U(1) symmetry and elimination of spin-0 gravitons in Horava-Lifshitz gravity without the projectability condition

0.15 MB·English

by  Tao Zhu

U(1) symmetry and elimination of spin-0 gravitons in Horava-Lifshitz gravity without
the projectability condition
Tao Zhu, Qiang Wu, and Anzhong Wang
∗
Institute for Advanced Physics & Mathematics, Zhejiang University of Technology, Hangzhou 310032, China
Fu-Wen Shu
College of Mathematics & Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
(Dated: October21, 2011)
In this paper, we show that the spin-0 gravitons appearing in Horava-Lifshitz gravity without
the projectability condition can be eliminated by extending the gauge symmetries of the foliation-
preservingdiffeomorphismstoincludealocalU(1)symmetry. Asaresult,theproblemsofstability,
ghost,strongcoupling,anddifferentspeedsinthegravitationalsectorareautomaticallyresolved. In
1
addition, with the detailed balance condition softly breaking, the number of independent coupling
1
0 constantscanbesignificantlyreduced(from morethan70downto15),whilethetheoryisstillUV
2 completeand possesses ahealthyIR limit, wherebytheprediction powers of thetheory areconsid-
erably improved. The strong coupling problem in the matter sector can be cured by introducing
ct an energy scale M∗, so that M∗ < Λω, where M∗ denotes the suppression energy of high order
O derivativeterms, and Λω thewould-be strong coupling energy scale.

0 PACSnumbers: 04.60.-m;04.50.Kd
2

] I. INTRODUCTION z must be z d [2, 14]. The associatedgauge symmetry
h ≥
nowis brokenfromthe generaldiffeomorphismsdownto
t
- Quantization of gravitational fields has been one of the foliation-preserving ones,
p
e the main driving forces in Physics in the past decades δt= f(t), δxi = ζi(t,x), (2)
h by following several different paths [1]. Recently, Ho- − −
[ ravaproposeda theoryof quantumgravityin the frame- to be denoted by Diff(M, ).
2 work of quantum field theory, with the spacetime metric AbandoningthegeneralFdiffeomorphisms,ontheother
v as the elementary field in the language of the standard hand, gives rise to a proliferation of independently cou-
7 path-integral formulas [2]. Applied to cosmology, it re- pling constants [2, 15], which could potentially limit the
3 sults in various remarkable features and has attracted a prediction powers of the theory. In particular, only the
2
great deal of attention [3]. In particular, high order spa- sixth-orderspatialderivativetermsaremorethan60[16].
1
tial derivative terms cangive rise to a bouncing universe To reduce the number of the independent coupling con-
.
8 [4, 5]; the anisotropic scaling solves the horizon problem stants, two conditions were introduced, the projectibility
0 and leads to scale-invariant perturbations without infla- and detailed balance [2]. The former requires that the
1
tion [6, 7]; the lack of the local Hamiltonian constraint lapse function N be a function of t only, while the latter
1
: leadsto“darkmatterasanintegrationconstant”[8];the requires that the gravitational potential should be ob-
v darksectorcanhaveits purelygeometricorigins[9]; and tained from a superpotential W , where W is given by
Xi theinclusionofaU(1)symmetry(withtheprojectability an integral of the gravitational Cghern-Simogns term over
condition) [10, 11] not only eliminates the spin-0 gravi- a 3-dimensional space, W ω (Γ). With these two
ar tons but also leads to a flat universe [12]. conditions, the general acgtio∼n nΣow3contains only five in-
With the perspective that Lorentz symmetry may ap- dependent coupling constantsR[2]. The detailed balance
pearonlyasanemergentoneatlowenergies,butcanbe condition has several remarkable features [2, 17]. For
fundamentallyabsentathighenergies,Horavaconsidered example, it is in the same spirit of the AdS/CFT cor-
a gravitational system whose scaling at short distances respondence [18], where a string theory and gravity de-
exhibits a strong anisotropy between space and time, fined on one space is equivalent to a quantum field the-
ory without gravity defined on the conformal boundary
x b 1x, t b zt. (1)
→ − → − of this space, which has one or more lower dimension(s).
