[Paper Review] Discreteness of Black Hole Microstates

Gerard ’t Hooft|arXiv (Cornell University)|Sep 14, 2018

Black Holes and Theoretical Physics5 references4 citations

TL;DR

This paper proposes that black hole microstates exhibit a discrete quantum spectrum due to gravitational back reaction and antipodal identification at the horizon, without requiring string theory. By modeling in- and out-particles via spherical harmonic modes with non-commuting coordinates, it derives a discrete spectrum through quantized momentum transfer, preserving unitarity and avoiding singularities.

ABSTRACT

It is explained that, for black holes much heavier than the Planck mass, black hole microstates can be well understood without string theory. It is essential to understand the antipodal identification at the horizon. We show why the microstates exhibit a discrete spectrum, and how they relate to the particles outside the hole. Version 2 has some minor typos corrected and a few sentences added for clarity.

Motivation & Objective

• To establish a discrete spectrum for black hole microstates without relying on string theory or AdS/CFT.
• To resolve the black hole information paradox by enforcing unitarity through antipodal identification at the horizon.
• To demonstrate that quantum gravity effects, particularly gravitational back reaction, lead to discrete energy levels in black hole states.
• To show that the Hartle-Hawking vacuum state naturally leads to entanglement between hemispheres, explaining thermal Hawking radiation.
• To provide a framework for black hole microstates in 3+1 dimensions with flat asymptotics, avoiding exotic assumptions like supersymmetry or the BCS limit.

Proposed method

• Uses Kruskal-Szekeres coordinates to define past and future horizons as $u^+ = 0$ and $u^- = 0$, with antipodal identification $S^2/\mathbb{Z}_2$ at the intersection.
• Models in- and out-particles via momentum distributions $p^{-}( heta,\varphi)$ and position functions $u^{-}( heta,\varphi)$ on the horizon.
• Applies quantum commutation relations: $[u^{\mp}_{\ell m}, p^{\pm}_{\ell' m'}] = i \delta_{\ell\ell'} \delta_{mm'}$ for spherical harmonic modes.
• Derives back-reaction relations: $u^{-}_{\ell m} = \frac{8\pi G}{\ell^2 + \ell + 1} p^{-}_{\ell m}$, linking in- and out-states.
• Constructs the non-commutative algebra $[u^{+}_{\ell m}, u^{-}_{\ell' m'}] = \frac{8\pi G i}{\ell^2 + \ell + 1} \delta_{\ell\ell'} \delta_{mm'}$, implying discrete spectrum.
• Imposes antipodal identification to eliminate cusp singularities and ensure causality, while preserving unitarity in the scattering process.

Experimental results

Research questions

• RQ1 How can black hole microstates be described without invoking string theory or extra dimensions?
• RQ2 What is the origin of the discrete spectrum of black hole microstates in quantum gravity?
• RQ3 How does antipodal identification at the horizon resolve the singularity and preserve unitarity?
• RQ4 What role does gravitational back reaction play in generating the discrete spectrum of black hole states?
• RQ5 How does the Hartle-Hawking state emerge from the entanglement of two hemispheres under antipodal identification?

Key findings

• The microstates of a Schwarzschild black hole have a discrete spectrum due to non-commutativity of horizon coordinates $u^{\pm}_{\ell m}$, arising from gravitational back reaction.
• The maximum angular momentum quantum number is bounded by $\ell_{\mathrm{max}}^2 \log \ell_{\mathrm{max}} = \mathcal{O}(M_{\mathrm{BH}}^2)$, indicating a finite number of microstates.
• The antipodal identification $S^2/\mathbb{Z}_2$ at the horizon prevents cusp singularities and restores unitarity in black hole evolution.
• The Hartle-Hawking wave function is interpreted as an entangled state $|\psi\rangle_{HH} = C \sum_{E,n} e^{-\frac{1}{2}\beta E} |E,n\rangle_I |E,n\rangle_{II}$, explaining thermal Hawking radiation.
• The discrete spectrum arises from the quantization of momentum transfer via $u^{-}_{\ell m} \propto p^{-}_{\ell m}$, with coupling strength $\propto G/(\ell^2 + \ell + 1)$.
• The framework avoids string theory, supersymmetry, and AdS/CFT, working in 3+1 dimensions with flat asymptotics and no need for the BCS limit.