The Routh of the Attractor Mechanism

TL;DR

This paper clarifies the effective dynamics of extremal black holes in Maxwell-Einstein-scalar theories using the Routhian formalism, linking various functionals to black hole entropy calculations.

Contribution

It introduces the Routhian formalism to analyze the attractor mechanism, connecting the black hole potential, entropy functional, and Routhian in a unified framework.

Findings

Routhian formalism effectively describes extremal black hole dynamics.

Critical points of functionals determine black hole entropy.

Interplay of functionals clarifies entropy calculations.

Abstract

We investigate and clarify various aspects of the effective dynamics of Maxwell-Einstein-scalar theories in the background of static, spherically symmetric and asymptotically flat extremal black holes in four space-time dimensions. This rigorously places the one-dimensional effective radial dynamics governed by the Attractor Mechanism, through the critical points of the Ferrara-Gibbons-Kallosh effective black hole potential $V_{BH}$, into the Routhian formalism, a framework which is intermediate between the Lagrange and Hamilton ones, based on a partial Legendre transform, and especially relevant in presence of cyclic variables. We elucidate and analyze the interplay of a trio of effective functionals: the aforementioned $V_{BH}$, Sen's entropy functional $\mathcal{E}$, and the relevant effective Routhian functional $\mathcal{R}$. Through their critical values at the event horizon, such functionals determine the Bekenstein-Hawking and the Wald entropy of the extremal black hole.