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arXiv:1304.2305v2

[gr-qc]10Apr2013

arXiv:1304.2305 KATP082013 (v2)

Black hole solution without curvature singularity

F.R. Klinkhamer

Institute for Theoretical Physics,

Karlsruhe Institute of Technology (KIT),

76128 Karlsruhe, Germany

AbstractAn exact solution of the vacuum Einstein field equations over a nonsimply connected manifold is

presented, which is spherically symmetric and has no curvature singularity. This solution can be

considered to be a regularization of the singular SchwarzschildKruskalSzekeres solution over R4.

A heuristic discussion is given of how spherically symmetric collapse of matter in R4 may result in

this nonsingular black hole solution, if quantum gravity effects allow for topology change near the

center.

PACS numbers: 04.20.Cv, 02.40.Pc

Keywords: general relativity, topology

Electronic address: frans.klinkhamer@kit.edu

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The vacuum Einstein field equations over M4 = R4 have a spherically symmetric static

solution, the SchwarzschildKruskalSzekeres solution [1? 4]. Recently, a related solution

has been found for the vacuum Einstein field equations over M4, a particular nonsimply-connected manifold [5]. The goal of the present article is to discuss the relation between

these two solutions.

The SchwarzschildKruskalSzekeres metric [13] is given by the following line element

(GN = c = 1):

ds2r>0M4=R4

= 32 M3er/(2M)

r

dv2 + du2

+ r2

d2 + sin2 d2

, (1)

with r > 0 expressed in terms of the coordinates u and v by the relation

r

2M 1

er/(2M) = u2 v2 > 1 . (2)

The solution (1) has M > 0 as a free parameter (the solution for the case M = 0 is the

standard Minkowski metric). The solution approaches a genuine singularity for r 0,

as shown by the divergence of the Kretschmann scalar (defined in terms of the Riemann

curvature tensor),

RR

M4

= 48M2

r6. (3)

Further details can be found in, e.g., Ref. [4].

Consider, now, the vacuum Einstein field equations overM4 = R M3 , (4)where the 3-space M3 is a noncompact, orientable, nonsimply-connected manifold withoutboundary. In fact, there are the following homeomorphisms:

M3 RP3 {point} SO(3) {point} . (5)The explicit construction of M3 has been given in Ref. [5]: surgery is performed on R3by removal of the interior of a ball with radius b and identification of antipodal points on

the boundary of the ball (2-sphere with radius b). In Ref. [5], also appropriate coordinates

are reviewed, for example, the coordinate Y (, +) instead of the standard radial

coordinate r [b, ).

The new exact vacuum solution [5] over M4 involves two parameters, b and M, which,for the present purpose, are taken to be related as follows:

0 < b < 2M . (6)

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Inspection of the solution as given by Eqs. (4.1), (5.1b), and (5.1c) in Ref. [ 5] shows it to be

precisely of the form of the standard Schwarzschild solution in the exterior region (r > 2M)

if Y2 + b2 and r2 are identified. This observation agrees with Birkhoffs theorem [4] and,

in turn, allows us to obtain the extended solution by a straightforward adaptation of the

KruskalSzekeres procedure. The extended metric is then given by the following line element:

ds2M4

= 32 M3e/(2M)

dV2 + dU2

+ 2

d2 + sin2 d2

, (7)

with b > 0 expressed in terms of the coordinates U and V by the relation

2M 1

e/(2M) = U2 V2

b

2M 1

eb/(2M) . (8)

The previous coordinates Y, T (, +) from Eqs. (2.6) and (2.8) in Ref. [5] are recov-

ered as follows:

Y2 = 2 b2 , (9a)

T

4M=

tanh1(V /U) for |U| > |V| ,

tanh1(U/V) for |U| |V| ,(9b)

where the top row on the right-hand side of (9b) applies to the exterior regions ( > 2M)

and the bottom row to the interior regions ( 2M).

The UV spacetime diagram is similar to Fig. 31.4.b of Ref. [4] but with the r = 0

hyperbolae relabeled = b. Most importantly, these = b hyperbolae are included in the

spacetime manifold M4 because M3 is a manifold without boundaries and without curvaturesingularities. In fact, Ref. [5] already gave the result

RR

M4

= 48M2

6, (10)

which remains finite because > 0, according to (8) with b > 0. Note that, even without

curvature singularity, the UV spacetime diagram mentioned above has event horizons at

= 2M.1

Purely mathematically, the nonsingular solution (7) with parameter b > 0 can be con-

sidered to be a regularization of the singular solution (1). But it is also possible that this

nonsingular solution appears in a physical context.

1 The geodesics near the core (= b or Y= 0) from the vacuum solution (7) will be discussed elsewhere.

It is, however, clear that, for the issue of geodesic completeness (i.e., having a traversable defect), the

inclusion of matter will be essential; see also the remark in Footnote 2 below.

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Start from a nearly flat spacetime (metric approximately equal to the Minkowski metric),

where a large amount of matter with total mass M is arranged to collapse spherically

symmetrically. Within the realm of classical Einstein gravity, one expects to end up with

part of the singular solution (1); see, for example, Fig. 32.1.b of Ref. [4]. But, very close

to the final curvature singularity, the local spacetime integral of the action density from ( 1)

differs from that from (7) by an amount and, as argued by Wheeler (cf. Secs. 34.6

and 43.4 of Ref. [4]), the local topology of the manifold may change by a quantum jump. If

that possibility is indeed available for appropriate matter content,2 there is a chance that

the manifold changes from the structure (1) to that of (7), thereby removing the classical

curvature singularity.

ACKNOWLEDGMENTS

It is a pleasure to thank S. Antoci, D. Grumiller, and C. Rahmede for discussions.

[1] K. Schwarzschild, Ueber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen

Theorie, Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fuer

Mathematik, Physik, und Technik (1916), S. 189 [available from http://de.wikisource.org/wiki].

[2] M.D. Kruskal, Maximal extension of Schwarzschild metric, Phys. Rev. 119, 1743 (1960).

[3] G. Szekeres, On the singularities of a Riemannian manifold, Publ. Math. Debrecen 7, 285

(1960).

[4] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman, New York, 1973).

[5] F.R. Klinkhamer and C. Rahmede, Nonsingular spacetime defect, arXiv:1303.7219.

[6] D. Gannon, Singularities in nonsimply connected space-times, J. Math. Phys. 16, 2364

(1975).

2 Most likely, negative energy densities must be available, in order to avoid a singularity theorem [6] appli-

cable to nonsimply-connected manifolds (see Ref. [5] for a related discussion).

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