Conserved Killing charges of quadratic curvature gravity theories in arbitrary backgrounds

0.14 MB·English

by  Deniz Olgu Devecioglu

Conserved Killing charges of quadratic curvature gravity theories in arbitrary
backgrounds
Deniz Olgu Devecio˘glu∗ and O¨zgu¨r Sarıog˘lu†
Department of Physics, Faculty of Arts and Sciences,
Middle East Technical University, 06531, Ankara, Turkey
(Dated: January 5, 2011)
1 Abstract
1
0
WeextendtheAbbott-Deser-Tekinprocedureofdefiningconservedquantitiesofasymptoticallyconstant-
2
curvature spacetimes, and give an analogous expression for the conserved charges of geometries that are
n
a
solutionsofquadraticcurvaturegravitymodelsingenericD-dimensionsandthathavearbitraryasymptotes
J

4 possessingatleastoneKillingisometry. Weshowthattheresultingchargeexpressioncorrectlyreducestoits

] counterpartwhenthebackgroundistakentobeaspaceofconstantcurvatureand,moreover,isbackground
h
t gauge invariant. As applications, we compute and comment on the energies of two specific examples: the
-
p
three dimensional Lifshitz black hole and a five dimensional companionof the first, whose energy has never
e
h
been calculated beforehand.
[

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PACS numbers: 04.20.Cv,04.50.-h
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∗ [email protected]
† [email protected]
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There has been an ongoing interest in gravitational models that involve quadratic curvature
terms which naturally emerge in various string theories or quantum gravity models. A recent ex-
ample of these with physically interesting properties is the three dimensional New Massive Gravity
(NMG) of [1]. There are also a growing number of exact analytic solutions to these models which
asymptote to rather exotic geometries other than the more familiar spaces of constant curvature.
An interesting family of such geometries involve the so called Lifshitz black holes in various dimen-
sions [2, 3] of late, which were inspired by the four-dimensional Lifshitz black hole solution of [4].
These spacetimes have emerged as gravity duals of some nonrelativistic scale invariant condensed
matter theories via a generalization of the AdS/CFT correspondence to such systems (see [2, 3]
andthereferences thereinfor details). Itis obviously of importanceto have abetter understanding
of the physical properties of these solutions.
Themain aim of the present work is to provide a step further in this direction. For this purpose
we generalize the Abbott-Deser-Tekin (ADT) procedure [5–7] originally developed for working
with spacetimes that asymptote to spaces of constant curvature to spacetimes that have arbitrary
asymptotic behavior and that are solutions of quadratic curvature gravity theories in generic D-
dimensions. We show how the general formula reduces to the original ADT expression of [7] in
the limit when the background is a space of constant curvature. To illustrate the idea, we apply
this new charge definition to compute the energy of the three dimensional Lifshitz black hole [2],
which was calculated by other methods beforehand [8] and comment on the result. As another
application, we also findthe energy of a five dimensional Lifshitz black hole [3], which has not been
computed before.
Let us first review the original ADT approach for finding the conserved gravitational charges
of a generic gravity model. One should start with a local gravity action whose field equations read
Φ = κτ , (1)
ab ab
where τ describes the energy-momentum of a covariantly conserved source and κ is basically a
ab
gravitational coupling constant. Now suppose that there exists a background metric g¯ which
ab
satisfies Φ¯ (g¯) = 0 for τ = 0. One can linearize a generic metric g which asymptotically
ab ab ab
approaches to the background g¯ in the usual fashion as
ab
g =g¯ +h . (2)
ab ab ab
Here the deviation h should vanish sufficiently rapidly as one approaches the background at
ab
2
infinity. Linearizing the full field equations (1) about the background, one obtains
ΦL = κT , (3)
ab ab
where ΦL is linear in h and the right hand side of (3) contains all the remaining nonlinear parts
ab ab
as well as τ . One should keep in mind that the background metric g¯ is responsible for raising
ab ab
and lowering indices and is used in defining the covariant derivative ¯ . Let us now assume that
a
∇
the background metric g¯ admits at least one globally defined1 Killing vector ξ¯a such that
ab
¯ ξ¯ + ¯ ξ¯ = 0. (4)
a b b a
∇ ∇
AswiththeoriginalADTapproach[5–7],onenowneedsthecondition ¯ Tab = 0,whichfollows
a
∇
from the Bianchi identity of the full theory, just as in the case of a constant curvature background.
