Static and spherically symmetric solutions in a scenario with quadratic curvature contribution

0.47 MB·

by  F. A. Silveira

Static and spherically symmetric solutions in a scenario
with quadratic curvature contribution
F. A. Silveira1,2∗ R. F. Sobreiro1† A. A. Tomaz1,3‡
1UFF − Universidade Federal Fluminense,
Instituto de F´ısica, Campus da Praia Vermelha,
Avenida General Milton Tavares de Souza s/n, 24210-346,
Niter´oi, RJ, Brazil.
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1
0 2UERJ - Universidade Estadual do Rio de Janeiro, Departamento de F´ısica Te´orica,
2
Rua S˜ao Francisco Xavier 524, 20550-013, Maracan˜a,
n
a Rio de Janeiro, RJ, Brasil.
J

9 3CBPF − Centro Brasileiro de Pesquisas F´ısicas,
1
Rua Dr. Xavier Sigaud, 150 , Urca, 22290-180

] Rio de Janeiro, RJ, Brazil
c
q
-
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g
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Abstract
1
v
Inthisworkweinvestigateanalyticstaticandsphericallysymmetricsolutionsofageneralized
5
theory of gravity in the Einstein-Cartan formalism. The main goal consists in analyzing the
1
4 behavior of the solutions under the influence of a quadratic curvature term in the presence of
5 cosmological constant and no torsion. In the first incursion we found an exact de Sitter-like
0 solution. This solution is obtained by imposing vanishing torsion in the field equations. On the
.
1 otherhand,byimposingvanishingtorsiondirectlyintheaction,weareabletofindaperturbative
0 solutionaroundtheSchwarzschild-deSitterusualsolution. Webrieflydiscussclassicalsingularities
7
for each solution and the event horizons. A primer discussion on the thermodynamics of the
1
geometrical solutions is also addressed.
:
v
i
X
1 Introduction
r
a
In the present work we consider a generalization of general relativity (GR) [1] with cosmological
constant in the Einstein-Cartan (EC) formalism [2, 3, 4, 5], i.e., where the fundamental variables are
the vierbein and the spin-connection instead the metric tensor and the affine-connection as in the
Palatini formalism [6, 7]. In this generalization, the usual Einstein-Hilbert action [8] is supplemented
by a quadratic curvature term and a quadratic torsion term.
∗[email protected]
†[email protected]ff.br
‡[email protected]ff.br
1
Specifically, we study vacuum static and spherically symmetric solutions of this model by con-
sidering the case of vanishing torsion. Because we are considering the EC formalism, the action
provides two field equations, one for the vierbein and another for the spin-connection. First, we
show that a de Sitter spacetime is an exact vacuum solution of these equations. This is a non-trivial
result since the system of equations is over-determined for vanishing torsion. After that, we obtain a
perturbative solution around the Schwarzschild-de Sitter solution by neglecting the spin-connection
equation, which is equivalent to impose vanishing torsion at the action level. It is important to be
clear that, to obtain such solution, the curvature squared term is treated as perturbation around
the usual Einstein term with cosmological constant. Hence, it the the generalized Einstein equation
which is perturbed instead a perturbed solution around a fixed background.
Static and spherically symmetric solutions in alternative gravity models is a recurrent subject of
investigation [9, 10, 11, 12]. In particular, K. Stelle investigated this subject in a quite general higher
derivative scenario in the metric formalism [9]. Hence, Stelle’s work encompasses the contributions
of quadratic curvature terms. Nevertheless, we call attention to the main differences between [9] and
the present paper: First, our work is performed in the EC formalism instead of the metric formalism.
Second, as already explained, our perturbed solution is a deformation around the Schwarzschild-de
SitterspacetimeobtainedbyconsideringthequadraticcurvaturetermasaperturbationwhileStelle’s
result is a perturbed solution around the Minkowski background obtained by imposing a perturbation
on the solution.
