Black hole for the Einstein-Chern-Simons gravity

1.2 MB·

by  Cristian Quinzacara

Black hole for the Einstein-Chern-Simons gravity
C.A.C. Quinzacara∗ and P. Salgado†
Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile
Abstract
We consider a 5-dimensional action which is composed of a gravitational sector and a sector of
matter, where the gravitational sector is given by a Einstein-Chern-Simons gravity action instead
of the Einstein-Hilbert action.
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1 We obtain the Einstein-Chern-Simons (EChS) field equations together with its spherically sym-
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2 metric solution, which lead, in certain limit, to the standard five dimensional solution of the

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a Einstein-Cartan field equations.
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8 ItisfoundtheconditionsunderwhichtheEChS fieldequationsadmitsblackholetypesolutions.

] The maximal extension and conformal compactification are also studied
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r PACS numbers: 04.50.+h, 04.20.Jb, 04.90.+e
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∗ Electronic address: [email protected]
† Electronic address: [email protected]
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I. INTRODUCTION
According to the principles of general relativity (GR), the spacetime is a dynamical
object which has independent degrees of freedom, and is governed by dynamical equations,
namely the Einstein field equations. This means that in GR the geometry is dynamically
determined. Therefore, the construction of a gauge theory of gravity requires an action that
does not consider a fixed space-time background. An five dimensional action for gravity
fulfilling these conditions is the five-dimensional Chern–Simons AdS gravity action, which
can be written as
(cid:18) (cid:19)
1 2 1
L(5) = κ (cid:15) ea1···ea5 + (cid:15) Ra1a2ea3···ea5 + (cid:15) Ra1a2Ra3a4ea5 , (1)
AdS 5l5 a1···a5 3l3 a1···a5 l a1···a5
where ea corresponds to the 1-form vielbein, and Rab = dωab + ωaωcb to the Riemann
c
curvature in the first order formalism [1], [2], [3].
If Chern-Simons theories are the appropriate gauge-theories to provide a framework for
the gravitational interaction, then these theories must satisfy the correspondence principle,
namely they must be related to General Relativity.
In ref. [4] was recently shown that the standard, five-dimensional General Relativity
(without a cosmological constant) can be obtained from Chern-Simons gravity theory for
a certain Lie algebra B. The Chern-Simons Lagrangian is built from a B-valued, one-form
gauge connection A which depends on a scale parameter l which can be interpreted as a
coupling constant that characterizes different regimes within the theory. The B algebra, on
the other hand, is obtained from the AdS algebra and a particular semigroup S by means
of the S-expansion procedure introduced in refs. [5], [6]. The field content induced by B
includes the vielbein ea, the spin connection ωab and two extra bosonic fields ha and kab.
The five dimensional Chern-Simons Lagrangian for the B algebra is given by [4]:
(cid:18) (cid:19)
2
L(5) = α l2ε RabRcdee +α ε Rabecedee +2l2kabRcdT e +l2RabRcdhe , (2)
ChS 1 abcde 3 abcde 3
where we can see that (i) if one identifies the field ea with the vielbein, the system consists of
the Einstein-Hilbert action plus nonminimally coupled matter fields given by ha and kab; (ii)
it is possible to recover the odd-dimensional Einstein gravity theory from a Chern-Simons
2
gravity theory in the limit where the coupling constant l equals to zero while keeping the
effective Newton’s constant fixed.
It is the purpose of this article to find a spherically symmetric solution for the EChS field
equations, which are obtained from the so called Einstein-Chern-Simons action (2) studied
in Refs, [4], [7]. It is shown that the standard five dimensional solution of the Einstein-
Cartan field equations can be obtained, in a certain limit, from the spherically symmetric
solution of EChS field equations. The conditions under which these equations admits black
hole type solutions are found and the maximal extension and conformal compactification
are also studied.
This paper is organized as follows: In section 2 we find a spherically symmetric solution
for the Einstein-Chern-Simons field equations and then it is shown that the standard five
dimensional solution of the Einstein-Cartan field equations can be obtained, in a certain
limit, from the spherically symmetric solution of EChS field equations. In section 3 we
find the conditions under which the field equations admits black hole type solutions and we
studied the maximal extension and conformal compactification of such solutions. A brief
comment and three appendices conclude this work.