NowusingthiswiththedefiningpropertyofaKillingvector (4), onearrivesattheusualexpression
¯ (Tabξ¯)= 0 = ∂ (√ g¯Tabξ¯), (5)
a b a b
∇ −
which one can use to construct a conserved Killing charge as in [5–7] provided Tabξ¯ can be cast
b
as a divergence, i.e. Tabξ¯ = ¯ ab for an antisymmetric tensor ab modulo terms that vanish
b b
∇ F F
on-shell. In that case the required charge expression takes the form
Qa(ξ¯) = dD−1x√ g¯Φabξ¯ = dS¯ ab, (6)
Z − L b Z bF
Σ ∂Σ
where Σ denotes the (D 1)-dimensional hypersurface of the D-dimensional spacetime and ∂Σ is
−
its (D 2)-dimensional boundary with the surface element dS¯ 2.
−
Letusnowseehow onecan defineconserved Killingcharges (6)of ageneric quadraticcurvature
gravity theory described by the action3
1
I = dDx√ g (R+2Λ )+αR2+βR Rab+γ(R Rabcd 4R Rab+R2) , (7)
Z − (cid:16)κ 0 ab abcd − ab (cid:17)
whereΛ isthebarecosmological constantandκisrelatedtotheD-dimensionalNewton’sconstant
0
G . The field equations following from the variation of the action I yield
D
1
Φ +αA +βB +γC , (8)
ab ab ab ab ab
≡ κG
1 Hereby“globallydefined”,wemeanthattheKillingvectorissmoothandwell-definedintheentirelocalcoordinate
chart that themetric g¯ab works.
2 Here we refrain from delving into delicate issues involving mathematical rigor and assume implicitly the validity
of all steps taken in arriving at (6).
3 Throughout weuseconventionsin which thesignatureof metricsis (−,+,+,...),all covariant derivativessatisfy
[∇a,∇b]Vc = RabcdVd and the Ricci tensor is related to the Riemann tensor as Rab = Rcacb. We also take
the symmetric and the antisymmetric parts of any 2-index tensor Wab as W(ab) ≡ 21(Wab+Wba) and W[ab] ≡
1(Wab−Wba), respectively.
2
3
where
1
R g R Λ g , (9)
ab ab ab 0 ab
G ≡ − 2 −
1
A 2RR 2 R+g (2(cid:3)R R2), (10)
ab ab a b ab
≡ − ∇ ∇ − 2
1
B 2R Rcd R+(cid:3)R + g ((cid:3)R R Rcd), (11)
ab acbd a b ab ab cd
≡ −∇ ∇ 2 −
1
C 2RR 4R Rcd+2R R cde 4R R c g (R Rcdef 4R Rcd+R2),(12)
ab ab acbd acde b ac b ab cdef cd
≡ − − − 2 −
where (cid:3) c. For D < 5, C , coming from the Gauss-Bonnet term, automatically vanishes.
c ab
≡ ∇ ∇
Asdiscussedbefore,thefieldequations(8)-(12)needtobelinearizedaboutageneralbackground
to first order in h . One simply finds
ab
1
ΦL L +αAL +βBL +γCL , (13)
ab ≡ κGab ab ab ab
analogous to (8), where now
1 1
L RL g¯ R h R¯ Λ h , (14)
Gab ≡ ab− 2 ab L− 2 ab − 0 ab
1
AL 2(R¯RL +R¯ R ) 2( R) +g¯ (2((cid:3)R) R¯R )+h (2(cid:3)¯R¯ R¯2), (15)
ab ≡ ab ab L − ∇b∇a L ab L − L ab − 2
1
BL 2(R Rcd) ( R) +((cid:3)R ) + g¯ (((cid:3)R) (R Rcd) )
ab ≡ acbd L− ∇b∇a L ab L 2 ab L − cd L
1
+ h ((cid:3)¯R¯ R¯ R¯cd), (16)
ab cd
2 −
CL 2(R¯RL +R¯ R ) 4(R Rcd) +2(R R cde) 4(R R c)
ab ≡ ab ab L − acbd L acde b L − ac b L
1 1
g¯ (R Rcdef) 4(R Rcd) +2R¯R h (R¯ R¯cdef 4R¯ R¯cd+R¯2).(17)
ab cdef L cd L L ab cdef cd
−2 − − 2 −
(cid:0) (cid:1)
At this stage we refer the reader to appendix A, where it is shown how the ingredients of equations
(14)-(17) can be calculated recursively using the Cadabra software [9, 10], to obtain the final form
of the linearized field equations Φab.
