Ho\v{r}ava Gravity at a Lifshitz Point: A Progress Report

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by  Anzhong Wang

Hoˇrava Gravity at a Lifshitz Point: A Progress Report
Anzhong Wang∗
Institute for Advanced Physics & Mathematics, Zhejiang University of Technology, Hangzhou 310032, China
GCAP-CASPER, Department of Physics, Baylor University, Waco, Texas 76798-7316, USA
(Dated: April 5, 2017)
Hoˇrava gravity at a Lifshitz point is a theory intended to quantize gravity by using techniques
of traditional quantum field theories. To avoid Ostrogradsky’s ghosts, a problem that has been
plaguing quantization of general relativity since the middle of 1970’s, Hoˇrava chose to break the
Lorentz invariance by a Lifshitz-type of anisotropic scaling between space and time at the ultra-
high energy, while recovering (approximately) the invariance at low energies. With the stringent
observationalconstraintsandself-consistency,itturnsoutthatthisisnotaneasytask,andvarious
modifications have been proposed, since the first incarnation of the theory in 2009. In this review,
we shall provide a progress report on the recent developments of Hoˇrava gravity. In particular, we
first present four most-studied versions of Hoˇrava gravity, by focusing first on their self-consistency
7 and then their consistency with experiments, including the solar system tests and cosmological
1 observations. Then, we provide a general review on the recent developments of the theory in three
0 different but also related areas: (i) universal horizons, black holes and their thermodynamics; (ii)
2 non-relativistic gauge/gravity duality; and (iii) quantization of the theory. The studies in these
areas can be generalized to other gravitational theories with broken Lorentz invariance.
r
p
A
CONTENTS The last particle predicted by SM, the Higgs boson,

4 was finally observed by Large Hadron Collider in 2012
I. Introduction 1 [1, 2], after 40 years search. On the other hand, general
]
c relativity (GR) describes the fourth force, gravity, and
q II. Hoˇrava Theory of Quantum Gravity 3 predicts the existence of cosmic microwave background
- A. The Minimal Theory 6 radiation (CMB), black holes, and gravitational waves
r
g B. With Projectability & U(1) Symmetry 7 (GWs),amongotherthings. CMBwasfirstobservedac-
[ C. The Healthy Extension 10 cidentallyin1964[3],andsincethenvariousexperiments

3 D. With Non-projectability & U(1) Symmetry 11 have remeasured it each time with unprecedented preci-
v E. Covariantization of Hoˇrava Gravity 13 sions [4–8]. Black holes have attracted a great deal of
7 attention both theoretically and experimentally [9], and
8 III. Black Holes & Thermodynamics 15 various evidences of their existence were found [10]. In
0 particular, on Sept. 14, 2015, the LIGO gravitational
6
IV. Non-relativistic Gauge/Gravity Duality 19 wave observatory made the first-ever successful observa-
0
tion of GWs [11]. The signal was consistent with theo-
.
1 V. Quantization of Hoˇrava Gravity 20 retical predictions for the GWs produced by the merger
0 of two binary black holes, which marks the beginning of
7 VI. Concluding Remarks 21 a new era: gravitational wave astronomy.
1
v: Acknowledgements 23 Despite these spectacular successes, we have also been
facing serious challenges. First, observations found that
i
X Appendix A: Lifshitz Scalar Theory 23 ouruniverseconsistsofabout25%darkmatter(DM)[8].
r It is generally believed that such matter should not be
a
References 25 made of particles from SM with a very simple argument:
otherwiseweshouldhavealreadyobservedthemdirectly.
Second, spacetime singularities exist generically [12], in-
I. INTRODUCTION cluding those of black hole and the big bang cosmology.
At the singularities, GR as well as any of other physics
In the beginning of the last century, physics started laws are all broken down, and it has been a cherished
withtwotriumphs,quantum mechanics and general rela- hope that quantum gravitational effects will step in and
tivity. Ononehand,basedonquantummechanics(QM), resolve the singularity problem.
the Standard Model (SM) of Particle Physics was devel- However,whenapplyingthewell-understoodquantum
oped,whichdescribesthreeofthefourinteractions: elec- field theories (QFTs) to GR to obtain a theory of quan-
tromagnetism, and the weak and strong nuclear forces. tum gravity (QG), we have been facing a tremendous
resistance [13–15]: GR is not (perturbatively) renormal-
izable. Power-counting analysis shows that this happens
because in four-dimensional spacetimes the gravitational
∗ Anzhong [email protected] coupling constant G has the dimension of (mass)−2 (in
N
2
units where the Planck constant (cid:126) and the speed of light Nowthenon-degeneracymeansthat∂2L/∂x¨2 (cid:54)=0,which
c are one), whereas it should be larger than or equal to implies that Eq.(1.5) can be cast in the form [19],
zero in order for the theory to be renormalizable pertur-
batively [16]. In fact, the expansion of a given physical d4x(t) ...
