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.

2.
More generally, a survey of the conceptual difficulties encountered by students when learning about the equivalence principle is found in
A.
Bandyopadhyay
and
A.
Kumar
, “
Probing students' ideas of the principle of equivalence
,”
Eur. J. Phys.
32
,
139
159
(
2011
).
3.
T. P.
Sotiriou
,
V.
Faraoni
, and
S.
Liberati
, “
Theory of gravitation theories: A no-progress report
,”
Int. J. Mod. Phys. D
17
,
399
423
(
2008
); e-print arXiv:0707.2748[gr-qc].
4.
D. B.
Malament
,
Topics in the Foundations of General Relativity and Newtonian Gravitation Theory
(
University of Chicago Press
,
Chicago
,
2012
), Chap. 4.
5.
K. S.
Thorne
,
D. L.
Lee
, and
A. P.
Lightman
, “
Foundations for a theory of gravitation theories
,”
Phys. Rev. D
7
,
3563
3578
(
1973
).
6.
C. M.
Will
,
Theory and Experiment in Gravitational Physics
, Revised ed. (
Cambridge U.P.
,
Cambridge
,
1993
).
7.
Usually, this happens for weak gravitational fields and slow motions. A notable exception to this rule is Bekenstein's tensor-vector-scalar theory of gravity (TeVeS), which reduces to Milgrom's modified Newtonian dynamics (MOND). However, the deviations of MOND from Newton's theory are relevant only for very small accelerations, and there is still a wide regime where Newtonian dynamics apply. For an accessible account of both theories, and for their motivations, see
J. D.
Bekenstein
, “
The modified Newtonian dynamics—MOND and its implications for new physics
,”
Contemp. Phys.
47
,
387
403
(
2006
); e-print arXiv:astro-ph/0701848.
8.

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.

9.
M.
Berry
,
Principles of Cosmology and Gravitation
(
Cambridge U.P.
,
Cambridge
,
1976
), pp.
27
28
.
10.
Although the identity between mi and mg was taken for granted in the two centuries separating Newton and Einstein, the implausibility of such a “coincidence” did not escape the scrutiny of a deep thinker like Hertz. See
J. P.
Blaser
, “
Remarks by Heinrich Hertz (1857–94) on the equivalence principle
,”
Class. Quantum Grav.
18
,
2393
2395
(
2001
).
11.
I.
Newton
,
Philosophiæ Naturalis Principia Mathematica
(
Streater
,
London
,
1687
), pp.
408
409
.
12.
A. M.
Nobili
 et al, “
On the universality of free fall, the equivalence principle, and the gravitational redshift
,”
Am. J. Phys.
81
,
527
536
(
2013
).
13.

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.

14.
E. Di
Casola
,
S.
Liberati
, and
S.
Sonego
, “
Weak equivalence principle for self-gravitating bodies: A sieve for purely metric theories of gravity
,”
Phys. Rev. D
89
,
084053-1
16
(
2014
); e-print arXiv:1401.0030[gr-qc].
15.

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.

16.
H. C.
Ohanian
, “
What is the principle of equivalence?
,”
Am. J. Phys.
45
,
903
909
(
1977
).
17.
A.
Papapetrou
, “
Spinning test-particles in general relativity. I
,”
Proc. Roy. Soc. London A
209
,
248
258
(
1951
).
18.
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.,
S.
Liberati
, “
Tests of Lorentz invariance: a 2013 update
,”
Class. Quantum Grav.
30
,
133001-1
50
(
2013
); e-print arXiv:1304.5795[gr-qc].
19.
B.
Bertotti
and
L. P.
Grishchuk
, “
The strong equivalence principle
,”
Class. Quantum Grav.
7
,
1733
1745
(
1990
).
20.
For instance, in Bohmian dynamics the equation of motion for a particle in the presence of a gravitational field is mix¨=mgΦQ, where Q is the so-called quantum potential, which contains mi but is not proportional to it; see, e.g., D. Bohm and B. J. Hiley, The Undivided Universe (Routledge, London, 1993). Then, the WEP is violated even if NEP holds. Actually, this is a particular case of a more severe violation, because even the behavior of particles that are not subject to forces other than Q (and which would then usually be regarded as free) turns out to depend on their mass. For details, see
S.
Sonego
, “
Is there a spacetime geometry?
,”
Phys. Lett. A
208
,
1
7
(
1995
).
For a critical review of the issue, see
E.
Okon
and
C.
Callender
, “
Does quantum mechanics clash with the equivalence principle—and does it matter?
,”
Eur. J. Philos. Sci.
1
,
133
145
(
2011
); e-print arXiv:1008.5192[gr-qc].
21.
K.
Nordtvedt
, “
Equivalence principle for massive bodies. II. Theory
,”
Phys. Rev.
169
,
1017
1025
(
1968
).
22.

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.

