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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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ds2

S= ds2

M−f(R)

Σ

Σ

(kµdxµ)2,(22)

kµ=

?

1 ,−

a2+ R2,0 ,−asin2ϑ

?

(23)

One recognizes the above line element as a Kerr-like metric written in the

Kerr-Schild form. We are left only with the unknown scalar function f(R)

to be found solving the Einstein equations. It turns out that the Einstein

equation for ∆ is particularly simple in form and reads

d2∆

dR2− 2 = −16π gϑϑ

?

TR

R+ Tϑ

ϑ

?

,GN≡ 1(24)

from eq.(24) one deduces, in view of (19), that TR

R+ Tϑ

ϑmust be of the form

d2f

dR2= 16πgϑϑ

?

TR

R+ Tϑ

ϑ

?

(25)

We hope to have paved a relatively simple way to the generalized Kerr metric

without the need to solve complicated equations. Now, we shall concentrate on

the form of the matter source which produces the above metric and determine

the explicit form of the function f(R) through (25).

3 Energy-momentum tensor

The question of a proper matter source for Kerr (or any other) metric is of

paramount importance to give the physical input to Einstein equations. Due

to the original “vacuum approach“, there have been many attempts to “en-

gineer” a suitable form of a matter source to cure the geometry anomalies.

In particular, a general a posteriori form of Tµνcan be found by inserting

the generalised axially symmetric, stationary, metric into the l.h.s. of the field

equations [33,39]. In this approach energy density and pressures remain un-

specified. Various attempts to guess suitable matter distributions reproducing

Kerr solution outside the source and possibly regularizing its inner singular

behavior were made [42,43,44], [45,46,47], [48,39]. However, this has always

led to different geometries which have to be glued together. Against this back-

ground we shall present a unique solution of the field equations, free of any

pathology, and smoothly interpolating between ordinary Kerr at large dis-

tance, and a new regular “Kerrr” at short-distance.

In order to determine the energy-momentum tensor we start from the result

in the spherically symmetric case [2].

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Tµ

uµ=√−grrδµ

lµ= −

ν= (ρ + pθ)(uµuν− lµlν) − pϑδµ

0,

1

√−grrδµ

ν,(26)

(27)

r, (28)

(29)

where, the mass distribution is described by a Gaussian density ρ as

ρ(r) =

M

(4π)3/2l3

0

e−r2/4l2

0

(30)

The total mass M is defined as the volume integral

M = 2π

?∞

0

dr

?π

0

dθ sinθρ(r)(31)

The above energy momentum tensor is describing an anisotropic fluid and has

the same form as in [33]. The pressure pθis determined from the vanishing of

the covariant divergence for the energy momentum tensor [2], which gives

∂rTr

r=

?

θ− Tφ

Tθ

θ− Tr

r

?

∂rlngθθ⇒ pθ= ρ +r

= 0 ⇒ Tθ

2∂rρ (32)

cotθ

?

Tθ

φ

?

θ= Tφ

φ

(33)

where, the equation of state pr= −ρ is understood. This form of the energy-

momentum tensor can be extended to the axial symmetry by maintaining its

form but changing the explicit expression for pressures and density. Further-

more, the four velocity uµdevelops a non-vanishing component uφdescribing

the rotation of the source. Additional component of uµcan be obtained from

uµuµ= 1 and gives

uφ

u0=

a

a2+ R2≡ ω(R)(34)

This is the angular velocity of fluid layers rotating around z axis. The mass

density ρM(R) is now chosen following the reasoning in [33,38] to pass from

non-rotating to rotating physical situation

ρM(R ,ϑ) ≡R2

ΣρG(R) =

M

8π3/2l3

0

R2

Σe−R2/4l2

0

(35)

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With the choice (35) the total mass M, defined as the spheroidal volume

integral, is found to be

M ≡ 2π

?∞

0

dR

?π

0

dϑ sinϑΣρM(R ,ϑ) = 4π

?∞

0

dRR2ρG(R) (36)

Finally, the energy-momentum tensor for the generalized Kerr metric is given

by

Tµ

ν= (ρ + pϑ)(uµuν− lµlν) − pϑδµ

uµ=

−gRR

1

√−gRRδR

ν,(37)

?

