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.
Highlights
Since the early stage of the thermodynamical description of black holes, the Smarr formula [1] has been an intensive subject of study as an analogous of the Euler equation for black hole mechanics
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 relation states the energy as a bilinear form of the global charges of the black hole along with their corresponding chemical potentials, as long as the entropy is a homogeneous function of a definite degree in the conserved charges
Summary
Since the early stage of the thermodynamical description of black holes, the Smarr formula [1] has been an intensive subject of study as an analogous of the Euler equation for black hole mechanics. III, we prove that the conservation law associated to the scale invariance of the action for the aforementioned class of configurations, allows us to obtain the Smarr formula as long as a special set of holographic boundary conditions is satisfied This particular set is singled out by requiring compatibility of the integrability conditions for the energy with the scale invariance of the reduced action principle.
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