=F(x,x˙,x¨,x;t). (1.6)
quantity F in terms of the small gravitational coupling dt4
constant G must be in the form
N Clearly, in order to determine a solution uniquely four
∞ initial conditions are needed. This in turn implies that
F = (cid:88)an(cid:0)GNE2(cid:1)n, (1.1) there must be four canonical coordinates, which can be
n=0 chosen as,
where E denotes the energy of the system involved, so ∂L d ∂L
that the combination (cid:0)GNE2(cid:1) is dimensionless. Clearly, X1 ≡x, P1 ≡ ∂x˙ − dt∂x¨,
when E2 (cid:38) G−1, such expansions diverge. Therefore, ∂L
N X ≡x˙, P ≡ . (1.7)
it is expected that perturbative effective QFT is broken 2 2 ∂x¨
downatsuchenergies. ItisinthissensethatGRisoften
The assumption of non-degeneracy guarantees that
said not perturbatively renormalizable.
Eq.(1.7) has the inverse solution x¨ = A(X ,X ,P ), so
An improved ultraviolet (UV) behavior can be ob- 1 2 2
that
tained by including high-order derivative corrections to
(cid:12)
the Einstein-Hilbert action, ∂L(cid:12)
S =(cid:90) d4x√−gR, (1.2) ∂x¨(cid:12)(cid:12)x=X1,x˙=X2,x¨=A =P2. (1.8)
EH
Then, the corresponding Hamiltonian is given by
such as a quadratic term, RµνRµν [17]. Then, the gravi- (cid:88)2 dix
tational propagator will be changed from 1/k2 to H = P −L=P X +P A−L, (1.9)
idti 1 2 2
1 1 1 1 1 1 i=1
+ G k4 + G k4 G k4 +....
k2 k2 N k2 k2 N k2 N k2 which is linear in the canonical momentum P1 and im-
1 plies that there are no barriers to prevent the system
= . (1.3)
k2−G k4 from decay, so the system is not stable generically.
N
Itisremarkabletonotehowpowerfulandgeneralthat
Thus, at high energy the propagator is dominated by
the theorem is: It applies to any Lagrangian of the form
the term 1/k4, and as a result, the UV divergence can
L(x,x˙,x¨). The only assumption is the non-degeneracy of
be cured. Unfortunately, this simultaneously makes the
the system,
modified theory not unitary, as now we have two poles,
∂2L(x,x˙,x¨)
1 1 1
(cid:54)=0, (1.10)
= − , (1.4)
k2−G k4 k2 k2−G−1 ∂x¨2
N N
so the inverse x¨=A(X ,X ,P ) of Eq.(1.7) exists. The
andthefirstone(1/k2)describesamasslessspin-2gravi- 1 2 2
aboveconsiderationscanbeeasilygeneralizedtosystems
ton, while the second one describes a massive one but
with even higher order time derivatives [19].
with a wrong sign in front of it, which implies that the
Clearly, with the above theorem one can see that any
massive graviton is actually a ghost (with a negative ki-
higher derivative theory of gravity with the Lorentz in-
neticenergy). Itistheexistenceofthisghostthatmakes
variance (LI) and the non-degeneracy condition is not
the theory not unitary, and has been there since its dis-
stable. Taking the above point of view into account, re-
covery [17].
cently extensions of scalar-tensor theories were investi-
Theexistenceoftheghostiscloselyrelatedtothefact
gated by evading the Ostrogradsky instability [20–22].
that the modified theory has orders of time-derivatives
Another way to evade Ostrogradsky’s theorem is to
higher than two. In the quadratic case, for example, the
break LI in the UV and include only high-order spatial
field equations are fourth-orders. As a matter of fact,
derivative terms in the Lagrangian, while still keep the
there exists a powerful theorem due to Mikhail Vasile-
timederivativetermstothesecondorder. Thisisexactly
vich Ostrogradsky, who established it in 1850 [18]. The
what Hoˇrava did recently [23].