Itisworthmentioningthat, althoughourmainmotivationistheactionoriginatedasanemergent
gravity in a quantum gauge theory scenario [13, 14, 15, 16], our analysis and discussions are general
enough to encode any gravity theory as above described. Nevertheless, we will refer to the model
[13, 14, 15, 16] whenever we find it elucidative. Perhaps, the most important general motivation
would be the recent detection of gravitational waves by the Laser Interferometer Gravitational-wave
Observatory (LIGO) collaboration [17]. Such discover brings a new era on black hole physics as
well as the possibility to test alternative solutions derived of modified theories of gravity that should
expand the horizons of the well-known Einstein’s gravitational theory.
Thisworkisorganizedasfollows: InSec.2wedefinethemodelandtherespectivefieldequations
and specify them for static and spherically symmetric variables. The exact solution is discussed in
Sect. 3. In Sec. 4 we provide the perturbative solution. In Sect. 5, a the thermodynamical aspects
of the solutions are briefly discussed. Finally, our final considerations are displayed in Sect. 6.
2 Action and field equations
Let us consider the following gravity action,
1 (cid:90) (cid:18) 3 (cid:15) Λ˜2 (cid:19)
S = Ra (cid:63)R b+Ta(cid:63)T − ε Rabeced+ ε eaebeced . (2.1)
grav 16πG 2Λ2 b a a 2 abcd 12 abcd
where Ra = dωa +ωaωc is the curvature 2-form, Ta = dea +ωa eb is the torsion 2-form, ωa is
b b c b b b
the spin connection 1-form, and ea is the vierbein 1-form. Moreover, (cid:63) stands for the Hodge dual
operator, G is the Newton constant, Λ(cid:101) is the cosmological constant, and Λ is a mass parameter.
The corresponding vacuum field equations are easily obtained. For the vierbein, we get
(cid:32) (cid:33)
3 Λ˜2
Rbc(cid:63)(R e )+Tb(cid:63)(T e )+D(cid:63)T −ε Rbced− ebeced = 0 , (2.2)
2Λ2 bc a b a a abcd 3
2
while for the spin connection, we obtain,
3
D(cid:63)R +e (cid:63)T −e (cid:63)T −ε Tced = 0 . (2.3)
Λ2 ab b a a b abcd
Eqs. (2.2) and Eq. (2.3) are coupled nonlinear differential equations and thus, highly difficult to solve
without any special insight. For this reason, we proceed with the simplest case where torsion is set
to vanish. Hence, Eqs. (2.2) and Eq. (2.3) reduce to1
(cid:32) (cid:33)
3 Λ˜2
Rbc(cid:63)(R e )−ε Rbced− ebeced = 0 , (2.4)
2Λ2 bc a abcd 3
3
D(cid:63)R = 0 . (2.5)
Λ2 ab
Our aim is to solve equations (2.4) and (2.5) for static and spherically symmetric conditions. To
do so, we will consider two different situations:
• First, we consider both equations (2.4) and (2.5) and find an exact solution, corresponding to
a strong influence of the quadratic curvature term. The result we find is the usual de Sitter
solution with an effective cosmological constant given by a mix between Λ and Λ(cid:101).
• Second, by taking Λ as a huge quantity2 when compared to the quadratic curvature term,
we treat this term as a perturbation. The perturbed solution is a deformation of the usual
Schwarzschild-deSittersolution. Toobtainsuchsolution, theequation(2.5)isneglectbecause
it is a pure perturbation in Λ−2. This situation is equivalent to set T = 0 at the action (2.1)
before the computation of the field equations.