II. EINSTEIN-CHERN-SIMONS FIELD EQUATIONS FOR A SPHERICALLY
SYMMETRIC METRIC
In this section we consider the field equations for the lagrangian L = L +L , where
g M
L is the Chern-Simons gravity lagrangian L(5) and L is the corresponding matter
g ChS M
lagrangian.
In the presence of matter described by the langragian L = L (ea,ha,ωab), we have that
M M
the field equations obtained from the action (2) are given by [7]:
ε RcdTe = 0,
abcde
δL
α l2ε RbcRde = − M,
3 abcde δha
δL
ε (cid:0)2α Rbcedee +α l2RbcRde +2α l2D kbcRde(cid:1) = − M,
abcde 3 1 3 ω δea
δL
2ε (cid:0)α l2RcdTe +α l2D kcdTe +α ecedTe +α l2RcdD he +α l2Rcdke ef(cid:1) = − M.
abcde 1 3 ω 3 3 ω 3 f δωab
(3)
3
If Ta = 0 and kab = 0, the equation (3) can be written in the form
dea +ωaeb = 0,
b
ε RcdD he = 0,
abcde ω
(cid:18) (cid:19)
δL
α l2Y = −(cid:63) M ,
3 a δha
α l2Y +2α X = κT eb, (4)
1 a 3 a ab
where
(cid:18) (cid:19)
δL
X = (cid:63)(cid:0)ε Rbcedee(cid:1), Y = (cid:63)(cid:0)ε RbcRde(cid:1), T = −(cid:63) M (5)
a abcde a abcde ab δea
and where “(cid:63)” is the Hodge star operator.
T is the energy-momentum tensor of matter fields and κ is the coupling constant. In
ab
the equations (4) are present the fields ea, ωab (through Rab) and ha. If we wish to find a
spherically-and static-symmetric solution, then we must demand that the three fields satisfy
this conditions. Since a static space-time is one which posseses a timelike Killing vector
orthogonal to the spacelike hypersurfaces. These conditions are satisfied by the metric (6).
A. Spherically symmetric metric in five dimensions
We consider first the fields ea and ωab (through Rab). In five dimensions the static and
spherically symmetric metric is given by
ds2 = −e2f(r)dt2 +e2g(r)dr2 +r2dΩ2 = η eaeb (6)
3 ab
where dΩ2 = dθ2 +sin2θ dθ2 +sin2θ sin2θ dθ2 and η = diag(−1,+1,+1,+1,+1).
3 1 1 2 1 2 3 ab
Introducing an orthonormal basis, we have
eT = ef(r)dt, eR = eg(r)dr, e1 = rdθ , e2 = rsinθ dθ , e3 = rsinθ sinθ dθ . (7)
1 1 2 1 2 3
Taking the exterior derivatives, we get:
e−g
deT = −f(cid:48)e−geTeR, deR = 0, de1 = eRe1,
r
1 e−g 1 1 e−g
de2 = e1e2 + eRe2, de3 = e1e3 + e2e3 + eRe3, (8)
rtanθ r rtanθ rsinθ tanθ r
1 1 1 2
4
where a prime “ (cid:48) ” denotes derivative with respect to r. The next step is to use Cartan’s
first structural equation
Ta = dea +ωa eb = 0
b
and the antisymmetry of the connection forms (ωab = −ωba) to find the non-zero connection
forms. The calculations give:
e−g 1
ω = −f(cid:48)e−geT, ω = − ei, ω = − e2,
TR Ri 12
r rtanθ
1
1 1
ω = − e3, ω = − e3; i = 1,2,3. (9)
13 23
rtanθ rsinθ tanθ
1 1 2
From Cartan’s second structural equation
Ra = dωa +ωa ωc ,
b b c b
we can calculate the curvature matrix. The non-zero components are
(cid:16) (cid:17) f(cid:48)e−2g
