Black holes in modified gravity theories

0.34 MB·

by  Alvaro de la Cruz-Dombriz

Black holes in modified gravity theories
A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto,
0 Departamento deF´ısica Te´orica I,Universidad Complutense deMadrid, 28040 Madrid, Spain.
1 E-mail: [email protected]
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Abstract. In the context of f(R) gravity theories, the issue of finding static and spherically
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symmetric black hole solutions is addressed. Two approaches to study the existence of such
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solutions are considered: first, constant curvature solutions, and second, the general case
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(without imposing constant curvature) is also studied. Performing a perturbative expansion
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aroundtheEinstein-Hilbertaction,itisfoundthatonlysolutionsoftheSchwarzschild-(Anti-)de
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Sittertypearepresent(uptosecondorderinperturbations)andtheexplicitexpressionsforthese

] solutions are provided in terms of the f(R) function. Finally we consider the thermodynamics
c of black holes in Anti-deSitterspace-time and studytheir local and global stability.
q
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r
g
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1 1. Introduction
v In the last years, increasing attention has been paid to modified theories of gravity in order to
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understand several open cosmological issues such as the present accelerated expansion of the
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universe [1]. Some of those theories modify general relativity by adding higher powers of the
4
2 scalar curvature R, i.e. f(R) theories, the Riemann and Ricci tensors or their derivatives [2]. It
1. is therefore quite natural to ask about black hole (BH) features in those gravitational theories
0 since,ontheonehand,someBHsignaturesmaybepeculiartoEinstein’sgravity andothersmay
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berobust features of all generally covariant theories of gravity. On the other hand, the obtained
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results may lead to rule out some models which will be in desagreement with expected physical
:
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results. For thoses purposes, research on thermodynamical quantities of BH is of particular
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X interest.
r In this work we will restrict ourselves to the so called f(R) gravity theories in metric
a
formalism in Jordan’s frame where the gravitational Lagrangian is given by R + f(R) and
modified Einstein’s equations are usually fourth order in the metric. Using the euclidean action
method [3, 4] in order to determine different thermodynamical quantities, Anti-de Sitter (AdS)
BHs in f(R) models and their stability will be studied [5].
2. Black Holes in f(R) gravities
The most general static and spherically symmetric D ≥ 4 dimensional external metric for the
gravitational field produced by a nonrotating object in f(R) gravity theories can be written as:
ds2 = e−2Φ(r)A(r)dt2−A−1(r)dr2−r2dΩ2 ≡ λ(r)dt2−µ−1(r)dr2−r2dΩ2 (1)
D−2 D−2
where dΩ2 is the metric on the SD−2 sphere. In this case the scalar curvature R in D
D−2
dimensions depends only on r and for the first parametrization in equation (1) is given by
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R = (D−2)[(D−3)(1−A)+2r(−A′+AΦ′)]+r2[3A′Φ′−A′′−2A(Φ′2−Φ′′)] . (2)
r2
n o
where the prime denotes derivative with respect to r from now on. Therefore the most general
static and spherically symmetric metrics with constant scalar curvatureR can befoundsolving
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the equation R = R . For a constant Φ(r)= Φ it is straightforward to see that the solution is:
0 0
R
A(r) = 1+a r3−D +a r2−D − 0 r2 (3)
1 2
D(D−1)
with a being arbitrary integration constants. For the particular case D = 4, R = 0 and
1,2 0
Φ = 0, the metric can be written exclusively in terms of the function A(r) = 1+a r−1+a r−2
0 1 2
where by establishing the identifications a = −2G M and a = Q2, this solution corresponds
1 N 2
to a charged massive BH solution with mass M and charge Q (Reissner-Nordstr¨om solution).