Eqs. (2.4) and (2.5), in Schwarzschild coordinates,
e0 = eα(r)dt , e1 = eβ(r)dr , e2 = rdθ , e3 = rsinθdφ , (2.6)
can be recasted as
(cid:34) (cid:18)e−2β∂ β(cid:19)2 (cid:18)1−e−2β(cid:19)2(cid:35) (cid:18)e−2β∂ β(cid:19) 1−e−2β
r r
σ 2 + +2 + +3λ = 0 , (2.7)
r r2 r r2
(cid:34) (cid:18)e−2β∂ α(cid:19)2 (cid:18)1−e−2β(cid:19)2(cid:35) (cid:18)e−2β∂ α(cid:19) 1−e−2β
r r
σ 2 + −2 + +3λ = 0 , (2.8)
r r2 r r2
(cid:40)(cid:104) (cid:16) (cid:17)(cid:105)2 (cid:18)e−2β∂ α(cid:19)2 (cid:18)e−2β∂ β(cid:19)2(cid:41)
σ e−(α+β)∂ e−β∂ eα + r + r +
r r
r r
(cid:16) (cid:17) e−2β∂ α e−2β∂ β
−e−(α+β)∂ e−β∂ eα − r + r +3λ = 0 , (2.9)
r r
r r
1It is worth mentioning that the action (2.1) describes an emergent gravity associated to an SO(5) Yang-Mills
theory [13]. However, the field equations (2.4) and (2.5) are general enough to describe any gravity theory with a
Riemann squared curvature in the action. Whenever is relevant, we will comment about the model developed in [13]
and the results here found.
2This situation is consistent with the results of [18, 16] where 1- and 2-loop explicit computations predict a huge
value for Λ in the effective gravity model constructed in [13].
3
∂ R = 0 , (2.10)
r
for a = 0 , a = 1 and a = 2, respectively. In Eq. (2.10), R stands for the scalar curvature. We
notice that differential equations obtained for a = 2 and a = 3 are identical. The constants in
Eqs. (2.7), (2.8) and (2.9) are σ ≡ −3/(2Λ2) and λ ≡ −Λ˜2/3. The combination of Eqs. (2.7) and
(2.8) leads to the following constraint
(cid:110) r (cid:111)
∂ [α(r)+β(r)] ∂ [β(r)−α(r)]+ e2β(r) = 0 , (2.11)
r r
σ
which allows two possibilities:
• ∂ (α(r)+β(r)) = 0 ⇒ α(r)+β(r) = f(t) - Thus, we have freedom to re-scale the time
r
coordinate with simply f(t) = 0.
• ∂ β(r) + ∂ α(r) + σ−1rexp(2β(r)) = 0 - This condition is unusual and quite complicated
r r
since it is a coupled non-linear differential equation for α(r) and β.
We will stick to the first possibility, which is the usual relation in General Relativity and impose
α(r) = −β(r).
3 Exact solution
In order to solve analytically the system of equations (2.7)-(2.10), we start by subtracting Eq. (2.8)
from Eq. (2.9), obtaining3,
(cid:32) (cid:33)(cid:34) (cid:32) (cid:33) (cid:35)
h¨ 1−h h¨ 1−h
+ σ − −1 = 0 , (3.1)
2 r2 2 r2
where, the condition α + β = 0 was employed. Moreover, we have defined e−2β(r) ≡ h and
h˙ ≡ dh/dr. Eq. (3.1) can be decomposed in two independent differential equations where only one
must be zero. First, we have
(cid:18) r2(cid:19)
r2h¨ +2h−2 1+ = 0 , (3.2)
σ
whose solution is
(cid:34) (cid:32)√ (cid:33) (cid:32)√ (cid:33)(cid:35)
r2 √ 7 7
h(r) = 1+ + r c cos lnr +c sin lnr , (3.3)
1 2
2σ 2 2
where c and c are integration constants. It turns out that this solution does not satisfy the whole
1 2
differential equation system (2.7)-(2.10).
The second possibility is
r2h¨ −2h+2 = 0 , (3.4)
whose solution is given by
c
h(r) = 1+c r2+ 4 , (3.5)
3
r
3The partial derivatives were changed to ordinary ones, since β is only r-dependent.
