Aharon Davidson1, 2 and Uzi Paz1. Received September 10, 1999. Removing a black hole conic singularity by means of Kruskal representation is equivalent to ...
Foundations of Physics, Vol. 30, No. 5, 2000
Extensible Embeddings of Black-Hole Geometries Aharon Davidson 1, 2 and Uzi Paz 1 Received September 10, 1999 Removing a black hole conic singularity by means of Kruskal representation is equivalent to imposing extensibility on the KasnerFronsdal local isometric embedding of the corresponding black hole geometry. Allowing for globally nontrivial embeddings, living in KaluzaKlein-like M 5_S 1 (rather than in standard Minkowski M 6) and parametrized by some wave number k, extensibility can be achieved for apparently ``forbidden'' frequencies | in the range | 1(k) | | 2(k). As k 0, | 1, 2(0) | H (e.g., | H =14M in the Schwarzschild case) such that the HawkingGibbons limit is fully recovered. The various Kruskal sheets are then viewed as slices of the KaluzaKlein background. Euclidean k discreteness, dictated by imaginary time periodicity, is correlated with flux quantization of the underlying embedding gauge field.
1. INTRODUCTION Local isometric embeddings (1) of a curved d-dimensional manifold, within some D-dimensional flat background, have been traditionally invoked to classify (2) the variety of general relativity solutions. Such an embedding is fully characterized by (i) its induced metric, (ii) the associated YangMills gauge field, and (iii) the extrinsic curvature. Together, on consistency grounds, they are subject to Gauss, Codazzi, and Ricci equations. Worth recalling are interesting attempts to v Interpret the embedding functions as alternative canonical variables for gravity, (3)
1
Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel; e-mail: davidsonbgumail.bgu.ac.il. 2 To whom correspondence should be addressed. 785 0015-9018000500-0785 18.000 2000 Plenum Publishing Corporation
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v Relate the normal space symmetries with the electronuclear interactions, (4) and v View the Universe as an (3+1)-dimensional bubble. (5) Depending on the differentiable nature of the embedding functions, D 12 d(d+1) for analytic embeddings, (6) whereas D 12 d(d+3) if integrability is required. (7) Of particular interest are the (3+1)-dimensional radially symmetric solutions; they fall categorically into the D=6 embedding class. This latter fact was already known to Kasner (8) who presented a (4+2)-dimensional embedding of the exterior Schwarzschild geometry. This embedding has, however, the double drawback of being non-causal and nonextensible. It is only the Fronsdal (5+1)-dimensional embedding (9) which has the advantage of being one-to-one correlated with the KruskalSzekeres manifold. (10) As far as black holes geometries are concerned, it is well known that the removal of apparent horizon singularities in the Lorentzian (Kruskal) representation can be translated into discharging conic singularities in the Euclidean regime by means of the HawkingGibbons (11) periodicity condition. However, less familiar is the fact that this can also be achieved by imposing extensibility in the KasnerFronsdal approach. In this paper, following a short pedagogical introduction, we attempt to carry the concept of extensibility one step further. While respecting the Hawking Gibbons limit, we explicitly construct extensible Schwarzschild embeddings for apparently ``forbidden'' periodicities (which deviate from the Hawking Gibbons limit). This may (or maynot) have some non-trivial impacts on black hole physics. (12) To be more specific, but keeping a certain amount of generality, consider the radially symmetric 4-metric ds 2 =&A(r) dt 2 +
1 dr 2 +r 2 d0 B(r)
(1)
where d0#d 2 +sin 2 d. 2. For this metric describe a black hole, - AB must well behave near the critical radius r h , the largest root of A(r h )=0. Indeed, invoking a set of Kruskal coordinates u=C(r) cosh |t v=C(r) sinh |t
=
(2)
one is led to an extremely compelling requirement, namely that A exp(&2| dr- AB ) must approach a non-zero finite value as r r h .
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This comes to assure a non-singular scale factor in front of the Kruskal light-cone combination (&dv 2 +du 2 ). In turn, the parameter | gets fixed |=
1 2
AB dAdr }
(3) r=rh
Upon Euclidization t it E , one faces an imaginary time periodicity of 2?|. The latter is known to be the HawkingGibbons (11) key to BekensteinHawking black hole thermodynamics. 2. FRONSDAL EMBEDDING An alternative formalism, less familiar though, involves the embedding of the black hole geometry within a flat M 6 background 5
ds 2 =&dy 20 + : dy 2n
(4)
n=1
Apart from the usual assignments y 1 =r cos y 2 =r sin cos . y 3 =r sin sin .
=
(5)
which define the radial marker, one further introduces y 0 = f (r) sinh |t y 4 = f (r) cosh |t y 5 = g(r)
=
(6)
This coordinate system is supposed to cover the (say) | y 0 y 4 | 1 section of the 4-manifold characterized by y 24 & y 20 =A| 2. After some algebra we arrive at f (r)= dg dr
\ +
2
1 -A |
1 dA 1 = &1+ & 2 B 4| A dr
\ +
2
(7)
One may now verify that dgdr remains finite at the horizon for that | given by Eq. (3). This establishes the correspondence between the Kruskal and the Fronsdal schemes.
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On pedagogical grounds, let us focus attention on the Schwarzschild geometry specified by A(r)=B(r)=1&(2mr). Studying carefully the details of the embedding, one infers that v For |>14m, the embedding does not cover the interior strip (m2| 2 ) 13
