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The Routh of the Attractor Mechanism

Arghya Chattopadhyay, Alessio Marrani, Sourav Roychowdhury

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.

The Routh of the Attractor Mechanism

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 , 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 , Sen's entropy functional , and the relevant effective Routhian functional . Through their critical values at the event horizon, such functionals determine the Bekenstein-Hawking and the Wald entropy of the extremal black hole.
Paper Structure (32 sections, 133 equations)

This paper contains 32 sections, 133 equations.

Table of Contents

  1. Introduction
  2. Effective dynamics of extremal black holes
  3. A naı̈ve approach
  4. $\mathcal{R}$ and the Routhian approach
  5. The Hamiltonian constraint
  6. Résumé on the Routhian formalism
  7. The partial Legendre transform
  8. Partial regularity and solvability
  9. Definition of the Routhian
  10. Mixed equations of motion and variational principle
  11. Reduced Euler--Lagrange equations
  12. Reconstruction (Hamilton-type) equations
  13. Attractors and entropy
  14. Attractors as critical points of $V_{BH}$
  15. $S_{BH}=S_{W}$ for two-derivative gravity
  16. ...and 17 more sections