane in space-time, or, more precisely, those for which the planes determined by velocity vector-acceleration vector pairs at a point coincide.



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Only such a set of trajectories, intuitively, could reduce to the corresponding set on Newtonian space-time.. .as the light cones of relativistic space-time flatten out to produce planes of absolute simultaneity, and the (equal magnitude) acceleration vectors concom- itantly collapse on top of one another to a common acceleration vector on Newtonian space-time.6

The property of velocity vector-acceleration vector pairs at a point that picks out such an equivalence class was noticed by Abhay Ashtekar at the University of Chicago several years ago.7 Each such pair, as I said, determines a timelike plane which intersects the light-cone at the point in two null rays. The appropriate equivalence class of velocity-acceleration vector pairs are simply those for which these null rays (or either one, it doesn't matter) are identical. So, succinctly,

R4 Two timelike trajectories defined on the manifold of a relativistic model and intersecting at some point on the manifold are relatively non-accelerating at their point of intersection if the Lorentz norms of their acceleration vectors are identical and they determine the same null rays.

As a foundation for a relativity principle, this condition on trajectories intersecting at a point obviously gives the same results as that considered previously: general relativity satisfies the special principle of relativity (trajectories inertial at a point are obviously relatively non-accelerating), but no general relativistic theory satisfies the general principle.

4. Other Intuitions

Let me sum up what I have said here so far. The fundamental intuition that underlies relativity principles in "classical" con- texts, that of the "active indistinguishability" of reference frames in states of uniform motion, is very difficult to apply in general relativistic theories in general. And even for those theories in which it might be applied, it certainly does not lead to a generalization of the special principle of relativity. Far from leading to a generalization of indistinguishability to include arbitrary states of motion then, one might better say that no states of motion would be counted as indistinguishable in the general case. And while certain technical problems in the formalization of active indistinguishability are avoided by a criterion of local indistinguishability within a model of a theory (a kind of "passive indistinguishability"), the results are the same: in general, no states of motion in general relativistic theories are (locally) indistinguishable. Only with an infinitesimal criterion does one find indistinguishability for arbitrary models of arbitrary general relativistic theories, and that is indistinguishability only for trajectories inertial at a point. That is, general relativistic theories satisfy the special principle of relativity infinitesimally, but certainly not the



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general principle.

But isn't this somewhat technical result fundamentally at odds with important intuitions about relativity principles? Actually, it is not at all so. There are certainly no observations of the sort that Galileo describes, gedanken or not, that suggest that unaccelerated states of motion are physically indistinguishable from accelerated states. No one doubts, for instance, that should Galileo's ship have been accelerated out to sea on a rip tide, or have entered a whirl- pool, the trajectories of butterflies, and the dynamical state of the observer, would have been markedly different.

But wait. What about Einstein's famous elevator? Remember the observer enclosed in an elevator, equipped, say, with an accelerometer? Surely such an observer in a stationary elevator acted on by a gravitational field will have the same accelerometer reading as one "really accelerated" in an appropriate way in field-free space-time. Isn't this a sense in which "unaccelerated motion is indistinguishable from accelerated motion"?

It is certainly a sense, and an important sense. Indeed, an analogous gedanken observation was used by Einstein (1916, p. 114) to argue for the necessity of a theory satisfying the general principle of relativity. One should be clear, though, about the difference between the intuitive content of this observation and that of Galileo's. Einstein is not claiming the indistinguishability of an observer waiting in an elevator on the ground floor of this hotel and one accelerating upward to his room. Nor does he make this claim for an elevator occupant stranded in field-free space and one therein accelerating off to an important appointment. The comparison is between the accelerating observer in field-free space and the stationary observer in the presence of the field. This is a comparison between models in two different theories in a sense, rather than between models within the same theory.

This sort of comparison is part of the intuitive foundation of the equivalence principle which I mentioned at the beginning of the paper. And though the equivalence principle itself might be invoked as an argument for a general principle of relativity, it really has to do with different features of space-time theories. In its "strong" form, the equivalence principle declares that all matter fields on space-time respect the geometrical structure associated with the gravitational field. This, as far as has been determined, is true in the general theory of relativity (Misner, Thorne, and Wheeler 1973, sec. 38.6). And it is why one cannot tell which one of Einstein's elevators one is in. For every effect (or observable) in the "really accelerating" elevator--certainly the accelerometer reading, or, for instance, the deflection of the paths of light rays from "straight line' behavior--would be duplicated in the stationary elevator in the presence of the gravitational field: light respects the curvature of space-time associated with the gravitational field. But one must be careful. As a general claim about all possible observa-



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tions--not just, say, accelerometers and light--it is important that this equivalence claim be understood only infinitesimally; one must consider "elevators at a point", or "along a world-line", as it were. And this is the sense of the formulation of the equivalence principle that says "locally--really infinitesimally--general relativity has the same structure as special relativity"; space-time can be treated as flat at a point (Misner, Thorne, and Wheeler 1973, p. 386).

