The Spacetime Hole and the EPR Arguments

There a fascinating parallel between the “hole” argument used by Einstein in an effort to prove the impossibility of generally covariant equations providing a complete description of the spacetime metric, and the “entanglement” scenario used by Einstein (with Podolsky and Rosen) in an effort to prove that the wave function of quantum mechanics cannot be a complete description of physical systems. It’s interesting to speculate on what he might have concluded if he had applied to the latter argument the same reasoning that led to the resolution of the former.

Recall that, in 1913, Einstein put forward an argument which he believed (at the time) proved that the field equations of gravitation could not possibly be generally covariant. This was his famous “hole” argument, so called because it was based on the assumed existence of a region of spacetime perfectly devoid of mass-energy and stress, so the stress-energy tensor T_μν equals zero throughout this “hole” region, while being non-zero in other regions of spacetime. In brief, his argument was that, for any given T_μν and metric field g_μν satisfying the posited generally covariant field equations f(g_μν,T_μν) = 0, we could apply an arbitrary diffeomorphic transformation δ to both the metric and the stress-energy tensor within the hole region (merging smoothly with the identity transformation outside the hole region), and the field equations f(δ(g_μν),δ(T_μν)) = 0 would still be satisfied. However, in the hole region where the tensor equation T_μν = 0 applies, we also have δ(T_μν) = 0, and so the overall stress-energy tensor field is unchanged, even though we apply an arbitrary diffeomorphic transformation to the metric field. Thus we have f(δ(g_μν),T_μν) = 0. Now, Einstein argued (in 1913) that this is physically unacceptable, because it signifies that a given distribution of stress-energy corresponds to multiple distinct metric fields. Later he realized that there is nothing unacceptable, nor even surprising, about this lack of uniqueness, because the transformation represents a change of coordinate systems, accompanied by the corresponding changes in metric coefficients, but this doesn’t affect the observable attributes of spacetime. We have four continuous degrees of freedom in the choice of coordinates, so it’s hardly surprising that any given physical situation can be described in terms of many different coordinate systems, each of which will have its own field of metric coefficients.

Of course, we don’t really need the complicated “hole” construction to make this case. We can simply point out that, in a completely “empty” Minkowski spacetime, one can establish infinitely many different systems of coordinates, including various Cartesian and polar coordinate systems, each with their own corresponding metric fields, but all describing exactly the same flat Minkowski spacetime. Admittedly in the early years of general relativity Einstein was inclined to reject the possibility of a totally empty solution of the field equations (he argued with de Sitter over this very possibility), but one would have thought that this example would at least dispel any surprise over the existence of multiple possible metric fields (due to the freedom of choice for coordinate systems) for a given spacetime. Nevertheless, in 1913, Einstein convinced himself (largely on Machian grounds) that any non-degenerate solution of the field equations must contain some mass-energy, and that the metric field resulting from a given distribution of mass-energy must be unique. If this is accepted, then the only way of blocking the “hole” argument is to limit the allowable transformations to linear transformations (rather than arbitrary diffeomorphisms). In other words, his rationale was that the “hole” argument relies on non-linear transformations to enable the smooth transition from outside to inside the hole, which is presumably surrounded on all sides by mass-energy. The restriction to linear transformations thus (according to Einstein in 1913) preserves the uniqueness of the metric field associated with any given distribution of mass energy.

As soon as Einstein succeeded (in 1915) in finding generally covariant field equations, and understood that they do in fact reduce to Newton’s law in the first approximation (contrary to what he and Grossmann had concluded in 1913), he returned to the “hole” argument and realized his mistake, which was the failure to recognize that the degrees of freedom in the metric field correspond to the freedom in choosing the coordinate system. Hence there is nothing “unphysical” about the fact that the same observable situation can be represented by distinct metric fields, just as there is nothing “unphysical” about the fact that we can choose distinct coordinate systems. To support this view, Einstein echoed a positivist comment of Kretschmann, who had noted that the only observables in physics are expressible as spacetime coincidences (i.e., as interactions between entities), and these coincidences – as well as the proper intervals between them – are preserved under any diffeomorphic transformation of the coordinates. Thus the “proof” of the impossibility of general covariance was unfounded.

It’s worth noting that the “hole” argument is perfectly correct in it’s implication that, under the assumption of general covariance, a given physical situation (i.e., a given distribution of mass-energy) possesses multiple formally distinct representations. The error was in the conclusion that this fact violates some principle of objective physicality. The principle in question was essentially determinism. Einstein’s firm belief (at that time) was that the distribution of mass-energy must uniquely determine the field. He mistakenly construed the distinct descriptions of a field as actually distinct fields, forcing him to conclude that determinism could not be maintained under the assumption of general covariance. Hence he rejected (“with a heavy heart”) general covariance – until he was able to disentangle the fields from their descriptions.