4
where c and c are integration constants. It is a straightforward computation to show that we must
3 4
have c = 0 and c (cid:54)= 0 in order to the solution (3.5) satisfy the system (2.7)-(2.10). Moreover,
4 3
there are two possible values for the constant c , namely Υ and Υ ,
3 p m
 (cid:115) 
Λ2 Λ˜2
Υp = 3 1+ 1−2Λ2 ,
 (cid:115) 
Λ2 Λ˜2
Υm = 3 1− 1−2Λ2 . (3.6)
Hence
e−2βp = 1−Υ r2 ,
p
e−2βm = 1−Υ r2 . (3.7)
m
Thesolutions(3.7)satisfythesystemofdifferentialequations(2.7)-(2.10),simultaneously. Thisisan
important and necessary verification since this system of equations is over-determined. From (3.6),
it is clear that Λ˜2 cannot exceed Λ2/2, otherwise, the solution founded is inconsistent. Moreover, if
2Λ˜2 = Λ2, only one solution is allowed Υ = Υ .
m p
Now, if we assume that Λ2 has a large value and Λ˜2 has a small value4, we can expand (3.6) to
find
1
Υ ≈ Λ˜2 ,
s
3
2
Υ ≈ Λ2 . (3.8)
b
3
The first case, Υs has a narrow value if we take Λ(cid:101) as its observational value. On the other hand, the
second case stems for a de Sitter-like space with a very small radius, since Λ2 is very big. Hence, we
have a weak curvature regime for Υ and a strong one for Υ .
s b
Obviously, all usual properties of de Sitter spacetime are maintained5.
4Thisisconsistent,forinstance,withtheexplicitvaluesΛ2 ≈7.665×1031TeV2 (cid:29)Λ˜2 ≈1.000×10−92TeV2 found
in [18, 16].
5An alternaltive and simple way to find the solution (3.7) is to directly deal with equation (2.2) in form notation
and try an ansatz solution of the form
Rab =ζeaeb , (3.9)
where ζ is a constant mass parameter. The solution (3.9) is a natural choice since we have the usual cosmological
constant term in (2.2). The direct substitution of (3.9) in (2.2) leads to the characteristic equation for ζ,
3 Λ˜2
ζ2−ζ+ =0, (3.10)
2Λ2 3
providing ζ = Υ . Hence, solution (3.9) is an alternative covariant form of the curvature associated with solution
p,m
(3.7).
5
4 Perturbative solution
From this point, we consider the quadratic curvature to be a small perturbation in Eq. (2.7). For
that, we multiply Eq. (2.7) by λ, then
(cid:34) (cid:18)e−2β∂ β(cid:19)2 (cid:18)1−e−2β(cid:19)2(cid:35) (cid:20) (cid:18)e−2β∂ β(cid:19) 1−e−2β (cid:21)
r r
η 2 + +λ 2 + +3λ = 0 . (4.1)
r r2 r r2
In this form, Eq. (4.1) can be solved analytically through the employment of perturbation theory
if η ≡ σλ ≡ Λ˜2/2Λ2 is a very small dimensionless parameter. Hence, the quadratic term can be
treated as a perturbation around the term proportional to λ. It is evident that the term proportional
to λ is the usual Einstein equation with cosmological constant Λ(cid:101). For simplicity, let u(r) = 1−e−2β
and take all derivatives as ordinary ones. So, we rewrite Eq. (4.1) as
(cid:20) (cid:21)
η 1u˙2+(cid:16) u (cid:17)2 +λ(cid:0)ru˙ +u+3λr2(cid:1) = 0 . (4.2)
2 r2
A perturbative solution of Eq. (4.2) has the general form
u(r) = u (r)+ηu (r)+η2u (r)+η3u (r)+··· . (4.3)
0 1 2 3
Substituting (4.3) in the Eq. (4.2), and split order by order in η, we find an infinite set of descent
coupled equations
d
(ru )+3λr2 = 0 ,
0
dr
d 1 (cid:18)du (cid:19)2 u2
(ru )+ 0 + 0 = 0 ,
dr 1 2λ dr λr2
(cid:18) (cid:19)
d 1 du du 2u u
0 1 0 1
(ru )+ + = 0 ,
dr 2 λ dr dr λr2
d 1 (cid:18)du (cid:19)2 u2 1 (cid:18)du du (cid:19) 2u u
(ru )+ 1 + 1 + 0 2 + 0 2 = 0 ,
dr 3 2λ dr λr2 λ dr dr λr2
(cid:18) (cid:19) (cid:18) (cid:19)
d 1 du du 2u u 1 du du 2u u
0 3 0 3 1 2 1 2
(ru )+ + + + = 0 ,
dr 4 λ dr dr λr2 λ dr dr λr2
.
