Thermodynamic route to field equations in Lanczos-Lovelock gravity

Abstract

Spacetimes with horizons show a resemblance to thermodynamic systems and one can associate the notions of temperature and entropy with them. In the case of Einstein-Hilbert gravity, it is possible to interpret Einstein's equations as the thermodynamic identity T dS = dE + P dV for a spherically symmetric spacetime and thus provide a thermodynamic route to understand the dynamics of gravity. We study this approach further and show that the field equations for LanczosLovelock action in a spherically symmetric spacetime can also be expressed as T dS = dE +P dV with S and E being given by expressions previously derived in the literature by other approaches. The Lanczos-Lovelock Lagrangians are of the form L = Q bcd a R a bcd with ∇ b Q bcd a = 0. In such models, the expansion of Q bcd a in terms of the derivatives of the metric tensor determines the structure of the theory and higher order terms can be interpreted quantum corrections to Einstein gravity. Our result indicates a deep connection between the thermodynamics of horizons and the allowed quantum corrections to standard Einstein gravity, and shows that the relation T dS = dE + P dV has a greater domain of validity that Einstein's field equations.

Emergent perspective of gravity and dark energy

“…the field equations become T dS = dE + P dV ) continues to hold for a very wide class of theories! In the more general class of theories, one can define a natural entropy for the horizon called the Wald entropy [17] and we again get the same result with correct Wald entropy (for a sample of results see [18][19][20][21][22][23][24][25][26]). …”

“…In the still simpler context of spherical symmetry, the integration over dA becomes multiplication by 4πR 2 where R is the radius of the equipotential surface under consideration and we can write the equipartition law as: (19) where…”

Einstein's equations as a thermodynamic identity: The cases of stationary axisymmetric horizons and evolving spherically symmetric horizons

“…In that case, the near horizon structure of the field equation can be cast as a thermodynamic identity T dS = dE + P dV arising from the virtual displacement of the horizon normal to itself. Recently this result has been extended to the spherically symmetric horizons in Lanczos-Lovelock gravity [14] and it has been shown in that case also that the field equations can be interpreted as a thermodynamic relation. Since the extra, higher derivative terms in Lanczos-Lovelock gravity can be thought of as quantum corrections to GR, this non-trivial result suggests the possibility that the thermodynamic interpretation of the field equations remains valid even if one includes possible quantum corrections to Einstein-Hilbert action functional motivated by quantum gravity models.…”

Lanczos-Lovelock Gravity from a Thermodynamic Perspective

“…All these features are shown to stem from the deep connection between gravitational dynamics and horizon thermodynamics. Even though they first emerged in the context of general relativity showing that Einstein's field equations near a horizon become a thermodynamic identity [4][5][6][7][8][9][10][11][12][13][14][15][16][17], this result transcends general relativity and extends naturally to horizons in spherically symmetric and static spacetimes within LanczosLovelock theories of gravity [18,19]. Horizons in general (and black holes in particular) possess thermodynamic attributes like entropy [20,21] and temperature [22][23][24][25].…”