Conserved current for the Cotton tensor, black hole entropy and equivariant Pontryagin forms

TL;DR

This paper develops an equivariant conserved current for the Cotton tensor using the first equivariant Pontryagin form, and links it to black hole entropy, energy, and angular momentum in the context of Chern-Simons gravity.

Contribution

It introduces a method to derive an equivariant conserved current for the Cotton tensor and connects it to black hole thermodynamics via the Chern-Simons term.

Findings

Constructed an equivariant conserved current for the Cotton tensor.

Linked the Chern-Simons contribution to black hole entropy and conserved charges.

Provided a Hamiltonian current framework for these physical quantities.

Abstract

The Chern-Simons lagrangian density in the space of metrics of a 3-dimensional manifold M is not invariant under the action of diffeomorphisms on M. However, its Euler-Lagrange operator can be identified with the Cotton tensor, which is invariant under diffeomorphims. As the lagrangian is not invariant, Noether Theorem cannot be applied to obtain conserved currents. We show that it is possible to obtain an equivariant conserved current for the Cotton tensor by using the first equivariant Pontryagin form on the bundle of metrics. Finally we define a hamiltonian current which gives the contribution of the Chern-Simons term to the black hole entropy, energy and angular momentum.