We discuss what we take to be three possible misconceptions in the foundations of general relativity, relating to: (a) the interpretation of the weak equivalence principle and the relationship between gravity and inertia; (b) the connection between gravitational redshift results and spacetime curvature; and (c) the Einstein equivalence principle and the ability to “transform away” gravity in local inertial coordinate systems.
References
1.
On the recent detection of gravitational waves, see
B. P.
Abbot
et al, “Observation of gravitational waves from a binary black hole merger
,” Phys. Rev. Lett.
116
, 061102
(2016
).2.
Albert
Einstein
, letter to R. W. Lawson, January 22, 1920
. (Reprinted as Vol. 7, Doc. 31 CPAE, p. 265
.)3.
Newton's own appreciation of this fact is seen in his application of Corollary VI of the laws of motion to gravitation in the Principia, particularly in relation to the discussion of the system of Jupiter and its moons. For a recent discussion, see
Simon
Saunders
, “Rethinking Newton's principia
,” Philos. Sci.
80
, 22
–48
(2013
).4.
Albert
Einstein
, “A brief outline of the development of the theory of relativity
,” Nature
106
(2677
), 782
–784
(1921
).5.
Albert
Einstein
, Über Friedrich Kottlers Abhandlung “Über Einstein's äquivalenzhypothese und die gravitation
,” Ann. Phys.
356
(22
), 632
–642
(1916
); reprinted as Vol. 6, Doc. 40 CPAE.6.
For a detailed discussion of Einstein's interpretation of GR, see
Michel
Janssen
, “The twins and the bucket: How Einstein made gravity rather than motion relative in general relativity
,” Stud. Hist. Philos. Mod. Phys.
43
, 159
–175
(2012
);Dennis
Lehmkuhl
, “Why Einstein did not believe that general relativity geometrizes gravity
,” Stud. Hist. Philos. Mod. Phys.
46
, 316
–326
(2014
). It is noteworthy that the covariant form of Maxwell's equations also played an important role as a template in the technical development of the field equations themselves;see
Michel
Janssen
and Jürgen
Renn
, “Untying the Knot: How Einstein found his way back to field equations discarded in the Zurich notebook
,” in The Genesis of General Relativity
, Einstein's Zurich Notebook. Commentary and Essays Vol. 2, edited by Jürgen
Renn
(Springer
, Dordrecht
, 2007
), pp. 839
–925
.7.
As Peter Havas noted in 1967, starting with coordinates in which the connection coefficients vanish at a spacetime point, the mere transformation to non-Cartesian coordinates may make the term in the geodesic equation containing the connection reappear at that point. See p. 134 of
Peter
Havas
, “Foundation problems in general relativity
,” in Delaware Seminar in the Foundations of Physics
, edited by Mario
Bunge
(Springer-Verlag
, Berlin/Heidelberg
, 1967
), pp. 124
–148
. And according to Einstein's suggestion, this makes gravity appear too, even though no acceleration is involved relative to the initial local inertial frame.8.
See Einstein Ref. 5.
9.
Albert
Einstein
, Vier Vorlesungen Über Relativitätstheorie Gehalten im Mai 1921 an der Universitt Princeton
(F. Vieweg
, Braunschweig
, 1922
); reprinted as Vol. 7, Doc. 71 CPAE; and in various editions as “The Meaning of Relativity” by Princeton U.P.10.
See p. 218 of
Eric
Poisson
and Clifford M.
Will
, Gravity. Newtonian, Post-Newtonian, Relativistic
(Cambridge U.P.
, Cambridge, UK
, 2014
).For a review of the experimental evidence in favor of WEP 1, see chap. 3 of
Ignazio
Ciufolini
and John
Archibald Wheeler
, Gravitation and Inertia
(Princeton U.P.
, Princeton
, 1995
), andDomenico
Giulini
, “Equivalence principle, quantum mechanics, and atom-interferometric tests
,” in Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework
, edited by F.