A part of this claim was manifested by my infinitesimal formaliza- tions of the indistinguishability of observers: trajectories inertial at a point in models of general relativistic theories are indistinguishable; general relativity satisfies the special principle of relativity infinitesimally. And the "observation" bearing most particularly on this aspect of the principle of equivalence is simply that of a pair of elevators converse to those just considered--one stranded in field-free space, the other freely-falling, pity the poor observer, from the 10th floor of the hotel to the lobby. Along most of their trajectories--taken as single world-lines in their respective space-times, these elevators are indistinguishable by all possible observations.

There is one final "observation" that ought to be mentioned here, for it too led Einstein to a belief in the need for an extension of the special principle of relativity. Einstein (1916, p. 112) pictured two fluid masses rotating with constant relative angular velocity about the line joining them. Then he supposed one of them, based on measurements undertaken in its own rest frame, to be ellipsoidal in shape, while the other was judged on a similar basis to be spherical. How, he then asked, could this difference in shape be explained? Certainly not by an appeal to a privileged state of rest--a rest frame, that is, with respect to which the spheroidal mass is at rest. For that would invoke an unobservable entity as a "causal agent", an account that Einstein rejected as "epistemologically unsatisfactory".

Einstein drew two conclusions from this rejection of a "privileged state of rest". First, the explanation of the difference in shape of the two masses must lie in their relation to some third observable entity--some other physical object or class of physical objects, like "distant masses". This notion, that local inertial effects are, in some sense, "caused by" the gross mass distribution of the universe is part of the intuitive content of the Mach's principle I mentioned earlier, and will not mention again. But the second conclusion that Einstein drew was that no reference frame could be treated as privileged for any explanatory purpose within physics, which is to say, according to Einstein, that "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion." (1916, p. 113). That, indeed, is the general principle of relativity.

But something is wrong here. For, as we have seen, a true general principle of relativity would declare the two states of motion represented by Einstein's two spheres indistinguishable: no differential



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effects could arise between bodies in relatively accelerated motion, e.g., rotation, any more than they arise between bodies moving with constant (linear) velocity. That is, far from being associated with an intuitive foundation for a general principle of relativity, Einstein's imagined spheres are inconsistent with one.

In fact, it appears to me that Einstein was just basically confused about the relation of these "Machian" ideas, those associated with relativity principles, and those I referred to earlier associated with the principle of general covariance. Of course, sorting these intuitions out in the absence of a mathematically well-defined theoretical framework was far from a simple matter. And Einstein's struggles to incorporate them in such a framework, as described, for example, by Earman and Glymour (1978), surely constitute one of the truly engros- sing episodes in the history of natural philosophy.

But for now I just reiterate my former claim. There are no observations anything like the classical sort Galileo describes which suggest that unaccelerated states of motion are indistinguishable from accelerated states. The general principle of relativity, understood in this way, is false in the real world; and it is false in the general theory of relativity. The general theory is not generally relativistic, in this sense.

5. Epilogue

I would like to close with an even stronger claim, if you can imagine it, one suggested to me recently by David Malament. Recall from the discussion of an "intra-model" criterion for indistinguishability of observers that observers are indistinguishable at first order. Their distinguishability emerges at second, and of course higher, orders.9 That is, observers are distinguishable in terms of properties of their trajectories expressed by their second derivatives. Now the forces an observer feels, the vector-valued readings of an accelerometer, are identified in general relativity (and in certain formulations of Newtonian gravitation theory) with a second-order property of an observer world-line--its curvature. And this identi- fication is non-trivial. It relates, in "observational terms", space and time measurements--those needed to determine the metric structure of space-time and hence the curvature of a world-line--to acceleration measurements--read off an accelerometer. And there are theories of gravitation in which it does not hold, theories which posit the existence of some other tensor field on space-time which combines with the curvature of a world-line to produce this physically measured acceleration.

But the details of this relationship between "geometry" and "dynam- ics" are not important here. Nor, for purposes of this point, is the order at which this relationship is posited to hold. It might be only



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a property of an observer trajectory at 23rd order that is claimed to bear some relation to an observer's "dynamical state". But in any theory that could possibly be characterized as a dynamical theory, a theory that accounted for the dynamical properties of physical systems, some such relationship would have to hold.

And yet it is precisely such a relationship that the general principle of relativity rejects. All states of motion, all trajectories on a space-time manifold, must be dynamically indistinguishable. Since such trajectories clearly do differ in curvature, or in higher order properties, there must be no relation between properties of an observer trajectory at any order, and the dynamical state of an observer.

