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Holographic description of quantum black hole on a computer

Masanori Hanada, Yoshifumi Hyakutake, Goro Ishiki, Jun Nishimura

arXiv:1311.5607v1hep-thgr-qchep-lathep-phquant-ph

TL;DR

Black-hole evaporation raises the information-loss paradox, while gauge/gravity duality proposes a quantum-mechanical description through a dual gauge theory. This paper tests that description with Monte Carlo simulation and finds agreement with quantum-gravity predictions for an evaporating black hole, including the leading corrections.

Abstract

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The discovery of the fact that black holes radiate particles and eventually evaporate led Hawking to pose the well-known information loss paradox. This paradox caused a long and serious debate since it claims that the fundamental laws of quantum mechanics may be violated. A possible cure appeared recently from superstring theory, a consistent theory of quantum gravity: if the holographic description of a quantum black hole based on the gauge/gravity duality is correct, the information is not lost and quantum mechanics remains valid. Here we test this gauge/gravity duality on a computer at the level of quantum gravity for the first time. The black hole mass obtained by Monte Carlo simulation of the dual gauge theory reproduces precisely the quantum gravity effects in an evaporating black hole. This result opens up totally new perspectives towards quantum gravity since one can simulate quantum black holes through dual gauge theories.

Introduction

Hawking radiation and evaporation frame the information-loss paradox: blackbody radiation appears to erase information, conflicting with quantum mechanics. The paper presents gauge/gravity duality as a route to describing an evaporating black hole and reports quantitative evidence for that description.

Introduction

  • The information-loss paradox asks whether information vanishes during black-hole evaporation, contrary to quantum mechanics’ requirement that initial-state information persist.The thought experiment is a book falling into a black hole, with only blackbody radiation observed as the hole evaporates.Source: Introduction · p. 2
  • D-brane bound states provide microscopic black-hole states whose multiplicity explains the entropy of extremal black holes, but not yet a complete account of evaporation.Extremal black holes are a special class that do not evaporate.Source: Introduction · p. 2
  • Gauge/gravity duality relates D-brane black holes to strongly coupled gauge theories, and the paper reports Monte Carlo evidence that the dual theory reproduces quantum-gravity predictions for an evaporating hole.The figure presents the conjecture that black holes in superstring theory are described by a dual gauge theory.Source: Introduction · p. 2, Figure 1 · p. 3

D-particles and the gauge/gravity duality

The section describes how D-particle dynamics can be represented by a gauge theory and how gauge/gravity duality connects that theory to black holes. The finite-N gauge theory is presented as a route to the quantum-gravity regime relevant to information loss.

D-particles and the gauge/gravity duality

  • D-branes are solitonic objects on which open strings end, and their dynamics, including open-string oscillations, is described by a gauge theory.Closed strings mediate gravity, while open strings mediate gauge interactions.Source: D-particles and the gauge/gravity… · p. 3
  • A system of N D-particles is described by a time-dependent gauge theory whose fields are N × N matrices.The D-particles are point-like in nine-dimensional space; the discussion considers uncompactified superstring theory.Source: D-particles and the gauge/gravity… · p. 3
  • At large N, the black hole is large and weakly curved, while quantum-gravity effects become important at small N and can make its specific heat negative.In this small-N regime, the black hole heats as it loses energy and may evaporate completely.Source: D-particles and the gauge/gravity… · p. 4
  • Gauge/gravity duality equates the D-particle superstring system with its gauge theory, which is well-defined at arbitrary N and makes information loss absent if the conjecture is correct.The section contrasts substantial evidence at N = ∞ with limited knowledge at the quantum-gravity level.Source: D-particles and the gauge/gravity… · p. 4

Analysis on the gravity side

The gravity-side analysis incorporates leading 1/N^2 quantum corrections to the black-hole geometry and derives an energy–temperature relation. Those corrections yield a low-temperature negative specific heat associated with instability and evaporation.

Analysis on the gravity side

  • The classical metric describes a D-particle black hole with a horizon at U = U0, where F(U) changes sign.The near-horizon geometry is spherically symmetric in nine-dimensional space.Source: Analysis on the gravity side · p. 4
  • The corrected near-horizon metric shifts the horizon and produces a repulsive force on a test particle, interpreted as Hawking-radiation back-reaction.The quantum corrections are leading-order 1/N^2 effects arising from corrections to the supergravity action.Source: Analysis on the gravity side · p. 5
  • The black-hole energy as a function of temperature is obtained by calculating entropy with Wald’s formula and integrating the first law using the Hawking temperature.The energy is the thermal non-extremal black-hole mass minus the extremal mass, corresponding to internal energy in the dual gauge theory.Source: Analysis on the gravity side · p. 5
  • At sufficiently low temperature, the leading quantum-gravity correction makes the specific heat negative, so the black hole becomes unstable and evaporates.The energy is normalized by λ^1/3 as effective energy, and the expression is given up to O(1/N^4).Source: Analysis on the gravity side · p. 5
  • The analysis omits α′ string-oscillation corrections, while the cited large-N gauge-theory fit had already supported the classical terms and this study targets the 1/N^2 corrections.The α′-corrected expression includes unknown constants.Source: Analysis on the gravity side · pp. 5–6