Finster
et al, DOI 10.1007/978-3-0348-0043-3-16, Springer Basel AG
, 2012
, pp. 345
–370
; in the latter the principle is called The Universality of Free Fall (UFF).11.
This identification is sometimes called Einstein's strong equivalence principle. See, e.g., p. 601 of
Kip S.
Thorne
and Clifford M.
Will
, “Theoretical frameworks for testing relativistic gravity. I. Foundations
,” Astrophys. Soc. J.
163
, 595
–610
(1971
), not to be confused with the similarly named principle discussed in Sec. IV. Initially, Einstein introduced such geodesic motion as a postulate, but by 1927 he and Grommer realized what others like Eddington had already seen that it can be regarded as a consequence of the Einstein field equations of GR. An important qualification to this claim will be indicated in Sec. IV. For a history of the geodesic theorem for test bodies, seePeter
Havas
, “The early history of the ‘problem of motion’ in general relativity
,” in Einstein and the History of General Relativity
, Einstein Studies Vol. 1, edited by D.
Howard
andJ.
Stachel
(Birkhäuser
, Boston/Basel/Berlin
, 1989
), pp. 234
–276
.Note that the extent to which the geodesic principle can be derived from first principles in GR depends to some extent on the chosen formulation of GR. In the case of the trace-free version of the field equations, the vanishing of the covariant divergence of the stress-energy tensor, a crucial element in the standard proof of the geodesic theorem, is itself a postulate and no longer a consequence of the field equations involving the metric; see
George F. R.
Ellis
, “The trace-free Einstein equations and inflation
,” Gen. Relativ. Gravitation
46
(1
), 1619
–1633
(2014
).12.
Albert
Einstein
, letter to R. W. Lawson, January 22, 1920
, reprinted as Vol. 7, Doc. 31 CPAE, p. 265
.13.
See p. 113 of
James
Hartle
, Gravity: An Introduction to Einstein's General Relativity
(Addison Wesley
, San Francisco
, 2003
).14.
Though WEP 1 and WEP 2 are both referred to as the “weak” equivalence principle, we stress that we do not conflate the two. In fact, in what follows we will argue broadly in favor of WEP 1, and against WEP 2, from the perspective of GR.
15.
See, e.g., p. 14 of Øyvind Grøn;
Sigbjørn
Hervick
, Einstein's General Theory of Relativity
(Springer
, New York
, 2007
),p. 68 of
Steven
Weinberg
, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity
(John Wiley & Sons
, New York
, 1972
);and p. 49 of
Sean
Carroll
, Spacetime and Geometry: An Introduction to General Relativity
(Addison Wesley
, San Francisco
, 2004
).Such language can also be found in foundationally oriented papers, such as
A. M.
Nobili
, D. M.
Lucchesi
, M. T.
Crosta
, M.
Shao
, S. G.
Turyshev
, R.
Peron
, G.
Catastini
, A.
Anselmi
, and G.
Zavattini
, “On the universality of free fall, the equivalence principle, and the gravitational redshift
,” Am. J. Phys.
81
, 527
–536
(2013
).16.
See, e.g.,
Charles W.
Misner
, Kip S.
Thorne
, and John Archibald
Wheeler
, Gravitation
(W. H. Freeman and Company
, San Francisco
, 1973
);and
Robert
Wald
, General Relativity
(University of Chicago Press
, Chicago
, 1984
).17.
op. cit.
18.
See p. 221 of Poisson and Will, Ref. 10;
Rohrlich [p. 186 of
F.
Rohrlich
, “The principle of equivalence
,” Ann. Phys.
22
, 169
–191
(1963
)] likewise considered WEP 2 to be without empirical content, but he saw it as a definition of inertial frames.19.
See p. 602 of Thorne and Will, Ref. 11;
and pp. 1055ff. of Misner et al., Ref. 16. In the former, it is argued that despite not relying on the details of the Einstein field equations, the redshift experiment should not be regarded as a “weak test” of GR.
