Table Of ContentAccelerated FRW Solutions in Chern-Simons Gravity
Juan Cris´ostomo, Fernando Gomez, and Patricio Salgado
∗ † ‡
Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile
Cristian Quinzacara
§
Departamento de F´ısica, Universidad de Concepci´on,
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Casilla 160-C, Concepci´on, Chile and
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Facultad de Ingenier´ıa y Tecnolog´ıa, Universidad San Sebasti´an,
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Campus Las Tres Pascualas, Lientur 1457, Concepci´on, Chile
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9 Mauricio Cataldo¶
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Departamento de F´ısica, Universidad del B´ıo-B´ıo, Casilla 5-C, Concepci´on, Chile.
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- Sergio del Campo∗∗
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Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso,
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3 Av. Universidad 300, Campus Curauma, Valpara´ıso, Chile.
v
8
(Dated: August 20, 2014)
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Abstract
We consider a five-dimensional Einstein-Chern-Simons action which is composed of a gravita-
tional sector and a sector of matter, where the gravitational sector is given by a Chern-Simons
gravity action instead of the Einstein-Hilbert action and where the matter sector is given by the
so called perfect fluid. It is shown that (i) the Einstein-Chern-Simons (EChS) field equations
subject to suitable conditions can be written in a similar way to the Einstein-Maxwell field equa-
tions; (ii) these equations have solutions that describe accelerated expansion for the three possible
cosmological models of the universe, namely, spherical expansion, flat expansion and hyperbolic
expansion when α, a parameter of theory, is greater than zero. This result allow us to conjeture
that this solutions are compatible with the era of Dark Energy and that the energy-momentum
tensor for the field ha, a bosonic gauge field from the Chern-Simons gravity action, corresponds to
a form of positive cosmological constant.
It is also shown that the EChS field equations have solutions compatible with the era of matter:
(i) In the case of an open universe, the solutions correspond to an accelerated expansion (α > 0)
with a minimum scale factor at initial time that, when the time goes to infinity, the scale factor
behaves as a hyperbolic sine function. (ii) In the case of a flat universe, the solutions describing
an accelerated expansion whose scale factor behaves as a exponencial function when time grows.
(iii) In the case of a closed universe it is found only one solution for a universe in expansion, which
behaves as a hyperbolic cosine function when time grows.
∗ [email protected]
† [email protected]
‡ [email protected]
§ [email protected]
¶ [email protected]
∗∗ [email protected]
2
I. INTRODUCTION
Some time ago was shown that the standard, five-dimensional General Relativity can be
obtained fromChern-Simons gravity theory for a certain Lie algebra B[1], whose generators
J ,P ,Z ,Z satisfy the commutation relationships
ab a ab a
{ }
[J ,J ] = η J η J +η J η J ,
ab cd ad bc ac bd bc ad bd ac
− −
[P ,J ] = η P η P ,
a bc ab c ac b
−
[J ,Z ] = η Z η Z +η Z η Z ,
ab cd ad bc ac bd bc ad bd ac
− −
[Z ,J ] = η Z η Z ,
a bc ab c ac b
−
[P ,P ] = Z ,
a b ab
[P ,Z ] = η Z η Z .
a bc ab c ac b
−
This algebra was obtained from the anti de Sitter (AdS) algebra and a particular semi-
group S by means of the S-expansion procedure introduced in Refs. [2], [3].
In order to write down a Chern–Simons lagrangian for the B algebra, we start from the
one-form gauge connection
1 1 1 1
A = ωabJ + eaP + kabZ + haZ , (1)
ab a ab a
2 l 2 l
and the two-form curvature
1 1 1 1
F = RabJ + TaP + D kab + eaeb Z
2 ab l a 2 ω l2 ab
(cid:18) (cid:19)
1
+ D ha +ka eb Z . (2)
l ω b a
(cid:0) (cid:1)
Consistency with the dual procedure of S-expansion in terms of the Maurer-Cartan forms
[3] demands that ha inherits units of length from the fu¨nfbein; that is why it is necessary to
introduce the l parameter again, this time associated with ha.
It is interesting to observe that J are still Lorentz generators, but P are no longer
ab a
AdS boosts; in fact, [P ,P ] = Z . However, ea still transforms as a vector under Lorentz
a b ab
transformations, as it must be in order to recover gravity in this scheme.
