Schwarzschild's metric and the Landau-Lifshitz formulation of general relativity
Date
2023
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Hill, J.M.
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Zeitschrift fur Angewandte Mathematik und Physik, 2023; 74(6):1-16
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Fromholz, Poisson and Will (Am J Phys 82:295–300, 2013) examine Schwarzschild’s original 1916 derivation of the now famous metric associated with his name, along with a derivation using the Landau–Lifshitz formulation of the Einstein field equations of general relativity. These authors exploit the Landau–Lifshitz formulation to derive a nonlinear second-order ordinary differential equation which they were unable to fully solve, the general solution of which, may or may not, produce further metrics which are distinct from the Schwarzschild line element.
In this paper, we examine in some detail the particular Landau–Lifshitz formulation given in Fromholz, Poisson and Will (Am J Phys 82:295–300, 2013), and we obtain both special and general solutions of their nonlinear second-order ordinary differential equation. Clearly, knowledge of the general solution gives rise to the prospect of determining new solutions which may be other than the Schwarzschild line element in disguise.
However, the present writer questions the validity of the analysis of Fromholz, Poisson and Will (Am J Phys 82:295–300, 2013) and two specific objections are raised: one related to the harmonic gauge condition and the other related to the determinants connecting the line element metric to the gothic metric.
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Copyright 2023 The Author(s), under exclusive licence to Springer Nature Switzerland AG. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.