L
Continuing in the fashion described beforehand, one finds that
1 1
Φabξ¯ = ¯ ab +habΦ¯ ξ¯c+ ξ¯aΦ¯ hbc hΦ¯abξ¯ , (18)
L b ∇bF bc 2 bc − 2 b
where
1
ab = ab+α ab +β ab +γ ab, (19)
F κFE Fα Fβ Fγ
4
with
1
ab ξ¯ ¯[ahb]c+ξ¯[b¯ ha]c+hc[b¯ ξ¯a]+ξ¯[a¯b]h+ h¯[aξ¯b], (20)
FE ≡ c∇ ∇c ∇c ∇ 2 ∇
ab 2R¯ ab +2ξ¯[bha]c¯ R¯ +4ξ¯[a¯b]R +2R ¯[aξ¯b], (21)
Fα ≡ FE ∇c ∇ L L∇
ab ξ¯[a¯b]R +2ξ¯c¯[b(Ra] ) +2(R[b ) ¯a]ξ¯c+h ξ¯[b¯a]R¯cd+2hc[aξ¯ ¯ R¯b]d+2R¯c[aξ¯ ¯ hb]d
Fβ ≡ ∇ L ∇ c L c L∇ cd ∇ d∇c d∇c
+hξ¯ ¯[bR¯a]c+2R¯c[bha]d¯ ξ¯ +hR¯c[b¯a]ξ¯ +2R¯c[bξ¯d¯ ha] +ξ¯[aR¯b]c¯ h+R¯cdξ¯[a¯b]h
c c d c d c c cd
∇ ∇ ∇ ∇ ∇ ∇
+2ξ¯dR¯ ¯[bha]c+2R¯c[aξ¯b]¯dh +2ξ¯dR¯c[b¯a]h , (22)
cd cd cd
∇ ∇ ∇
ab 2R¯ ab +2R¯[ba]cdξ¯ ¯ h+4ξ¯R¯c[b¯a]h+4R¯c[aξ¯b]¯ h+2hR¯c[ab]d¯ ξ¯ +4hR¯c[a¯b]ξ¯
Fγ ≡ FE d∇c c ∇ ∇c ∇d c ∇ c
+4ξ¯ R¯dec[a¯ hb] +4ξ¯ R¯dec[a¯ hb] +4ξ¯ R¯dec[b¯a]h +4ξ¯[aR¯b]cde¯ h +2R¯[ab]cdξ¯e¯ h
d c e d e c d ce d ce c de
∇ ∇ ∇ ∇ ∇
+2ξ¯ h ¯cR¯[ab]de+4ξ¯ R¯d[a¯ hb]c+4h R¯dec[a¯b]ξ¯ +4R¯[ab]cdh ¯ ξ¯e+4R¯ ξ¯d¯[bha]c
d ce d c ce d ce d cd
∇ ∇ ∇ ∇ ∇
+4ξ¯dR¯c[a¯b]h +4ξ¯dR¯ [b¯ ha]c+4ξ¯dR¯c[b¯ ha] +8R¯cdξ¯[a¯ hb] +4R¯cdξ¯[b¯a]h
cd c d c d c d cd
∇ ∇ ∇ ∇ ∇
+2R¯cdh ¯[bξ¯a] +4ξ¯ hc[a¯ R¯b]d+4hcdξ¯[b¯ R¯a] +4hc[aR¯b]d¯ ξ¯ +8h R¯c[b¯a]ξ¯d
cd d c c d c d cd
∇ ∇ ∇ ∇ ∇
+2ξ¯[ahb]c¯ R¯, (23)
c
∇
and one should recall that the background was chosen such that Φ¯ (g¯) = 0 so that (6) follows.