20.
K.
Hentschel
, “Measurements of gravitational redshift between 1959 and 1971
,” Ann. Sci.
53
, 269
–295
(1996
), which casts some doubt on whether the earlier experiment performed by J. P. Schiffer and his collaborators at Harwell in England was as inconclusive as Pound claimed at the time. For more recent and more accurate versions of the redshift experiment,see the review at p. 103 of Ciufolini and Wheeler, Ref. 10.
21.
See, e.g., pp. 133ff. of Hartle, Ref. 13.
22.
As noted in
P. T.
Landsberg
and N. T.
Bishop
, “Equivalence principle: 60 Years of a misuse?
,” Nature
252
, 459
–460
(1974
); and P. T.
Landsberg
and N. T.
Bishop
, “Gravitational redshift and the equivalence principle
,” Found. Phys.
6
(6
), 727
–737
(1976
), in some textbooks this result is erroneously derived from models in which the detector moves with respect to the source;see, e.g., p. 190 of Misner et al., Ref. 16. In fact, this is not an example of a gravitational redshift-type result, but rather of the relativistic Doppler effect. In a correct derivation of Eq. (1), source and accelerator must be considered at rest with respect to one another, but accelerating with respect to an inertial frame.
23.
Note that if the source A is the higher sample, then received signals at B will in fact be blueshifted. If the positions of emitter and receiver are swapped, then the situation resembles the original Pound–Rebka experiment;
R. V.
Pound
and G. A.
Rebka
,Jr. , “Apparent weight of photons
,” Phys. Rev. Lett.
4
(7
), 337
–341
(1960
), and the received signals (at the higher sample) will indeed be red shifted;see, e.g., pp. 187 of Misner et al., Ref. 16.
For details concerning later more accurate versions of the redshift experiment, see chap. 3 of Ciufolini and Wheeler, Ref. 10.
24.
Albert
Einstein
, “Die Grundlage der allgemeinen relativitätstheorie
,” Ann. Phys.
49
(7), 769
–822
(1916
).For discussion, see chap. 5 of
Harvey R.
Brown
, Physical Relativity: Spacetime Structure from a Dynamical Perspective
(Oxford U.P.
, Oxford
, 2007
). But note that the inertial frame is now freely falling.25.
See, e.g., p. 602 of Thorne and Will, Ref. 11.
26.
See
Alfred
Schild
, “Equivalence principle and red-shift measurements
,” Am. J. Phys.
28
, 778
–780
(1960
).27.
See p. 126 of Hartle, Ref. 13.
28.
See pp. 53–54 of Carroll, Ref. 15.
For similar conclusions, see, e.g., p. 210 of
Hans
Ohanian
, Gravitation and Spacetime
(Cambridge U.P.
, Cambridge, UK
, 1976
);p. 27 of
Wolfgang
Rindler
, Relativity: Special, General, and Cosmological
(Oxford U.P.
, Oxford
, 2001
);p. 189 of Misner et al., Ref. 16.
29.
In what follows, we use the Einstein summation convention, with Greek indices ranging from 0 to 3.
30.
See pp. 602–603 of Thorne and Will, Ref. 11;
p. 166 of Misner et al., Ref. 16;
p. 132 of Hartle, Ref. 13;
and p. 285 of Poisson and Will, Ref. 10.
31.
We do not agree, however, with a long-standing notion that the Pound–Rebka effect can be understood by combining special relativity with WEP 2; see Sec. IV of
A. M.
Nobili
, D. M.
Lucchesi
, M. T.
Crosta
, M.
Shao
, S. G.
Turyshev
, R.
Peron
, G.
Catastini
, A.
Anselmi
, and G.
Zavattini
, “On the universality of free fall, the equivalence principle, and the gravitational redshift
,” Am. J. Phys.