A Chern-Simons lagrangian in d = 5 dimensions is defined to be the following local
function of a one-form gauge connection A:
1 1
L(5) (A) = k AF2 A3 F+ A5 , (3)
ChS − 2 10
(cid:28) (cid:29)
3
where denotes a invariant tensor for the corresponding Lie algebra, F = dA+AA is
h···i
the corresponding the two-form curvature and k is a constant [4].
Using theorem VII.2 of Ref. [2], it is possible to show that the only non-vanishing com-
ponents of a invariant tensor for the B algebra are given by
4l3
J J P = α ǫ , (4)
h a1a2 a3a4 a5i 1 3 a1···a5
4l3
J J Z = α ǫ ,
h a1a2 a3a4 a5i 3 3 a1···a5
4l3
J Z P = α ǫ ,
h a1a2 a3a4 a5i 3 3 a1···a5
where α1 and α3 are arbitrary independient constants of dimensions [length]−3.
Using the extended Cartan’s homotopy formula as in Ref. [5], and integrating by parts, it
is possible to write down the Chern-Simons Lagrangian in five dimensions for the algebra
B
as
L(5) = α l2ǫ eaRbcRde
EChS 1 abcde
2
+α ǫ Rabecedee +2l2kabRcdTe +l2RabRcdhe
3 abcde
3
(cid:18) (cid:19)
(4)
+dB (5)
EChS
(4)
where the suface term B is given by
EChS
2 1
B(4) = α l2ǫ eaωbc dωde + ωd ωfe
EChS 1 abcde 3 2 f
(cid:18) (cid:19)
2 1
+α ǫ l2 haωbc +kabec dωde + ωd ωfe
3 abcde 3 2 f
"
(cid:18) (cid:19)
(cid:0) (cid:1)
2 1 1
+l2kabωcd dee + ωd ee + eaebecωde (6)
3 2 f 6
#
(cid:18) (cid:19)
and where α , α are parameters of the theory, l is a coupling constant, Rab = dωab+ωa ωcb
1 3 c
corresponds to the curvature 2-form in the first-order formalism related to the 1-form spin
connection [4], [6], [7], and ea, ha and kab are others gauge fields presents in the theory [1].
From (5) we can see that the third term is a surface term and can be removed from this
Lagrangian. So that,
L(5) = α l2ε RabRcdee
EChS 1 abcde
2
+α ǫ Rabecedee +2l2kabRcdTe +l2RabRcdhe (7)
3 abcde
3
(cid:18) (cid:19)
4
is the Einstein-Chern-Simons Lagrangian studied in Ref [1].
It should be noted the absence of kinetic terms for the fields ha and kab in equation (7).
The term kinetic for the ha and kab fields are present in the surface term of the Lagrangian
(5) given by (6).
The Lagrangian (7) show that standard, five-dimensional General Relativity emerges as
the l 0 limit of a CS theory for the generalized Poincar´e algebra B. Here l is a length
→
scale, a coupling constant that characterizes different regimes within the theory. The B
algebra, on the other hand, is constructed from the AdS algebra and a particular semigroup
S by means of the S-expansion procedure. The field content induced by the B algebra
includes the vielbein ea, the spin connection ωab and two extra bosonic fields ha and kab,
which can be interpreted as boson fields coupled to the field curvature and the parameter l2
can be interpreted as a kind of coupling constant.
Recently was found [8] that the standard five-dimensional FRW equations and some of
their solutions can be obtained, in a certain limit, from the so-called Chern-Simons-FRW
field equations, which are the cosmological field equations corresponding to a Chern-Simons
gravity theory.
It is the purpose of this paper to show that the Einstein-Chern-Simons (EChS) field
equations, subject to (i) the torsion-free condition (Ta = 0) and (ii) the variation of the
matter Lagrangian with respect to (w.r.t.) the spin connection is zero (δL /δωab = 0) can
M
be written in a similar way to the Einstein-Maxwell field equations. The interpretation of
theha fieldasa perfect fluid allowus toshow that theEinstein-Chern-Simons fieldequations
have an universe in accelerated expansion as a of their solutions.
This paper is organized as follows: In Section II we briefly review the Einstein-Chern-
Simons field equations. In Section III we study the Einstein-Chern-Simons field equations in
the range of validity of general relativity. In Section V we consider accelerated solutions for
Einstein-Chern-Simons field equations. We try to find solutions that describes accelerated
expansion for cases of open universes, flat universes and closed universes. In Section VI
we consider the consistency of the solutions with the ”Era of Matter”. A summary and an
appendix conclude this work.