?

δ0

µ+

a

(a2+ R2)δφ

µ

?

,(38)

lµ= −

µ,(39)

It is important to keep track of different “ρ” functions present in this case. In

particular, ρ in (37) is an invariant energy density ρ = Tµνuµuν. It is given in

terms of ρGas

ρ(R ,ϑ) =R4

Σ2ρG(R) (40)

in agreement with [38].

Again, the pressure pϑand pφare determined from the vanishing covariant

divergence condition for the energy momentum tensor:

Tϑ

ϑ= TR

R+Σ

2R∂RTR

R→ −pϑ= ρ +Σ

2R∂Rρ (41)

0 = ∂ϑTϑ

ϑ+ 2cotϑ

?

Tϑ

ϑ− Tφ

φ

?

⇒ pφ= pϑ+tanϑ

2

∂ϑpϑ

(42)

(43)

which reproduces the corresponding quantities in (32), (33) in the limit a → 0.

4Regular Kerr-like solution

Our “Kerrr” metric reads

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ds2=

?

∆dR2− Σdϑ2−sin2ϑ

∆ = R2− 2M (R)R + a2

1 −2RM(R)

Σ

?

dt2+4aM(R)R

Σ

sin2ϑdtdφ +

−Σ

Σ

??

R2+ a2?2− a2sin2ϑ∆

?

dφ2,(44)

(45)

where, M(R) is found to be

M (R) ≡ 4π

?R

0

dxx2ρG(x) =

M

Γ(3/2)γ

?

3/2 ;R2/4l2

0

?

(46)

γ (3/2 ;x) is the lower incomplete gamma function defined as

γ (b ;x) ≡

?x

0

dttb−1e−t

(47)

We see that the solution for M(R) has the same form as in the spherically

symmetric case with the substitution r → R.

In the above formula l0is a minimal length which, in our approach [30,30,31],

is reminiscent of the underlying non-commutativity of spacetime coordinates

leading to the Gaussian matter distribution.

On more general ground, l0can be considered as the width of the Gaussian

matter distribution of the source. Thus, in spite of the origin of l0, the model

is applicable both micro black holes and astrophysical objects.

Horizons in (45) are real solutions of the equation

R2

H+ a2−2MRH

Γ(3

2)

γ

?3

2;R2

H

4l2

0

?

= 0(48)

This equation cannot be solved explicitly for RH= RH(M ;a) as it is possible

for ordinary Kerr solution. Thus, we follow an alternative approach: we solve

the parameter M in equation (46) as a function of the horizon radius RH. The

plot is given in figure(1). As in ordinary Kerr solution for any assigned value

of a we find three possible situations

(1) M > Mextr.there are two distinct horizons R±and the solution represents

a non-extremal black hole;

(2) M = Mextr.there is a single degenerate horizon and the solution corre-

spods to an extremal black hole of mass given by the minimum value of

the curve in figure(1);

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(3) M < Mextr.there are no horizons. Black hole cannot be formed but there

is no curvature singularity, nontheless.

2468 10 12 14

R?

2.5

5

7.5

10

12.5

15

17.5

20

M

Fig. 1.

The dotted curve corresponds to a = 0 (regular Schwarzschild solution), and the

continuous curve to a = 6. For any a there is a curve whose intersections with the

line M = const. determines the position of horizons. The minimum corresponds to

the extremal black hole. Increasing a lifts the minimum upwards.

Plot of the function M(RH) for different values of a in l0 = 1 units.

It is important to consider the asymptotic form of our metric (45) in ϑ = π/2,

and R → 0 where ordinary Kerr solution exhibits the infamous ring singularity.