ab
There are a few nontrivial requirements that the gravitational charge Qa(ξ¯) given by (6) (with
(19)-(23)) must satisfy. The first one is that it has to reduce to its counterpart given in [6, 7] when
the background g¯ is taken to be a space of constant curvature. In that case one simply sets
ab
2Λ 2Λ 2DΛ
R¯ = (g¯ g¯ g¯ g¯ ), R¯ = g¯ , R¯ = ,
abcd ac bd ad bc ab ab
(D 2)(D 1) − D 2 D 2
− − − −
where Λ is the effective cosmological constant (see [7] for details), in (20) to (23). We have verified
explicitly that this is indeed the case and Qa(ξ¯) reduces to equation (31) of [7] as it should.
The second requirement is that Qa(ξ¯) has to be background gauge invariant and this is shown in
appendix B below.
We should also make a few remarks regarding the tensor potentials (20) to (23). Equations
(18) to (20), in the case α = β = γ = 0, have already appeared in [11] in the context of defining
the conserved Killing charges of Topologically Massive Gravity (TMG) [12] around an arbitrary
background. Similarly equations (18) to (21), in the case β = γ = 0, have been presented in [13]
where an analogous task was undertaken for the NMG theory of [1]. In [13], the authors have also
given an expression (see equation (27) of [13]) in the case γ = 0 which should correspond to our
tensor potential ab (22). However the expression they have is considerably different than ours.
Fβ
We strongly believe that our ab is correct since it already passes the two nontrivial requirements
Fβ
mentioned above. Further work is required to understandthe discrepancy between the two results.
5
As a final remark, the tensor potential ab (23), to our knowledge, has not appeared elsewhere
Fγ
beforehand.
Let us now consider a few illustrative examples where the charge definition Qa(ξ¯) (6) can be
employed to demonstrate its practical use. For this purpose we resort to the Lifshitz black holes
that we have alluded to earlier.
The three dimensional Lifshitz black hole of [2], which reads
r6 mℓ2 dr2 r2
ds2 = 1 dt2+ + dx2, (24)
−ℓ6(cid:16) − r2 (cid:17) r2 m ℓ2
ℓ2 −
is a solution to the cosmological NMG theory of [1] with
13 2ℓ2 3ℓ2
Λ = , β = , α =
0 2ℓ2 κ −4κ
in our conventions. Taking the background to be the Lifshitz spacetime
r6 ℓ2 r2
ds2 = dt2+ dr2+ dx2,
−ℓ6 r2 ℓ2
which is obtained by the m 0 limit in (24), employing the timelike Killing vector ξ¯a = ( 1,0,0),
→ −
setting κ = 16πG in accordance with the conventions of [2] and noting that the angular coordinate
x is periodic with 2πℓ, one finds the energy of (24) to be
2πℓ m2r2( 9ℓ4m2+24ℓ2mr2 7r4) 2πℓ 7m2 7m2
E = lim − − dx = dx = , (25)
r→∞Z ℓκ(ℓ2m r2)3 Z ℓκ 8G
0 0
−
where G denotes the three dimensional Newton’s constant.
The energy of (24) has also been computed in [8] by using the boundary stress tensor method,
and found to be E = m2/4G. Clearly there is a discrepancy between this and our result, which
deserves further attention.
Another nontrivial example involving a five dimensional Lifshitz black hole can also be given.