RTR = e−g f(cid:48)g(cid:48) −f(cid:48)(cid:48) −(f(cid:48))2 eTeR, RTi = − eTei
r
g(cid:48)e−2g 1−e−2g
RRi = eRei, Rij = eiej; i,j = 1,2,3. (10)
r r2
Introducing (7), (10) into (5) we find
e−2g
(cid:0) (cid:1)
X = 12 g(cid:48)r+e2g −1 eT,
T r2
e−2g
(cid:0) (cid:1)
X = 12 f(cid:48)r−e2g +1 eR,
R r2
e−2g(cid:16) (cid:17)
X = 4 −f(cid:48)g(cid:48)r2 +f(cid:48)(cid:48)r2 +(f(cid:48))2r2 +2f(cid:48)r−2g(cid:48)r−e2g +1 ei, (11)
i r2
e−2g
(cid:0) (cid:1)
Y = 24 g(cid:48) 1−e−2g eT,
T r3
e−2g
(cid:0) (cid:1)
Y = 24 f(cid:48) 1−e−2g eR,
R r3
e−2g(cid:16) (cid:17)
Y = 8 f(cid:48)(cid:48) +(f(cid:48))2 −f(cid:48)g(cid:48) −e−2gf(cid:48)(cid:48) −e−2g(f(cid:48))2 +3e−2gf(cid:48)g(cid:48) ei. (12)
i r2
Introducing (7), (11), (12) into the third equation (4) and considering the energy-
momentum tensor as the energy-momentum tensor of a perfect fluid at rest, i.e., T = ρ(r)
TT
and T = T = P(r), where ρ(r) and P(r) are the energy density and pressure (for the
RR ii
5
perfect fluid), we find
e−2g e−2g κ
(cid:0) (cid:1) (cid:0) (cid:1)
α l2 g(cid:48) 1−e−2g +α g(cid:48)r+e2g −1 = ρ (13)
1 r3 3 r2 24
e−2g e−2g κ
(cid:0) (cid:1) (cid:0) (cid:1)
α l2 f(cid:48) 1−e−2g +α f(cid:48)r−e2g +1 = P (14)
1 r3 3 r2 24
e−2g (cid:16) (cid:17)
α l2 f(cid:48)(cid:48) +(f(cid:48))2 −f(cid:48)g(cid:48) −e−2gf(cid:48)(cid:48) −e−2g(f(cid:48))2 +3e−2gf(cid:48)g(cid:48)
1 r2
e−2g (cid:16) (cid:17) κ
+α −f(cid:48)g(cid:48)r2 +f(cid:48)(cid:48)r2 +(f(cid:48))2r2 +2f(cid:48)r−2g(cid:48)r−e2g +1 = P (15)
3 r2 8
Now consider the equation (13). After multiplying by 4r3 we find
(cid:110)(cid:0) (cid:1)(cid:16) (cid:0) (cid:1) (cid:17)(cid:111)(cid:48) κ
1−e−2g α l2 1−e−2g +2α r2 = ρr3. (16)
1 3
6
Integrating we have
(cid:16) (cid:17) κ (cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
1−e−2g α l2 1−e−2g +2α r2 = M(r)−M , (17)
1 3 12π2 0
where M is an integration constant and M(r) is the Newtonian mass, which is defined as
0
(cid:90) r
M(r) = 2π2 ρ(r¯)r¯3dr¯. (18)
0
From equation (17) we can see that
(cid:114)
r2 r4 K (cid:16) (cid:17)
e−2g = 1+α ± α2 + M(r)−M , (19)
l2 l4 12π2l2 0
where α = α /α , K = κ/α .
3 1 1
In order to make contact with the solutions of the Einstein-Cartan theory, consider the
limit l → 0:
(cid:32) (cid:114) (cid:33)
r2 r4 K (cid:16) (cid:17)
lime−2g = lim 1+α ± α2 + M(r)−M . (20)
l→0 l→0 l2 l4 12π2l2 0
If we consider the case of small l2 limit, we can expand the root to first order in l2. In
fact,
r2 (cid:26) (cid:18) Kl2 (cid:16) (cid:17) (cid:19)(cid:27)
e−2g ≈ 1+ α±|α| 1+ M(r)−M +O(l4)
l2 12π2l2α2r4 0
r2 K (cid:16) (cid:17)
≈ 1+ (α±|α|)± M(r)−M +O(l4). (21)
l2 24π2|α|r2 0
From (21) we can see that for this expression to be finite when l → 0, is necessary that
(α±|α|) = 0.
