Bohr's Correspondence Principle and the Area Spectrum of Quantum Black Holes
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During the last twenty-five years evidence has been mounting that a black-hole surface area has a discrete spectrum. Moreover, it is widely believed that area eigenvalues are uniformly spaced. There is, however, no general agreement on the spacing of the levels. In this Letter we use Bohr's correspondence principle to provide this missing link. We conclude that the area spacing of a black hole is $4\ensuremath{\Elzxh}\mathrm{ln}3$. This is the unique spacing consistent both with the area-entropy thermodynamic relation for black holes, with the Boltzmann-Einstein formula in statistical physics, and with Bohr's correspondence principle.
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The eigenvalues of p are discrete, exhibiting electric charge quantization as integer multiples of an elementary charge.
“Furthermore, they are discrete exhibiting electric charge quantization as integer multiples of an elementary charge (this fact has already been observed in [42] in the asymmetric theory).”
From Symmetry reduction for quantized diffeomorphism-invariant theories of connections (2000) · cites this paper as [42]
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Cites
Bekenstein, J. D. (1973) · Physical Review D · doi:10.1103/physrevd.7.2333 · Cited by 7,555 · see citing works
Stability of a Schwarzschild Singularity
Regge, T., & WHEELER, J. A. (1957) · Physical Review · doi:10.1103/physrev.108.1063 · Cited by 2,737 · see citing works
An analytic representation for the quasi-normal modes of Kerr black holes
Leaver, E. W. (1985) · Proceedings of the Royal Society of London A Mathematical and Physical Sciences · doi:10.1098/rspa.1985.0119 · Cited by 1,315 · see citing works
Sources of Gravitational Radiation
McCarthy, P. J. (1980) · Physics Bulletin · doi:10.1088/0031-9112/31/5/025 · Cited by 748 · see citing works
Reversible and Irreversible Transformations in Black-Hole Physics
Christodoulou, D. (1970) · Physical Review Letters · doi:10.1103/physrevlett.25.1596 · Cited by 705 · see citing works
Scattering of Gravitational Radiation by a Schwarzschild Black-hole
Vishveshwara, C. V. (1970) · Nature · doi:10.1038/227936a0 · Cited by 676 · see citing works
The quantum mass spectrum of the Kerr black hole
Bekenstein, J. D. (1974) · Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento · doi:10.1007/bf02762768 · Cited by 584 · see citing works
Reversible Transformations of a Charged Black Hole
Christodoulou, D., & RUFFINI, R. J. (1971) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.4.3552 · Cited by 493 · see citing works
Long Wave Trains of Gravitational Waves from a Vibrating Black Hole
Press, W. H. (1971) · The Astrophysical Journal · doi:10.1086/180849 · Cited by 446 · see citing works
Spectroscopy of the quantum black hole
Bekenstein, J. D., & Mukhanov, V. F. (1995) · Physics Letters B · doi:10.1016/0370-2693(95)01148-j · Cited by 432 · see citing works
Nollert, H.-P. (1993) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.47.5253 · Cited by 403 · see citing works
Gravitational Radiation from a Particle Falling Radially into a Schwarzschild Black Hole
Davis, M., RUFFINI, R. J., Press, W. H., et al. (1971) · Physical Review Letters · doi:10.1103/physrevlett.27.1466 · Cited by 382 · see citing works
Sources of gravitational radiation
Smarr, L. L. (1979) · Cambridge University Press eBooks · Cited by 334 · see citing works
Area spectrum of the Schwarzschild black hole
Louko, J., & Mäkelä, J. (1996) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.54.4982 · Cited by 148 · see citing works
On the asymptotic distribution of quasinormal-mode frequencies for Schwarzschild black holes
Andersson, N. (1993) · Classical and Quantum Gravity · doi:10.1088/0264-9381/10/6/001 · Cited by 108 · see citing works
The quantum levels of isolated spherically symmetric gravitational systems
Kastrup, H. A. (1996) · Physics Letters B · doi:10.1016/0370-2693(96)00846-5 · Cited by 101 · see citing works
Loustó, C. O. (1995) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.51.1733 · Cited by 88 · see citing works
Gravitational Collapse with Asymmetries
de la Cruz, V., Chase, J. E., & Israel, W. (1970) · Physical Review Letters · doi:10.1103/physrevlett.24.423 · Cited by 80 · see citing works
Exact physical black hole states in generic 2-D dilaton gravity
Barvinsky, A. O., & Kunstatter, G. (1996) · Physics Letters B · doi:10.1016/s0370-2693(96)01261-0 · Cited by 68 · see citing works
Black holes as quantum membranes
Maggiore, M. (1994) · Nuclear Physics B · doi:10.1016/s0550-3213(94)80047-2 · Cited by 66 · see citing works
The spectrum of quantum dust black holes
Peleg, Y. (1995) · Physics Letters B · doi:10.1016/0370-2693(95)00874-k · Cited by 43 · see citing works
Schroedinger Equation of the Schwarzschild Black HolePreprint
Mäkelä, J. (1996) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9602008 · Cited by 4 · see citing works
Flagged first, then by citations
Research works printed but not linked: 5
- [2][2] See for example M. Born, Atomic Physics (Blackie, London, 1969), eighth edition. -
- [7][7] J. D. Bekenstein in XVII Brazilian National Meeting on Particles and Fields, eds. A. J. da Silva et. al. (Brazilian Physical Society, Sao Paulo, 1996), J. D. Bekenstein in Proceedings of the VIII Marcel Grossmann Meeting on General Relativity, eds. T. Piran and R. Ruffini (World Scientific , Singapore, 1998). -
- [11][11] M. Davis, R. Ruffini, W. H. Press and R. H. Price, Phys. Rev. Lett. 27 , 1466 (1971). -
- [12][12] W. H. Press, Astrophys. J. 170 , L105 (1971). -
- [14][14] E. W. Leaver, Proc. R. Soc. A 402 , 285 (1985). -
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Sorted by citations
Symmetry reduction for quantized diffeomorphism-invariant theories of connections
Bojowald, M., & Kastrup, H. A. (2000) · Classical and Quantum Gravity · doi:10.1088/0264-9381/17/15/311 · Cited by 168 · see citing works
Hod, S. (1998) · Physical Review D · doi:10.1103/physrevd.59.024014 · Cited by 70 · see citing works
Self-citationQuantum mechanics of a black hole
Gour, G. (2000) · Physical Review D · doi:10.1103/physrevd.61.124007 · Cited by 20 · see citing works
- Authors
- Hod, S.
- Affiliations
- Hebrew University of Jerusalem (IL)
- Corresponding author
- Hod, Shahar
- ORCID
- Hod, Shahar
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1998-11-16
- Volume
- 81
- Issue
- 20
- Pages
- 4293-4296
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W1973092187
- Cited by
- 729
- References
- 22
- Citations by year
- 2026: 24 · 2025: 43 · 2024: 13 · 2023: 19 · 2022: 24 · 2021: 20 · 2020: 22 · 2019: 22 · 2018: 36 · 2017: 16 · 2016: 20 · 2015: 21 · 2014: 32 · 2013: 47 · 2012: 44
- FWCI
- 3.71
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
- Keywords
- quantum black holes, black hole area spectrum, Bohr correspondence principle, level spacing, black hole thermodynamics, black hole entropy, quantum gravity
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