The Black Hole, 25 Years After

The following sections are included:

• CONTENTS

• PREFACE

The following sections are included:

• Introduction

• Finite Boundaries

• The Action

• Lorentzian and Euclidean Notation

• Functional Integrals

• Black Hole Entropy

• Acknowledgments

• References

All the causally regular geometries obtained from (2+1)-anti-de Sitter space by identifications by isometries of the form P → (exp 27πξ)P, where ξ is a self- dual Killing vector of so(2,2), are explicitely constructed. Their remarkable symmetry properties (Killing vectors, Killing spinors) are listed. These solutions of Einstein gravity with negative cosmological constant are also shown to be invariant under the string duality transformation applied to the angular translational symmetry ϕ → ϕ + a. The analysis is made particularly convenient through the construction of global coordinates adapted to the identifications.

The following sections are included:

• Problems of Quantum Gravity

• Possible Approaches

• Our String Approach

• Two-dimensional stringy black holes

• Evaporation and decay of black holes [4]

• More comments on 4 dimensions [5]

• References

The emergence from vacuum fluctuations of the Hawking radiation of an incipient black hole is analyzed in conventional field theories. These fluctuations travel in that space-time and carry transplanckian energies sharply localized on cisplanckian distances. We argue that such features cannot accommodate gravitational nonlinearities. We discuss the possibility of generating Hawking quanta from tamed vacuum fluctuations and point out the relevance of this alternative for the back-reaction problem and for the unitary issue.

John Wheeler has described the issue of gravitational collapse as “the greatest crisis in physics of all time”. Coming to terms with the limitless crushability of matter and the exposure of solidity as a hollow facade has also been psychologically traumatic. Acceptance and understanding of superdense objects were long impeded by blind spots and emotional resistance. This article recalls part of the story.

Determining the internal structure of a black hole is really an evolutionary problem, with precisely known initial data. In principle, the evolution can be followed to within Planck times of the singularity at the inner horizon, using only well-established physical laws. This article is a progress report and a survey of open questions.

The classical Bekenstein entropy of a black hole is argued to arise from configurations of strings with ends which are frozen on the horizon. Quantum corrections to this entropy are probably finite unlike the case in quantum, field theory. Finally it is speculated that all black holes are single string states. The level density of strings is of the right order of magnitude to reproduce the Bekenstein entropy.

This report follows closely an article published in The Physiscal Review D [1]. The opportunity is taken to correct herein an error in that article.

We will be concerned with the action and entropy of extreme and nonextreme black holes [2,3]. To put into evidence as clearly as possible the distinction between extreme and non extreme holes, we first perform the analysis for the non-extreme case and then see how it is modified in the extreme case…

The S-matrix Ansatz for the construction of a quantum theory of black holes is further exploited. We first note that treating the metric tensor g_μν as an operator rather than a background allows us to use a setting where information is not lost. But then we also observe that the ‘trans-Planckian’ particles (particles with kinetic energies beyond the Planck energy) need to be addressed. It is now postulated that they can be transformed into ‘cis-Planckian particles’ (having energies less that the Planck energy). This requires the existence of a delicate algebra of operators defined at a black hole horizon. Operators describing ingoing particles are mapped onto operators describing outgoing ones, preserving their commutator algebra. At short distance, the transverse gravitational back reaction dictates a discrete lattice of data points, and at large distance the algebra must reproduce known interactions of the Standard Model of elementary particles. It is suggested that further elaboration of these ideas requiring complete agreement with general relativity and unitarity should lead to severe restrictions concerning the inter-particle interactions.

The following sections are included:

• Crunch Inescapable

• The Mistery Of The Missing Mass

• How Many Gravity-Wave MACHOS (GMs) Today?

• A Cosmic Corridor

• No Escape Hatch

• Black Hole Evaporation

• It From Bit

• Acknowledgment

• References

The following sections are included:

• Introduction and Summary

• Form of black holes

• Thermal interpretation
  • Temperature and entropy
  • Grey-body factors
  • Breakdown of thermal description

• Formulation of perturbation equations
  • The axial equations
  • The polar equations
  • Particular Integral and Reduction of the Polar Equations
  • Magnetically Charged Dilaton Black Holes
  • Remark on stability

• Qualitative Features of the Potentials
  • Spectator Scalar Field
  • Axial and Polar potentials

• Emission Spectrum

• The Callan-Rubakov Mode

• Final Comments

• REFERENCES

• FIGURE CAPTIONS

The following sections are included:

• Introduction
  • Challenges for the quantum theory of black holes
  • Black holes and elementary particle properties

• Discrete Gauge Hair: Preliminary Discussion
  • No-hair theorems
  • Discrete gauge symmetry
  • Discrete electric gauge hair
  • Discrete magnetic hair

• The Reissner–Nordström Black Hole
  • The Euclidean Path Integral
  • The Semiclassical Limit
  • The Electric Charge Projection
  • The Schwarzschild Black Hole
  • The Electrically Charged Black Hole
  • The Extreme Solutions
  • Magnetically Charged Black Hole
  • Magnetically Charged Black Holes in Yang–Mills Theory

• Screened Electric Charge
  • Z_N Electric Charge Projection
  • Semiclassical Thermodynamics
  • The Thin String Limit
  • The Thick String Limit
  • The Electric Field
  • The Vortex and the No-Hair Theorem

• Screened Magnetic Charge
  • Z_N Magnetic Charge Projection
  • Electric-Magnetic Duality
  • The Thin String Limit
  • The Thick String Limit
  • Screened Magnetic Charge vs. Screened Electric Charge

• Dual Formulation of Broken Symmetry Phases
  • Goldstone boson
  • Gauge coupling
  • Axion

• Duality and Hair
  • Dual charge
  • Dual picture of the lasso process
  • Global symmetry and hair

• Primary and Secondary Hair

• REFERENCES

• FIGURE CAPTIONS