theorem basically states that a system is not (kinemati-
It must be emphasized that this has to be done with
cally) stable if it is described by a non-degenerate higher
great care. First, LI is one of the fundamental princi-
time-derivative Lagrangian. To be more specific, let us
ples of modern physics and strongly supported by ob-
consider a system whose Lagrangian depends on x¨, i.e.,
servations. In fact, all the experiments carried out so
L=L(x,x˙,x¨),wherex˙ =dx(t)/dt,etc. Then,theEuler-
far are consistent with it [24], and no evidence to show
Lagrange equation reads,
that such a symmetry must be broken at certain energy
∂L d ∂L d2 ∂L scales, although the constraints in the gravitational sec-
− + =0. (1.5) tor are much weaker than those in the matter sector
∂x dt∂x˙ dt2 ∂x¨
3
[25]. Second, the breaking of LI can have significant variantization of these models, which can be considered
effects on the low-energy physics through the interac- as the IR limits of the corresponding versions of Hoˇrava
tions between gravity and matter, no matter how high gravity. In Sec. III, we present the recent developments
the scale of symmetry breaking is [26]. Recently, it was ofuniversalhorizonsandblackholes,anddiscussthecor-
proposed a mechanism of SUSY breaking by coupling a respondingthermodynamics,whileinSec. IV,wediscuss
Lorentz-invariant supersymmetric matter sector to non- thenon-relativisticgauge/gravityduality,bypayingpar-
supersymmetric gravitational interactions with Lifshitz ticular attention on spacetimes with Lifshitz symmetry.
scaling,andshownthatitcanleadtoaconsistentHoˇrava In Sec. V, we consider the quantization of Hoˇrava grav-
gravity [27] 1. Another scenario is to go beyond the per- ity, and summarize the main results obtained so far in
turbativerealm,sothatstronginteractionswilltakeover the literature. These studies can be easily generalized to
atanintermediatescale(whichisinbetweentheLorentz other gravitational theories with broken LI. The review
violationandtheinfrared(IR)scales)andacceleratethe isendedinSec. VI,inwhichwelistsomeopenquestions
renormalization group (RG) flow quickly to the LI fixed of Hoˇrava’s quantum gravity and present some conclud-
point in the IR [31–33]. ing remarks. An appendix is also included, in which we
Withtheaboveinmind,inthisarticleweshallgivean give a brief introduction to Lifshitz scalar theory.
updated review of Hoˇrava gravity. Our emphases will be Before proceeding to the next section, let us men-
on: (i) the self-consistency of the theory, such as free of tion some (well studied) theories of QG. These include
ghosts and instability; (ii) consistency with experiments, string/M-Theory[42–44],LoopQuantumGravity(LQG)
mainly the solar system tests and cosmological observa- [45–48] 4, Causal Dynamical Triangulation (CDT) [52],
tions; and (iii) predictions. One must confess that this andAsymptoticSafety[16,53],tonameonlyfewofthem.
is not an easy task, considering the fact that the field For more details, see [54]. However, our understanding
hasbeenextensivelydevelopedinthepastfewyearsand oneachofthemisstillhighlylimited. Inparticularwedo
there have been various extensions of Hoˇrava’s original not know the relations among them (if there exists any),
minimal theory(tobedefinedsoonbelow)2. So, oneway andmoreimportantly,ifanyofthemisthetheorywehave
or another one has to make a choice on which subjects been looking for 5. One of the main reasons is the lack
thatshouldbeincludedinabriefreview,likethecurrent of experimental evidences. This is understandable, con-
one. Such a choice clearly contains the reviewer’s bias. sidering the fact that quantum gravitational effects are
In addition, in this review we do not intend to exhaust expected to become important only at the Planck scale,
all the relevant articles even within the chosen subjects, whichcurrentlyiswellabovetherangeofanyman-made
as in the information era, one can simply find them, for terrestrial experiments. However, the situation has been
example, from the list of the citations of Hoˇrava’s paper changingrecentlywiththearrivalofprecisioncosmology
3. With all these reasons, I would like first to offer my [56–63]. Inparticular,theinconsistencyofthetheoretical
sincerethanksandapologiestowhomhis/herworkisnot predictions with current observations, obtained by using
mentionedinthisreview. Inaddition,therehavealready thedeformedalgebraapproachintheframeworkofLQC
existed a couple of excellent reviews on Hoˇrava gravity [64], has shown that cosmology has indeed already en-
and its applications to cosmology and astrophysics [34– teredanerainwhichquantumtheoriesofgravitycanbe
40], including the one by Hoˇrava himself [41]. There- tested directly by observations.
fore, fortheworkspriorto thesereviews, thereadersare
strongly recommended to them for details.
