Thermodynamic route to field equations in Lanczos-Lovelock gravity

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 the Lanczos-Lovelock action in a spherically symmetric spacetime can also be expressed as $TdS=dE+PdV$ with $S$ and $E$ given by expressions previously derived in the literature by other approaches. The Lanczos-Lovelock Lagrangians are of the form $\mathcal{L}=Q_a{}^{bcd}R^a{}_{bcd}$ with ${\nabla }_{b}Q_a{}^{bcd}=0$. In such models, the expansion of $Q_a{}^{bcd}$ in terms of the derivatives of the metric tensor determines the structure of the theory and higher order terms can be interpreted as 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 than Einstein’s field equations.