Thermodynamic route to Field equations in Lanczos-Lovelock Gravity

TL;DR

The paper extends the thermodynamic reinterpretation of gravitational dynamics, previously established for Einstein gravity, to the Lanczos-Lovelock class by analyzing static, spherically symmetric horizons. Using the near-horizon (Rindler) limit, it derives a TdS = dE + PdV structure for Gauss-Bonnet and general LL actions, with explicit horizon entropy S and energy E expressions built from the LL densities. The results reproduce known entropy formulas (Wald entropy) and horizon energies, suggesting a universal thermodynamic route to gravitational dynamics that naturally incorporates quantum corrections. This supports a holographic perspective where horizon thermodynamics encodes the bulk gravitational equations and their quantum extensions.

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 TdS = dE + PdV 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 Lanczos-Lovelock action in a spherically symmetric spacetime can also be expressed as TdS = dE + PdV 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_a^{bcd}R^a_{bcd} with \nabla_b Q^{abcd}=0. In such models, the expansion of Q^{abcd} 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 TdS = dE + PdV has a greater domain of validity that Einstein's field equations.