Quasi-Local Conserved Charges of Spin-3 Topologically Massive Gravity

0.23 MB·

by  M. R. Setare

Quasi-Local Conserved Charges of Spin-3
Topologically Massive Gravity
M. R. Setare 1 , H. Adami 2
6 Department of Science, University of Kurdistan, Sanandaj, Iran.
1
0 Abstract
2
In this paper we obtain conserved charges of spin-3 topologically massive

n
gravity by using a quasi-local formalism. We find a general formula to
u
J calculate conserved charge of the spin-3 topologically massive gravity

3 which corresponds to a Killing vector field ξ. We show that this general
1 formula reduces to the previous one for the ordinary spin-3 gravity

presented in [18] when we take into account only transformation under
]
h
diffeomorphism, without considering generalized Lorentz gauge
t
- transformation (i.e. λ =0), and by taking 1 0. Then we obtain a
p ξ µ →
e general formula for the entropy of black hole solutions of the spin-3
h topologically massive gravity. Finally we apply our formalism to calculate
[
energy, angular momentum and entropy of a special black hole solution

3
and we find that obtained results are consistent with previous results in
v
the limiting cases. Moreover our result for energy, angular momentum and
1
7 entropy are consistent with the first law of black hole mechanics.
1
0
0 1 Introduction
.
1
0
Higher spin gravity was formulated by Vasiliev and collaborators in papers
6
1 [1]. Initssimplestformitisanextension ofordinarygravity thatincludesa
: masslessscalar andmasslessfieldswithspinsS = 3,4,.... In[2]Vasiliev pro-
v
i posed a system of gauge invariant nonlinear dynamical equations for totally
X
symmetric massless fields of all spins in (A)dS backgrounds. According to
r
a the result of this paper, ”in the framework of gravity, unbroken higher spin
gauge symmetries require a non-zero cosmological constant”. In paper [3]
the authors have considered the coupling of a symmetric spin-3 gauge field
ϕ to 3-dimensional gravity in a second order metric-like formulation. In
µνλ
the context of frame-like approach the gravitational coupling of a symmet-
ric tensor of rank 3 in the presence of negative cosmological constant can
be given by SL(3,R) SL(3,R) Chern-Simons theory [4, 5, 6]. The emer-
×
gence of W-algebras as asymptotic symmetries of higher-spin gauge theories
1E-mail: [email protected]
2E-mail: [email protected]
1
coupled to three-dimensional Einstein gravity with a negative cosmological
constant has been discussed in [4].
Higher-spin theories in AdS , like ordinary gravity, they possess no prop-
3
agating degrees of freedom [4, 7]. Pure Einstein-Hilbert gravity in three
dimensions exhibits no propagating physical degrees of freedom [8, 9]. But
addingthegravitationalChern-Simonstermproducesapropagatingmassive
graviton [10]. The resulting theory is called topologically massive gravity
(TMG). The authors of [11] have done the generalization of topologically
massive gravity to higher spins, specifically spin-3 (see also [12]). In this
paper we would like to obtain conserved charges of black hole solutions of
spin-3 topologically massive gravity by using a quasi-local formalism. Here,
we just consider diffeomorphism covariance and we find energy, angular mo-
mentum, and entropy, but we don’t investigate the higher-spin charges.
A method to calculate the energy of asymptotically AdS solution was given
by Abbott and Deser [13]. Deser and Tekin have extended this approach to
thecalculation of theenergyof asymptotically dSorAdSsolutions inhigher
curvature gravity models and also to TMG [14]. The authors of [15] have
obtained the quasi-local conserved charges for black holes in any diffeomor-
phically invariant theory of gravity. By consideringan appropriatevariation
of the metric, they have established a one-to-one correspondence between
the ADT approach and the linear Noether expressions. They have extended
this work to a theory of gravity containing a gravitational Chern-Simons
term in [16], and have computed the off-shell potential and quasi-local con-
served charges of some black holes in TMG. In the first order formalism,
quasi-local conserved charges of Lorentz-diffeomorphism covariant theories
of gravity are investigated in [17]. But there are theories which are not
Lorentz-diffeomorphism covariant so for such theories a method for which
one can calculate conserved charges of Lorentz-diffeomorphism covariant
theories are useless. In previous paper [19] by introducing the total varia-
tion of a quantity due to an infinitesimal Lorentz-diffeomorphism transfor-
mation, we have obtained the conserved charges in Lorentz-diffeomorphism
non-covariant theories. The formalism proposed in [19] is for on-shell case.