The rest part of the review is organized as follows: II. HORˇAVA THEORY OF QUANTUM
In the next section (Sec. II), we first give a brief in- GRAVITY
troduction to the gauge symmetry that Hoˇrava gravity
adoptsandthegeneralformoftheactionthatcanbecon- According to our current understanding, space and
structed under such a symmetry. Then, we state clearly timearequantizedinthedeepPlanckregime,andacon-
the problems with this incarnation. To solve these prob- tinuous spacetime only emerges later as a classical limit
lems, various modifications have been proposed. In this of QG from some discrete substratum. Then, since the
review, we introduce four of them, respectively, in Sec. LI is a continuous symmetry of spacetime, it may not
II.A-D,whichhavebeenmostintensivelystudiedsofar.
Attheendofthissection(Sec. II.E),weconsidertheco-
4 It is interesting to note that big bang singularities have been
intensively studied in Loop Quantum Cosmology (LQC) [49],
1 A supersymmetric version of Hoˇrava gravity has not been suc- andalargenumberofcosmologicalmodelshavebeenconsidered
cessfullyconstructed,yet[28–30]. [50]. In all of these models big bang singularity is resolved by
2 Uptothemomentofwritingthisreview,Hoˇrava’sseminalpaper quantumgravitationaleffectsinthedeepPlanckregime. Similar
[23] has been already cited about 1400 times, see, for example, conclusionsarealsoobtainedforblackholes[51].
https://inspirehep.net/search?p=find+eprint+0901.3775. 5 One may never be able to prove truly that a theory is correct,
3 A more complete list of articles concerning Hoˇrava gravity can but rather disprove or more accurately constrain a hypothesis
be found from the citation list of Hoˇrava’s seminal paper [23]: [55]. The history of science tells us that this has been the case
https://inspirehep.net/search?p=find+eprint+0901.3775. sofar.
4
exist quantum mechanically, and instead emerges at the so that their dimensions are
low energy physics. Along this line of arguing, it is not
unreasonable to assume that LI is broken in the UV but N =0, (cid:2)Ni(cid:3)=2, [gij]=0. (2.7)
recovered later in the IR. Once LI is broken, one can
include only high-order spatial derivative operators into UndertheDiff(M, F),ontheotherhand,theytransform
the Lagrangian, so the UV behavior can be improved, as,
while the time derivative operators are still kept to the
δN =ξk∇ N +N˙ξ +Nξ˙ ,
second-order, in order to evade Ostrogradsky’s ghosts. k 0 0
This was precisely what Hoˇrava did [23]. δNi =Nk∇iξk+ξk∇kNi+gikξ˙k+N˙iξ0+Niξ˙0,
Of course, there are many ways to break LI. But, δg =∇ ξ +∇ ξ +ξ g˙ , (2.8)
ij i j j i 0 ij
Hoˇrava chose to break it by considering anisotropic scal-
ing between time and space, where Ni ≡ gijNj, and in writing the above we had as-
sumed that ξ (t) and ξk(cid:0)t,xi(cid:1) are small, so that only
0
t→b−zt, xi →b−1x(cid:48)i, (i=1,2,...,d) (2.1) their linear terms appear. Once we know the transforms
(2.8), we can construct the basic operators of the funda-
where z denotes the dynamical critical exponent, and LI
mental variables (2.4) and their derivatives, which turn
requiresz =1,whilepower-countingrenomalizibalityre- out to be 8,
quires z ≥ d, where d denotes the spatial dimension of
the spacetime [23, 65–69]. In this review we mainly con- R , K , a , ∇ , (2.9)
ij ij i i
sider spacetimes with d = 3 and take the minimal value
z = d, except for particular considerations. Whenever where a ≡N /N, and ∇ denotes the covariant deriva-
i ,i i
this happens, we shall make specific notice. Eq.(2.1) is a tive with respect to g , while R is the 3-dimensional
ij ij
reminiscent of Lifshitz’s scalar fields in condensed mat- Ricci tensor constructed from the 3-metric g and gij
