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arXiv:astro-ph/0510051v2 19 Feb 2008
Evolution of a Kerr-Newman black hole in a dark energy universe
Jos´ e A. Jim´ enez Madrid1,2, aand Pedro F. Gonz´ alez-D´ ıaz1, b
1Instituto de Matem´ aticas y F´ ısica Fundamental Consejo Superior
de Investigaciones Cient´ ıficas, Serrano 121, 28006 Madrid, Spain
2Instituto de Astrof´ ısica de Andaluc´ ıa, Consejo Superior de Investigaciones Cient´ ıficas,
Camino Bajo de Hu´ etor 50, 18008 Granada, Spain
(Dated: February 19, 2008)
This paper deals with the study of the accretion of dark energy with equation of state p = wρ
onto Kerr-Newman black holes. We have obtained that when w > −1 the mass and specific angular
momentum increase, and that whereas the specific angular momentum increases up to a given
plateau, the mass grows up unboundedly. On the regime where the dominant energy condition is
violated our model predicts a steady decreasing of mass and angular momentum of black holes as
phantom energy is being accreted. Masses and angular momenta of all black holes tend to zero when
one approaches the big rip. The results that cosmic censorship is violated and that the black hole
size increases beyond the universe size itself are discussed in terms of considering the used models
as approximations to a more general descriptions where the metric is time-dependent.
PACS numbers: 04.20.Dw, 04.70.-s, 98.80.-k
Keywords: accretion, Kerr-Newman black holes, dark energy.
I.INTRODUCTION
Several astronomical and cosmological observations,
ranging from observations of distant supernovae Ia[1] to
the cosmic microwave background anisotropy[2], indicate
that the universe is currently undergoing an accelerat-
ing stage. It is assumed that this acceleration is due to
some unknown stuff usually dubbed dark energy, with a
positive energy density ρ > 0 and with negative pres-
sure p < −(1/3)ρ. There are several candidate mod-
els for describing the dark energy, being the cosmolog-
ical constant, Λ, by far the simplest and most popular
candidate[3]. Other interesting models are based on con-
sidering a perfect fluid with given equation of state like
in quintessence[4], K-essence[5] or generalized Chaply-
gin gas models[6, 7, 8, 9, 10]. Note that there are also
other candidates for dark energy based on brane-world
models[11] and modified 4-dimensional Einstein-Hilbert
actions[12], where a late time acceleration of the universe
may be achieved, too.
One of the peculiar properties of the resulting cosmo-
logical models is the possibility of occurrence of a cosmic
doomsday, also dubbed big rip[13, 14, 15, 16, 17, 18]. The
big rip appears in models where dark energy particular-
izes as the so-called phantom energy for which p+ρ < 0.
In these models the scale factor blows up in a finite time
because its cosmic acceleration is even larger than that
induced by a positive cosmological constant. In these
models every component of the universe goes beyond the
horizon of all other universe components in finite cosmic
time. It should be noted, that the condition p + ρ < 0
is not enough for the occurrence of a big rip[19]. In re-
aElectronic address: madrid@imaff.cfmac.csic.es
bElectronic address: p.gonzalezdiaz@imaff.csic.es
cent papers[20, 21], it has been shown that the mass of
a Schwarzschild black hole decreases with accretion of
phantom energy, in such a way that the black hole disap-
pears at the time of the big rip. Therefore, it is interest-
ing to study how dark energy is accreted by more general
black holes, that is to say, black holes bearing charge
and angular momentum. The interest of this study is
enhanced by the eventual competition or joint contribu-
tion that may arise between the dark energy accretion
process and super-radiance which tends to decrease the
rotational (or charge) energy of the hole, so lowering its
spin (or charge), such as one would expect phantom en-
ergy induced as well. For this reason, in the present pa-
per we shall investigate how distinct forms of dark energy
can be accreted onto Kerr-Newman black holes. We in
fact obtain that Kerr-Newman black holes progressively
increase their mass and angular momentum as a result
from dark energy accretion when the equation of state
allows p+ρ > 0. That increase of mass and angular mo-
mentum is either unbounded or tends to a given plateau,
depending on the dark energy model being considered. If
p+ρ < 0 then both the mass and the angular momentum
of black hole rapidly decrease until disappearing at the
big rip, or tend to constant values in the absence of a
future singularity. It is seen that the latter process pre-
vails over both the Hawking evaporation process and spin
super-radiance. Our quantitative results appear to indi-
cate, on the other hand, that whereas phantom energy
does not violate cosmic censorship conjecture[26], dark
energy with w > −1 does.
The paper can be outlined as follows.
section, we will generalize the solution obtained by
Babichev, Dokuchaev and Eroshenko[20, 21] to the case
of dark energy accretion onto a charged, rotating black
hole, and present the general equations for the rate of
mass and momentum. In the next section we apply such
a formalism to quintessence and K-essence cosmological
In the next
Page 2
2
fields, so as to the generalized Chaplygin gas model, an-
alyzing the corresponding evolution of the black hole. In
section IV we discuss the results that cosmic censorship is
violated and that the black hole size grows up unbound-
edly beyond the universe size in terms of considering the
used models as approximations to a more general descrip-
tion where the metric is not static. Finally, we briefly
summarize and discuss our results in section V
II.GENERAL ACCRETION FORMALISM FOR
KERR-NEWMAN BLACK HOLES
In this section we shall follow the accretion for-
malism, first considered by Babichev, Dokuchaev and
Eroshenko[20, 21], generalizing it to the case in which
the black hole has an angular momentum and charge.
