Bernar, Rafael P

Crispino, Luis C. B.

Higuchi, Atsushi

Lima, Haroldo C. D.

English

White Rose Research Online

This is a repository copy of Comment on "The equivalence principle in the Schwarzschild geometry" [Am. J. Phys. 62, 1037 (1994)].White Rose Research Online URL for this paper:https://eprints.whiterose.ac.uk/162538/Version: Accepted VersionArticle:Higuchi, Atsushi orcid.org/0000-0002-3703-7021, Crispino, Luis C. B., Bernar, Rafael P et al. (1 more author) (Accepted: 2020) Comment on "The equivalence principle in the Schwarzschild geometry" [Am. J. Phys. 62, 1037 (1994)]. American Journal of Physics. ISSN 0002-9505 (In Press) eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Comment on “The equivalence principle in the Schwarzschild geometry”[Am. J. Phys. 62, 1037 (1994)]Rafael P. Bernar,1, ∗ Lúıs C. B. Crispino,1, † Atsushi Higuchi,2, ‡ and Haroldo C. D. Lima Junior1, §1Faculdade de F́ısica, Universidade Federal do Pará, Belém, Pará, 66075-110, Brazil2Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom(Dated: 15 de junho de 2020)In the article “The equivalence principle in the Schwarzschild geometry” [Am. J. Phys. 62,1037-1040 (1994)], some manifestation of the equivalence principle in Schwarzschild spacetime wasstudied. We point out that the result for the Riemann-tensor component in this article is incorrectbecause only the first order terms in the metric expansion are taken into account. We show thecorrected result, illustrating the importance of the second order terms in the approximate metric incomputing curvature quantities.PACS numbers:In Ref. 1, the authors provide a concrete example of manifestation of the equivalence principle in General Relativity,using the Schwarzschild solution. It is shown, by making a suitable coordinate transformation, that in the vicinity of adisplaced Cartesian coordinate system, the Schwarzschild line element is equivalent to the line element associated witha uniformly accelerated observer in flat spacetime, except for some off-diagonal components. In order to show thisequivalence, the components of the metric tensor, written in displaced Cartesian coordinates, are expanded up to firstorder in x/R, y/R and z/R, where R is the Schwarzschild radial coordinate of the origin of the displaced Cartesiancoordinates. The result is given in Eq. (7) of Ref. 1. Under the approximation considered, the off-diagonal terms inthe metric do not affect the motion along the acceleration direction.2 In Sec. IV of Ref. 1, the authors compute thezeroth-order expression of the Riemann tensor using the results of Eq. (7), i.e., the components of the metric tensorexpanded up to first order. We point out that these results are not correct.Although the off-diagonal terms may be linked to tidal effects by showing that they contribute to the Riemanntensor and, thus, to the geodesic deviation equation, we note that, to correctly compute the zeroth-order terms of theRiemann tensor, the second order terms in x/R, y/R and z/R of the metric components are needed. The componentsof the Riemann tensor are mainly comprised of second derivatives of the metric. Hence, quadratic terms in x/R, y/Rand z/R will contribute to the zeroth order expression of the Riemann tensor. In Ref. 1, the (incorrect) componentR0101 of the Riemann tensor was computed using the linear approximation, where (x0, x1, x2, x3) = (ct, x, y, z) labelsthe displaced coordinate system. The z-direction is parallel to the radial direction pointing from the center of thegravitational source to the origin of this displaced coordinate system, whereas the x- and y-directions are orthogonalto it. The results are given in Eq. (17) of Ref. 1.We compute the component R0101 of the Riemann tensor using the second order approximation of the metric tensor.We first find that the nonzero components of the metric tensor up to second order in x/R, y/R and z/R are given byg00 = −F (R)− F′(R)z −F ′(R)2R(x2 + y2)−F ′′(R)2z2, (1)g11 = 1 +G(R)− 1R2x2, (2)g22 = 1 +G(R)− 1R2y2, (3)g33 = G(R) +G′(R)z +H(R)2R2(x2 + y2)+G′′(R)2z2, (4)g12 =G(R)− 1R2xy, (5)g13 =G(R)− 1Rx+1−G(R) +RG′(R)R2xz, (6)g23 =G(R)− 1Ry +1−G(R) +RG′(R)R2yz, (7)2whereF (R) =1G(R)= 1−2mR, (8)H(R) = RG′(R)− 2G(R) + 2. (9)The correct3 zeroth-order expression of the Riemann-tensor component R0101, computed with the metric-tensorcomponents up to second order in x/R, y/R and z/R, is given byR01 0 1 = −GMc2R3. (10)We also note that the zeroth-order expression of the Ricci scalar vanishes, as expected for the Schwarzschild solution.In summary, when computed in the first-order approximation of the metric-tensor components, the zeroth-orderexpressions of the curvature quantities are incorrect due to missing information from the second order coefficients inthe metric. The Ricci scalar, for example, is nonvanishing in the linear approximation.AcknowledgmentsThe authors thank Richard Neutze for email correspondence. The authors acknowledge the partial financial supportof Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento dePessoal de Nı́vel Superior (CAPES) - Finance Code 001, and Fundação Amazônia Paraense de Amparo à Pesquisa(FAPESPA), from Brazil. This research has also received funding from the European Union’s Horizon 2020 researchand innovation programme under the H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. We also acknowledgethe support from the Abdus Salam International Centre for Theoretical Physics through Visiting Scholar/ConsultantProgramme. One of the authors (A.H.) also thanks Federal University of Pará for kind hospitality.∗ Electronic address: rafael.bernar@icen.ufpa.br† Electronic address: crispino@ufpa.br‡ Electronic address: atsushi.higuchi@york.ac.uk§ Electronic address: haroldo.ufpa@gmail.com1 W. Moreau, R. Neutze, D. K. Ross, “The equivalence principle in the Schwarzschild geometry,” Am. J. Phys. 62, 1037-1040(1994).2 The off-diagonal terms in this case are directly related to the geodesic deviation equation – we can think of x and y as beingrelated to the components of the separation vector between two nearby geodesics directed towards the source of gravity.3 Although, in Ref. 1, they do not give details to the definition of the Riemann-tensor components used, our results here wouldbe different by, at most, a change of sign.

Comment on "The equivalence principle in the Schwarzschild geometry" [Am. J. Phys. 62, 1037 (1994)]

https://eprints.whiterose.ac.uk/162538/1/Rafael_P_Bernar.pdf

Comment on "The equivalence principle in the Schwarzschild geometry" [Am. J. Phys. 62, 1037 (1994)]

Abstract

Similar works

Full text

Available Versions

White Rose Research Online