Equivalence principles played a central role in the development of general relativity. Furthermore, they have provided operative procedures for testing the validity of general relativity, or constraining competing theories of gravitation. This has led to a flourishing of different, and inequivalent, formulations of these principles, with the undesired consequence that often the same name, “equivalence principle,” is associated with statements having a quite different physical meaning. In this paper, we provide a precise formulation of the several incarnations of the equivalence principle, clarifying their uses and reciprocal relations. We also discuss their possible role as selecting principles in the design and classification of viable theories of gravitation.

1.

There are notable exceptions. Some authors prefer to replace heuristic principles by sound mathematical postulates, on the ground that the former are not well-defined. A champion of this attitude was the Irish mathematical physicist John Lighton Synge, who wrote with characteristic wit: “When, in a relativistic discussion, I try to make things clearer by a spacetime diagram, the other participants look at it with polite detachment and, after a pause of embarrassment as if some childish indecency had been exhibited, resume the debate in their own terms. Perhaps they speak of the Principle of Equivalence. If so, it is my turn to have a blank mind, for I have never been able to understand this Principle. Does it mean that the signature of the space-time metric is +2 (or −2 if you prefer the other convention)? If so, it is important, but hardly a Principle. Does it mean that the effects of a gravitational field are indistinguishable from the effects of an observer's acceleration? If so, it is false. In Einstein's theory, either there is a gravitational field or there is none, according as the Riemann tensor does not or does vanish. This is an absolute property; it has nothing to do with any observer's world-line. Space-time is either flat or curved…. The Principle of Equivalence performed the essential office of midwife at the birth of general relativity, but, as Einstein remarked, the infant would never have got beyond its long-clothes had it not been for Minkowski's concept. I suggest that the midwife be now buried with appropriate honours and the facts of absolute space-time faced.” See J. L. Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1960), pp. IX–X. Although this point of view is logically possible for a full-fledged theory like general relativity, we argue in this paper that the equivalence principle could still play an important role when building alternative theories of gravity.

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As is common in classical mechanics, by “particle” we mean a body whose extension is irrelevant for what concerns its dynamical behavior. Moreover, we exclude the possibility that the mentioned particle may carry any spin, quadrupole moment, or any other higher multipole moment whatsoever.

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For example, r could be defined as the radius of the smallest sphere containing the body, evaluated as r = (A/4π)1∕2, where A is the area of the sphere.

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In practice, for each finite resolution of the measuring device, there is a size below which tidal effects become indistinguishable from noise. This, however, does not make EEP valid—not even “for all practical purposes.” The correct statement is that, for experiments performed at a given size in a gravitational field, there is always a sufficiently high resolution that allows one to detect tidal effects. This is, of course, true if spacetime curvature at an event has to be an observable quantity.

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The validity of local Lorentz invariance has been recently questioned in the context of quantum theories of gravity. Investigations of possible departures from exact relativistic behavior at high energies have been proposed as a relic of the existence of new physics at the Planck scale. For a comprehensive review on tests of Lorentz violations see, e.g.,
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Incidentally, this implies that there is no need to include the WEP explicitly in the statement of EEP. The condition that a gravitational field be locally eliminable (usually implemented through the requirements of local position invariance and local Lorentz invariance) is entirely sufficient.

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Strictly speaking, one should distinguish between path, projective, and affine structures on spacetime. Universality of free-fall only defines a path structure, and the identification of an affine connection requires further hypotheses and labor. We believe, however, that such technical niceties are not appropriate for our heuristic discussion. For a rigorous approach to the issue, see, in particular:
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It should also be pointed out, however, that the issue of whether gravity is a manifestation of physics or of geometry is, at this stage, not amenable to experimental verification, and might then not be very relevant. As repeatedly stressed by Poincaré, whether a universal behavior is due to physics or to geometry is ultimately a matter of convention. And perhaps it is simply wrong to regard “geometry” as something intrinsically different from “physics”: if one defines “geometry” as the “physics with a universal character” (but still able to make experimentally verifiable predictions), the subject of the debate becomes just a semantic issue.

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Of course, this is applicable only to physical laws that can be formulated in an entirely local way, so one must be very cautious when dealing, e.g., with quantum field theory in a curved spacetime, whose formulation relies on the global notion of a quantum vacuum. For an apparent violation of EEP within this context, see
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For example, Nordström's scalar theory of gravity, which contains only one field equation for a relativistic gravitational potential, is too simple for that. The equation of motion for test particles in such a theory can—and must—be postulated independently, exactly as it happens in Newton's theory. This can be traced to the fact that this theory contains a non-dynamical object: the Minkowski metric. In scalar theories of gravity, the WEP and the GWEP constrain only the equation of motion, not the field equations.

38.
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In Einstein's general relativity, κ = 8πG/c4 and $jabc=∇ bTac−∇ aTbc+12(gbc∇ aT−gac∇ bT)$, where Tab is the stress-energy-momentum tensor and T = gabTab is its trace.