Since α = α /α we can distinguish two cases:
3 1
6
(a) If α > 0 and α > 0 or if α < 0 and α < 0 we have
3 1 3 1
(cid:114)
r2 r4 K (cid:16) (cid:17)
e−2g = 1+α − α2 + M(r)−M
l2 l4 12π2l2 0
K (cid:16) (cid:17)
≈ 1− M(r)−M
24π2|α|r2 0
κ (cid:16) (cid:17)
≈ 1− M(r)−M . (22)
24π2α r2 0
3
(b) If α > 0 and α < 0 or if α < 0 and α > 0 we have
3 1 3 1
(cid:114)
r2 r4 K (cid:16) (cid:17)
e−2g = 1+α + α2 + M(r)−M
l2 l4 12π2l2 0
K (cid:16) (cid:17)
≈ 1+ M(r)−M
24π2|α|r2 0
κ (cid:16) (cid:17)
≈ 1− M(r)−M . (23)
24π2α r2 0
3
This means that whatever the choice of the sign of the constant α and α we obtain
1 3
κ (cid:16) (cid:17)
lime−2g = 1− M(r)−M . (24)
l→0 24π2α3r2 0
From (24) we can see that if κ/2α = κ and M = 0 we recover the usual 5-dimensional
3 E 0
expresion for e−2g (see A17).
B. The Exterior Solution
The third equation (4) can be rewritten in the form
1
(cid:0) (cid:1) (cid:0) (cid:1)
(cid:63) ε Rbcedee + l2 (cid:63) ε RbcRde = κ T eb, (25)
abcde abcde E ab
2α
where α = α /α and κ = κ/2α .
3 1 E 3
(cid:112)
Rescaling the parameter l in the form l −→ l(cid:48) = l/ |α| we have
l2
(cid:0) (cid:1) (cid:0) (cid:1)
(cid:63) ε Rbcedee +sgn(α) (cid:63) ε RbcRde = κ T eb. (26)
abcde abcde E ab
2
If ρ(r) = P(r) = 0 and δL /δha (cid:54)= 0, the field equations are given by:
M
e−2g sgn(α)e−2g
(cid:0) (cid:1) (cid:0) (cid:1)
g(cid:48) 1−e−2g + g(cid:48)r+e2g −1 = 0, (27)
r3 l2 r2
e−2g sgn(α)e−2g
(cid:0) (cid:1) (cid:0) (cid:1)
f(cid:48) 1−e−2g + f(cid:48)r−e2g +1 = 0, (28)
r3 l2 r2
e−2g (cid:16) (cid:17)
f(cid:48)(cid:48) +(f(cid:48))2 −f(cid:48)g(cid:48) −e−2gf(cid:48)(cid:48) −e−2g(f(cid:48))2 +3e−2gf(cid:48)g(cid:48)
r2
sgn(α)e−2g (cid:16) (cid:17)
+ −f(cid:48)g(cid:48)r2 +f(cid:48)(cid:48)r2 +(f(cid:48))2r2 +2f(cid:48)r−2g(cid:48)r−e2g +1 = 0. (29)
l2 r2
7
Following the usual procedure, we find that the equation (27) has the following solution:
(cid:114)
r2 r4 κ
e−2g = 1+sgn(α) −sgn(α) +sgn(α) E M, (30)
l2 l4 6π2l2
where M is a constant of integration. From (30) is straightforward to see that in the limit
l → 0 we obtain the solution (A22) to Einstein’s gravity.
Adding equations (27) and (28) we find
e2f = e−2g. (31)
This solution satisfies the equation (29).