In this paper we want to extend this formalism to the spin-3 topologically
massive gravity. We find an expression for ADT conserved current which
is off-shell conserved for a given Killing vector field. Then we can find the
generalized ADT conserved charge by virtue of the Poincare lemma. In ref.
[18] the energy of the higher spin black hole solutions of ordinary higher
spin gravity has obtained by canonical formalism. Here we not only obtain
the energy but also we find the angular momentum of black hole solutions
of spin-3 topologically massive gravity. Furthermore we obtain a general
2
formula for entropy of black hole solutions of spin-3 topologically massive
gravity, where by substituting e ab = ω ab = 0, this formula reduces to
µ µ
the entropy formula in the ordinary topologically massive gravity which we
obtained previously in the paper [19].
2 First-order formalism of 3D spin-3 gravity
In this section, we will briefly review relevant aspects of spin-3 gravity in
three dimensions in a slightly different way. We can describe the spin-3
gravity in three dimensions by the generalized dreibein and spin-connection
[4]. We want to work in the first order formalism, so we take generalized
spin-connection as an independent quantity as well as generalized dreibein.
The algebra sl(3,R) have 3 generators J and 5 generators T with the
a ab
following commutation relations 3
[J ,J ]= ε cJ , [J ,T ] = 2εd T ,
a b ab c a bc a(b c)d
(1)
[T ,T ]= 2 η ε e+η ε e J ,
ab cd a(c d)b b(c d)a e
−
where T are symmetric and(cid:0) trace-less in the Lo(cid:1)rentz indices. In the repre-
ab
sentation we have considered the inner product of the generators are [20]
tr(J ) = 0, tr(T ) = 0, tr(J T ) = 0,
a ab a bc
4 (2)
tr(J J )= 2η , tr(T T ) = 4η η η η .
a b ab ab cd a(c d)b ab cd
− 3
The generalized dreibein and the generalized spin-connection take values in
the Lie algebra sl(3,R) as follows respectively:
e = e aJ +e abT , (3)
µ µ a µ ab
ω = ω aJ +ω abT , (4)
µ µ a µ ab
a,b = 0,1,2. Notice that e ab is symmetric and trace-less with respect to
µ
Lorentz indices. It is convenient to write the generalized dreibein one-form
and the generalized spin-connection one-form as [3, 21]
e= e AJ dxµ, ω = ω AJ dxµ, (5)
µ A µ A
respectively, and A = 1,...,8. In the above J denotes a full set of sl(3,R)
A
generators,
[J ,J ] = f CJ , (6)
A B AB C
3Inthispaper,weusetheordinarysymmetrizationbyapairofparentheses,forinstance
A(µBν) = 12(AµBν+AνBµ),i.e. wedividebythenumberoftermsinthesymmetrization.
3
where f C are anti-symmetric structure constants. The Killing form in
AB
the fundamental representation of sl(3,R) is defined as
1
K = tr(J J ), (7)
AB A B
2
andtheanti-symmetricandsymmetricstructureconstantsoftheLiealgebra
are given by
1 1
f = tr([J ,J ]J ), d = tr( J ,J J ). (8)
ABC A B C ABC A B C
2 2 { }
Translating the frame-like formalism to the metric-like formalism given by
the following transformations [4]
1
g = tr(e e )
µν 2! (µ ν)
(9)
1
ϕ = tr(e e e ).
µνλ (µ ν λ)
3!
where g is the metric and ϕ is the spin-3 field and it is totally symmet-
µν µνλ
ric. By substituting e from Eq.(5) into Eq.(9) and using Eq.(7) and Eq.(8)
µ
we have
g = K e Ae B
µν AB µ ν
1 (10)
ϕ = d e Ae Be C.
µνλ ABC µ ν λ
3!
