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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

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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)

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3

The integration of Eq. (6) gives the second integral of

motion that we shall use in what follows

u

M2

?r2+ a2cos2θ?exp

where u < 0 in the case of a fluid flow directed toward the

black hole, and AMis a positive dimensionless constant.

Eq. 7 gives us the energy flux induced in the accretion

process. From Eqs. (4) and (7) one can easily get:

??ρ

ρ∞

dρ′

p(ρ′) + ρ′

?

= −AM, (7)

?

1 +

Q2− 2Mr

r2+ a2cos2θ+r2+ a2cos2θ + Q2− 2Mr

r2+ a2+ Q2− 2Mr

×(p + ρ)exp

ρ∞

u2

?1/2

?

−

?ρ

dρ′

p(ρ′) + ρ′

?

= C2M, (8)

where C2M= −CM/AM= p(ρ∞) + ρ∞.

The rate of change of the black hole mass due to ac-

cretion of dark energy can be derived by integrating over

the surface area the density of momentum T0r, that is[23]

˙ M = −

?

T0rdA,(9)

with dA = r2sinθdθdφ, and r constant. Using Eqs. (2),

(7) and (8) this can be rewritten as

˙M =4πAMM3r

J

arctan

?

J

Mr

?

[p(ρ∞) + ρ∞], (10)

with r and J constants. It is worth noticing that Eq. (10)

consistently reduces to the corresponding rate equation

for a Schwarzschild black hole derived by Babichev,

Dokuchaev and Eroshenko in Refs. [20] and [21] when

one lets J to become very small. One has the following

integral expression that governs the evolution of the mass

of the Kerr-Newman black hole

?M

M0

JdM

M3rarctan?

?t

J

Mr

?

= 4πAM

t0

[p(ρ∞) + ρ∞]dt.(11)

Now, the integration in the left-hand-side of Eq. (11)

gives

I(M) =

?M

M0

JdM

M3rarctan?

2J×

arctan?

4

arctan3?

?

J

Mr

?

= −r

?

1 +

J2

M2r2

J

Mr

? +

J

Mrarctan2?

J

Mr

?

+

J

Mr

?

×

∞

k=1

?22k− 1?arctan2k?

J

Mr

?

π2k(2k − 3)

ζ (2k)

?M

M0

, (12)

where ζ is the Riemann zeta function. The integration of

the right-hand-side of Eq. (11) will be performed in the

next section. We turn now to consider r, instead of θ, as

a constant with which the integration of Eq. (3) yields

Ca =

ω

asinθ?r2+ a2cos2θ?(p + ρ)

?

?r2+ a2cos2θ + Q2− 2Mr?ω2?1/2,

where ω = dθ/ds, and Cais another integration constant.

The second integral of motion for the energy flux

in this case is also obtained from the projection of

the energy-momentum tensor conservation law along the

four-velocity; then the integration of Eq. (6) gives that

second integral of motion

× 1 +

Q2− 2Mr

r2+ a2cos2θ

+ (13)

1

aω sinθ?r2+ a2cos2θ?exp

where ω < 0 in the case of a fluid flow directed toward the

black hole, and Aais a positive dimensionless constant.

From Eqs. (13) and (14) one can easily get:

??ρ

ρ∞

dρ′

p(ρ′) + ρ′

?

= −Aa,

(14)

?

?r2+ a2cos2θ + Q2− 2Mr?ω2?1/2

× (p + ρ)exp

ρ∞

1 +

Q2− 2Mr

r2+ a2cos2θ

+

?

−

?ρ

dρ′

p(ρ′) + ρ′

?

= C2a, (15)

where C2a= −Ca/Aa= p(ρ∞) + ρ∞.