From (30) and (31) we can see that the line element for the outer region is given by
dr2
ds2 = −F(r)dt2 + +r2dΩ2, (32)
F(r) 3
where
(cid:114)
r2 r4 κ
E
F(r) = 1+sgn(α) −sgn(α) +sgn(α) M. (33)
l2 l4 6π2l2
III. BLACK-HOLE SOLUTION OF EINSTEIN-CHERN-SIMONS FIELD EQUA-
TIONS
Let us consider now the conditions under which the equation (26) admits black hole type
solutions.
A. Case α > 0: Black Holes
In this case the exterior solution is given by (32) with
(cid:114)
r2 r4 κ
E
F(r) = 1+ − + M. (34)
l2 l4 6π2l2
This solution shows an anomalous behaviour at
(cid:114)
r2 r4 κ
F(r ) = 1+ 0 − 0 + E M = 0,
0 l2 l4 6π2l2
i.e., at
(cid:114)
κ l2
E
r = M − (35)
0 12π2 2
8
so that
(cid:114)
r2 r4 +2r2l2 +l4
F(r) = 1+ − 0 . (36)
l2 l4
From the equations (32) and (35) we can see that if
κ
E M > l2, (37)
6π2
then the metric (32) shows an anomalous behaviour at r = r . A first elementary anomaly
0
is that we have at r = r
0
g = g11 = 0; g00 = g = ∞. (38)
00 11
A more serious anomaly is the following. One can verify that the parametric lines of
the coordinate r, i.e. the lines on which the coordinates t,θ ,θ ,θ have constant values,
1 2 3
are geodesics. But these geodesics are space-like for r > r and time-like for r < r . The
0 0
tangentvectorofageodesicundergoesparalleltransportalongthegeodesicandconsequently
itcannotchangefromatimeliketoaspace-likevector. Itfollowsthatthetworegions r > r
0
and r < r do not joint smoothly on the surface r = r .
0 0
This can be see in a more striking manner if we consider the radial null directions, on
which dθ = dθ = dθ = 0. We have then
1 2 3
dr2
ds2 = −F(r)dt2 + = 0. (39)
F(r)
Consequently the radial null directions satisfies the relations
dr
= ±F(r). (40)
dt
If we take into account the fact that the time-like directions are contained in the light-
cone, we find that in the region r > r the light cones have, in the plane (r,t), the orientation
0
shown on the figure 1.
The opening of the light cone, which is nearly equal to π/4 for r (cid:29) r , decreases with r
0
and tends to zero when r → r . On the contrary, in the region r < r the parametric lines
0 0
of the coordinate t are space-like and consequently the light cones are oriented as shown on
the left-hand side of figure 1, the opening of the cone increasing from the value zero at r = 0
to π/2 at r = r . Comparing the two diferent forms of the light cones on figure (1), we see
0
that the regions on either side of the surface r = r do not join smoothly on this surface.
0
9
FIG. 1. Space-time diagram in Schwarzschild-like coordinates for l2 = 2 and κ (6π2)−1M = 20,
EC
so that r = 3. Some future light cone has been drawn.
0
B. Eddington-Finkelstein and Kruskal-Szekeres coordinates
Let us define a radial coordinate
(cid:90)
dr
r∗ = , (41)
F(r)
we obtain (see appendix C2)
(cid:32) (cid:33)
r r2 +l2 (cid:18) (r−r )2 (cid:19)
r∗ = + 0 ln 0 +Z (r)
α>0
2 4r r (r+r )
0 0 0
(cid:115) (cid:32)(cid:115) (cid:33)
ir2 i i
− 0 F r,i
(cid:112) (cid:112)
2 2r2l2 +l4 2r2l2 +l4
0 0
(cid:114) (cid:40) (cid:32)(cid:115) (cid:33) (cid:32)(cid:115) (cid:33)(cid:41)
1 (cid:113) i i
+ i 2r2l2 +l4 F r,i −E r,i . (42)
2 0 (cid:112)2r2l2 +l4 (cid:112)2r2l2 +l4
0 0
In these coordinates the equation of the null geodesic (40) takes the form
d(t±r∗) = 0. (43)
10