Electrically charged black hole on AdS$_3$: scale invariance and the Smarr formula

TL;DR

This paper derives the Smarr formula for electrically charged BTZ black holes in AdS3 using scale invariance and holographic boundary conditions, revealing deep connections between symmetries and black hole thermodynamics.

Contribution

It demonstrates two distinct methods to derive the Smarr relation from scale invariance in three-dimensional Einstein-Maxwell theory with negative cosmological constant.

Findings

Smarr formula obtained from scale invariance and boundary conditions.

Conservation law linked to scale symmetry yields the Smarr relation.

Explicit derivation using Euler theorem confirms the Smarr formula.

Abstract

The Einstein-Maxwell theory with negative cosmological constant in three spacetime dimensions is considered. It is shown that the Smarr relation for the electrically charged BTZ black hole emerges from two different approaches based on the scaling symmetry of the asymptotic behaviour of the fields at infinity. In the first approach, we prove that the conservation law associated to the scale invariance of the action for a class of stationary and circularly symmetric configurations, allows to obtain the Smarr formula as long as a special set of holographic boundary conditions is satisfied. This particular set is singled out making the integrability conditions for the energy compatible with the scale invariance of the reduced action. In the second approach, it is explicitly shown that the Smarr formula is recovered through the Euler theorem for homogeneous functions, provided the same set of holographic boundary conditions is fulfilled.