.
. . (4.4)
With such hierarchy of equations (4.4), we can solve, iteratively, all the equations above. Starting
with the zeroth order, we obtain
Λ˜2 2GM
u = r2+ , (4.5)
0
3 r
which is the usual Schwarzschild-de Sitter solution [19, 20, 21], obviously, since this is the situation
for η = 0. Hence, the integration constant at the 1/r term is obtained from the Newtonian limit.
Solving iteratively the rest of the equations (4.4), we find the following solution (at fourth order),
2GM Λ˜2 (cid:18)C C (cid:19) (cid:18)C C C (cid:19)
e−2β ≈ 1− − r2−η 12 +C r2+ 13 −η2 22 +C r2+ 23 + 24
r 3 r 11 r4 r 21 r4 r7
(cid:18) (cid:19) (cid:18) (cid:19)
C C C C C C C C C
− η3 32 +C r2+ 32 + 34 + 35 −η4 42 +C r2+ 43 + 44 + 45 + 46 +...
r 31 r4 r7 r10 r 41 r4 r7 r10 r13
(4.6)
6
where the constants C can be arranged as6
k(cid:96)
 Λ˜2 2GM 
3
 Λ˜2 C 6G2M2 
 3 12 Λ˜2 
Ck(cid:96) ≡  2Λ˜32 C22 6GΛ˜2MΩ1 −36GΛ˜34M3  (4.8)
 5Λ˜2 C 9 Ω 54G2M2Ω 312G4M4 
 3 32 2Λ˜2 2 Λ˜4 4 Λ˜6 
14Λ˜2 C − 3 Ω 3GMΩ 54G2M2Ω −3564G5M5
3 42 Λ˜2 3 Λ˜2 5 Λ˜4 6 Λ˜8
where the index k (line) and (cid:96) (column) run along the discrete intervals [0,∞] and [1,k + 2],
respectively, and
Ω = C −2GM ,
1 12
Ω = (cid:2)C2 +4GM (6GM −2C −C )(cid:3) ,
2 12 12 22
Ω = [C (C +C −12GM)−2GM (2C +C −20GM)] ,
3 12 12 22 22 32
Ω = (8GM −3C ) ,
4 12
Ω = (cid:2)3C2 −2GM (12C +3C −24GM)(cid:3) ,
5 12 12 22
Ω = 2(C −3GM) . (4.9)
6 12
The differential equation Eq. (2.9) can be solved for α once we have β. The result is consistent
with the condition α = −β. Moreover, as discussed before, Eq. (2.10) can be neglected. Simple
consistency checks are: the limits η → 0 and Λ˜2 = 0, providing a pure Schwarzschild solution; the
limits M = 0 and η → 0 result in a de Sitter spacetime solution; and, as already seen, the limit
η → 0 provides the Schwarzschild-de Sitter solution.
It is interesting to take the limit r (cid:29) 2GM in the solution (4.6). The result is immediate,
(cid:16) (cid:17)
e−2β ≈ 1−Υ˜r2 , (4.10)
where
(cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33)
Λ˜2 Λ˜2 Λ˜2 Λ˜2 Λ˜2
Υ˜ ≈ +η +η2 2 +η3 5 +η4 14 +... , (4.11)
3 3 3 3 3
which is a perturbatively asymptotically de Sitter spacetime, as expected. Remarkably, the relation
(4.11) can be written as
(cid:32)(cid:88)∞ (cid:33) Λ˜2
Υ(cid:101) = ηwaw , (4.12)
3
w=0
where
(2w)!
a = (4.13)
w
(w+1)!w!