5
II. EINSTEIN-CHERN-SIMONS FIELD EQUATIONS
In Ref. [8] was found that in the presence of matter the lagrangian is given by
(5)
L = L +κL (8)
ChS M
(5)
where L is the five-dimensional Chern-Simons lagrangian given by (7),
ChS
L = L (ea,ha,ωab) is the matter Lagrangian and κ is a coupling constant related to
M M
the effective Newton’s constant. The variation of the lagrangian (8) w.r.t. the dynamical
fields vielbein ea, spin connection ωab, ha and kab, leads to the following field equations
ε 2α Rabeced +α l2RabRcd
abcde 3 1
(cid:16) δL
+2α l2D kabRcd = κ M, (9)
3 ω δee
(cid:17)
δL
α l2ε RabRcd = κ M, (10)
3 abcde δhe
δL
2α l2ε RcdTe = κ M, (11)
3 abcde
δkab
2ε α l2RcdT e +α l2D kabTe
abcde 1 3 ω
(cid:16)
+α ecedTe +α l2RcdD he
3 3 ω
(cid:17) δL
+2α ε l2Rcdke ef = κ M. (12)
3 abcde f δωab
For simplicity, we will assume that the torsion vanishes (Ta = 0) and kab = 0. In this
case the Eqs.(9 - 12) takes the form
δL
ε 2α Rabeced +α l2RabRcd = κ M, (13)
abcde 3 1
δee
(cid:0) α l2ε RabRc(cid:1)d = κδLM, (14)
3 abcde δhe
δL
M
= 0 (15)
δkab
δL
2α l2ε RcdD he = κ M. (16)
3 abcde ω δωab
6
This field equations system can be written in the form
δL δL
ε Rabeced = 4κ M +α M , (17)
abcde 5 δee δhe
(cid:18) (cid:19)
δL
l2ε RabRcd = 8κ M, (18)
abcde 5 δhe
δL
l2ε RcdD he = 4κ M (19)
abcde ω 5
δωab
where we introduce κ = κ/8α and α = α /α .
5 3 1 3
−
The field equation (9) contains three terms. The first one, proportional to the Einstein
tensor. The second one corresponds to a quadratic term in the curvature, and a third one,
a term that describes the dynamics of the field kab. Since we asume kab = 0 the last term
in left side of Eq. (9) vanishes.
InordertowritethisfieldequationmanneranalogoustoEinstein’s equations, onechooses
to leave the term proportional to the Einstein tensor on the left side of Eq. (9)
κ δL α
ǫ Rbcedee = M 1 l2ǫ RbcRde
abcde 2α δea − 2α abcde
3 3
and using the Eq. (14) we obtain Eq. (17).
This result allows us to interpret δL /δha as the energy momentum tensor for a second
M
type of matter, not ordinary. Henceforth we will say that δL /δha corresponds to the
M
energy-momentum tensor for the field ha.
The equation of motion for the ha-field is given by Eq.(19). The condition δL /δωab = 0
M
(usual in gravity theories), imposed for consistency with the condition Ta = 0, leads to
the equation of motion (22) for the ha-field . This means that ha-field is governed by the
following field equations
δL δL
ε Rabeced = 4κ M +α M , (20)
abcde 5 δee δhe
(cid:18) (cid:19)
l2 δL
ε RabRcd = M, (21)
8κ abcde δhe
5
ε RcdD he = 0. (22)
abcde ω
This means that the Einstein-Chern-Simons field equations, subject to the conditions
Ta = 0, kab = 0 and δL /δωab = 0, can be re-written in a way similar to the Einstein-
M
Maxwell field equations.
7
From (20-22) we can see that if L = 0, then in five dimensions there is no solution of
M
Schwarzschild type [1], [9].
III. EINSTEIN-CHERN-SIMONS EQUATIONS IN THE RANGE OF VALIDITY
OF GENERAL RELATIVITY
From (20-21) we can see that general relativity is valid when (i) the curvature Rab takes
values not excessively large (ii) the parameter l takes small values (l 0) [1]; (iii) the
−→
constant α takes values not excessively large. In fact, in this case we have that (21) takes
the form
δL
M
0. (23)
δhe ≈
Introducing (23) into (20) we obtain the Einstein’s field equation
δL
ε Rabeced 4κ M. (24)
abcde 5
≈ δee
If Rab is not large then δL /δea is also not large. This means that General Relativity
M
can be seen as a low energy limit of Einstein-Chern-Simons gravity. So that, in the range
of validity of the General Relativity, the equations (20-22) are given by
δL
ε Rabeced = 4κ M, (25)
abcde 5 δee
ε RcdD he = 0. (26)
abcde ω
On the another hand, if Rab is large enough, so that when it is multiplied by l2 (which
is very small) will have a non-negligible results, then we will find that δL /δha is not
M
negligible. This means that, in this case, we must consider the entire system of equations
(20-22).
