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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)

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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

Page 4

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)

Page 6

6

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Specific angular momentum

Time

w=-0.9 r=3

w=-0.8 r=3

w=-0.9 r=6

Figure 2: This figure shows the behaviour of the specific an-

gular momentum 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 observe on the figure that the

larger w or r, the quicker the increase of specific angular mo-

mentum. In the inset we can see that a tends to a constant

value for large times in all studied cases.

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120

R3(1+α)=

B

?√ρ0−

?

6G

πI(J)

?2(1+α)

− Ach

, (36)

for M, a and J, respectively. Again for the case where

the dominant energy condition is preserved, i.e. B > 0,

we obtain that M, a and J all increase with time, M

and a tending to constant values for moderately large B.

If B is large enough, then whereas M tends to infinity,

a approaches a larger but still finite constant value. On

the other hand, M and a are both seen to increase more

rapidly as parameter α is made smaller, with M tending

once again to infinity, if α is taken to be sufficiently small.

As to the accretion dependence on r for B > 0, it has

been checked that as r is made very small, M and a are

nearly frozen into their original values. On the contrary,

for large r, the evolution of M will tend to match that in

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8

Time

1 1.2 1.4 1.6

Angular momentum

w=-0.9 r=3 M=10

w=-0.9 r=6 M=1

w=-0.8 r=3 M=1

w=-0.9 r=3 M=1

Figure 3: This figure shows the behaviour of the angular mo-

mentum of a Kerr-Newman black hole as a function of the

cosmic time in the presence of dark energy with w = −0.8

and w = −0.9. One can also see on the figure that the larger

w, r or M, the quicker the increase of angular momentum. In

the inset we can see that J tends to a constant value for large

times in all studied cases.

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500 600 700 800

the Schwarzschild case, while a increases now again up to

a given constant value. If the dominant energy condition

is assumed to be violated, i.e. B < 0, then M, a and J

all decrease with time, with M and a always tending to

minimum, nonzero constant values. Making |B| larger,

or α smaller, makes the evolution quicker and the final

minima values for M and a smaller but still nonzero. The

dependence of the evolution process on r in this case is

quite similar to what we already described for B > 0,

that is to say, M and a nearly keep their initial values

for very small r, but both decrease each time quicker as

r is increased. Also common for B > 0 and B < 0 is the

feature that the evolution of M is damped as we choose

larger values of the angular momentum J.

All these behaviours have been checked by numerical

calculations which provides plots that are actually quite

the same those corresponding to the quintessence case

Page 7

7

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

Mass

Time

w=-1.1 J=0.01

w=-1.2 J=0.01

w=-1.1 J=1

Figure 4: This figure shows the behaviour of the mass of a

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

presence of phantom energy with w = −1.1 and w = −1.2.

One can also see on the figure that the larger |w < −1| or

smaller J, the quicker the decrease of mass.

[see Figs. (1)-(6)] except (i) for the behaviour of M vs

time for moderate B > 0 and α far from −1 (in which

case M tends to a constant at large t [see Fig. (7)] and

(ii) for the behaviour of M and a for moderate B < 0

and α ?= −1 (in which cases the studied parameters tend

to nonzero constant values at large t [see Fig. (8)]).

C. Super-radiance and cosmic censorship

In the case of a Kerr-Newman metric, the cosmic cen-

sorship conjecture[26] holds provided that

Q2+ a2≤ M2. (37)

Otherwise, the Kerr-Newman black hole will show a

naked singularity. It is interesting to study if dark en-

ergy accretion can produce a naked singularity in this

case. Since accretion of dark energy is a gravitatorial pro-

cess, whereas angular momentum is affected by it, elec-

tric charge is invariant under accretion. We have pointed

0

0.002

0.004

0.006

0.008

0.01

0 0.05 0.1

Time

0.15 0.2

Specific angular momentum

w=-1.1 r=3

w=-1.2 r=3

w=-1.1 r=6

Figure 5: This figure shows the behaviour of the specific an-

gular momentum of a Kerr-Newman black hole as a function

of the cosmic time in the presence of phantom energy with

w = −1.1 and w = −1.2. One can also observe on the figure

that the larger |w < −1| or r, the quicker the decrease of

specific angular momentum.

