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arXiv:1003.3918v1 [hep-th] 20 Mar 2010

“Kerrr” black hole: the Lord of the String

Anais Smailagic

Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy

Euro Spallucci

Dipartimento di Fisica dell’ Universit` a di Trieste, and Istituto Nazionale di Fisica

Nucleare, Sezione di Trieste, Trieste, Italy

Abstract

Kerrr in the title is not a typo. The third “r” stands for regular, in the sense

of pathology-free rotating black hole. We exhibit a long search-for, exact, Kerr-

like, solution of the Einstein equations with novel features: i) no curvature ring

singularity; ii) no “anti-gravity” universe with causality violating timelike closed

world-lines ; iii) no “super-luminal” matter disk.

The ring singularity is replaced by a classical, circular, rotating string with Planck

tension representing the inner engine driving the rotation of all the surrounding

matter.

The resulting geometry is regular and smoothly interpolates among inner Minkowski

space, borderline deSitter and outer Kerr universe. The key ingredient to cure all

unphysical features of the ordinary Kerr black hole is the choice of a “noncommu-

tative geometry inspired” matter source as the input for the Einstein equations, in

analogy with spherically symmetric black holes described in earlier works.

1Introduction

Among the several black hole solutions of the Einstein equations, the Kerr

geometry is without any doubts the most appropriate to fit the observational

data showing that collapsed objects exhibit high angular momenta. Therefore,

the complete and through understanding of its properties is crucial for cor-

rect description of astrophysical objects. Furthermore, recent expectations are

1email: anais@ts.infn.it

2email: spallucci@trieste.infn.it

Preprint submitted to Elsevier23 March 2010

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related to the possible outcome of LHC experiments, including production of

micro black holes.

On the other hand, the history of Kerr solution is studded with technical

difficulties in solving Einstein equations, accompanied with a complete ig-

norance of the appropriate matter source. The textbook procedure is based

on the so-called “vacuum solution” method consisting in assuming an “ad-

hoc“ symmetry for the metric and solving field equations with no source on

the r.h.s. Integration constants are then determined comparing the weak-field

limit of the solution with known Newtonian-like forms. While mathematically

correct this approach is physically unsatisfactory especially in General Relativ-

ity, where basic postulate is that geometry is determined by the mass-energy

distribution. Furthermore, insisting on the vacuum nature of the solution leads

to the presence of curvature singularities where the whole classical theory, i.e.

General Relativity, fails. In the Kerr geometry, there are further complications

such as: an anti-gravity region and causality violating closed time-like curves.

These pathologies should not be present in a physically meaningful gravita-

tional field. A simple way out, is to replace the pathological vacuum region

with a regular matter source. In a series of recent papers we have presented

a spherically symmetric, regular, matter distribution leading to both neutral

and charge black hole solutions with no curvature singularity [1,2,3,5,6,4],

[7,9,10,11] and traversable wormholes [8]. Global structure and inner horizon

stability for such a kind of geometries are currently under investigation [12,13].

The regularity of the metric follows from the presence of a minimal length pro-

viding a universal cut-off for short-distance physics. The idea that there should

be a minimal distance is supported by many results in different approaches to

quantum gravity [14,15,16], [17,18,19], [20,21], [22,23,24,25,26], [27,28]. This

new parameter enters the Einstein equations through the energy-momentum

tensor, and represents the degree of delocalization of the matter distribution

[29,30,31,32].

In this paper we are going to apply the same approach to the axially symmet-

ric problem attempting to remove not only the curvature singularity but all

the pathologies quoted above.

2 Preliminary remarks

It is known that both Schwartzschild and Kerr solutions of general relativ-

ity belong to the same class of metrics ([33]) with some common proper-

ties. Firstly, combinations of the metric components can be brought to the

simple form by an appropriate gauge choice [34]. Secondly, both metrics can

be put in the so called Kerr-Schild form [35] i.e. it can be written in terms

of the Minkowski metric plus terms involving a specific null vector kµ. This

parametrization has the advantage that the Einstein equations are linearized in

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a true sense (not an approximation) which renders them much more tractable.

In the course of the paper, we shall exploit these common properties, when

needed, in order to establish analogy between the two metrics. This could be

particularly helpful having in mind that solving field equations for the Kerr

metric is either extremely cumbersome [34], or based on some mathematical

procedure without clear physical input [36,33,37].

Textbook approach introduces Schwarzschild and Kerr geometry as “vacuum

solutions” of the Einstein equations, where the resulting spacetime symmetry

is an initial assumption. Against this background, our approach follows the

basic Einstein’s idea that spacetime is curved due to the presence of matter.