6The solution (4.6) can be generalized to all orders in a very concise form given by
∞ k+2
e−2β =1−(cid:88)ηk(cid:88)C r5−3(cid:96) , (4.7)
k(cid:96)
k=0 (cid:96)=1
where the general constants C depend on the previous order constants. In particular, the C , with k =0,1,2,...,
k(cid:96) k2
stand for the actual integration constants.
7
are the Catalan numbers7. Moreover, by setting η = 0 in (4.11) we find Υ˜ = Υ . Hence, we have an
s
asymptotic relation between the perturbative solution and the exact one. Further, by expanding the
constant Υp in (3.6) for small Λ(cid:101)/Λ, we get the same expression (4.12). Hence, we have consistency
between the exact and perturbative solution for small Λ(cid:101)/Λ.
For the next sections, for the sake of simplicity, we attain ourselves to the first order correction
in η. Thus, the solution (4.6) is reduced to
(cid:18) (cid:19)
C C C
e−2β ≈ 1− 02 −C r2−η 12 +C r2+ 13 . (4.14)
r 01 r 11 r4
where the explicit form of the constants are displayed in (4.8).
4.1 Horizons
Eq. (4.14) can be used to determine the horizons by solving e−2β = 0 [20, 21, 23, 24]. Thus, we
must solve the following perturbed algebraic equation
r3(cid:0)r−C −C r3(cid:1)−η(cid:0)C r6+C r3+C (cid:1) = 0 , (4.15)
02 01 11 12 13
where the constants C’s are listed in (4.8). The solution can be taken as a perturbative one of the
form
r ≈ r +ηr . (4.16)
0 1
Substituting Eq. (4.16) in Eq. (4.15) we obtain a system of two algebraic equations. At zeroth order,
already with the substitution of constants Cs, we have
3 6GM
r3− r + = 0 , (4.17)
0 Λ˜2 0 Λ˜2
and, at first order,
r3(cid:0)1+3C r2(cid:1)r −C r3−C r6−C = 0 , (4.18)
0 01 0 1 13 0 11 0 13
which we left with the constants C’s for the sake of simplicity. The polynomial discriminant of the
Eq. (4.17) is easily computed
108 (cid:16) (cid:17)
∆ = 1−9G2M2Λ˜2 , (4.19)
Λ˜6
which is important to determine the nature of the roots of Eq. (4.17). If, and only if ∆ > 0,
Eq. (4.17) has three real roots. Such condition implies that 3GMΛ˜ < 1, since Λ˜, G and M are
positive quantities. By applying the trigonometric method to find all the roots of Eq. (4.17), it is
found only two different positive roots, namely,
1 (cid:16) √ (cid:17)
r = c + 3s ,
01 Λ˜ ξ ξ
1 (cid:16) √ (cid:17)
r = c − 3s , (4.20)
02 Λ˜ ξ ξ
(cid:16) (cid:17)
where c ≡ cosξ, s ≡ sinξ and ξ = 1/3arccos 3GMΛ˜ . The third root is r = −(r +r ),
ξ ξ 03 01 02
(cid:16) (cid:17)
whichisessentiallynegativeanditisnotphysical. Since0 < 3GMΛ˜ < 1 ⇒ 0 < arccos 3GMΛ˜ <
7Named after the discovery of the sequence of natural numbers by the Belgian mathematician Eug`ene C. Catalan
(1814−1894), which made several contributions to combinatorial mathematics [22].
8
π/2 we have r > r > 0. Accordingly, r stands for the cosmological horizon and r stands for
01 02 01 02
the mass distribution event horizon.