We take the rate of change of the specific angular mo-

mentum of the Kerr-Newman black hole originating from

accretion of dark energy to be now given by[23]

˙ a = −

?

rT0θdA,(16)

with dA = r2sinθdθdφ, and θ constant. Using Eqs. (14)

and (15) this can be rewritten as

˙ a =2π2Aaar2[ρ∞+ p(ρ∞)]

√r2+ a2

(17)

with r constant. Therefore, one has the following integral

expression that reports about the evolution of the specific

angular momentum

√r2+ a2

ar2

?a

a0

da = 2π2Aa

?t

t0

[p(ρ∞) + ρ∞]dt.(18)

Then, the integration in the left-hand-side of Eq. (18)

gives rise to the following expression

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4

I(a) =

?a

1

r2

a0

√r2+ a2

ar2

da (19)

=

??

a2+ r2−1

rln

?2r

a

?

r +

?

a2+ r2???a

a0

.

The integration of the right-hand-side of Eq. (18) will

again be calculated in the next section for the distinct

dark energy models.

Now, we study the influence of dark energy accretion

in the angular momentum J. Using J = Ma and Eqs. (9)

and (16) we can obtain the rate of change of the angular

momentum of the black hole performing the following

integral

˙J = −

??MrT0θ+ aT0r?dA, (20)

with dA = r2sinθdθdφ, and r constant. So, we obtain

˙J =π [p(ρ∞) + ρ∞]

×

?

2JπAar

q

1+

J2

M2r2

+ 4AMM2rarctan?

J

Mr

?

?

, (21)

with M and r constants. Therefore, one has the follow-

ing integral expression that governs the evolution of the

angular momentum of a black hole

?J

J0

dJ

2JπAar

q

1+

J2

M2r2

+ 4AMM2rarctan?

= π

t0

J

Mr

?

?t

[p(ρ∞) + ρ∞]dt.(22)

Once again the integration of the right-hand-side of the

equation will be carried out in the next section. Here, we

have been however unable to perform the integration of

the left-hand side (L) in closed form.

proceed as follows. The integral L, in Eq. (22) can be

recast in the form

Thus, we have

L =

?x

x0

dx

cos2x(2rπAasinx + 4AMMx), (23)

where 0 ≤ x ≤ π/2 and x = arctan(J/Mr). It can be

noticed that, since 0 ≤ sinx ≤ x, we have

?x

=

2πAar + 4AMM

?

xx2

k=2

L ≥

x0

dx

(2πAar + 4AMM)xcos2x

1

×

tanx

+ lnx +

1

∞

?

?22k− 1?x2k

(k − 1)π2kζ (2k)

?x

x0

=

1

2πAar + 4AMM

?

Mrarctan?

1

arctan2?

?

where ζ is again the Riemann zeta function.

L ≥ I(J) which in turn implies that if we use I(J) for

studying the evolution of the Kerr-Newman black hole

and the cosmic censorship is taken to be physically pre-

served, then L should respect this conjecture, too. This

argument entitles us to use I(J) as a suitable expres-

sion to study the evolution of J during accretion of dark

energy.

×

J

J

Mr

? + lnarctan

?

J

Mr

?

+

J

Mr

?

(24)

×

∞

k=2

?22k− 1?arctan2k?

J

Mr

?

(k − 1)π2k

ζ (2k)

?J

J0

≡ I(J),

Thus,

III.COSMOLOGICAL MODELS

In order to obtain exact integrated expressions for the

right-hand-side of Eqs. (11), (18) and (22), we shall

use in this section two different models for dark en-

ergy, namely, quintessence and generalized Chaplygin gas

models. It can be seen that the results obtained by using

the quintessence model are the same as those derived if

one used the so-called K-essence model for dark energy.

A. Quintessence models

Starting with the equation of state p = wρ, where w

is assumed constant, we can use the conservation of the

energy-momentum tensor to get

ρ = ρ0

?R0

R

?3(1+w)

, (25)

where R ≡ R(t) is the scale factor, with ρ0and R0con-

stants. Hence

?t

t0[p(ρ∞) + ρ∞]dt

= (1 + w)ρ0R3(1+w)

0

?t

t0R−3(1+w)dt.(26)

We then have for the scale factor[25] corresponding to

a general flat quintessence universe

R(t) = R0

?

