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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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