Bekenstein–Hawking entropy
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| Bekenstein–Hawking entropy | |
|---|---|
| Name | Bekenstein–Hawking entropy |
| Field | Theoretical physics |
| Discovered by | Jacob Bekenstein; Stephen Hawking |
| Year | 1970s |
Bekenstein–Hawking entropy is the proportionality relation assigning an entropy to a black hole proportional to the area of its event horizon and inversely proportional to Planck's constant, uniting ideas from Jacob Bekenstein, Stephen Hawking, Albert Einstein, Max Planck, and John Wheeler. It establishes a bridge between thermodynamics, quantum mechanics, general relativity, statistical mechanics, and information theory as discussed by researchers at institutions such as Princeton University, Cambridge University, University of Maryland, Harvard University, and Institute for Advanced Study. The result underpins modern work involving string theory, loop quantum gravity, AdS/CFT correspondence, and observational programs at facilities like Event Horizon Telescope and LIGO Scientific Collaboration.
Introduction
Bekenstein–Hawking entropy arose from efforts by Jacob Bekenstein to assign entropy to black holes and by Stephen Hawking to compute quantum particle emission from black holes, connecting concepts used by Ludwig Boltzmann, Rudolf Clausius, Satyendra Nath Bose, Enrico Fermi, and Paul Dirac. Early reactions involved discussions at seminars attended by figures including Roger Penrose, Kip Thorne, John Wheeler, and David Hawking and were influenced by results from Karl Schwarzschild solutions, Roy Kerr metrics, and research at Caltech and CERN. The relation reframed problems previously studied by Hermann von Helmholtz, Gibbs, and Maxwell, integrating insights from Andrei Sakharov on induced gravity and prompting work by Gerard 't Hooft and Leonard Susskind.
Derivation and Formula
The formula S = k_B A / (4 l_P^2) (with c and ħ set appropriately) was presented following semiclassical analysis by Stephen Hawking and thermodynamic arguments by Jacob Bekenstein, combining results from Albert Einstein's field equations, Subrahmanyan Chandrasekhar limits, and quantum field theory in curved spacetime developed by Niels Bohr-era successors. Derivations employ techniques originally refined by Paul Dirac for quantum theory, by Richard Feynman in path integrals, and by Julian Schwinger in effective actions, and they invoke boundary terms studied by G. W. Gibbons and Stephen Fulling. Alternative derivations use the Euclidean path integral method associated with Edward Witten and thermodynamic partition functions developed by Lars Onsager analogues in curved backgrounds. The entropy-area relation connects to constants from Max Planck and geometric inputs like surface gravity introduced in work by Subrahmanyan Chandrasekhar and refined by Roger Penrose and Kip Thorne.
Physical Interpretation and Properties
Physically, the entropy counts degrees of freedom associated with the horizon as proposed in debates involving John Wheeler's "it from bit" idea and Rolf Landauer's principle relating information and thermodynamics. This interpretation motivated cross-disciplinary dialogue among researchers at Los Alamos National Laboratory, Bell Labs, IBM Research, and university groups led by Gerard 't Hooft, Leonard Susskind, Ashoke Sen, and Juan Maldacena. Properties such as the generalized second law trace conceptual lineage to Ludwig Boltzmann and Rudolf Clausius and were tested conceptually through gedankenexperiments by James Bardeen, Ted Jacobson, and William Unruh. The area-proportional scaling contrasts with volume-scaling expectations familiar from Gibbs ensembles and stimulated the holographic principle advocated by Gerard 't Hooft and Leonard Susskind.
Quantum and Statistical Mechanical Approaches
Statistical accounts attempt to reproduce the entropy by counting microstates in frameworks proposed by String Theory groups centered at Institute for Advanced Study, Harvard University, and Princeton University and by loop-quantum-gravity researchers at Penn State University, Pennsylvania State University, and MIT. Notable microstate counts include work by Andrew Strominger and Cumrun Vafa in supersymmetric black holes, and loop quantum gravity computations led by Carlo Rovelli and Abhay Ashtekar. Approaches also involve techniques from conformal field theory researchers such as Alexander Zamolodchikov, Paul Ginsparg, and John Cardy and utilize holographic dualities like AdS/CFT correspondence developed by Juan Maldacena. Quantum corrections computed using renormalization group methods draw on tools refined by Kenneth Wilson, Giovanni Jona-Lasinio, and Miguel Virasoro-related algebra in string contexts discussed by Michael Green and John Schwarz.
Applications and Implications
Applications extend to discussions of information loss raised by Stephen Hawking and responded to by proposals from Leonard Susskind, Gerard 't Hooft, Juan Maldacena, Joseph Polchinski, and Samir Mathur including fuzzball models and complementarity debates. Implications reach cosmology through considerations by Alan Guth, Andrei Linde, Alexander Vilenkin, and Stephen Hawking on inflationary entropy budgets, and to quantum information theory developed by groups at MIT, Stanford University, and Caltech exploring entanglement entropy and quantum error correction codes influenced by Peter Shor and Daniel Gottesman. Observational relevance ties to programs at Event Horizon Telescope for horizon imaging and to gravitational-wave astronomy via LIGO Scientific Collaboration and Virgo Collaboration, which probe black hole parameters entering entropy expressions.
Open Problems and Research Directions
Outstanding problems include the microscopic origin of entropy for general black holes debated by researchers at Perimeter Institute, CERN, Institute for Advanced Study, and university groups led by Raphael Bousso, Gary Horowitz, Samir Mathur, Juan Maldacena, and Edward Witten; the reconciliation of unitarity with semiclassical evaporation considered by Stephen Hawking, Don Page, Joseph Polchinski, and Ahmed Almheiri; precise computations of logarithmic corrections pursued by Ashoke Sen and Raju-affiliated teams; and tests of holography in non-AdS spacetimes investigated by Eva Silverstein and Xiao Liu. Future directions connect to quantum gravity programs at Perimeter Institute, experimental probes by Event Horizon Telescope successors, theoretical developments inspired by Claude Shannon's information theory, and interdisciplinary work at institutions like Santa Fe Institute exploring emergent descriptions bridging statistical mechanics and fundamental theories.