For this purpose we consider the spacetime given in section 3.1 (in the case z = 2) of [3] which
reads
r4 mℓ5/2 ℓ2 mℓ5/2 −1 r2
ds2 = 1 dt2+ 1 dr2+ d~x2, (26)
−ℓ4(cid:16) − r5/2 (cid:17) r2(cid:16) − r5/2 (cid:17) ℓ2
with
2197 16ℓ2 1584ℓ2 2211ℓ2
κ = 1, Λ = , α = , β = , γ =
0 551ℓ2 − 725 13775 11020
in our conventions. As in the previous example we take the background to be given by the Lifshitz
spacetime obtained by taking the m 0 limit in (26) and the timelike Killing vector to be
→
6
ξ¯a = ( 1,0,0,0,0). With these choices the energy of (26) turns out to be
−
536m2Ω
E = , (27)
2755ℓ
whereΩdenotesthecontributionofthethreedimensionalintegrationovertherangesoftheangular
variables~x. Toour knowledgethis is thefirsttimethat theenergy of (26)has ever beencalculated.
As a summary, we have generalized the ADT charge definition to work with spacetimes that
have arbitrary asymptotic behavior for the case of quadratic curvature gravity theories in generic
D-dimensions. We have checked that this definition reduces to the original ADT expression when
the background is a space of constant curvature. We have also shown that it is background gauge
invariant. We have seen how it works on two illustrative examples.
Clearly there are various open problems that can be attacked using our work. It would be
interesting to compute the charges of other exotic solutions of NMG that have not been considered
hereusingourQa(ξ¯)(6). Herewehaveconsideredonly oneexamplefromthesolutionslisted in[3].
The others certainly deserve further attention. It may be of interest to apply the ADT procedure
generalized for generic quadratic curvature gravity models here to the so called Lovelock gravity
theories of late.
ACKNOWLEDGMENTS
We would like to thank Bayram Tekin for useful discussions and his careful reading of this
manuscript. We would also like to thank the anonymous referee whose useful comments led us
to correct a mistake in an earlier version of this paper. This work is partially supported by the
Scientific and Technological Research Council of Turkey (TU¨BI˙TAK).
Appendix A: The linearized field equations Φab
L
Here we list some useful identities that have been used in the linearization of the full field
equations (8)-(12) and describe how one can obtain Φab.
L
The inverse of the metric is given by gab = g¯ab hab to (h2). Using this, one can find the
− O
linearized Christoffel symbols as
1
(Γa ) = g¯ad ¯ h + ¯ h ¯ h . (A1)
bc L b cd c bd d bc
2 ∇ ∇ −∇
(cid:0) (cid:1)
A straightforward calculation shows that the linearized Riemann tensor is given by
(Ra ) = ¯ (Γa ) ¯ (Γa ) , (A2)
bcd L c bd L d bc L
∇ −∇
7
which leads to
1
RL = ¯c¯ h + ¯c¯ h (cid:3)¯h ¯ ¯ h (A3)
ab 2 ∇ ∇b ac ∇ ∇a bc− ab−∇a∇b
(cid:0) (cid:1)
for the linearized Ricci tensor, where h g¯cdh and (cid:3)¯ ¯ ¯c. Since the curvature scalar is
cd c
≡ ≡ ∇ ∇
defined as R gabR , its linearized version follows as
ab
≡
R = ¯a¯bh (cid:3)¯h habR¯ . (A4)
L ab ab
∇ ∇ − −
However the covariant derivatives and the contractions of the curvature tensors also show up in
the fullfield equations (8)-(12), and these are to belinearized accordingly. Thesecan becalculated
inasimilarfashion. Weherebylistsomeofthemostrelevantexpressionsinvolvingthelinearization
of such terms:
( R ) = ¯ (R ) (Γe ) R¯ (Γe ) R¯ ,
a cd L a cd L ac L ed ad L ec
∇ ∇ − −
( R ) = ¯ ( R ) ¯ R¯ (Γe ) ¯ R¯ (Γe ) ¯ R¯ (Γe ) ,
b a cd L b a cd L e cd ba L a ed bc L a ce bd L
∇ ∇ ∇ ∇ −∇ −∇ −∇
((cid:3)R ) = g¯ab( R ) hab¯ ¯ R¯ ,
cd L b a cd L b a cd
∇ ∇ − ∇ ∇
( R) = g¯cd( R ) hcd¯ ¯ R¯ , (A5)
b a L b a cd L b a cd
∇ ∇ ∇ ∇ − ∇ ∇
((cid:3)R) = g¯ab( R) hab(¯ ¯ R¯),
L b a L b a
∇ ∇ − ∇ ∇
(R Rcd) = g¯ (Re ) R¯cd+h R¯e R¯cd+R¯ c d(R ) hceR¯ R¯ d hdeR¯ R¯c ,
acbd L ae cbd L ae cbd a b cd L acbd e acbd e
− −
(R Rab) = 2R¯ab(R ) 2habR¯ R¯ c.