81
, 527
–536
(2013
).Apart from our concerns about the significance of WEP 2 outlined in Sec. II B, we would also emphasise that in the 1905 version of special relativity, Einstein was explicit that the inertial frames were the same as Newton's: They are not freely falling. In this case, no redshift is predicted; see p. 602 of Thorne and Will, Ref. 11.
32.
The arguments leading to our Eq. (8) above are essentially those given by Thorne and Will, Ref. 30, pp. 602–603; the analysis there is given in the context of a curved spacetime geometry, however, with the curvature playing no part in the analysis. Given Hartle's claim above that redshift is ultimately connected to spacetime curvature, it is striking that in Problem 6, p. 132 (op. cit.), Hartle uses the transformations to accelerating coordinates to conclude that accelerating clocks in special relativity are affected in accordance with our Eq. (8), but leaves how this is related to “the equivalence principle idea” as a question. Exercise 5.1 in the Poisson-Will textbook (p. 285, op. cit.) almost exactly reproduces Hartle's Problem 6, even repeating the same question. This section of the present paper can be read as an attempt to answer this question.
33.
The importance, from the point of view of establishing spacetime curvature, of considering multiple redshift experiments spread out on the surface of Earth was stressed by Schild in 1962 (Sec. 19 of
Alfred
Schild
, “Gravitational theories of the whitehead type
,” in Proceedings of the School of Physics “Enrico Fermi,” Course XX, Evidence for Gravitational Theories
(Academic Press
, New York
, 1962
), pp. 69
–115
). But Schild defines the WEP in an unusually complicated fashion because the laboratories in his account are, confusingly, accelerating with respect to Earth, not with respect to the free-fall frames.34.
J. W.
Brault
, “The Gravitational Redshift in the Solar Spectrum
,” Ph.D. dissertation, Princeton University, 1962
; a summary is “Gravitational redshift of solar lines,” Bull. Am. Phys. Soc. 8, 28 (1963).An interesting account of Brault's work and its reception is found in Hentschel, Ref. 20.
35.
See p. 603 of Thorne and Will, Ref. 11.
36.
See p. 64 of
Arthur
Eddington
, Space, Time and Gravitation. An Outline of the General Theory of Relativity
(Cambridge U.P.
, Cambridge
, 1966
).37.
C. W.
Sherwin
, “Some recent experimental tests of the ‘clock paradox’
,” Phys. Rev. D
120
, 17
–24.
(1960
).38.
J.
Bailey
, K.
Borer
, F.
Combley
, H.
Drumm
, F.
Krienen
, F.
Lange
, E.
Picasso
, W.
von Ruden
, F. J. M.
Farley
, J. H.
Field
, W.
Flegel
, and P. M.
Hattersley
, “Measurements of relativistic time dilation for positive and negative muons in a circular orbit
,” Nature
268
, 301
–305
(1977
).39.
Anton
Eisele
, “On the behaviour of an accelerated clock
,” Helv. Phys. Acta
60
, 1024
–1037
(1987
).40.
See pp. 395–396 of Misner et al., Ref. 16;
see also in this connection
Domenico
Giulini
, “Equivalence principle, quantum mechanics, and atom-interferometric tests
,” in Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework
, edited by F.
Finster
et al, DOI 10.1007/978-3-0348-0043-3-16, Springer Basel AG
, 2012
, pp. 345
–370
, Remark 2.1. Note that this important insight is lost from view when an ideal clock in GR is defined as one satisfying the clock hypothesis, rather than on the basis of its performance when moving inertially (freely falling).41.
Similar considerations hold for the “proper distance” read off by accelerating rigid rulers. For further discussion of the role of rods and clocks in GR, see Harvey R. Brown, “The behavior of rods and clocks in general relativity, and the meaning of the metric field,” forthcoming in the Einstein Studies Series, e-print arXiv: 0911.4440.
42.
The metric is of course also surveyed by the proper distance read off by rigid rulers.
43.
