Bekenstein bound
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| Name | Bekenstein bound |
| Field | Theoretical physics |
| Proposer | Jacob Bekenstein |
| Year | 1981 |
| Related | Black hole thermodynamics; Hawking radiation; entropy bounds |
Bekenstein bound The Bekenstein bound is a proposed limit on the maximum entropy, or information, that can be contained within a finite region of space having finite energy. It was introduced in the context of Jacob Bekenstein's work on black hole thermodynamics and is connected to concepts in statistical mechanics, quantum mechanics, and general relativity. The bound has been influential in discussions involving Stephen Hawking's discovery of Hawking radiation and broader debates about the nature of information in quantum gravity.
Introduction
The proposal originated from attempts to reconcile the generalized second law of thermodynamics with processes involving black holes and was motivated by thought experiments such as lowering a thermodynamic system toward the event horizon considered in analyses by John Wheeler and critiques engaged by peers including William Unruh and Robert Wald. Bekenstein framed the limit using concepts familiar from thermodynamics and information theory as applied to systems characterized by a total energy and a circumscribing radius, invoking analogies with Bekenstein–Hawking entropy of black holes and with limits discussed in work by Claude Shannon and Rolf Landauer.
Formulation
Bekenstein proposed that for a physical system of energy E contained within a sphere of radius R, the entropy S satisfies an inequality of the form S ≤ 2πRE/ħc (in units with Planck's constant ħ and the speed of light c). The formulation appeals to comparisons with the entropy of a Schwarzschild black hole, drawing on earlier results by Stephen Hawking and the entropy-area relation developed by Bekenstein himself and collaborators such as James M. Bardeen and Ted Jacobson. The bound is often expressed in information-theoretic terms by converting entropy to bits via the Boltzmann constant and referencing limits similar to those discussed in Shannon entropy contexts and in operational analyses by Rolf Landauer.
Derivations and theoretical foundations
Several heuristic and more formal arguments underpin the bound. Bekenstein's original argument used thought experiments involving lowering a box toward a black hole and invoking the generalized second law, engaging with analyses by Robert Wald that scrutinize entropy accounting in curved spacetime. More formal derivations attempt to connect the bound to the covariant entropy bound proposed by Raphael Bousso and to proofs leveraging properties of quantum field theory in curved backgrounds examined by groups associated with Harvard University and Princeton University research programs. Work linking the bound to properties of modular Hamiltonians and entanglement entropy cites developments in conformal field theory and techniques from researchers such as Alexei Kitaev and Edward Witten, and draws on insights from AdS/CFT correspondence investigations by Juan Maldacena and collaborators.
Applications and implications
The Bekenstein bound has been invoked in arguments about the maximum information storage in physical media, influencing speculative engineering discussions among communities interested in quantum computing and limits on memory density informed by research from institutions like IBM and Google. It also plays a conceptual role in debates about the black hole information paradox debated by figures including Leonard Susskind and Gerard 't Hooft, and informs cosmological entropy accounting in studies by teams at CERN and national laboratories. In theoretical cosmology, the bound intersects with constraints considered in inflationary scenarios and with entropy estimates in the context of the observable universe discussed by astrophysicists at observatories such as Hubble Space Telescope teams.
Criticisms and limitations
Critics have pointed to ambiguities in defining the radius R and the energy E for arbitrary systems, debates emphasized in critiques by Robert Wald and analyses by researchers linked to Perimeter Institute and Institute for Advanced Study. Counterexamples and loopholes have been proposed using systems with significant self-gravitation, long-range interactions, or exotic boundary conditions examined in papers by groups at Cambridge University and MIT, and some arguments suggest that the bound requires additional hypotheses (e.g., weak gravity, isolation) to hold universally. Discussions also consider alternative entropy measures, with contributions from Juan Maldacena-related research on entanglement entropy and from studies in quantum field theory that reveal regimes where naive application of the bound is problematic.
Related bounds and extensions
Related proposals include the Bousso bound (covariant entropy bound), the holographic principle championed by Gerard 't Hooft and Leonard Susskind, and the Bekenstein–Hawking entropy relation for black holes developed in collaboration with figures such as Stephen Hawking and James Bardeen. Extensions explore entropy bounds in anti-de Sitter space and in frameworks motivated by AdS/CFT correspondence, with technical links to results by Maldacena and studies of entanglement wedges by groups at universities including Princeton University and Harvard University. Other related inequalities appear in quantum information theory literature connected to Rolf Landauer's principle and to entropy-energy relations studied by researchers affiliated with Bell Labs and Los Alamos National Laboratory.