ab L ab L ac b
−
As explained in the text, we have used these and a number of such expressions recursively in the
Cadabra software [9, 10], which is rather handy for such symbolic manipulations, and obtained the
final form of the linearized field equations Φab. However, its final form is rather cumbersome and
L
hardly illuminating, so it is best not displayed here!
Appendix B: The background gauge invariance of Qa(ξ¯)
Here we show how the gravitational charge Qa(ξ¯) given by (6) (with (19)-(23)) is indeed back-
ground gauge invariant.
As is well known, the deviation h transforms as
ab
δ h = ¯ ζ¯ + ¯ ζ¯ (B1)
ζ¯ ab ∇a b ∇b a
under an infinitesimal diffeomorphism generated by a vector ζ¯a. To see that Qa(ξ¯) is background
gauge invariant, we apply the gauge transformation (B1) to Φab, the invariance of which implies
L
8
the background gauge invariance of ab (19) and hence of Qa(ξ¯). This is obviously easier since one
F
can work with the linearized field equations Φab rather than the gravitational charge Qa(ξ¯) which
L
was not easy to find at the first place.
Some of the most basic expressions that are needed are as follows:
δ (Γa ) = ζ¯eR¯ a + ¯ ¯ ζ¯a,
ζ¯ bc L ec b ∇c∇b
δ (Ra ) = R¯a ¯ ζ¯e ζ¯e¯ R¯a +R¯a ¯ ζ¯e+ζ¯e¯ R¯a +R¯ e ¯ ζ¯a+R¯ae ¯ ζ¯,
ζ¯ bcd L − bde∇c − ∇c bde bce∇d ∇d bce b cd∇e cd∇b e
δ RL = ζ¯c¯ R¯ +R¯ ¯ ζ¯c+R¯ ¯ ζ¯c,
ζ¯ ab ∇c ab ac∇b bc∇a
δ R = ζ¯a¯ R¯.
ζ¯ L ∇a
These easily follow considering the action of (B1) on the expressions (A1) to (A4). One can
similarly calculate the analogous expressions regarding the action of (B1) on (A5) (and on those
ones that have not been listed). Once again we have employed the Cadabra software [9, 10] at this
stage and found that
δ ΦL = ζ¯c¯ Φ¯ +Φ¯ ¯ ζ¯c+Φ¯ ¯ ζ¯c,
ζ¯ ab ∇c ab ac∇b bc∇a
which vanishes thanks to Φ¯ (g¯) = 0. As explained by the reasoning above, this shows that the
ab
gravitational charge Qa(ξ¯) is indeed background gauge invariant.
Appendix C: Useful identities
Throughout the calculation we have used a number of identities involving the Killing vectors.
The proofs of these are very simple, so we simply state them:
¯ ¯ ξ¯ = R¯ ξ¯d, ¯b¯ ξ¯ = R¯ ξ¯b, ξ¯a¯ R¯ = 0, ξ¯c¯ R¯ = (R¯ ¯ +R¯ ¯ )ξ¯c.
a b c cbad a b ab a c ab ca b cb a
∇ ∇ ∇ ∇ ∇ ∇ − ∇ ∇
Thefollowing are also usefulidentities involving the Riemann tensor. We state them withoutproof
since they follow from elementary properties of the Riemann tensor:
¯ R¯a = ¯ R¯ ¯ R¯ , R¯acdeR¯ = 2R¯acdeR¯ , R¯abefR¯ = R¯afebR¯ .
a bcd c bd d bc bcde bdce cfed cdef
∇ ∇ −∇
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