Thermodynamic structure of the Einstein tensor

Dawood Kothawala
Phys. Rev. D 83, 024026 – Published 20 January 2011

Abstract

We analyze the generic structure of Einstein tensor projected onto a 2D spacelike surface S defined by a unit timelike and spacelike vectors u and n, respectively, which describe an accelerated observer (see text). Assuming that flow along u defines an approximate Killing vector ξ, we then show that near the corresponding Rindler horizon, the flux ja=Gbaξb along the ingoing null geodesics k, i.e., j·k, has a natural thermodynamic interpretation. Moreover, change in the cross-sectional area of the k congruence yields the required change in area of S under virtual displacements normal to it. The main aim of this paper is to clearly demonstrate how, and why, the content of Einstein equations under such horizon deformations, originally pointed out by Padmanabhan, is essentially different from the result of Jacobson, who employed the so-called Clausius relation in an attempt to derive Einstein equations from such a Clausius relation. More specifically, we show how a very specific geometric term (reminiscent of Hawking’s quasilocal expression for energy of spheres) corresponding to change in gravitational energy arises inevitably in the first law: dEG/dλHd2xσR(2) (see text)—the contribution of this purely geometric term would be missed in attempts to obtain area (and hence entropy) change by integrating the Raychaudhuri equation.

  • Received 22 October 2010

DOI:https://doi.org/10.1103/PhysRevD.83.024026

© 2011 American Physical Society

Authors & Affiliations

Dawood Kothawala*

  • Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada E3B 5A3

  • *dawood@iucaa.ernet.in

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Issue

Vol. 83, Iss. 2 — 15 January 2011

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