Twenty years later, in 1935, Einstein (in collaboration with Boris Podolsky and Nathan Rosen) put forward another impossibility proof, this one aimed at showing that the wave function of quantum mechanics cannot be considered as providing a complete representation of objective physical systems. There is a remarkable formal similarity between this so-called “EPR” argument of 1935 and Einstein’s “hole” argument from twenty years earlier. In simple terms, We are asked to consider a system S consisting of two space-like separated components S₁ and S₂. Let ψ denote the wave function representing the overall system. Now, according to quantum mechanics, if a “complete” measurement is performed on the component S₁, the component S₂ can be represented by a wave function ψ₂ that depends only on the original wave function ψ and the results of the measurement on S₁. The objectionable point, according to Einstein et al, is that there are multiple distinct “complete measurements” that one can choose to perform on S₁, and these lead to multiple distinct wave functions ψ₂ for the space-like separated component S₂.  Einstein summarized his conclusion in his Autobiographical Notes

On one assumption we should, in my opinion, insist without qualification: the real state of the system S₂ is independent of any manipulation of the system S₁, which is spatially separated from the former. However, depending on the type of measurement we perform on S₁, we get a very different wave function ψ₂ for the second partial system. Thus for the same real state of S₂ it is possible to find different types of wave functions ψ₂. If we accept this reasoning as valid, we must give up the position that the wave function constitutes a complete description of a real state, because if it did, it would be impossible for two distinct wave functions to be assigned to the same real state.

The similarity of this argument to the “hole” argument of 1913 is obvious. In both cases, there is presumed to be a single objective physical entity that is characterized by some formal representation (the metric field or the wave function), and in both cases we have a simple chain of reasoning leading to the conclusion that multiple distinct formal representations can be assigned to the single physical entity. Carrying the analogy still further, we might say that in both cases the multiplicity of representations corresponds to a choice of measurements. In the case of spatio-temporal relations we have considerable freedom of choice as to how we parameterize those relations, and, in effect, we select a type of (implicit) measurement when we choose a system of coordinates. This is somewhat comparable to the choice of measurements that we might perform on a physical system such as S₁.

Einstein acknowledged two possible “escapes” from the EPR argument. One possibility is to assert that (contrary to his fundamental assumption) the real state of S₂ is altered “telepathically” (to use Einstein’s word) by manipulations of S₁ at a space-like separated location. The other is to deny that spatially separate systems such as S₁ and S₂ possess independent real states. This second possibility seems to be a variation of the first, since it implies some telepathic unity between separate systems. Not surprisingly, he wrote that “both alternatives appear to me entirely unacceptable.” Oddly enough, he didn’t mention the fact that, in a completely deterministic universe, there is no “free choice” as to the measurement performed on S₁, nor any other free choice, so there is always a unique wave function for every real state (or at least we can’t possibly prove that there isn’t). It’s especially surprising that Einstein neglected to mention this alternative, considering that the unified spacetime and strict determinism of classical general relativity is already committed to the “block universe” model, according to which all places and times must claim equal existence. If, as he wrote to his friend Max Born, God does not throw dice, then we have Laplacian determinism, and counter-factual “what ifs” have no meaning.

In a sense, Einstein’s dilemma can be traced back to the dichotomy in physics that comes about because we mix partial and total differential equations. The physics of particles is expressed in terms of total differential equations, with derivatives evaluated along the specific identified paths of the individual particles. All the values and rates of change are actually realized in the behavior of the particles. In contrast, fields and their governing laws are expressed in terms of partial differential equations, which involve “transverse” derivatives over the putative continuous spacetime manifold – this manifold being (at least arguably) a purely hypothetical entity. Einstein always viewed the mixing of these two modes of expression (not to say modes of existence) as highly unnatural. Nature, he believed, ought to be one or the other, and after some ambivalence in his early years, he firmly settled on the field view of physics, with its partial differential equations. Thereafter his goal was always to eliminate total differential equations, and reduce everything to a continuous singularity-free field. But this automatically committed him to a program that was inherently based on transverse “what if” partial derivatives. The hypothetical spacetime manifold has infinitely many more degrees of freedom than even a countably infinite set of discrete particles, so it is a world overflowing with free choices. Einstein worked hard to find some way of imposing “over-determination” onto the field, in hope that this would lead to particle-like solutions, but never really came close to success. In retrospect, his convictions about absolute determinism might have had a better chance of finding their full expression in the context of a field-free theory of retarded action at a distance between discrete particles, such as in the classical electrodynamics of Weber and Wheeler/Feynman. (Of course this is a “what if” proposition of the kind that is eschewed by these approaches!)