On the other hand, if the constant curvature case R = R is considered, the modified
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Einstein’s equation from f(R) theories for the function A(r) reduces to:
(D−2)
R+f(R)+(1+f ) A′′+ A′−2AΦ′ −3A′Φ′+2A Φ′2−Φ′′ = 0 (4)
R
(cid:20) r (cid:0) (cid:1) (cid:16) (cid:17)(cid:21)
wheref holds for df(R)/dR . As it is explained in [5], the vacuum constant curvature solutions
R
of f(R) gravities are given by:
Df(R )
0
R = (5)
0
2(1+f )−D
R0
′
If the hypothesis Φ(r)= 0 is made, then equation (4) , by using (5), reduces in this case to
′
A 2 R
A′′+(D−2) = − R → A(r) = c + c r3−D − 0 r2 (6)
0 1 2
r D D(D−1)
where c are two arbitrary constants. However this solution has no constant curvature in the
1,2
general case since, as it can be seen in expression (3), the constant curvature requirement
demands c = 1. Then, for negative R , this solution is basically the D dimensional
1 0
generalization obtained by Witten [4] of the BH in AdS space-time solution considered by
Hawking and Page [3]. With the natural choice Φ = 0 the solution can be written as:
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A(r) = 1−(R /r)D−3+r2/l2 (7)
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where R and l have the usual interpretations found in the literature.
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Thuswehaveconcludedthattheonlystaticandsphericallysymmetricvacuumsolutionswith
constant (negative) curvature of any f(R) gravity is just the Hawking-Page BH in AdS space.
However this kind of solution is not the most general static and spherically symmetric metric
with constant curvature as can be seen by comparison with the solutions given by expression
(3).
3. Perturbative results
In Einstein-Hilbert (EH) theory the most general static solution with spherical symmetry is the
onewithconstantcurvature. However,thisisnotguaranteedtobethecaseinf(R)theories. The
problem of finding the general static spherically symmetric solution in arbitrary f(R) theories
without imposing the constant curvature condition is in principle too complicated. For that
reason a perturbative analysis of the problem is performed here, assuming that the modified
action is a small perturbation around EH theory of the form f(R) = −(D −2)Λ +αg(R)
D
where α ≪ 1 is a dimensionless parameter and g(R) is assumed to be analytic. Assuming that
the λ(r) and µ(r) functions appearing in the second parametrization of the metric (1) are also
analytical in α, they can be written as follows
∞ ∞
λ(r) = λ (r)+ αiλ (r) ; µ(r) = µ (r)+ αiµ (r) (8)
0 i 0 i
Xi=1 Xi=1
where{λ (r), µ (r)}aretheunperturbedsolutionsfortheEHactionwithcosmological constant
0 0
Λ given by µ (r) = 1+ C1 − ΛD r2 and λ (r) = −C (D−2)(D−1)µ (r) which are the
D 0 rD−3 (D−1) 0 2 0
standard BH solutions in a D dimensional AdS spacetime with constant curvature R ≡ DΛ .
0 D
By inserting the proposed f(R) expansion and equation (8) in modified Einstein’s equations
the first and second order equations in α are obtained, whose solutions are
C C −C C C Λ C A(g; D, Λ )
λ (r)= (D−1)(D−2) C + 1 4 2 3 − 4 D + 2 D r2
1 (cid:26) 4 rD−3 (cid:20)D−1 (D−1)(D−2)(cid:21) (cid:27)
C A(g; D, Λ )
µ (r) = 3 + D r2
1 rD−3 (D−2)(D−1)
C +C C C Λ 2C R g
λ (r)= C + 5 6 1 − 6 D −A(g; DΛ ) C +C g − 2 0 R0R0 r2
2 6 rD−3 (cid:20)D−1 D (cid:18) 4 2 R0 D−2 (cid:19)(cid:21)
−C C−1 A(g; D, Λ )(2R g −(D−2)g )
µ (r) = C C + 5 2 + D 0 R0R0 R0 r2(9)
2 (cid:18) 3 4 (D−2)(D−1)(cid:19)rD−3 (D−2)2(D−1)
where g ≡ dg(R)/dR| , g ≡ d2g(R)/dR2| , and A(g; D, Λ ) ≡ g(R )−2Λ g .
