The thermodynamic structure of Einstein tensor

TL;DR

This paper explores the thermodynamic interpretation of Einstein tensor projections near horizons, emphasizing the geometric contributions to energy and area changes that differentiate this approach from Jacobson's Clausius relation-based derivation.

Contribution

It demonstrates how Einstein equations' horizon deformations inherently include a geometric energy term, distinguishing this from previous thermodynamic derivations.

Findings

Identifies a geometric energy term in Einstein equations near horizons.

Shows the difference between horizon deformation approaches and Clausius relation methods.

Highlights the importance of specific geometric contributions to area and energy changes.

Abstract

We analyze the generic structure of Einstein tensor projected onto a 2-D spacelike surface S defined by unit timelike and spacelike vectors u_i and n_i respectively, which describe an accelerated observer (see text). Assuming that flow along u_i defines an approximate Killing vector X_i, we then show that near the corresponding Rindler horizon, the flux j_a=G_ab X^b along the ingoing null geodesics k_i normalised to have unit Killing energy, given by j . k, has a natural thermodynamic interpretation. Moreover, change in cross-sectional area of the k_i congruence yields the required change in area of S under virtual displacements \emph{normal} to it. The main aim of this note 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 \emph{very specific geometric term} [reminiscent of Hawking's quasi-local expression for energy of spheres] corresponding to change in \emph{gravitational energy} arises inevitably in the first law: dE_G/d{\lambda} \alpha \int_{H} dA R_(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.