Lanczos-Lovelock gravity from a thermodynamic perspective

The deep connection between gravitational dynamics and horizon thermodynamics leads to several intriguing features both in general relativity and in Lanczos-Lovelock theories of gravity. Recently in arXiv: 1312.3253 several additional results strengthening the above connection have been established within the framework of general relativity. In this work we provide a generalization of the above setup to Lanczos-Lovelock gravity as well. To our expectation it turns out that most of the results obtained in the context of general relativity generalize to Lanczos-Lovelock gravity in a straightforward but non-trivial manner. First, we provide an alternative and more general derivation of the connection between Noether charge for a specific time evolution vector field and gravitational heat density of the boundary surface. This will lead to holographic equipartition for static spacetimes in Lanczos-Lovelock gravity as well. Taking a cue from this, we have introduced naturally defined four-momentum current associated with gravity and matter energy momentum tensor for both Lanczos-Lovelock Lagrangian and its quadratic part. Then, we consider the concepts of Noether charge for null boundaries in Lanczos-Lovelock gravity by providing a direct generalization of previous results derived in the context of general relativity. Another very interesting feature for gravity is that gravitational field equations for arbitrary static and spherically symmetric spacetimes with horizon can be written as a thermodynamic identity in the near horizon limit. This result holds in both general relativity and in Lanczos-Lovelock gravity as well. In a previous work [arXiv: 1505.05297] we have shown that, for an arbitrary spacetime, the gravitational field equations near any null surface generically leads to a thermodynamic identity. In this work, we have also generalized this result to Lanczos-Lovelock gravity by showing that gravitational field equations for Lanczos-Lovelock gravity near an arbitrary null surface can be written as a thermodynamic identity. Our general expressions under appropriate limits reproduce previously derived results for both the static and spherically symmetric spacetimes in Lanczos-Lovelock gravity. Also by taking appropriate limit to general relativity we can reproduce the results presented in arXiv: 1312.3253 and arXiv: 1505.05297.