23.
A. P.
Lightman
and
D. L.
Lee
, “
Restricted proof that the weak equivalence principle implies the Einstein equivalence principle
,”
Phys. Rev. D
8
,
364
376
(
1973
).
24.
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:
J.
Ehlers
,
F. A. E.
Pirani
, and
A.
Schild
, “
The geometry of free fall and light propagation
,” in
General Relativity: Papers in Honour of J. L. Synge
, edited by
L.
O'Raifeartaigh
(
Clarendon Press
,
Oxford
,
1972
), pp.
63
84
,
republished in
J.
Ehlers
,
F. A. E.
Pirani
, and
A.
Schild
,
Gen. Relativ. Gravit.
44
,
1587
1609
(
2012
)
with an
editorial note
by A. Trautman, pp. 1581–1586;
R. A.
Coleman
and
H.
Korte
, “
Constraints on the nature of inertial motion arising from the universality of free fall and the conformal causal structure of space-time
,”
J. Math. Phys.
25
,
3513
3526
(
1984
);
G. S.
Hall
and
D. P.
Lonie
, “
Projective structure and holonomy in general relativity
,”
Class. Quantum Grav.
28
,
083101-1
17
(
2011
).
25.
H.
Reichenbach
,
The Philosophy of Space & Time
(
Dover
,
New York
,
1958
).
26.

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.

27.
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
S.
Sonego
and
H.
Westman
, “
Particle detectors, geodesic motion, and the equivalence principle
,”
Class. Quantum Grav.
21
,
433
444
(
2004
); e-print arXiv:gr-qc/0307040.
28.
S.
Sonego
and
V.
Faraoni
, “
Coupling to the curvature for a scalar field from the equivalence principle
,”
Class. Quantum Grav.
10
,
1185
1187
(
1993
).
29.
J.
Norton
, “
What was Einstein's principle of equivalence?
,”
Stud. Hist. Philos. Sci.
16
,
203
246
(
1985
).
30.
For a different opinion, see
G.
Muñoz
and
P.
Jones
, “
The equivalence principle, uniformly accelerated reference frames, and the uniform gravitational field
,”
Am. J. Phys.
78
,
377
383
(
2010
); e-print arXiv:1003.3022[gr-qc]. However, since the alleged “uniform gravitational field” in this reference is just Minkowski spacetime in disguise, one may object that it is not a gravitational field at all. For the same reason, the proof offered of the equivalence between a uniformly accelerated frame and a uniform gravitational field is invalid, because it relies on the fact that the same metric is first obtained within a special-relativistic context, and then as a particular solution of Einstein's field equations in vacuum. But such a solution is also required to satisfy the stronger supplementary condition Rabcd=0 in order to avoid singularities, so the coincidence between the two metrics is only to be expected.
31.
P. A. R.
Ade
 et al (Planck Collaboration), “
Planck 2013 results. XVI. Cosmological parameters
,”
Astron. Astrophys.
(forthcoming); e-print arXiv:1303.5076[astro-ph.CO].
32.
A. V.
Ramallo
, “
Introduction to the AdS/CFT correspondence
,” e-print arXiv:1310.4319[hep-th].
33.
T.
Padmanabhan
,
Gravitation: Foundations and Frontiers
(
Cambridge U.P.
,
Cambridge
,
2010
).
34.
R.
Infeld
and
A.
Schild
, “
On the motion of test particles in general relativity
,”
Rev. Mod. Phys.
21
,
408
413
(
1949
).
35.
J. N.
Goldberg
, “
The equations of motion
,” in
Gravitation: An Introduction to Current Research
, edited by
L.
Witten
(
Wiley
,
New York
,
1962
), pp.
102
129
.
36.
D. L.
Lee
,
L. P.
Lightman
, and
W.-T.
Ni
, “
Conservation laws and variational principles in metric theories of gravity
,”
Phys. Rev. D
10
,
1685
1700
(
1974
).
37.

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.
This view has been recently challenged in
J. O.
Weatherall
, “
On the status of the geodesic principle in Newtonian and relativistic physics
,”
Stud. Hist. Philos. Mod. Phys.
42
,
276
281
(
2011
); e-print arXiv:1106.2332[physics.hist-ph]. The key element of this criticism is that, since the equation of motion follows from local energy-momentum conservation, which is a very general postulate, its derivation in Einstein's theory has nothing peculiar. It remains true, however, that in theories like general relativity, where energy-momentum conservation can be derived from the field equations, the WEP and the GWEP provide a constraint on the latter. Furthermore, in order to derive the GWEP, energy-momentum conservation for the matter fields is not enough; one should also have a contribution from the gravitational field, which in a linear theory (like Newton's) is missing.
39.
J.-M.
Gérard
, “
The strong equivalence principle from a gravitational gauge structure
,”
Class. Quantum Grav.
24
,
1867
1877
(
2007
); e-print arXiv:gr-qc/0607019.
40.
J.-M.
Gérard
, “
Further issues in fundamental interactions
,” in
Proceedings of the 2008 European School of High-Energy Physics
, edited by
N.
Ellis
and
R.
Fleischer
(
CERN
,
Geneva
,
2009
), pp.
281
314
; e-print arXiv:0811.0540[hep-ph].
41.
For a simple presentation with a philosophical twist, see
S.
Hacyan
, “
Geometry as an object of experience: the missed debate between Poincaré and Einstein
,”
Eur. J. Phys.
30
,
337
343
(
2009
); e-print arXiv:0712.2222[physics.hist-ph].
42.

In Einstein's general relativity, κ = 8πG/c4 and jabc=bTacaTbc+12(gbcaTgacbT), where Tab is the stress-energy-momentum tensor and T = gabTab is its trace.

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