1 +3

2(1 + w)C1/2(t − t0)

?2/[3(1+w)]

, (27)

where C = 8πGρ0/3. Integration of the right-hand-side

of Eqs. (11), (18) and (22), can then be performed using

Eq. (27). We respectively get

t = t0+

I(M)

(1 + w)?4πAMρ0−3

2C1/2I(M)?,(28)

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5

t = t0+

I(a)

(1 + w)?2π2Aaρ0−3

2C1/2I(a)?, (29)

t = t0+

I(J)

(1 + w)?πρ0−3

2C1/2I(J)?, (30)

where I(M),I(a) and I(J) are defined in Eqs. (12), (20)

and (25), respectively. These are three parametric equa-

tions from which one can obtain how the mass, specific

angular momentum and angular momentum evolve in the

accelerating universe. Thus, if w > −1 we see that M,a,

and J will all progressively increase with time from their

initial values, with a tending to a finite constant value as

t → ∞, showing that the increase of M tends to become

proportional to the increase of J; however M goes to in-

finity in a finite time, but J tends to a finite constant

value as t → ∞. Notice that there is no contradiction

between the results of Figs. (2) and (3) as the plot in

Fig. (3) is obtained relative to a constant value of mass.

The larger w the quicker the increase of these parameters

[see Figs. (1), (2) and (3)].

If w < −1 we can observe that M,a, and J will all

progressively decrease from their initial values, tending

to zero as one approaches the big rip, where the black

holes will disappear independently of the initial values of

their mass and angular momentum [see Figs. (4), (5) and

(6)]. This generalizes the result obtained by Babichev,

Dokuchaev and Eroshenko[20, 21].

In the case of a charged black hole, the process of

super-radiance of charge allows the black hole to emit

the charge before it disappears. It has been checked as

well that the larger |w < −1| the quicker is the decrease

of M and J, and that for large r the evolution of the

mass nearly matches the evolution that was derived for

the Schwarzschild case. Also remarkable are the features

that the larger J, or the smaller r, the smaller the rate of

mass decrease. Accretion of phantom energy leads also

to a decreasing of a which becomes zero quickly, so that

J must decrease quite more rapidly than M does. For

any w, it has been finally seen that the rate of variation

(increase for w > −1 and decrease for w < −1) of J

speeds up as one makes r or M larger.

B. Generalized Chaplygin gas

We shall derive now the expression for the rates˙M, ˙ a,˙J

in the case of a generalized Chaplygin gas. This can be

described as a perfect fluid with the equation of state[6]:

p = −Ach/ρα, (31)

where Ach is a positive constant and α is a parameter.

In the particular case α = 1, the equation of state (31)

corresponds to a Chaplygin gas. The conservation of the

energy-momentum tensor implies

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Mass

Time

w=-0.9 J=0.01

w=-0.8 J=0.01

w=-0.9 J=1

Figure 1: This figure shows the behaviour of the mass of a

Kerr-Newman black hole as a function of the cosmic time in

presence of dark energy with w = −0.8 and w = −0.9. One

can also see on figure that the larger w or smaller J, the

quicker the increase of mass.

ρ =

?

− Ach

Ach+

B

R3(1+α)

?1/(1+α)

. Now, from the Fried-

, (32)

with B ≡

mann equation we can get

?ρα+1

0

?R3(α+1)

0

˙R =

?

8πG

3

R(t)

?

Ach+

B

R3(1+α)

?1/[2(1+α)]

.(33)

Hence,

R3(1+α)=

B

?√ρ0−

?

3G

8πA2

MI(M)

?2(1+α)

− Ach

,(34)

R3(1+α)=

B

?√ρ0−

?

3G

2π3A2

aI(a)

?2(1+α)

− Ach

, (35)