In any case, it’s interesting to consider whether the resolution of the spacetime “hole” argument has any implications for the EPR argument. A straightforward analogy would lead us to ask if the different wave functions ψ₂ really are describing intrinsically distinct states, or if they might be, in some sense, all describing the same real state, differing only in a formal way that has to do with our choice of coordinates. There is definitely a sense in which the real state of S₂ does not change when the selected measurement is performed on S₁, because nothing in the subsequent behavior of S₂ provides any information as to the measurement performed on S₁. This is known to be true in general: No spacelike conveyance of information (or energy) is entailed by the dependence of ψ₂ on the measurement of S₁. Thus, at least on this superficial level, we can indeed say that the EPR argument suffers from the very same fallacy as the “hole” argument, namely, the failure to realize that the multiple formally distinct wave functions (or metric fields) arising from different choices of the measurement basis do not represent intrinsically distinct real states. The differences are extrinsic, related to the chosen basis of measurement.

For this analogy to be valid on a deeper level, the choice of a spacetime coordinate system within the “hole” must correspond with the choice of a complete measurement of S₁. Here the analogy might seem to break down, because a complete quantum measurement is irreversible, whereas it isn’t clear whether a spacetime measurement is equally irreversible. To clarify this, we need to distinguish between the hypothetical consideration of possible measurements and the actual performance of a measurement. In the case of the “hole” argument, we don’t contemplate actually measuring anything, yet we feel free to talk about different coordinate systems and the metric fields that would correspond to them. The reason this seems to “reversible” is that we aren’t really doing anything, we are just considering various possibilities and their consequences. If we took this same approach to the EPR situation, we would simply note the different wave functions ψ₂ that would correspond to various choices and outcomes of measurements on S₁. Just as in the “hole” argument, the intrinsic state of S₂ is the same for all these choices, whereas (again, just as in the “hole” argument) the extrinsic state of S₂, relative to S₁ in particular, is different for each choice.

To be more precise, the way in which S₂ is correlated with S₁ seems to be affected by the complete measurement of S₁, and we can just as well say that the way the metric field is correlated with our measurements of space and time is affected by our (complete) measurement of a system of space and time coordinates. Admittedly by speaking in such generic terms we gloss over some distinctions that could be drawn between the two arguments. For example, it’s not clear what would constitute a “complete” measurement corresponding to the actual establishment of a space and time coordinate system. Also, we haven’t identified spacelike separated events in the “hole” argument. Instead, we have simply asserted (tacitly) that the establishment of a space-time coordinate system in a given region does not affect the “real” state of the metric field in that region (let alone any other region). This corresponds to the claim that a measurement of S₁ has no effect on the real (intrinsic) state of S₂.

Presumably none of this would have changed the minds of any of the participants in the debates over the spacetime “hole” and the EPR arguments, but it’s interesting, if only from a psychological perspective, for the insight it gives into the patterns in Einstein’s thought. These two arguments are by no means the only examples we could cite. Throughout his career he repeatedly critiqued various theoretical frameworks by identifying what he was convinced must be a single physical state or process, and then pointing out that this state or process had multiple distinct representations within the given theoretical framework. Consider his 1905 critique of electrodynamics:

It is well known that Maxwell's electrodynamics – as usually understood at present – when applied to moving bodies, leads to asymmetries that do not seem to be inherent in the phenomena. Take, for example, the electrodynamic interaction between a magnet and a conductor. The observable phenomenon here depends only on the relative motion of conductor and magnet, whereas the customary view draws a sharp distinction between the two cases, in which either the one or the other of the two bodies is in motion…

Even more striking, in a 1919 essay on the origins of special relativity (Fundamental Ideas and Methods of Relativity Theory, Presented in their Development), he recalled

The thought that one is dealing here with two fundamentally different cases was for me unbearable. The difference between these two cases could not be a real difference but rather, in my conviction, only a difference in the choice of the reference point… The phenomenon of electromagnetic induction forced me to postulate the special relativity principle.

Perhaps the greatest and most profound application of this peculiarly Einsteinian mode of thought was what he called “the happiest thought of my life”, when he drew from the coincidental equality of inertial and gravitational mass the conviction that, once again, the difference between the two could not be real, but must be only due to a different frame of reference. In Newtonian theory an object far from any gravitating body is in a very different intrinsic state than an object in free fall near a large gravitating body, but Einstein was convinced that they were intrinsically the same, and he sought a theoretical framework that would make this equivalence explicit.

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