First order flow for non-extremal AdS black holes and mass from holographic renormalization

TL;DR

This paper develops a first order formulation for non-extremal AdS black holes in 4D supergravity, identifying a superpotential, deriving flow equations, and computing the black hole mass through holographic renormalization.

Contribution

It introduces a superpotential-based first order flow for non-extremal AdS black holes and applies holographic renormalization to compute their mass.

Findings

Derived first order flow equations for black holes

Identified superpotential for electric and magnetic solutions

Computed black hole mass satisfying thermodynamic laws

Abstract

In this paper we present a first order formulation for non-extremal Anti-de Sitter black hole solutions in four dimensional $\mathcal{N}=2$ U(1)-gauged Supergravity. The dynamics is determined in terms of a quantity $\mathcal{W}$ which plays the role of a superpotential for the gauging potential in the action. We show how the first order flow arises from writing the action as a sum of squares and we identify the superpotential driving the first order flow for two classes of solutions (electric and magnetic) of the $t^3$ model. After identifying $\mathcal{W}$, we study the Hamilton-Jacobi holographic renormalization procedure in presence of mixed boundary conditions for the scalar fields. We compute the renormalized on-shell action and the mass of the black hole configurations. The expression obtained for the mass satisfies the first law of thermodynamics.