It follows that no theory which could be considered a "dynamical theory" could possibly satisfy the general principle of relativity, understood in this way, Let me put this another way: no system of ideas that satisfied this general principle of relativity could be considered a "dynamical theory". In fact, it's hard to think of such a system of ideas as a "theory" of anything. Seen from this per- spective, this general principle of relativity is almost worse than false: it expresses a requirement that is almost incoherent,

Notes

Most of the ideas in this paper arose in conversations with Robert Geroch and David Malament. Discussions with John Earman and Michael Friedman have also been helpful. An earlier version of this paper was read at Memphis State University (Jones 1981). This work was supported by the National Science Foundation under Grants SOC77-07264 and SES79-26725.

2lichael Friedman (1982) discusses at length the roles of the principles associated with relativity theory. Their historical roles are treated carefully in Earman and Glymour (1978).

The approach I take here follows closely, and was in fact motiva- ted by, that of Earman (1974).

Of the principles associated with relativity theory, it is perhaps Mach's principle about which there is most disagreement. See, for example, Goenner (1970).

This approach was pioneered by Cartan (1923, 1924), It serves as the foundation for most modern textbook treatments, as in Misner, Thorne, and Wheeler (1973, esp, Ch. 12).

6 The passage from a relativistic space-time to its "Newtonian

limit" is described rigorously in KInzle (1976) and Malament (1982).



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Ashtekar has not published this construction, but it is described in somewhat more detail in Jones (1981).

8 This gedanken observation is first referred to in Einstein and

Grossman (1913), in a passage translated in Misner, Thorne, and Wheeler (1973, p. 431). The Einstein and Grossman paper is discussed extensively in Earman and Glymour (1978).

See Adler (1980) for a recent discussion of the significance of this fact.



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References

Adler, C.G. (1980). "Why Is Mechanics Based on Acceleration?" Philosonhy of Science 47: 146-152.

Cartan, E. (1923). "Sur les Variet6s a Connexion Affine et la Theorie de la Relativite Generalisee (premiere partie)." Annales Scientifiaues de L'Ecole Normale SuDerieure 40: 325-412.

---------. (1924). "Sur les Varietes a Connexion Affine et la Th6orie dejla Relativite Generalisse (suite)." Annales Scientifiaue de L1'Ecole Normale SuDerieure 41: 1-25.

Earman, J. (1974). "Covariance, Invariance, and the Equivalence of Frames." Foundations of Physics 4: 267-289.

--------- and Glymour, C. (1978). "Lost in the Tensors: Einstein ' s Struggles with Covariance Principles 1912-1916." Studies in History and Philosonhv of Science 9: 251-278.

Einstein, A. (1916). "Die Grundlage der allgemeinen Relativitatsthe- ories." Annalen der Physik 49: 770-822. (As reprinted and translated in Perret, W. and Jeffrey, G.B. (1952). The Principle of Relativity. New York: Dover. Pages 111-164.)

----------- and Grossman, M. (1913). "Entwurf einer verallgemeinerten Relativit'atstheorie und einer Theorie der Gravitation." Zeitschrift fUr Mathematik und Physik 62: 225-261.

Friedman M. (1982). Foundations of Space-Time Theories. Princeton: Princeton University Press. In Press.

Galileo, G. (1632). Diologo. Florence: Batista Landini. (Translated as DialoRue Concerning The Two Chief World Systems (trans.) Stillman Drake. Berkeley and Los Angeles: University of California Press, 1962.)

Goenner, H. (1970). "Mach's Principle and Einstein's Theory of Gravitation." In Ernst Mach, Physicist and PhilosoDher. (Boston Studies in the PhilosoDhy of Science, Volume VI.) Edited by R.S. Cohen and R.J. Seeger. Dordrecht: D. Reidel. Pages 200-215.

Jones, R. (1981). "The Special and General Principles of Relativity: In After Einstein: Proceedings of the Einstein Centenary Con- ference. Edited by P. Barker and C.G. Shugart. Memphis: Memphis State University Press.

Kunzle, H.P. (1976). "Covariant Newtonian Limit of Lorentz Space-Times." General Relativity and Gravitation 7: 445-457.

Malament, D. (1982). "Newtonian Gravity, Limits, and the Geometry of Space." Forthcoming in Pittsburgh Studies in the PhilosoDhy of Science,.



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Messiah, A. (1960). Mecanioue Quantiaue. Paris: Dunod. ( As translated as Quantum Mechanics, Vol. 1. (trans.) G.M. Temmer. New York: Wiley, 1966).

Misner, C., Thorne, K., and Wheeler, J. (1973). Gravitation. San Francisco: W.H. Freeman.



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