out above that when P + ρ > 0, a and M increase with

time during accretion of dark energy. Even though we

have not a formal proof for the violation of cosmic cen-

sorship in this case, a numerical analysis performed for

most reasonable values of M and a appears to indicate

that the dark energy accretion process violates the in-

equality in (37) for most reasonable situations. Actually,

there always is a very small initial time interval where

the conjecture holds, except at the extreme case where

Q2+ a2= M2[see Fig. (9)], but as soon as the initial

value of a is taken to be significantly comparable with

that of M, the conjecture is almost immediately violated

[see Fig. (10)]. In the next section we shall discuss and

interpret the reason for that violation.

If accretion involves phantom energy, then a and M

both decrease. In this case, since accretion does not af-

fect the value of electric charge, at first sight, it could

be thought that when sufficiently small values of a and

Page 8

8

0

0.002

0.004

0.006

0.008

0.01

0 0.2 0.4 0.6 0.8 1

Angular momentum

Time

w=-1.1 r=3 M=1

w=-1.1 r=6 M=1

w=-1.2 r=3 M=1

w=-1.1 r=3 M=10

Figure 6: This figure shows the behaviour of the angular mo-

mentum of a Kerr-Newman black hole as a function of the

cosmic time in the presence of phantom energy with w = −1.1

and w = −1.2. One can also see on the figure that the larger

|w < −1|, r or M, the quicker the decrease of angular mo-

mentum.

M are reached, Eq. (37) would no longer hold too, and

cosmic censorship would be violated as well. However, it

may also be expected that super-radiance of charge would

act upon its value in such way that it decreased charge

during accretion of phantom energy so that Eq. (37)

would still be satisfied. Moreover, as M progressively

decreases the black hole temperature should rise up and

the charge super-radiance would correspondingly speed

up. Our numerical calculations appear to indicate that

this is actually the case as all the simultaneous effects on

M, a [see Fig. (11)] and Q due to dark energy accretion

and Q-super-radiance seem to be mutually concited in

such way that the cosmic censorship is preserved indeed.

Obtaining an explicit, accurate expression for the relation

between mass or temperature and electric charge, how-

ever is a task that contains some subtleties and therefore

requires further elaboration which is left for a future con-

sideration. We do not consider in this paper the process

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

1 1.5 2 2.5 3 3.5 4 4.5

Mass

Time

B=0.5 α=0.5

Figure 7: This figure shows the behaviour of the mass of a

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

presence of a generalized Chaplygin gas with B = 0.5 and

α = 0.5.

of super-radiance of spin because phantom energy clearly

prevails over it.

IV.AN APPROXIMATED ACCRETION

MODEL

Violation of cosmic censorship in black holes dark en-

ergy accretion (w > −1) is a very surprising result ac-

tually, but it is perhaps not so surprising as the features

coming about when both rotating and non-rotating black

holes continue accreting such type of dark energy at suffi-

ciently large times, according to the accretion model used

by Babichev, Dokuchaev and Eroshenko[20, 21] and gen-

eralized in section II One of such features results e.g.

from Eq. (28) where it can be seen that the black hole

mass blows up at a finite time in the future, when the size

of the universe is still finite. It follows that the grown-up

black hole will engulf the entire universe in a finite time in

the future, an implication which is rather bizarre indeed,

Page 9

9

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.5 2 2.5 3 3.5

Time

4 4.5 5 5.5 6

Mass

Specific angular momentum

Figure 8: This figure shows the behaviour of the mass and

the specific angular momentum of a Kerr-Newman black hole

as a function of the cosmic time in presence of a (phantom)

generalized Chaplygin gas with B = −0.5 and α = 0.5.

and that it is also present when non-rotating black holes

are considered[20, 21]. Nevertheless, all the predictions

that have been derived for large time could be regarded

as artifacts coming from the fact that the black hole met-

ric used in our accretion procedure is static. Really, that

procedure becomes in such a case just an approximate de-

scription which can only be valid for a sufficiently short

initial time interval. Therefore the results obtained in

the present paper would just mark the tendency of the

different involved parameters once the initial evolution

has been overcome, but cannot be taken for granted for

large times.