It is clear that g and ϕ both are invariant under the following Lorentz-
µν µνλ
like gauge transformation
e˜ = Le L−1, (11)
µ µ
where L SL(3,R) and we can write L = exp(λ) so that λ is the generator
∈
ofLorentz-liketransformationanditisasl(3,R)Liealgebravaluedquantity,
λ = λaJ +λabT = λAJ . (12)
a ab A
Now we can introduce the exterior covariant derivative by demanding that
it be covariant under the Lorentz-like transformation (11) as well as diffeo-
morphism covariance. To this end, we need to introduce the generalized
spin-connection in the form of (5). Thus, we can define exterior covariant
derivative as
D e = ∂ e +[ω ,e ] D e A = ∂ e A+fA ω Be C. (13)
µ ν µ ν µ ν µ ν µ ν BC µ ν
⇔
4
In order the above exterior derivative be covariant under the Lorentz-like
transformation, the generalized spin-connection must transforms as
ω˜ = LωL−1+LdL−1, (14)
where d denotes the ordinary exterior derivative. Now, one can define the
total derivative as follows:
D(T)e = e +[ω ,e ]
µ ν µ ν µ ν
∇ (15)
=∂ e Γλ e +[ω ,e ]
µ ν µν λ µ ν
−
whereΓλ canbeinterpretsasAffineconnectionand denotestheordinary
µν
∇
covariantderivativewhichisdefinedbyusingAffineconnection. Themetric-
connectioncompatibility condition g = 0leadstothetotalderivativeof
λ µν
∇
(T)
dreibein vanishes, i.e. D e = 0. Therefore, one can definethegeneralized
µ ν
torsion 2-form as
= e Γλ dxµ dxν = De. (16)
λ µν
T ∧
Another useful covariant quantity is the generalized curvature 2-form which
is given by
1
= dω+ω ω A = dωA+ fA ωB ωC. (17)
BC
R ∧ ⇔ R 2 ∧
Since the total derivative of e A is zero, then we have ϕ = 0.
µ σ µνλ
∇
FromEq.(11)andEq.(14),itiseasytocheckthatthevariationofgeneralized
dreibein and generalized spin connection under an infinitesimal Lorentz-like
gauge transformation are
δ e = [λ,e], (18)
λ
δ ω = Dλ, (19)
λ
−
where
D = ∂ λ+[ω ,λ]. (20)
µ µ µ
It is well-known that the ordinary Lie derivative of a diffeomorphism in-
variant quantity is diffeomorphism covariant but it may be not Lorentz-like
covariant. In the manner of papers [22, 23, 24, 25, 19], we can generalize
the Lie derivative so that it becomes covariant under Lorentz-like transfor-
mation as well as diffeomorphism. To this end, we consider the ordinary
Lie derivative of dreibein along a curve generated by the vector filed ξ,
£ e = ξλ∂ e +e ∂ ξλ. Now we generalize this by adding variation of e
ξ µ λ µ λ µ µ
with respect to an infinitesimal Lorentz-like gauge transformation
L e= £ e+[λ ,e]. (21)
ξ ξ ξ
5
If the generator of Lorentz-like gauge transformation λ transforms as
ξ
λ˜ = Lλ L−1+L£ L−1, (22)
ξ ξ ξ
under Lorentz-gauge transformation, then the definition of generalized Lie
derivative (21) will be covariant under the Lorentz-like transformations. It
is straightforward to extend this expression of the generalized Lie derivative
for e A to the case for which we have more than one Lorentz-like index. We
µ
demandthatλ dependsonthevector fieldξ generating thediffeomorphism
ξ
and it is not depends on fields in general but in section 5, we will fix it such
that Lie derivative of e along ξ vanishes explicitly when ξ is a Killing
µ
vector field. The generalized Lie derivative of generalized spin-connection,
L ω, is not covariant under Lorentz-like transformations, while L ω dλ
ξ ξ ξ
−
will be covariant under the Lorentz-like transformations. Thus, under the
generalized local translations we have
δ e = L e, (23)
ξ ξ
δ ω = L ω dλ , (24)
ξ ξ ξ
−
thereforetheyarecovariantundertheLorentz-liketransformationsaswellas
diffeomorphism. Hence, we can interpret δ as a generalized diffeomorphism
ξ
transformation. We can useEq.(23)to obtain behaviourof themetric under
this generalized diffeomorphism transformation as
δ g = £ g . (25)
ξ µν ξ µν
This equation is exactly what we expect. In a similar way, for spin-3 field
we obtain
δ ϕ = £ ϕ , (26)
ξ µνλ ξ µνλ
In the last step of the calculation we used fE d = 0. Since ϕ
D(A BC)E ∇σ µνλ
is zero, the equation (26) can be rewritten as
δ ϕ = 3 ϕ ξσ +3(i σ) ϕ , (27)
ξ µνλ ∇(λ µν)σ ξT (λ µν)σ
(cid:0) (cid:1)
thus in a torsion free theory this reduce to
δ ϕ = ξ , (28)
ξ µνλ (λ µν)
∇
where
ξ = 3ϕ ξσ = 3e ae be ξσ 4e abe e c ξσ. (29)
µν µνσ µ ν σab µ νbc σ a
−
6
The equation (28) is just the diffeomorphism part of the spin-3 field gauge
transformation which was introduced in [26, 27], so it is clear from Eq.(28)
that any transformation under diffeomorphism can be written as a spin-3
field gauge transformation for a torsion free theory.