Now, the substitution of Eq. (4.20) in Eq. (4.18) provides
(cid:32) (cid:33)
1 6G2M2 1 Λ˜2
r = − +C + r3 , (4.21)
1(cid:96) 3(cid:16)1− Λ˜2r2 (cid:17) Λ˜2 r03(cid:96) 12 3 0(cid:96)
3 0(cid:96)
with (cid:96) = 1 or (cid:96) = 2. Hence, the form of the horizons at first order are8
(cid:40) (cid:34) (cid:35)(cid:41)
1 (cid:16) √ (cid:17) sec(3ξ) 18G2M2Λ˜2 (cid:16) √ (cid:17) (cid:16) √ (cid:17)4
rb = Λ˜ cξ − 3sξ +η 6 −(cid:0)c −√3s (cid:1)2 +3C12 cξ − 3sξ + cξ − 3sξ ,
ξ ξ
(cid:40) (cid:34) (cid:35)(cid:41)
1 (cid:16) √ (cid:17) sec(3ξ) 18G2M2Λ˜2 (cid:16) √ (cid:17) (cid:16) √ (cid:17)4
rc = Λ˜ cξ + 3sξ +η 6 −(cid:0)c +√3s (cid:1)2 +3C12 cξ + 3sξ + cξ + 3sξ ,
ξ ξ
(4.22)
where r is the event horizon and r is the cosmological one. It is obvious that the limit η → 0
b c
recovers the two horizons for a Schwarzschild-de Sitter spacetime. The behavior of r and r are
b c
qualitatively displayed9 in Figure 1 and Figure 2, respectively. We observe that the increasing of M
implies on the increasing of r , as expected. On the other hand, r decreases as M increases.
b c
Figure1 Eventhorizonrelatedtothemasssphericaldistribution. r (x(M))isinunitsofΛ˜−1 and3GMΛ˜ ≡
b
x. The dashed curve represents the event horizon behavior of a standard black hole and the thick curve
represents the perturbative horizon. For the thick curve we adopted η =10−1.
8WepointoutthatthehorizonsdependontheintegrationconstantC ,whichmaydependexplicitlyonthemass
12
M. The qualitative/quantitative behaviour of such horizons and the respective thermodynamical quantities is directly
boundedbyC . InthenextstepsweareassumingthatC hasalineardependenceonM,soallgraphsinthispaper
12 12
areplottedunderthisassumption. ThisisanaturalhypothesissinceC appearasafactorofthepureSchwarzschild
12
term correction ∼ 1/r. Nevertheless, it may be ajusted by employing more sophisticated boundary conditions. A
complete analysis of this fine tuning is left for future investigation [25].
9The η parameter, according to [18, 16] is very small, providing no actual difference in the plots. For this reason
we overestimate η for the sake of comparison. The same observation holds for all the plots in this work.
9
Figure 2 Cosmological horizon. r (x(M)) is in units of Λ˜−1, 3GMΛ˜ ≡x and η =10−1. The dashed curve
c
represents the cosmological horizon obtained from the Schwarzschild-de Sitter geometry. The thick curve is
the cosmological horizon with the correction from the quadratic term contribution.
4.2 Singularities
It is a straightforward calculation to find the possible singularities in the perturbative solution (4.14)
by computing the Kretschmann invariant, the scalar curvature and the Ricci tensor squared, namely,
(cid:34) (cid:35)
48G2M2 8Λ˜4 (cid:18) 24G3M3(cid:19) 12 1440G3M3 16Λ˜2
RαβγδR = + +η 4G2M2+ + + ,
αβγδ r6 3 Λ˜2 r6 Λ˜2r9 r6
(cid:18) 36G2M2(cid:19)
R = 4Λ˜2+η 4Λ˜2+ ,
Λ˜2r6
(cid:18)72G2M2 (cid:19)
RαβR = 4Λ˜4+η +8Λ˜4 . (4.23)
αβ r6
It is then clear that a physical singularity exists at r = 0, as expected. Interestingly, the perturbative
contributions to R and RαβR are singular, while their zeroth order terms are not,
αβ
limR → η∞
r→0
limRαβR → η∞ . (4.24)
αβ
r→0
The physical singularity expressed by (4.23) and (4.24) are in agreement with the standard results
obtained in the Einsteinian gravity.
10