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Cited by 2,496Black Holes and Theoretical PhysicsAstrophysical Phenomena and ObservationsRelativity and Gravitational Theory

Black hole entropy is the Noether charge

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Wald, Robert M.·1993·Physical Review D
DOI 10.1103/physrevd.48.r3427
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Abstract

We consider a general, classical theory of gravity in $n$ dimensions, arising from a diffeomorphism-invariant Lagrangian. In any such theory, to each vector field ${\ensuremath{\xi}}^{a}$ on spacetime one can associate a local symmetry and, hence, a Noether current ($n\ensuremath{-}1$)-form j and (for solutions to the field equations) a Noether charge ($n\ensuremath{-}2$)-form Q, both of which are locally constructed from ${\ensuremath{\xi}}^{a}$ and the fields appearing in the Lagrangian. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon (with bifurcation surface $\ensuremath{\Sigma}$), and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply $2\ensuremath{\pi}$ times the integral over $\ensuremath{\Sigma}$ of the Noether charge ($n\ensuremath{-}2$)-form associated with the horizon Killing field. Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the "second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via "Euclidean methods" also is explained.

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  • The space of solutions has a natural symplectic structure equivalent to the standard symplectic structure on phase space.

    “the space of solutions has natural symplectic structure [175, 272], which can be shown to be equivalent to the standard symplectic structure on phase space.”

    From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [272]

    The passage labeled Abstract (p. 6) defines a symplectic current from variations of the dynamical fields, supporting the claim’s symplectic-structure component. The passage labeled Abstract (p. 9) relates the construction to Hamiltonian evolution on phase space, but our copy does not establish equivalence to the standard phase-space symplectic structure. (LLM, conf=0.78)

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Research works printed but not linked: 3

  1. [3][3] D. Sudarsky and R.M. Wald, Phys. Rev. D 46 , 1453 (1992); R.M. Wald, ' The first law of black hole mechanics' in Directions in General Relativity , vol. 1, ed. by B.L. Hu, M. Ryan, and C.V. Vishveshwara (Cambridge University Press, Cambridge, 1993). See also J.D. Brown and J.W. York Phys. Rev. D47 , 1407 and 1420 (1993). -
  2. [9][9] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973). -
  3. [19][19] S.W. Hawking, 'The Path Integral Approach to Quantum Gravity', in General Relativity, an Einstein Centennary Survey , ed. by S.W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979)

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Authors
Wald, R. M.
Affiliations
University of Chicago (US)
Corresponding author
Wald, Robert M.
ORCID
Wald, Robert M.
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
1993-10-15
Volume
48
Issue
8
Pages
R3427-R3431
ISSN
0556-2821 (ISSN-L)
DOI
10.1103/physrevd.48.r3427
OpenAlex
W2108734736
Cited by
2,496
References
2
Citations by year
2026: 140 · 2025: 148 · 2024: 137 · 2023: 148 · 2022: 152 · 2021: 130 · 2020: 136 · 2019: 118 · 2018: 123 · 2017: 111 · 2016: 120 · 2015: 99 · 2014: 90 · 2013: 91 · 2012: 84
FWCI
11.68
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • PubMed (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 › Astronomy and Astrophysics › Astrophysical Phenomena and Observations
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
Keywords
black hole entropy, Noether charge, conserved currents, black hole mechanics, first law of black hole thermodynamics, bifurcate Killing horizon, stationary black holes, positive energy

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