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Wiggling Throat of Extremal Black Holes

G. Compère, K. Hajian, A. Seraj, M. M. Sheikh-Jabbari

TL;DR

This work constructs a covariant classical phase space for near-horizon extremal geometries (NHEG) in vacuum gravity, parameterized by a single periodic wiggle function on the U(1) torus. It provides a consistent symplectic structure, identifies symplectic (bulk) symmetries with an infinite Virasoro-like algebra, and shows the central extension is proportional to the black hole entropy, thereby encoding microstate information in near-horizon data. Charges associated with these symmetries are expressible as Fourier modes of a Liouville-type stress tensor $T[\Psi]$ built from the wiggle function, offering a concrete route to quantize the phase space and obtain a quantum NHEG algebra \widehat{\mathcal{V}_{\vec{k},S}}$. In four dimensions this reduces to the standard Virasoro algebra with central charge $c=12S/(2\pi)$, while higher dimensions yield multiple Virasoro subalgebras, all tied to the entropy. The framework clarifies how near-horizon dynamics may illuminate black hole microstates and provides a basis for exploring holographic interpretations via Liouville-like theories on the torus of angular directions.

Abstract

We construct the classical phase space of geometries in the near-horizon region of vacuum extremal black holes as announced in [arXiv:1503.07861]. Motivated by the uniqueness theorems for such solutions and for perturbations around them, we build a family of metrics depending upon a single periodic function defined on the torus spanned by the $U(1)$ isometry directions. We show that this set of metrics is equipped with a consistent symplectic structure and hence defines a phase space. The phase space forms a representation of an infinite dimensional algebra of so-called symplectic symmetries. The symmetry algebra is an extension of the Virasoro algebra whose central extension is the black hole entropy. We motivate the choice of diffeomorphisms leading to the phase space and explicitly derive the symplectic structure, the algebra of symplectic symmetries and the corresponding conserved charges. We also discuss a formulation of these charges with a Liouville type stress-tensor on the torus defined by the $U(1)$ isometries and outline possible future directions.

Wiggling Throat of Extremal Black Holes

TL;DR

This work constructs a covariant classical phase space for near-horizon extremal geometries (NHEG) in vacuum gravity, parameterized by a single periodic wiggle function on the U(1) torus. It provides a consistent symplectic structure, identifies symplectic (bulk) symmetries with an infinite Virasoro-like algebra, and shows the central extension is proportional to the black hole entropy, thereby encoding microstate information in near-horizon data. Charges associated with these symmetries are expressible as Fourier modes of a Liouville-type stress tensor built from the wiggle function, offering a concrete route to quantize the phase space and obtain a quantum NHEG algebra \widehat{\mathcal{V}_{\vec{k},S}}c=12S/(2\pi)$, while higher dimensions yield multiple Virasoro subalgebras, all tied to the entropy. The framework clarifies how near-horizon dynamics may illuminate black hole microstates and provides a basis for exploring holographic interpretations via Liouville-like theories on the torus of angular directions.

Abstract

We construct the classical phase space of geometries in the near-horizon region of vacuum extremal black holes as announced in [arXiv:1503.07861]. Motivated by the uniqueness theorems for such solutions and for perturbations around them, we build a family of metrics depending upon a single periodic function defined on the torus spanned by the isometry directions. We show that this set of metrics is equipped with a consistent symplectic structure and hence defines a phase space. The phase space forms a representation of an infinite dimensional algebra of so-called symplectic symmetries. The symmetry algebra is an extension of the Virasoro algebra whose central extension is the black hole entropy. We motivate the choice of diffeomorphisms leading to the phase space and explicitly derive the symplectic structure, the algebra of symplectic symmetries and the corresponding conserved charges. We also discuss a formulation of these charges with a Liouville type stress-tensor on the torus defined by the isometries and outline possible future directions.

Paper Structure

This paper contains 64 sections, 1 theorem, 181 equations, 3 figures.

Table of Contents

  1. Introduction and Summary of Results
  2. Summary of results
  3. Outline
  4. Quick Review on NHEG
  5. NHEG isometries.
  6. NHEG examples in $4d$ and $5d$.
  7. Killing horizons
  8. Killing Horizon Generator.
  9. Gaussian null coordinates.
  10. Kruskal-type coordinates and causal structure.
  11. NHEG entropy
  12. Laws of NHEG mechanics
  13. Zeroth Law:
  14. The Entropy Law:
  15. Entropy Perturbation Law.
  16. ...and 49 more sections

Key Result

Theorem 1

If the presymplectic form is contracted with a gauge transformation $\delta_\chi\Phi$, there is a unique (up to a total derivative) $d-2$ form ${\boldsymbol k}_\chi [\delta\Phi,\Phi]$ satisfying the following identity provided that the fields $\Phi$ satisfy the equations of motion and the field variations $\delta\Phi$ satisfy the linearized equations of motion around $\Phi$. The form of ${\boldsy

Figures (3)

  • Figure 1: Penrose diagram for NHEG, suppressing the $\theta,\varphi^i$ directions. The positive and negative $r$ values of the coordinates used in \ref{['NHEG-metric']} respectively cover ${\mathbf{I}}$ and ${\mathbf{II}}$ regions in the above figure. The two boundaries are mapped onto each other by an $r$--$\vec{\varphi}$ inversion symmetry \ref{['r-phi-inversion']}. The arrows on the boundaries shows the flow of time $t$. Note also that flow of time is reversed between regions I and II.
  • Figure 2: Flow of the Killing vector $\zeta_{\mathcal{H}}$. The two black dots denote the codimension two bifurcation surfaces and the $45^\circ$ lines intersecting at them are the Killing horizons $\mathcal{N}$. Under $r$--$\vec{\varphi}$ inversion \ref{['r-phi-inversion']} the upper and lower triangles separated by $r=0$ line are mapped to each other. The Killing vector is mapped as $\zeta_{\mathcal{H}}\to -\zeta_{\mathcal{H}}$ under $r$--$\vec{\varphi}$ inversion.
  • Figure 3: A schematic depiction of the NHEG phase space $\mathcal{G}_{\{ p \}}[F]$. The vertical axis shows different background NHEG solutions of the form \ref{['NHEG-metric']} specified by different angular momenta $J_i$, and the horizontal plane shows the phase space constructed by the action of the finite coordinate transformation \ref{['finite ansatz']}. Each geometry in the phase space is identified by a periodic function $F(\vec{\varphi})$ and admits the same angular momenta $J_i$ and entropy.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • proof