Yet, in the non-equilibrium thermodynamics, the coun-
This is quite similar to Lifsitz’s scalar field [13], so the
terpart of the superpotential W plays the role of en-
theory is often referred to as the Horava-Lifshitz (HL) g
tropy, while δW /δg represents the corresponding en-
gravity. In (d+1)-dimensions, in order for the theory to g ij
tropic force [19]. This might shed further lights on the
be power-counting renormalizable, the critical exponent
nature of gravitational forces, as proposed recently by
Verlinde [20].
Despite of all the above remarkable features, the the-
∗On Leave from GCAP-CASPER, Physics Department, Baylor ory (with these two conditions) is plagued with several
University,Waco, TX76798-7316, USA problems [3], including the instability, ghost [2, 21], and
2
strongcoupling[16,22]. However,allofthem areclosely +2K ˆijlka α+2(1 λ)K 2α+ak α ,(6)
ij (l k) k
related to the existence of the spin-0 gravitons [3]. Be- G ∇ − ∇ ∇
cause of their presence, another important question also where ˆijkl = ijkl , ijkl(λ) (cid:0)(gikgjk+gilgjk(cid:1))o/2
raises: Their speeds are generically different from those G G λ=1 G ≡ −
λgijglk, G R Rg /2 and f = (f +f )/2.
of gravitational waves, as they are not related by any ij ≡ ij(cid:12)− ij (ij) ij ji
R (R) is the Ric(cid:12)ci tensor (scalar) of g . To have
symmetry. This poses a great challenge for any attempt ij ij
the U(1) symmetry, we introduce the gauge field A [10],
to restore Lorentz symmetry at low energies where it
which transforms as
has been well tested experimentally. In particular, one
needs a mechanism to ensure that in those energy scales δ A=α˙ Ni α. (7)
α i
all species, including gravitons, have the same effective − ∇
speed and light cones. With these in mind, Horava and Then, adding 1
Melby-Thompson (HMT) [10] extended the symmetry
(2) to include a local U(1), S = ζ2 dtd3x√gA(R 2Λ ), (8)
A g
− −
U(1)⋉Diff(M, ). (3) Z
F to Sˆ , one finds that its variation (with Λ = 0) with
g g
With this enlarged symmetry, the spin-0 gravitons are
respect to α exactly cancels the first term in Eq.(6). To
eliminated[10,23,24],andthetheoryhasthesamelong-
repairthe rest, weintroduce the Newtonianprepotential
distance limit as that of general relativity. This was ini-
ϕ [10], which transforms as
tially done in the special case λ = 1, and soon general-
ized to the case with any λ [11], where λ is a coupling δ ϕ= α. (9)
α
−
constant defined below. Although the spin-0 gravitons
Under Eq.(4), the variation of the term,
are also eliminated in the general case, the strong cou-
plingproblemraisesagain[12]. However,itcanbesolved
by the Blas-Pujolas-Sibiryakov (BPS) mechanism [15], Sϕ =ζ2 dtd3x√gN ϕ ij 2Kij + i jϕ+ai jϕ
G ∇ ∇ ∇
iMn w<hicΛh a, wnhewereenMergydesncoatleesMth∗eissuipnptrreosdsuiocnede,nseorgtyhaotf +(1 Zλ) 2ϕ+aniϕ 2(cid:0)+2 2ϕ+a iϕ K (cid:1)
ω i i
hig∗h order derivativ∗e terms of the theory, and Λ the − ∇ ∇ ∇ ∇
would-be strong coupling energy scale [25]. ω + 1 ˆijlk h4(cid:0)( ϕ)a (cid:1) ϕ+(cid:0)5 a ϕ a (cid:1) iϕ
3G ∇i∇j (k∇l) (i∇j) (k∇l)
Note that in [10, 11] the projectability condition was
h (cid:0) (cid:1)
assumed. In this paper, we shall study the case without +2 ϕ a ϕ+6K a ϕ , (10)
(i j)(k l) ij (l k)
∇ ∇ ∇
it, and our goals are twofold: (i) Extend the symmetry
(cid:0) (cid:1) io
(3) to the case without the projectability condition, so will exactly cancel the rest in Eq.(6) as well as the one
thatthe spin-0 gravitonsareeliminated. (ii) Reduce sig- from the term Λ A in Eq.(8), where G +Λ gij
g ij ij g
G ≡
nificantly the number of the coupling constants by the and a a . Hence, the total action
ij i j
≡∇
detailed balance condition, whereby the prediction pow-
ers of the theory can be improved considerably. To have Sg =Sˆg+SA+Sϕ, (11)
a healthy IR, we allow it to be broken softly by adding
is invariant under (3).