R0 R=R0 R0R0 R=R0 D 0 D R0
Higher order solutions {λ (r),µ (r)} can be obtained by inserting the previous results in
3,4,... 3,4,...
the corresponding equations order α3,4,... which become increasingly complicated.
Notice that from the obtained results up to second order in α, the corresponding metric
has constant scalar curvature for any value of the parameters C . As a matter of fact, this
1,...,6
metric is nothing but the standard Schwarzschild-AdS geometry, and can be easily rewritten
in the usual form by making a trivial time reparametrization. Therefore, at least up to second
order, theonly static, spherically symmetric solutions which areanalytical in α arethe standard
Schwarzschild-AdS space-times.
4. Black-hole thermodynamics
For simplicity we will concentrate only on constant curvature AdS BH solutions given by
expression (7) with Φ(r) ≡ 0 as a natural choice. Then, usual definitions of temperature lead
to β ≡ 1/T = 4πl2r /[(D −1)r2 +(D −3)l2] where r satisfies A(r ) = 0. Notice that the
H H H H
temperatureis a function of r only, i.e. it dependsonly on the BH size. It is easy to prove that
H
T has a minimum T at r and whose existence was established in [3] for D = 4. Notice that
0 H0
for any T > T , there are two possible BH sizes: one corresponding to the small BH phase with
0
r < r and the other corresponding to the large BH phase with r > r . The remaining
H H0 H H0
thermodynamicquantities aredeterminedbyusingtheEuclideanmethodsothatthefreeenergy
F is given by
F = −(R0+f(R0))µD−2(l2rD−3−rD−1) (10)
36π(D−1)G H H
D
where provided −(R +f(R )) > 0, which is the usual case in EH gravity, F > 0 for r < l and
0 0 H
F < 0 for r > l. On the other hand, the heat capacity C can be written as:
H
C = ∂E = ∂E ∂rH = −(R0+f(R0))(D−2)µD−2rHD−2l2(D−1)rH2 +(D−3)l2. (11)
∂T ∂r ∂T 8G (D−1) (D−1)r2 −(D−3)l2
H D H
Let us assume from now on that, like in the EH case, the condition (R +f(R )) < 0 holds.
0 0
This is the required condition, as explicitly shown in [5] to get both positive energy and entropy
in these theories and according to (5) it implies 1 +f > 0 in agreement with cosmological
R0
requirements [6]. With this assumption, we find C > 0 for r > r (the large BH region)
H H0
and C < 0 for r < r (the small BH region). For r ∼ r (T close to T ) C is divergent.
H H0 H H0 0
In any case, for f(R) theories with R +f(R ) < 0, we have found a similar scenario to the
0 0
one described in full detail in [3] for the EH case: for T > T two possible BH solutions are
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present, the small BH and the large BH: The small one has C < 0 and F > 0 as the standard
Schwarzschild BH. For the large BH two possibilities appear; if T < T < T(r = l) then C > 0
0 H
and F > 0 and the BH will decay by tunneling into radiation, but if T > T(r = l) then C > 0
H
and F < 0. In this case F of the large BH will be less than that of pure radiation.
Toillustratethepreviouscalculations, amodelf(R) = α(−R)β isconsideredwheretheheatca-
pacityC andthefreeenergyF signswillbestudiedtoknowthelocalandglobalstability ofBHs.
This model has a nonvanishing constant curvature solution R = −[(2−D)/α(2β −D)]1/(β−1)
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calculated by using (5) and therefore two separated regions, where R < 0 holds, have to be
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studied: Region 1 {α < 0, β > D/2} and Region 2 {α > 0, β < D/2}. Further models following
an analogous procedure were presented in [5].
Figure 1. Regions for C and F signs have been compared in the (α,β) plane for Model I in
D = 4. Region 1(left), Region 2 (right). Schwarzschild radius R = 2 was assumed.
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Acknowledgments
ThisworkhasbeensupportedbyMinisteriodeCienciaeInnovaci´on(Spain)projectnumbersFIS
2008-01323 and FPA 2008-00592, UCM-Santander PR34/07-15875 and UCM-BSCH GR58/08
910309.
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