It is well known that Einstein-Hilbert action in the presence of negative
cosmologicalconstantin3-dimensioncanbereformulatedasaChern-Simons
theory withgage groupSO(2,2) SL(2,R) SL(2,R) [28, 29]. Similarly, a
∼ ×
SL(3,R) SL(3,R)Chern-Simonstheorywiththefollowingactiondescribes
×
a three dimensional spin-3 gravity theory [5, 6] (see also [30]),
S = S [A+] S [A−], (30)
EH CS CS
−
where
l 2
S [A] = (A dA+ A A A), (31)
CS
8πG ∧ 3 ∧ ∧
Z
where l2 > 0 corresponds to a negative cosmological constant, and G is
Newton’s constant. To relate these to the first order formalism based on
dreibein and spin-connection we introduce following two independent con-
nection one-form A+ and A−,
1
±
A = ω e. (32)
± l
±
By substitutingA fromEq.(32)intoEq.(30)weobtainthefollowing action
which describes a three dimensional spin-3 gravity theory
1 1
S = (e + e e e), (33)
EH 16πG ∧R 3l2 ∧ ∧
Z
In the next section we will extend the above Einstein-Hilbert spin-3 gravity
actiontothespin-3topologically massivegravity theory. Thefieldequations
come from above action are
1
= 0, + e e = 0. (34)
T R l2 ∧
Sincei ω transformslike(22)thenonecanchooseλ = i ω. Noticethatthis
ξ ξ ξ
choice is not unique. Using the field equations and by choosing λ = i ω,
ξ ξ
Eq.(18) and Eq.(19) reduce to
1
δ e = DΞ, δ ω = [e,Ξ], (35)
Ξ Ξ l2
where Ξ = i e. These results are exactly the generalized local translations
ξ
which were already mentioned in [4, 3, 21].
7
3 Spin-3 topologically massive gravity
In this section, we consider an action describing the spin-3 field coupled to
TMG. Thisgeneralization has already beendone[11,12]. Inthat paper,the
authors have studied this model at the linearized level. In this paper, we
are interested to consider the Spin-3 Topologically Massive Gravity in non-
linear level and we want to find a general formula for quasi-local conserved
charges of the solutions of this model.
The Lagrangian of the spin-3 Topologically Massive Gravity is given by [11]
Λ 1 2
= tr σe + e e e+ ωdω+ ω ω ω +h . (36)
L {− ∧R 3 ∧ ∧ 2µ 3 ∧ ∧ ∧T}
(cid:18) (cid:19)
In the above Lagrangian h is an auxiliary sl(3,R) Lie algebra valued one-
form field. Also µ is a mass parameter, σ is a sign and Λ denotes the
cosmological parameter.
The general variation of the Lagrangian (36) is given by
δ = tr(δΦ E )+dΘ(Φ,δΦ). (37)
Φ
L ∧
We use the symbol Φ to denote all of fields e, ω and h. Here E denotes the
Φ
equation of motion associated with field Φ. Equations of motion are given
as follows:
E = σ +Λe e+Dh = 0, (38)
e
− R ∧
1
E = σ + +e h+h e = 0, (39)
ω
− T µR ∧ ∧
E = = 0. (40)
h
T
Also, the surface term Θ(Φ,δΦ) is
1
Θ(Φ,δΦ) = tr σδω e+ δω ω+δe h . (41)
− ∧ 2µ ∧ ∧
(cid:18) (cid:19)
If we take
Λ 1
l2 = and h= e, (42)
−σ 2µl2
then the equations of motion (38)-(40) reduce to
1
= 0, + e e = 0. (43)
T R l2 ∧
These are exactly what we are mentioned already in the equation (34). In
this way, all the solutions of the spin-3 gravity (for instance, see [20, 30, 31])
are solutions of the spin-3 topologically massive gravity.
8
4 Quasi-local conserved charges
The action (33) of spin-3 gravity in 3D is invariant under the generalized
local Lorentz translations (23) and (24). In other words, variation of the
Lagrangian of spin-3 gravity with respect to the generalized local Lorentz
translations is equals to the generalized Lie derivative of the Lagrangian,
i.e. δ = L . But, this is not the case for the spin-3 topologically
ξ EH ξ EH
L L
massive gravity. Consider the spin-3 topologically massive gravity (36).