IV. EINSTEIN-CHERN-SIMONS FIELD EQUATIONS FOR A FRIEDMANN-
ROBERTSON-WALKER-LIKE SPACETIME
Theshapeofthefieldea isobtainedfromoftheapplicationofthecosmologicalprincipleto
the metric tensor of spacetime: it is considered a splitting of the 5D-manifoldin a maximally
symmetric four-dimensional manifoldandonetemporaldimension (M = R Σ ). This leads
4
×
8
to five dimensional Friedmann-Robertson-Walker (FRW) metric. So that, the vielbein can
be chosen like in [8]:
e0 = dt,
a(t)
e1 = dr,
√1 kr2
−
e2 = a(t)rdθ ,
2
e3 = a(t)rsinθ dθ ,
2 3
e4 = a(t)rsinθ sinθ dθ (27)
2 3 4
where a(t) is the scale factor of the universe and k is the sign of the curvature of space (Σ ):
4
(i) +1 for a closed space (S4), (ii) 0 for a flat space (E4) and (iii) 1 for an open space
−
(hyperbolic).
The application of the cosmological principle to the metric tensor of the spacetime also
constrains the shape of the field ha (see for example [8]). A detailed discussion can be also
found in Ref. [10]. The bosonic field ha is given by
h0 = h(0)dt = h(0)e0,
a(t)
h1 = h(t) dr = h(t)e1,
√1 kr2
−
h2 = h(t)a(t)rdθ = h(t)e2,
2
h3 = h(t)a(t)rsinθ dθ = h(t)e3,
2 3
h4 = h(t)a(t)rsinθ sinθ dθ = h(t)e4 (28)
2 3 4
where h(0) is a constant and h(t) is a function of time t that must be determined. Sub-
stituting (28) into Eq. (22) we obtain the explicit form of the equations of motion for the
ha-field, which will be displayed in Eq.(39).
In accordance with the equation (20), we will consider a fluid composed of two perfect
fluids, thefirst onerelatedtoordinaryenergy-momentum tensor (T δLM)andthesecond
µν ∼ δea
one related to field ha (T(h) δLM). The energy-momentum tensors in the comoving frame,
µν ∼ δha
are given by
T = diag(ρ,p,p,p,p), (29)
µν
T(h) = diag ρ(h),p(h),p(h),p(h),p(h) , (30)
µν
(cid:16) (cid:17)
9
where ρ is the matter density and p is the pressure of fluid. Then, the energy-momentum
tensor for the composed fluid is
T˜ = T +αT(h) (31)
µν µν µν
= diag ρ+αρ(h),p+αp(h),
(cid:16)
p+αp(h),p+αp(h),p+αp(h) (32)
(cid:17)
= diag(ρ˜,p˜,p˜,p˜,p˜). (33)
In the torsion-free case, the energy momentum tensor of ordinary matter satisfies a con-
servation equation and the Einstein tensor has also zero divergence. In this case the energy
momentum tensor for the non-ordinary matter must also satisfy a conservation equation. In
fact, from Eq. (20) we find
Tµ = 0 , T(h)µ = 0 (34)
∇µ ν ∇µ ν
Introducing (27 - 33) into eqs. (20 - 22) we find the following field equations (see Ref. [8]
and Appendix A)
a˙2 +k
6 = κ ρ˜, (35)
a2 5
(cid:18) (cid:19)
a¨ a˙2 +k
3 + = κ p˜, (36)
a a2 − 5
(cid:20) (cid:18) (cid:19)(cid:21)
3l2 a˙2 +k 2
= ρ(h), (37)
κ a2
5 (cid:18) (cid:19)
3l2a¨ a˙2 +k
= p(h), (38)
κ a a2 −
5 (cid:18) (cid:19)
a˙2 +k a˙
˙
(h h(0)) +h = 0. (39)
a2 − a
(cid:18) (cid:19)(cid:20) (cid:21)
We should note that equation (35) was studied in Ref. [11] in the context of inflationary
cosmology .
The Equations (35) and (36) are very similar to the Friedmann equations in five dimen-
sions. However now ρ and p are subject to restrictions imposed by the remaining equations.
10