allthe lowdimensionalrelevantterms [17]. We alsonote
Equation (3) does not uniquely fix . To our sec-
that in [26] a U(1) extension of F(R) HL gravity was LV
ondgoal,weintroducethe“generalized”detailedbalance
recently studied.
condition,
ˆ = E ijklE g AiAj, (12)
II. THE MODEL LV ijG kl− ij
where
UndertheU(1)transformations,themetriccoefficients
1 δW 1 δW
transform as [10], Eij = g, Ai = a,
√g δgij √g δai
δαN =0, δαNi =N iα, δαgij =0, (4) 1 2
∇ W = Tr Γ dΓ+ Γ Γ Γ ,
where α is the U(1) generator, Ni the shift vector, and g w2 ZΣ (cid:16) ∧ 3 ∧ ∧ (cid:17)
i the covariant derivative of gij. Since [15] 1
∇ W = d3x√gai b ∆na , (13)
a n i
Sˆg =ζ2 dtdx3N√g(LK −LV(gij,ak)), (5) Z nX=0
Z
where a (lnN), , K Kij λK2 and K =
i i K ij ij
≡ L ≡ −
( g˙ + N + N )/(2N),thepotential isinvari-
a−ntiujnde∇ri(4)j, w∇hijle tihe kinetic part transfoLrmVs as 1 Notethedifferencebetweenthenotationsusedhereandtheones
used in [10, 11]. In particular, we have Kij =−KiHjMT, Λg =
δS =ζ2 dtd3xN√g α˙ Ni α R +2αG Kij ΩHMT, ϕ=−νHMT,Gij =ΘHijMT, where quantities with the
K − ∇i N ij superindices“HMT”wereusedin[10,11].
Z n(cid:0) (cid:1)
3
with ∆ gij and b being arbitrary constants. λ 1 ∂2B = 1 3λ ψ˙, (19)
i j n
≡ ∇ ∇ − −
Now, some comments are in order. First, the term ðφ=2℘ψ, (20)
a ∆1/2ai inprinciplecanbeincludedintoW ,whichwill (cid:0) (cid:1) (cid:0) (cid:1)
i a
give rise to fifth order derivative terms. In this paper we where ℘ 1 ζ 2β ∂2. Eq.(17) shows that ψ is not
− 7
shall discard them by parity. Second, it is well-known propagati≡ng, a−nd with proper boundary conditions, one
that with the detailed balance condition a scalar field is can set ψ = 0. Then, Eqs.(18)-(20) show that B, A,
not UV complete [4], and the Newtonian limit does not and φ are also not propagating and can be set to zero.
exist [27]. Following [17], we break it softly by adding Hence, we obtain ψ = B = A = φ = 0, that is, the
allthe lowdimensionalrelevanttermsto ˆ ,sothatthe scalar perturbations vanish identically in the Minkowski
V
L
potential finally takes the form, background, similar to that in general relativity. Thus,
withtheenlargedsymmetry(3),the spin-0gravitonsare
V = γ0ζ2 β0aiai γ1R +ζ−2 γ2R2+γ3RijRij indeed eliminated even without the projectability condi-
L − −
tion.