Consequently, the symmetry of the metric is determined by the symmetry of

the matter source. Having at hand the details of the spherically symmetric

solution [2] we trace the pattern to follow in this paper.

As an introduction of the idea, let us start from the simple Minkowski line

element written in a spherical basis

ds2

M= (dx0)2− dr2− r2dϑ2− r2sin2ϑdφ2

(1)

One notices that specific combinations of metric components can be built up

to give

ηϑϑηrr= r2

η00ηφφ= −r2sin2ϑ

(2)

(3)

Equations (2), (3) are elementary in Minkowski space, but turn out to be

very useful for black hole spacetime (see (18), (19), because they allow a very

simple generalization leading to a quite non-trivial metric.

When matter is present, Schwarzschild-like class of metrics, in the Kerr-Schild

form, read

ds2

S= ds2

M−f(r)

r2(kµdxµ)2,(4)

kµ= (1 ,−1 ,0 ,0)

kµkµ= 0

(5)

(6)

We allow f(r) to be an arbitrary function of the radial coordinate in order to

account for both ordinary (singular), as well as, our regular solution. Standard

“vacuum” solution is

f(r) = const. × r(7)

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At this point one cannot but assign a posteriori significance to the arbitrary

constant by comparing the weak-field with the Newtonian potential. On other

other hand, if one starts with a matter source, i.e. a proper energy momentum

tensor in the r.h.s. of the field equations [2,3], it is physics that determines

the solution free of any arbitrariness. We found

f(r) ≡ 2M(r)r = 2M rγ(3/2 ;r2/4l2

0)

Γ(3/2)

(8)

where, M is the total mass-energy of the source.

Furthermore, on general grounds one can always write Schwartzschild-like so-

lution in terms of the radius dependent mass M(r) defined as

M(r) = 4π

?r

0

dxx2ρ(x)(9)

where ρ(r) is energy density of matter. The combinations (2), (3) turn out to

general’s in a simple way, as follows

gϑϑgrr= r2− f(r) ≡ ∆(r) ,

g00gφφ= −∆(r)sin2ϑ

(10)

(11)

Now, let us proceed and change the symmetry of the problem. Instead of spher-

ical symmetry we choose an axially symmetric spheroidal geometry parametrized

by the coordinates

x =

√R2+ a2sinϑcosφ ,

√R2+ a2sinϑsinφ ,

z = Rcosϑ ,

(12)

y =

(13)

(14)

Although the above parametrization is found in many textbooks, its geo-

metrical meaning is seldom clear, mainly due to the habit to use a notation

which is often indistinguishable from the spherical one. For the reader’s ad-

vantage, it is worth clearing any possible misinterpretation. Notice that the

surfaces described by these coordinates are a family of confocal ellipsoids,for

R = const., and confocal hyperboloids, for ϑ = const, with foci on the ring

(0 ,acosφ,asinφ ,0). These surfaces are described by

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x2+ y2

R2+ a2+z2

x2+ y2

a2sin2ϑ−

R2= 1

z2

a2cos2ϑ= 1(15)

The asymptote of the hyperbola, taking y = 0 for simplicity, is

z = xcotϑ(16)

Therefore, ϑ is the angle between the z-axis and the asymptote. Like-

wise, coordinate R is the smaller semi-axis of the ellipse, and not the radial

coordinate r. Any function F(R) is not to be considered a radial function in

the usual sense. The parameter a appearing in the Kerr metric has the geo-

metrical meaning of a focal length.

With these preliminary introduction of the symmetry, we write the Minkowski

line element in spheroidal coordinates

ds2

M= (dx0)2−

Σ

R2+ a2dR2− Σdϑ2− (a2+ R2)sin2ϑdφ2

(17)

where, Σ ≡ R2+ a2cos2ϑ. Again, we look at specific combinations (2), (3)

and find spheroidal analogue

gϑϑgRR= R2+ a2≡ ∆(R) ,

g00gφφ= −∆(R)sin2ϑ

(18)

(19)

The passage from flat to curved space-time is obtained by adding a function

f(R) in the definition of ∆. Thus, we find

gϑϑgRR= R2+ a2− f(R) ≡ ∆(R) ,

g00gφφ= −∆(R)sin2ϑ

(20)

(21)

Using (20), (21) one is left only with two unknown functions g00and ∆, since

the component gϑϑis preserved by the spheroidal symmetry. These functions

are found using Einstein equations with appropriate matter source. However,

things can be further simplified exploiting the power of the Kerr-Schild de-

composition of the metric. In fact, null four-vector kµcan be found solely on

the basis of symmetry arguments as

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