First of all, we notice that, even though we shall use a
static Kerr-Newman metric, the time evolution induced
by accretion will be taken into account by the time depen-
dence of the scale factor entering the integrated conser-
vation laws and the rate equations for mass and angular
momentum.
The procedure is based on integrating the conservation
laws for energy-momentum tensor and its projection onto
the four-velocity, using as general definition of energy-
momentum tensor a perfect fluid where the properties
of the dark energy and those of the black hole metric
are both contained. By combining the results from these
integrations with assumed rate equations for black hole
mass, angular momentum and specific angular momen-
tum, we can derive final rate equations for these quan-
tities in terms of the dark pressure, p, and dark energy
density, ρ. Now, since the conservation of dark energy
and its state equation p = wρ lead to a unique relation
between p and ρ with the scale factor R(t), our final rate
equation will only depend on R(t).
Using a static metric nevertheless restrict in princi-
ple ourselves to deal with small accretion rates as the
mixed component of the energy-momentum tensor used
in this case to derive the metric is zero.
sight, this procedure becomes an approximate scheme
whose description can only be valid for a short initial
time. However, the use of a non-static metric for which
that energy-momentum component is no longer vanishing
does not generally amount to different results asymptot-
ically, which is the physically relevant situation we have
to consider. This question will be dealt with in more
detail in Sec. IV Throughout this paper we shall use nat-
ural units so that G = c = 1. Let us then consider the
stationary and axisymmetric Kerr-Newman space-time.
The metric in this case can be given by
So, at first
ds2=
?
2a?2Mr − Q2?sin2θ
1 +
Q2− 2Mr
r2+ a2cos2θ
?
dt2
+
r2+ a2cos2θ
dtdφ
−
−
r2+ a2cos2θ
r2+ a2+ Q2− 2Mrdr2
?r2+ a2cos2θ?dθ2
??
× sin2θdφ2?,
where M is the mass, Q is the electric charge, a = J/M is
the specific angular momentum of black hole, with J the
total angular momentum, r is the radial coordinate, and
θ and φ are the angular spherical coordinates. We model
the dark energy in the black hole by the test perfect fluid
with a negative pressure and an arbitrary equation of
state p(ρ), with the energy-momentum tensor
−r2+ a2+2Mra2sin2θ − Q2a2sin2θ
r2+ a2cos2θ
?
(1)
Tµν= (p + ρ)uµuν− pgµν,(2)
where p is the pressure, ρ is the energy density, and uµ=
dxµ/ds is the 4-velocity with uµuµ= 1. There is no loss
of generality in a restricting consideration to Tµν of this
form, as it is actually the most general form that Tµνcan
take consistent with homogeneity and isotropy[27].
Usingthegeneral expression
operator[22] applied to this case, we get that the zeroth
(time) component of the energy-momentum conservation
law Tµν;ν= 0 can then generally be written as
fora derivative
d
dr
?
(p + ρ)
?
1 +
Q2− 2Mr
r2+ a2cos2θ
?
Q2− 2Mr
r2+ a2cos2θ
?dt
Q2− 2Mr
r2+ a2cos2θ
?dt
ds
dr
ds
?
+
2r
r2+ a2cos2θ(p + ρ)
+d
dθ
??cosθ
?
This expression should now be integrated. We consider
two cases. First, we take θ as a constant. The integration
of Eq. (3) gives then,
1 +
?dt
ds
dr
ds
?
sinθ−2a2sinθcosθ
r2+ a2cos2θ
(p + ρ)
?
1 +
ds
dθ
ds
?
+
?
×(p + ρ) 1 +
Q2− 2Mr
r2+ a2cos2θ
?dt
ds
dθ
ds
?
= 0. (3)
CM=
u
M2
?r2+ a2cos2θ?(p + ρ)
r2+a2+Q2−2Mr
×
?
1 +
Q2−2Mr
r2+a2cos2θ+r2+a2cos2θ+Q2−2Mr
u2?1/2
, (4)
where u = dr/ds, and CMis an integration constant.
Another integral of motion can be derived by using the
projection of the conservation law for energy-momentum
tensor along the four-velocity, i.e. the flux equation
uµTµν;ν= 0.(5)
For a perfect fluid, this equation reduces to
uµρ,µ+ (p + ρ)uµ;µ= 0.(6)
Page 11
11
tendency of the considered parameters, and could well
be not valid for large times. Even in this case the re-
sult that cosmic censorship is violated when black hole
accrete dark energy with w > −1, cannot be justified,
since that violation occurs in the very beginning of the
evolution for extreme black holes. Thus, the accretion of
dark energy verifying p + ρ > 0 onto black holes seems
to produce rather surprising and unexpected results.
If accretion involves phantom energy, then a and M
both decrease from their initial values, tending to zero as
one approaches the big rip, where the black holes will dis-
appear, independently of the initial values of their mass
and angular momentum. In this case (P +ρ < 0), cosmic
censorship conjecture is preserved, since super-radiance
of charge and phantom energy accretion mutually inter-
related.
Whether or not the above features can be taken to im-
ply that phantom energy is a more consistent component
than normal dark energy with w > −1 is a matter that
will depend on both the intrinsic consistency of the mod-
els and the current observational data and those that can
be expected in the future.
Acknowledgments
We acknowledge Prof. E. Babichev for useful explana-
tions and correspondence. We are also grateful to V.
Aldaya, A. Ferrera, S. Robles and M. Rodr´ ıguez for con-
structive discussions and criticisms. This work was sup-
ported by DGICYT under Research Projects BMF2002-
03758 and BFM2002-00778.
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