First, notice that the mass asymptotic behavior is given by

M (R) ∼

M

6√π

R3

l3

0

≡Λ

6R3,R → 0(49)

and

∆ → ∆Λ= R2+ a2−Λ

3R4

(50)

leading to the rotating deSitter geometry

ds2= R2dR2

∆Λ

−

1

R2

?

adt −

?

R2+ a2?

dφ

?2+∆Λ

R2[dt − adφ]2, (51)

characterized by a scalar curvature

RdSr= 4ΛR2

Σ

(52)

The singular ring can be reached by sliding along ellipsoids R = const. > 0

until arriving on the equatorial plane ϑ = π/2 and then letting R → 0. In this

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region we find the Ricci scalar to be

lim

R→0+R(R ,ϑ = π/2) =

4M

√πl3

0

= 4Λ(53)

which is the constant curvature scalar of a regular rotating deSitter geometry.

It remains to check what happens when moving along a ϑ = const. < π/2

hyperbolae , which brings us down on equatorial disk with R = 0. In this

case, we find

lim

R→0+R(R ,ϑ0< π/2) = 0(54)

and

lim

R→0+ρ(R ,ϑ0< π/2) = 0(55)

Thus, the disk is a matter-free, zero curvature Minkowski flat spacetime, as it

is in the ordinary Kerr geometry:

ds2

M= −(dt)2+ cos2ϑ0dR2+ a2sin2ϑ0dφ2

(56)

The difference with ordinary Kerr is that the singular ring is replaced by

a regular deSitter, Saturn-like region of non-zero width, with inner radius

x2+y2= a2. Our model represents the first explicit example of a matter source

leading to a singularity-free metric that naturally interpolates between near-

by de Sitter and outer Kerr-like forms. No ad hoc conjectures, or patching, is

required.

5The stringy heart of the Kerrr solution

From (53) and (54) we see that there is a discontinuity in the Ricci scalar

as one approaches R → 0 ,ϑ → π/2 from two different directions. One may

wonder wether thia jump has a physical meaning? We shall try to answer this

question.

First notice that the metric induced on the equatorial plane is strongly rem-

iniscent of the spacetime geometry in the presence of a vacuum bubble [40].

To be more precise, we can intepret the Minkowski disk as a “true vacuum”

planar bubble surrounded by a deStter “false vacuum” and we can apply the

Israel matching condition [41] to give a physical meaning to the metric dis-

continuity. We write the flat metric (56) in terms of planar polar coordinates

r = asinϑ, φ, as

ds2

in= dt2− dr2− r2dφ2,r ≤ a (57)

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In the same way we write the outer equatorial plane,outer, near-ring deSitter

geometry as

ds2

out=

?

1 −Λ

3

?

r2− a2??

dt2− dr2− r2dφ2,r ≥ a(58)

The matching of the extrinsic curvatures along the static boundary r = a

ǫin√η00− ǫout

?

g00(r = a) = 4πGNσ (59)

where, ǫin/out= ±1 according with the choice of orientation of the normal to

the ring; σ > 0 is the linear energy density, i.e. the tension, of the ring. The

jump in the extrinsic curvature is non-zero for ǫin= 1, ǫout= −1 leading to

σ =

1

2πGN

≡

1

2πα′

(60)

From this expression for the tension we can recover the mass of the ring Mr

as

Mr≡ 2πaσ =a

α′

(61)

Now, we can compute the ring angular momentum which rotates with angular

velocity

ωr= ω(R = 0) =1

a

(62)

thus, we get

Jr= Mra2ωr= Mra (63)

By inserting (63) in (61) we find

Jr= α′M2

r

(64)

which is a classical Regge Trajectory with a planckian Regge slope α′=

MPl./lPl.!