Even in this case the result that cosmic censorship is

violated when dark energy with w > −1 is being accreted

cannot be justified, as that violation takes place from the

very beginning of the evolution for extreme black holes.

According to the results displayed in Figs. (1) and (2),

there appears to be a possibility to avoid incompatibility

of a simultaneous violation of cosmic censorship and a

black hole engulfing of the universe. Indeed, a black hole

1

2

3

4

5

6

7

0 0.05 0.1 0.15

Time

0.2 0.25 0.3

Specific angular momentum

Mass

Figure 9: Evolution of an extreme Kerr-Newman black hole

with dark energy. This figure shows the behaviour of the mass

and specific angular momentum of a extreme Kerr-Newman

black hole as a function of the cosmic time in the presence of

dark energy with w = −0.9. One can see on the figure that

the cosmic censorship conjecture is violated.

might have the following bizarre evolution when accretes

dark energy with w > −1, according to these figures. It

could violate cosmic censorship at the beginning of its

evolution and become a naked singularity. In this stage,

accretion of dark energy produces a bigger increase of

mass than specific angular momentum. Let us remember

now that specific angular momentum grows until a con-

stant value as t → ∞, whereas the mas blow up at finite

time. So, in a finite time the naked singularity becomes

again a black hole. Next, black hole can continue its evo-

lution ending in an universe engulfed by the black hole.

Thus, whereas the evolution of black holes induced by

accretion of phantom energy appears to be quite reason-

able at least on the early periods, in the case of satisfying

the dominant energy condition, the accretion onto black

holes seems to produce rather unexpected results along

the entire subsequent evolution.

Page 10

10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Time

Specific angular momentum

Mass

Figure 10: Evolution of a Kerr-Newman black hole with dark

energy. This figure shows the behaviour of the mass and spe-

cific angular momentum of a Kerr-Newman black hole as a

function of the cosmic time in the presence of dark energy

with w = −0.8. The separation between the two curves di-

minishes as the initial value of a is increased. One can also

see on the figure that there exists a small initial time interval

(running up to nearly t = 0.05s in this case) where the cosmic

censorship conjecture still holds.

V. CONCLUSIONS

In this paper we have studied the behaviour of ac-

cretion of dark energy onto a Kerr-Newman black

hole. First, we have generalized the accretion formal-

ism originally considered by Babichev, Dokuchaev and

Eroshenko[20, 21] for the case in which the black hole

has angular momentum and electric charge. We have

applied such a formalism to quintessence and K-essence

cosmological fields, so as to the generalized Chaplygin

gas model. The evolution of mass, specific angular mo-

mentum and angular momentum when dark energy with

w > −1 has been considered. It has been seen that all of

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6

Time

0.8 1 1.2

Specific angular momentum

Mass

Figure 11: Evolution of an extremal Kerr-Newman black hole

with phantom energy. This figure shows the behaviour of the

mass and specific angular momentum of an extreme Kerr-

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

ence of phantom energy with w = −1.2. The separation be-

tween the two curves increases as the initial value of a is

diminished or r is made larger.

these parameters (M,a and J) increase with cosmic time.

The specific angular momentum a grows up to reaching

a constant value whereas M is not bounded from above.

It is also checked, in this case, that the accretion of dark

energy verifying dominant energy condition usually leads

to a situation where the cosmic censorship is violated.

There is another feature even more surprising, i.e., the

mass of black hole blows up in a finite time and therefore

black holes will engulf the entire universe in a finite time.

These two predictions could be however regarded as ar-

tifacts coming from the fact that the black hole metric

used in our formalism is static. Really, the used proce-

dure could be seen just as an approximate description

which is valid only for a sufficiently short initial time

interval. Therefore the results obtained only mark the

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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