Bekenstein–Hawking formula

Bekenstein–Hawking formula
Eric Gaba (Sting - fr:Sting) · Public domain · source
Name	Bekenstein–Hawking formula
Field	Theoretical physics
Discoverer	Jacob Bekenstein; Stephen Hawking
Year	1970s

Bekenstein–Hawking formula The Bekenstein–Hawking formula relates the entropy of a black hole to the area of its event horizon, linking Jacob Bekenstein, Stephen Hawking, Albert Einstein, Karl Schwarzschild, and Roger Penrose–style research in gravitation and quantum theory. It sits at the crossroads of General relativity, Quantum field theory, Thermodynamics, Statistical mechanics, and later work by researchers associated with Gerard 't Hooft, Leonard Susskind, and Juan Maldacena. The formula catalyzed developments in Black hole thermodynamics, String theory, Loop quantum gravity, and discussions at institutions such as Princeton University, Cambridge University, and Institute for Advanced Study.

History and development

The historical arc begins with Karl Schwarzschild solutions to Einstein field equations and the classical notion of event horizons emerging in the context of Subrahmanyan Chandrasekhar and David Finkelstein analyses; Jacob Bekenstein proposed an entropy-area relation influenced by prior work of John Wheeler, Roger Penrose, and conceptual advances at Yale University and Hebrew University. Stephen Hawking’s semiclassical calculation, building on Hartle–Hawking state ideas and techniques from Paul Dirac-inspired quantum field methods, produced radiation from black holes that fixed the proportionality constant and connected to the Hawking temperature concept discussed at Cambridge University and in seminars involving Gary Gibbons and James Hartle. The development prompted responses across communities including Gerard 't Hooft’s holographic arguments, Leonard Susskind’s complementarity proposals, and Jacob Bekenstein’s later refinements; workshops at Perimeter Institute and collaborations with Edward Witten and Andrew Strominger further shaped the field.

Mathematical formulation

The formula equates the entropy S of a stationary black hole to one quarter of the area A of its event horizon in Planck units: S = k_B A / (4 l_P^2), where constants introduced in derivations involve Max Planck, Boltzmann's constant k_B, and the Planck length defined via Max Planck units and Isaac Newton’s gravitational constant G and Paul Dirac's ħ; the semiclassical temperature T is related via surface gravity κ as in formulations attributed to Stephen Hawking and James Bardeen. The expression appears in analyses using metrics named for Karl Schwarzschild, Roy Kerr, and Reissner–Nordström and is used in comparisons with thermodynamic laws originally articulated by Sadi Carnot-inspired formulations and black hole analogues to the Zeroth law of thermodynamics and laws advanced by James Bardeen and Brandon Carter. Mathematical precision has been pursued through techniques from Riemannian geometry, methods used by Bernard Schutz and Yvonne Choquet-Bruhat, and quantum state constructions connected to Hartle–Hawking state and Unruh effect analyses.

Physical interpretation and implications

Physically, the formula implies a deep link between geometry captured by horizons studied in Karl Schwarzschild and Roy Kerr spacetimes and microscopic degrees of freedom speculated in frameworks by Edward Witten, Juan Maldacena, Gerard 't Hooft, and Leonard Susskind. It suggests a holographic scaling that inspired the AdS/CFT correspondence developed by Juan Maldacena and exploited in work at Princeton University and Institute for Advanced Study. The entropy-area relation constrains proposals for quantum gravity from String theory labs led by Joseph Polchinski and canonical programs advocated by Abhay Ashtekar at Penn State University. It raises puzzles such as information loss debated by John Preskill, Don Page, and Stephen Hawking and motivates concepts like black hole complementarity and firewall proposals discussed by Almheiri, Marolf, Polchinski, Sully communities.

Derivations and approaches

Derivations include Hawking’s semiclassical quantum field calculation in curved spacetime relying on Bogoliubov transformations used in Paul Dirac-style quantization, Euclidean path-integral techniques influenced by Richard Feynman and Stephen Hawking’s Euclidean gravity, and statistical counts of microstates in String theory by Andrew Strominger and Cumrun Vafa. Canonical quantization approaches from Loop quantum gravity by Abhay Ashtekar and combinatorial counting of punctures on spin networks have produced the entropy proportionality with Immirzi-parameter discussions linked to Giorgio Immirzi. Holographic derivations leverage AdS/CFT correspondence and entropy bounds proposed by Gerard 't Hooft and Leonard Susskind, while path-integral and anomaly-based methods have roots in techniques used by Stephen Adler and L. Parker.

Generalizations and extensions

Extensions include generalized entropy formulae for charged and rotating solutions like Reissner–Nordström and Kerr–Newman black holes, higher-curvature corrections appearing in effective actions studied in John Wheeler-inspired quantum gravity programs and string compactifications explored by Michael Green and John Schwarz. Holographic entanglement entropy proposals by Ryu–Takayanagi (building on Juan Maldacena and Leonard Susskind ideas) relate boundary conformal field theories studied by groups at Institute for Advanced Study and Harvard University to bulk geometric areas, while quantum corrections and generalized second law discussions involve work by Ted Jacobson, Rafael Sorkin, and Raphael Bousso. Applications have been pursued in cosmological horizon contexts such as De Sitter space analyses linked to Alan Guth-inspired inflationary scenarios and in analogue systems following experiments influenced by Bill Unruh's proposals.

Experimental and observational status

Direct laboratory tests are beyond current capabilities given Planck-scale units introduced by Max Planck and technological limits faced by institutions like CERN and observatories such as Event Horizon Telescope and LIGO Scientific Collaboration; nevertheless, indirect support arises from astrophysical observations of accretion disks around candidates associated with Sagittarius A* and quasars studied by teams at Harvard–Smithsonian Center for Astrophysics and from analogue gravity experiments motivated by Bill Unruh and performed in condensed-matter labs connected to MIT and University of Chicago. Theoretical consistency checks occur in calculations by Edward Witten, Andrew Strominger, and Joseph Polchinski that match microscopic counts to the area law in specific String theory backgrounds and through thought experiments promoted by John Preskill and Don Page.

Category:Black hole thermodynamics