This result offers an exciting interpretation of the ring as a classical, rotating,

circular string leaving on a Regge trajectory. Therefore, we offer the folowing

physical interpretation of our solution. The “heart” of the Kerrr black hole is

a rotating string of finite tension replacing the standard Kerr singularity. The

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stringy interpretation is supported by the Regge realtion between the mass

and the angular momentum of the ring. The string is immerged in a cloud

of matter described by the Tµνdiscussed previously and the is inner engine

inducing rotation of the matter elipsoidal layers. The gaussian profile of the

matter cloud is instrumental to regularize the inifinte curvature jump, present

in the ordinary Kerr solution, to a finite value 4Λ, thanks to the presence of

the outer deSitter belt. Or, in other words, the infinte tension ring-like curva-

ture singularity is “renormalized“ to the the maximum physically acceptable

Planckia tension of a fundamental string.

6 Diving through the equatorial disk

There is another question long awaiting a satisfactory answer in the Kerr

metric. It is known that, unless it is forcefully cut-off, there is an “anti-gravity”,

negative R region, were causality violation takes place due to the existence of

closed time-like curves. We shall show that our metric resolves both problems.

Let us go to the z-axis by taking ϑ = 0 and R = |z|. Then, g00reads

g00= 1 −2|z|M(z)

z2+ a2

(65)

The problem of multiple Riemannian sheets arises whenever g00is a function

-4

-22

4

z

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

goo

Fig. 2.

(smooth curve) solutions.

Plot of g00 along the axis of rotation for Kerr (spiky curve) and Kerrr

of odd power of |z|. In the usual Kerr M(z) = const. and g00is discontinuous

in the first derivative at z = 0 leading to a jump in the extrinsic curvature. In

our case, the behavior of M(z) near the origin is given by

M(z) ∼ const. × z3

(66)

which gives an even function g00for small argument with a continuous first

derivative in z = 0. In the standard Kerr geometry ther are two different inter-

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pretations of the discontinuity of the gravitational field across the Minkowski

disk:

(1) discontinuity is justified by the presence of matter on the disk. Unfortu-

nately, this matter turns out to move in a super-luminal fashion and have

divergent density on the ring [42,49].

(2) Discontinuity is removed by interpreting the disk as a “branch cut” and

attaching a second Riemann sheet of negative R. This procedure restores

analiticity at the price of introducing a negative gravity sheet of the met-

ric, where closed time-like curves can exist as a consequence of allowing

R to be negative.

Our solution has no problems of this kind since it is analytic everywhere,

and thus it is meaningless to talk about analytic continuation of the metric.

In other words, geodesics can cross the Minkowski disk without any prob-

lem. Everything fits nicely together, as conjectured, following the same line of

reasoning already encountered in the regular spherically symmetric case.

7Conclusions

In this paper we presented the first example of a smooth matter distribution

which leads to a pathology-free Kerr solution. The form of the source is a

generalization of the corresponding Gaussian mass/energy distribution we in-

troduced for spherically symmetric sources to the case of a rotating object. For

both solutions, the same mechanism is at work: the curvature singularity is re-

placed by a deSitter vacuum domain. In the spherically symmetric case it is an

inner deSitter core, while in the Kerrr solution it turns out to be a Saturn-like

belt of rotating deSitter vacuum, surrounding an empty Minkowskian disk.

The novel feature of the Kerrr solution is that the Minkowski disk joins the

deSitter belt through a a rotating string with Planckian (finite!) tension.

Beside removing the nasty ring singularity the gaussian cloud of matter elimi-

nates the negative R sheet of the Kerr black hole by ensuring analyticity of the

metric across the disk. Positivity of R forbids the presence of closed time-like

curves.

To keep the length of the paper short enough to fit the journal format, we

must postpone a detailed study of Kerrr black hole thermodynamics to a next

article. We anticipate that as in the case of spherically symmetric regular black

holes, we find that the Hawking temperature is not unbounded but reaches a

maximum value and then drops to zero as the extremal configuration is ap-

proached.

In a forthcoming paper we shall present the extension of the present work to

the Kerr-Newman black hole.

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