Higher derivative gravity: Field equation as the equation of state

Abstract

One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. Extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.

Einstein’s equations from the stretched future light cone

“…Here we have used (19) and (21). Next we make use of the fact that ∇ a ξ b = −∇ b ξ a for the projection of ∇ a ξ b in the n − ξ plane, as we see from the first line of (8). Then defining…”

“…It has been a longstanding challenge to obtain the gravitational equations of motion for general, higher-curvature theories of gravity from thermodynamics. Broadly, we can divide earlier attempts into two categories: (i) those that aim to derive the equations of motion for f (R) theories of gravity via a nonequilibrium modification of the Clausius theorem to account for internal entropy production terms [5][6][7], and (ii) those that aim to derive the gravitational equations for general theories of gravity [8][9][10][11][12]. The approaches that fall into category (i) have been critically reviewed in [10], which points out that this nonequilibrium approach can never lead to theories beyond f (R) gravity.…”

“…The approaches that fall into category (i) have been critically reviewed in [10], which points out that this nonequilibrium approach can never lead to theories beyond f (R) gravity. The attempts that fall into category (ii) mainly use a "Noetheresque" approach, in which the local entropy is expressed as an integral of a Noether current [8][9][10][11] over spacelike sections of a local Rindler plane. Unfortunately, all the early papers using the Noetheresque approach contained technical errors, as reviewed in [10].…”

Noether current, black hole entropy and spacetime torsion

“…This started from the demonstration that one can arrive at Einstein's equations starting from Clausius relation [24], which was supplemented by the demonstration that Einstein's equations can also be casted in a thermodynamic language [25]. Subsequently, both these methods were generalized to higher curvature theories and for generic null surfaces with modified expressions for entropy [26][27][28][29][30][31][32][33]. In all these contexts the fact that the entropy-area relation is modified, played a crucial role in order to write down the respective expression associated with gravitational dynamics in a thermodynamic language.…”

Emergent cosmos in Einstein–Cartan theory

“…1 Note that the area law S = A/4L 2 P as the saturation of the Bekenstein limit [153] is completely justified solely in the context of general relativity and is not correct in general in modified theories; see [154,155]. However, one may argue that true gravitational degrees of freedom are those of GR only, and the effect of torsion is to modify the right-hand side and effectively acts as an additional energy-momentum tensor, restoring the A/4L 2 P law.…”