+ζ−2 β(cid:16)1 aiai 2+β2(cid:17)aii 2+(cid:16)β3 aiai ajj (cid:17)
+β4aihjaij(cid:0)+β5(cid:1) aiai R(cid:0) +(cid:1)β6aiajR(cid:0)ij +(cid:1)β7Raii IV. STRONG COUPLING AND THE
BLAS-PUJOLAS-SIBIRYAKOV MECHANISM
+ζ 4 γ C Cij(cid:0)+β (cid:1)∆ai 2 , (1i4)
− 5 ij 8
where β bh2, γ ζ4/w4(cid:0), β (cid:1) i ζ4b2, and C de- Since the spin-0 gravitons are eliminated, the ghost,
0 ≡ 0 5 ≡ 8 ≡ − 1 ij instability, strong coupling and different speed problems
notes the Cotton tensor, defined as Cij ≡ e√ikgl∇k Rlj − in the gravitational sector do not exist any longer. But,
1Rδj . Allthecoefficients,β andγ ,aredimensi(cid:16)onless the self-interaction of matter fields and the interaction
4 l n n between a matter field and a gravitational field can still
and a(cid:17)rbitrary, except β0 0,γ5 0 and β8 0, as indi- lead to strong coupling, as shown recently in [25] for the
≥ ≥ ≤
cated by their definitions. γ0 is related to the cosmolog- theorywiththeprojectabilitycondition. Inthefollowing
ical constant by Λ = ζ2γ0/2, while the IR limit requires we shall show that this is also the case here. Since the
γ1 = 1, ζ2 =1/(16πG),whereGistheNewtoniancon- proofisquitesimilartothatgivenin[25],inthefollowing
−
stant. From the above, we can see that with the “gen- we just summarize our main results, and for detail, we
eralized” detailed balance condition softly breaking, the refer readers to [25]. A scalar field χ with detailed bal-
numberofindependentcouplingconstantsissignificantly ance condition softly breaking is described by Eqs.(3.11)
reduced from more 70 to 15: G, Λ, λ, β , γ , (n =
n s and(3.12)of[17]. IntheMinkowskibackground,wehave
0,...,8; s = 2,3,5). In the following, we shall show that
χ¯ = 0 = V(0) = V (0). Considering the linear pertur-
′
such a setup is UV complete and IR healthy.
bations χ = δχ, we find that the quadratic part of the
scalar field action reads,
III. ELIMINATION OF THE SPIN-0
1 1 1
GRAVITONS S(2) = dtd3x fχ˙2 V χ2 (1+2V )(∂χ)2
χ "2 − 2 ′′ − 2 1
Z
To show this, it is sufficient to consider linear pertur-
bations in the Minkowski background, given by +c1A∂2χ−V2(∂2χ2)2−V4′χ∂4χ+σ32∂2χ∂4χ#, (21)
N = 1+φ, N =∂ B, g =(1 2ψ)δ +2E ,
i i ij ij ,ij
−
wheref isaconstant,usuallychosentobeone[17]. Since
A = δA, ϕ=δϕ. (15)
S(2) doesnotdependonψ, B andφexplicitly,thevaria-
χ
Choosing the gauge, E =0=δϕ, we find that tionsofthetotalactionS(2) =S(2)+S(2) withrespectto
χ g
them will give the same Eqs.(18) - (20), while Eq.(17) is
Sg(2) = ζ2 dtd3x (1−3λ)(3ψ˙2+2ψ˙∂2B) replaced by ψ =c1χ/(4ζ2). Integrating out φ, ψ, B, A,
Z n we obtain
+(1 λ)(∂2B)2 φð+4β7ζ−2∂2ψ ∂2φ
− −
−2(ψ−2φ+2A+(cid:0)α1ψ∂2)∂2ψo, (cid:1) (16) S(2) =M2Z dtd3x"χ˙2−α0 ∂χ 2−m2χχ2
whereα ζ 2(8γ +3γ )andð β +ζ 2(β +β )∂2 (cid:0) (cid:1)
1 ≡ − 2 3 ≡ 0 − 2 4 − ℘2χ
φζ−y4iβe8ld∂,4r.eVspaerciattivioenlys,ofSg(2) withrespecttoA, ψ, B,and −MA−2χ∂4χ+MB−4χ∂6χ+γχ∂2(cid:18) ð (cid:19)#, (22)
∂2ψ =0, (17) where M2 2πGc2/c 2 +f/2, γ 4πGc2/M2, α
ψ¨+ 1∂2B˙ + 2∂2(A+ψ+α1∂2ψ) = 2℘∂2φ , (18) 1+2V1−≡4πGc21 /1(2|Mψ|2), MA2 ≡M≡2/ 2πG1α1c21+V02≡+
3 3(1−3λ) 3(1−3λ) (cid:0)V4′ , MB4 ≡ M2/σ(cid:1)32, m2χ ≡ V′′/(2M2)(cid:0), and c2ψ = (1−
(cid:1)
4
λ)/(3λ 1). Clearly, the scalar field is ghost-free for M &M ,thesixth-orderderivativetermsbecomedom-
A B
−
f > 0, and stable in the UV and IR. In fact, it can be inant for processes with E & M , and the quadratic
madestableinallenergyscalesbyproperlychoosingthe action (22) now is invariant only u∗nder the anisotropic
coupling coefficients V . rescaling,
n
To consider the strong coupling problem, one needs to
calculate the cubic part of the total action, which takes t b 3t, xi b 1xi, χ χ. (26)
− −
→ → →
the form,
Then,onefindsthatunderthenewrescalingalltheterms
1 1 of λ in Eq.(23) become scaling-invariant, while the rest
S(3) = dtd3x λ χ¨ χ∂2χ+λ χ¨ χ χ,i s
1 ∂2 2 ∂2 ,i scale as b δ with δ > 0. The former are strictly renor-
Z ( (cid:18) (cid:19) (cid:18) (cid:19) −
malizable, while the latter are superrenormalizable [28].