Analysis on the gauge theory side

The gauge-theory analysis uses Monte Carlo sampling of a matrix model to identify metastable D-particle bound states and estimate their effective internal energy. In the continuum limit, that energy increases as temperature decreases for N = 3, 4, and 5.

Analysis on the gauge theory side

  • The gauge theory is a matrix action sampled after a frequency cutoff Λ is imposed, with continuum energies obtained by extrapolating to Λ = ∞.The finite-cutoff energies are fit using Egauge + const./Λ.Source: Analysis on the gauge theory side · pp. 6–7
  • At small N, diverging bosonic-matrix eigenvalues correspond to Hawking radiation on the gravity side, and the bound-state cluster becomes metastable.The analysis focuses on N = 3, 4, and 5 to probe quantum gravity effects.Source: Analysis on the gauge theory side · p. 7
  • The R2 histogram has a peak near 3.5 for N = 4, while its tail at 4 ≲ R2 ≲ 4.2 reflects runaway behavior.The peak identifies metastable bound states; the tail is associated with instability.Source: Analysis on the gauge theory side · p. 7, Figure 2 · p. 8
  • The energy estimate uses configurations with R2 < x, taking the plateau in E(x)/N^2 as the metastable-state internal energy.A potential term suppresses the instability during simulation.Source: Analysis on the gauge theory side · p. 7
  • The continuum-limit effective internal energy rises as temperature decreases for N = 3, 4, and 5, indicating negative specific heat for the metastable bound states.Figure 3 plots the energy against T and shows fits to behaviors expected from gravity.Source: Analysis on the gauge theory side · p. 9, Figure 3 · p. 9

Testing the gauge/gravity duality

The gauge-theory energies agree with gravity-side predictions including quantum-gravity corrections across the tested temperature range. Fits also support the predicted temperature dependence of the finite-N correction coefficients.

Testing the gauge/gravity duality

  • Straight-line fits through the origin show that (E_gauge − E_gravity)/N^2 scales with 1/N^4 at T = 0.08 and T = 0.11, consistent with the gravity-side result including quantum-gravity corrections.Similar accuracy was observed at other tested temperatures.Source: Figure 4 · p. 10, Testing the gauge/gravity duality · p. 10
  • The gravity-side fit for E_gauge/N^2 is 7.41 T^2.8 − 5.77 T^0.4/N^2 + const./N^4.The O(1/N^4) term is comparable to the O(1/N^2) term, while an O(1/N^6) term is not visible in the data.Source: Testing the gauge/gravity duality · p. 10
  • The fitted c1 coefficient agrees with the gravity prediction c1 = −5.77 T^0.4.The two-parameter fit leaves c1 and c2 free while fitting E_gauge/N^2 = 7.41 T^2.8 + c1/N^2 + c2/N^4.Source: Testing the gauge/gravity duality · p. 10, Figure 5 · p. 11
  • The fitted c2 temperature dependence is consistent with the gravity prediction c2 = c T^−2.6 + ···.A fit gives c = 0.0340(12), c̃ = 0.17(23) × 10^6, and p = 4.30(62) for c2 = c T^−2.6 + c̃ T^p.Source: Testing the gauge/gravity duality · pp. 10, 12

Summary and discussions

The study presents quantitative evidence for gauge/gravity duality at the quantum-gravity level. It identifies an evaporating black hole as an example in which information is not lost and points to gauge-theory simulations as a way to study quantum gravity.

Summary and discussions

  • The work gives quantitative evidence for gauge/gravity duality at the level of quantum gravity.It presents an evaporating black hole as an explicit example in which information is not lost.Source: Summary and discussions · p. 12
  • The evaporating black hole described by the dual gauge theory is based on fundamental principles of quantum mechanics.The authors identify it as an explicit example of information preservation.Source: Summary and discussions · p. 12
  • Monte Carlo simulation of the dual gauge theory offers an approach to studying issues involving quantum gravity.The paper compares this approach with simulating Quantum Chromodynamics on the lattice.Source: Summary and discussions · p. 12
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