See, e.g.,
David
Malament
, “A remark about the ‘geodesic principle’ in general relativity
,” in Analysis and Interpretation in the Exact Sciences: Essays in Honor of William Demopoulos
(Springer
, Dordrecht
, 2012
), pp. 245
–252
.44.
Special relativity implies the existence of a finite invariant speed c; whether it is photons which instantiate this speed is a separate question; recently circumstances in which the group velocity of light in vacuo is slightly less than c were demonstrated experimentally. See
Daniel
Giovannini
, Jacquiline
Romero
, Václav
Potoček
, Gergely
Ferenczi
, Fiona
Speirits
, Stephen M.
Barnett
, Daniele
Faccio
, and Miles J.
Padgett
, “Spatially structured photons that travel in free space slower than the speed of light
,” Science 20
347
(6224), 857
–860
(2015
).45.
Einstein, Ref. 24.
46.
Nomenclature varies here considerably; for example, EEP is related to the “medium strong equivalence principle” in Sec. 3.2.4 of Ciufolini and Wheeler, Ref. 10. In the philosophical literature, it is frequently referred to as the strong equivalence principle.
47.
See p. 386 of Misner et al., Ref. 16.
48.
See, e.g., pp. 182–183 of Hartle, Ref. 13;
pp. 230–242 of Poisson and Will, Ref. 10.
49.
For further discussion, see p. 352 of
Eleanor
Knox
, “Effective spacetime geometry
,” Stud. Hist. Philos. Mod. Phys.
44
, 346
–356
(2013
).50.
See Sec. 16.5 of Misner et al., Ref. 16.
51.
Hans
Ohanian
, “What is the principle of equivalence?
,” Am. J. Phys.
45
, 903
–909
(1977
).Similar formulations in the recent literature can be found at, e.g., p. 169 of
Harvey R.
Brown
, Physical Relativity: Spacetime Structure from a Dynamical Perspective
(Oxford U.P.
, Oxford
, 2007
)and p. 352 of Knox, Ref. 49.
52.
This latter assumption itself is remarkable; it is saying that all the non-gravitational forces are Lorentz-covariant with respect to the same family of coordinates; the non-triviality of such universality was emphasised by Anderson at Sec. 10.2 of
J. L.
Anderson
, Principles of Relativity Physics
(Academic Press
, New York
, 1967
).53.
See Sections II B and III B, and, e.g., p. 218 of Misner et al., Ref. 16.
54.
See p. 176 of
Arthur
Eddington
, The Mathematical Theory of Relativity
(Cambridge U.P.
, Cambridge
, 1923
).See also in this connection
Friedrich
Hehl
and Yuri
Obukhov
, “How does the electromagnetic field couple to gravity, in particular to metric, nonmetricity, torsion, and curvature?
,” in Testing Relativistic Gravity in Space: Gyroscopes, Clocks, Interferometers
, edited by Bad
Honnef
,C.
Laemmerzahl
et al (Springer
, Berlin
, 2000
), pp. 479
-504
; andYakov
Itin
and Friedrich
Hehl
, “Is the Lorentzian signature of the metric of spacetime electromagnetic in origin?
” Ann. Phys.
312
, 60
–83
(2004
).55.
Another example is the second-order wave equation for the 4-vector potential in vacuum electrodynamics, which picks up a term linear in the Ricci tensor; see p. 605 of Thorne and Will, Ref. 11;
and pp. 388–389 of Misner et al., Ref. 16.
56.
See p. 390 of Misner et al., Ref. 16;
p. 248 of Poisson and Will, Ref. 10.
57.
For further discussion see
James
Read
, Harvey R.
Brown
, and Dennis
Lehmkuhl
, “On the local validity of special relativity
,” (unpublished).58.
Pedro
Ferreira
, The Perfect Theory: A Century of Geniuses and the Battle over General Relativity
(Little Brown
, St. Ives
, 2014
).© 2016 American Association of Physics Teachers.
2016
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