2℘ 2℘ χ˙
+λ χ˙2 1 χ+λ χ˙∂i 1 χ∂ ThereforeEq.(25)makesthestrongcouplingproblemdis-
3 (cid:18) ð − (cid:19) 4 (cid:18) ð − (cid:19) i(cid:18)∂2(cid:19) appeared.
∂i∂j ∂ 2℘ V. CONCLUSIONS
i
+λ χ˙ χ˙ ∂ +3 χ
5 ∂2 ∂2 j ð
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
Inthis paper,wehaveshownthatthe spin-0gravitons
∂
+λ χ˙χ,i iχ˙ +... , (23) in the HL theory even without the projectability condi-
6 ∂2
(cid:18) (cid:19) ) tioncanbeeliminatedbytheenlargedsymmetry(3). An
immediateresultisthatalltheproblemsrelatedtothem
where “...” represents the terms that are independent of
disappear, including the ghost, instability, strong cou-
λ, so they are irrelevant to the strong coupling prob-
pling and different speeds in the gravitationalsector. In
lem. λ are functions of λ,f,ζ and c . In particular,
s i addition,therequirements[15], λ 1 β , 0<β <2,
λ =c3/(64ζ4 c 4), whichwill yieldthe strongcoupling | − |≃ 0 0
5 1 | ψ| now become unnecessary, which might help to relax the
energy Λ given below. The exact expressions of other
ω observationalconstraints. Moreover,itisexactlybecause
coefficients are not relevant to Λ , so will not be given
ω of this elimination that softly breaking detailed balance
here. For a process with energy E M ,M , the first
two terms in Eq.(22) are dominant,≪and SA(2) iBs invariant condition can be imposed, whereby the number of inde-
pendent coupling constants is significantly reduced from
undertherelativisticrescalingt b 1t,xi b 1xi,χ
bχ. Then, all the terms of λ →in S−(3) sca→le a−s b. As→a more than70 to 15. This considerablyimprovesthe pre-
s diction powers of the theory. Note that, without the
result, when the energy of a process is greater than a
elimination of the spin-0 gravitons, the strong coupling
certain value, say, Λ , the amplitudes of these terms are
ω problem will appear in the gravitational sector and can-
greater than one, and the theory becomes nonrenormal-
notbesolvedbytheBPSmechanism,becausethiscondi-
izable [28]. In the present case, it can be shown that
tionpreventstheexistenceofsixth-orderderivativeterms
(3)
M 3/2 in Sg .
Λ pl M c 5/2. (24) TheseresultsputtheHLtheorywith/withoutthepro-
ω pl ψ
≃ c | |
(cid:18) 1 (cid:19) jectability condition in the same footing, and provide
a very promising direction to build a viable theory of
However, if
quantum gravity, put forwards recently by HMT [10].
M <Λ , (25) Certainly, many challenging questions [3] need to be an-
ω
∗ swered before such a goal is finally reached.
where M = Min.(M ,M ), one can see that before Acknowledgements: This work was supported
A B
the stron∗g coupling energy Λ is reached, the high or- in part by DOE Grant, DE-FG02-10ER41692 (AW);
ω
der derivative terms in Eq.(22) become large, and their NNSFC No. 11005165 (FWS); and NNSFC No.
effects must be taken into account. In particular, for 11047008(QW, TZ).
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