This Lagrangian is not invariant under the generalized local Lorentz trans-
lations. In this model, variation of the Lagrangian due to the generalized
local Lorentz translations becomes
δ = L +dΨ , (44)
ξ ξ ξ
L L
where Ψ is given by
ξ
1
Ψ = tr(dλ ω). (45)
ξ ξ
2µ ∧
In the calculation of the equation (44), we use the fact that the generalized
Lie derivative do not commute with the exterior derivative,
[d,L ]e = dλ e+e λ . (46)
ξ ξ ξ
∧ ∧
Also, notice that λ does not depend on the fields. In a similar way, one
ξ
can show that the variation of surface term (41) due to the generalized local
Lorentz translations differs form its generalized Lie derivative as
δ Θ(Φ,δΦ) = L Θ(Φ,δΦ)+Π , (47)
ξ ξ ξ
where Π is given by
ξ
1
Π = tr(dλ δω). (48)
ξ ξ
2µ ∧
Now we consider variation of the Lagrangian Eq.(36) due to a generalized
local Lorentz translation,
δ = tr(δ Φ E )+dΘ(Φ,δ Φ). (49)
ξ ξ Φ ξ
L ∧
By using Eqs.(44), (49) and from L = £ = di , we can write
ξ ξ ξ
L L L
d(Θ(Φ,δ Φ) i Ψ ) = tr(δ Φ E ). (50)
ξ ξ ξ ξ Φ
− L− − ∧
We will not restrict our study to the on-shell case. One can rewrite Eq.(23)
as follows:
δ e = £ e+[λ ,e] = £ +[i ω,e]+[λ i ω,e] = D(i e)+i +[λ i ω,e],
ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
− T −
(51)
9
where i = (i σ) e dxν. Now we rewrite Eq.(24) as
ξ ξ ν σ
T T
δ ω = £ ω+[λ ,ω] dλ = i D(λ i ω). (52)
ξ ξ ξ ξ ξ ξ
− R− −
Also, we can show that
D2h= dDh+[ω,Dh] = h h . (53)
R∧ − ∧R
BysomecalculationsandusingEqs.(51)-(53),wecansimplifytherighthand
side of the equation (50) as
tr(δ Φ E ) =dtr(i eE (λ i ω)E +i hE )
ξ Φ ξ e ξ ξ ω ξ h
∧ − −
+tr σi eD i h(D +e e)
{ ξ R− ξ T ∧R−R∧ } (54)
1
+tr (λ i ω)[ σ(D +e e)+ D ] .
ξ ξ
{ − − T ∧R−R∧ µ R}
By substituting Eq.(54) into Eq.(50) we will have
dJ =tr σi eD +i h(D +e e)
ξ ξ ξ
{− R T ∧R−R∧ }
1 (55)
+tr (λ i ω)[σ(D +e e) D ] ,
ξ ξ
{ − T ∧R−R∧ − µ R}
where J is given by
ξ
J = Θ(Φ,δ Φ) i Ψ +tr(i eE (λ i ω)E +i hE ). (56)
ξ ξ ξ ξ ξ e ξ ξ ω ξ h
− L− − −
By virtue the following generalized Bianchi identities 4.
D = 0,
R (57)
D +e e = 0.
T ∧R−R∧
the current J is conserved off-shell, for arbitrary ξ, because the right hand
ξ
side of Eq.(55) is zero, so dJ = 0. Thus, by the Poincare lemma we find
ξ
that
J = dK . (58)
ξ ξ
4 WecanreadofftheBianchiidentitiesfromtheactionofconsideredtheoryofgravity.
To clear this matter, consider the action of a gravity theory in the metric formalism,
Sgravity = . Under an infinitesimal diffeomorphism generated by a vector field ξ, we
have δξSgraRviLty = √ g µνδξgµν+ surface term, where µν is generalized Einstein
tensor. Because δξ−gµRν =− Gµξν + νξµ then δξSgravity = G2 √ g µν µξν+ surface
term or δξSgravity = 2 √∇g µ µ∇νξν+ surface term. Dro−ppiRng s−urfGace∇term by some
boundary conditions, wRe se−e t∇hatGthe action is diffeomorphism invariant if µ µν = 0,
∇ G
for any non-zero vector field ξ, and this is exactly the Bianchi identities. Thus, using
the invariant property of the action under diffeomorphism, we can read off the Bianchi
identities. Here,weusethefirstorderandoff-shellversion of thisprescription toread off
the Bianchi identities in considered formalism
10