The First Law of Isolated Horizons via Noether Theorem

TL;DR

This paper applies a Noether theorem-based method to Einstein-Maxwell theory, deriving the first law of thermodynamics for rotating horizons by analyzing energy variations and boundary conditions within a geometric framework.

Contribution

It demonstrates that Einstein-Maxwell theory is natural rather than gauge-natural and derives the first law of thermodynamics for rotating horizons using this approach.

Findings

Einstein-Maxwell theory is shown to be natural, not gauge-natural.

A correction term is introduced for energy variation in this framework.

The first law of thermodynamics for rotating horizons is derived.

Abstract

A general recipe proposed elsewhere to define, via Noether theorem, the variation of energy for a natural field theory is applied to Einstein-Maxwell theory. The electromagnetic field is analysed in the geometric framework of natural bundles. Einstein-Maxwell theory turns then out to be natural rather than gauge-natural. As a consequence of this assumption a correction term \a la Regge-Teitelboim is needed to define the variation of energy, also for the pure electromagnetic part of the Einstein-Maxwell Lagrangian. Integrability conditions for the variational equation which defines the variation of energy are analysed in relation with boundary conditions on physical data. As an application the first law of thermodynamics for rigidly rotating horizons is obtained.