The Euclidean gravitational action as black hole entropy, singularities, and spacetime voids

Journal Article · Tue Apr 15 00:00:00 EDT 2008 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.2898115· OSTI ID:21100259

Castro, Carlos ^[1]

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1. Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, Georgia 30314 (United States)

We argue why the static spherically symmetric vacuum solutions of Einstein's equations described by the textbook Hilbert metric g{sub {mu}}{sub {nu}}(r) is not diffeomorphic to the metric g{sub {mu}}{sub {nu}}(|r|) corresponding to the gravitational field of a point mass delta function source at r=0. By choosing a judicious radial function R(r)=r+2G|M|{theta}(r) involving the Heaviside step function, one has the correct boundary condition R(r=0)=0, while displacing the horizon from r=2G|M| to a location arbitrarily close to r=0 as one desires, r{sub h}{yields}0, where stringy geometry and quantum gravitational effects begin to take place. We solve the field equations due to a delta function point mass source at r=0, and show that the Euclidean gravitational action (in ({Dirac_h}/2{pi}) units) is precisely equal to the black hole entropy (in Planck area units). This result holds in any dimensions D{>=}3. In the Reissner-Nordstrom (massive charged) and Kerr-Newman black hole case (massive rotating charged) we show that the Euclidean action in a bulk domain bounded by the inner and outer horizons is the same as the black hole entropy. When one smears out the point-mass and point-charge delta function distributions by a Gaussian distribution, the area-entropy relation is modified. We postulate why these modifications should furnish the logarithmic corrections (and higher inverse powers of the area) to the entropy of these smeared black holes. To finalize, we analyze the Bars-Witten stringy black hole in 1+1 dimension and its relation to the maximal acceleration principle in phase spaces and Finsler geometries.

• OSTI ID: 21100259
• Journal Information: Journal of Mathematical Physics, Vol. 49, Issue 4; Other Information: DOI: 10.1063/1.2898115; (c) 2008 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ACCELERATION
BLACK HOLES
BOUNDARY CONDITIONS
DELTA FUNCTION
EINSTEIN FIELD EQUATIONS
ENTROPY
EUCLIDEAN SPACE
GAUSS FUNCTION
GEOMETRY
GRAVITATIONAL FIELDS
HILBERT SPACE
MASS
MATHEMATICAL SOLUTIONS
PHASE SPACE
POINT CHARGE
QUANTUM GRAVITY
SINGULARITY
SPACE-TIME
STRING MODELS
STRING THEORY