ij
terphysics[70,71],henceintheliteratureHoˇravagravity where g gik = δk. K denotes the extrinsic curvature
ij j ij
is also called the Hoˇrava-Lifshitz (HL) theory. With the tensor of the leaves t= constant, defined as
scalingofEq.(2.1),thetimeandspacehave,respectively,
the dimensions 6, 1
K ≡ (−g˙ +∇ N +∇ N ), (2.10)
ij 2N ij i j j i
[t]=−z, (cid:2)xi(cid:3)=−1. (2.2)
with g˙ ≡ ∂g /∂t. It can be easily shown that these
ij ij
Clearly,suchascalingbreaksexplicitlytheLIandhence
basicquantitiesarevectors/tensorsunderthecoordinate
4-dimensional diffeomorphism invariance. Hoˇrava as-
transformations (2.3), and have the dimensions,
sumed that it is broken only down to the level
t→ξ (t), xi →ξi(cid:0)t,xk(cid:1), (2.3) [Rij]=2, [Kij]=3, [ai]=1, [∇i]=1. (2.11)
0
so the spatial diffeomorphism still remains. The above WiththebasicblocksofEq.(2.9)andtheirdimensions,
symmetry is often referred as to the foliation-preserving wecanbuildscalaroperatorsorderbyorder,sothetotal
diffeomorphism, denoted by Diff(M, F). To see how Lagrangian will finally take the form,
gravitationalfieldstransformundertheabovediffeomor-
2z
phism, let us first introduce the Arnowitt-Deser-Misner L = (cid:88)L(n)(cid:0)N,Ni,g (cid:1), (2.12)
(ADM) variables [72], g g ij
n=0
(cid:0)N,Ni,g (cid:1), (2.4)
ij
where L(n) denotes the part of the Lagrangian that con-
g
whereN, Niandg ,denote,respectively,thelapsefunc- tains operators of the nth-order only. In particular, to
ij
tion, shift vector, and 3-dimensional metric of the leaves each order of [k], we have the following independent
t= constant 7. Under the rescaling (2.1) N, Ni and g terms that are all scalars under the transformations of
ij
are assumed to scale, respectively, as [23], the foliation-preserving diffeomorphisms (2.3) [73],
N →N, Ni →b2Ni, gij →gij, (2.6) [k]6 :K Kij, K2, R3, RR Rij, RiRjRk, (∇R)2,
ij ij j k i
6 Inthisreviewwewillmeasurecanonicaldimensionsofallobjects
intheunitiesofspatialmomentak. But,inHoˇravagravitythelineelementdsisnotnecessarilygiven
7 IntheADMdecomposition,thelineelementdsisgivenbyds2= bythisrelation[Forexample,seeEqs.(2.41)and(2.42)]. Instead,
gµνdxµdxν with the 4-dimensional metrics gµν and gµν being one can simply consider (cid:0)N,Ni,gij(cid:1) as the fundamental quan-
givenby[72], tities that describe the quantum gravitational field of Hoˇrava
gµν =(cid:18)−N2N+iNiNi gNiji(cid:19), gµν =(cid:32)−NN1i2 gij−NNN2iiNj(cid:33). 8 gaNrsoadtvesit,tyhw,aialtlnwedmitthehretgihereirngeleatnhteieoranIlRsdtlioiffmetoihtm.eomrpahcirsomsc,oxpµic→quξaµn(cid:0)titt,ixesk,(cid:1),suthche
N2 N2 (2.5) fundamentalquantityistheRiemanntensorRµναβ.
5
(∇ R )(cid:0)∇iRjk(cid:1), (cid:0)a ai(cid:1)2R, (cid:0)a ai(cid:1)(cid:0)a a Rij(cid:1), notreducethetotalnumberofcouplingconstantssignif-
i jk i i i j
icantly.
(cid:0)a ai(cid:1)3,ai∆2a ,(cid:0)ai (cid:1)∆R,...,
i i i Tofurtherreducethenumberofindependentcoupling
[k]5 : K Rij, (cid:15)ijkR ∇ Rl, (cid:15)ijka a ∇ Rl, constants, Hoˇrava introduced two additional conditions,
ij il j k i l j k
a a Kij, Kija , (cid:0)ai (cid:1)K, the projectability and detailed balance [23]. The former
i j ij i requiresthatthelapsefunctionN beafunctionoftonly,
[k]4 : R2, R Rij, (cid:0)a ai(cid:1)2, (cid:0)ai (cid:1)2, (cid:0)a ai(cid:1)aj ,
ij i i i j
N =N(t), (2.17)
aija , (cid:0)a ai(cid:1)R, a a Rij, Rai ,
ij i i j i
[k]3 : ω3(Γ), so that all the terms proportional to ai and its deriva-
[k]2 : R, a ai, tives will be dropped out. This will reduce considerably
i
the total number of the independent terms in Eq.(2.12),
[k]1 : None,
considering the fact that a has dimension of one only.
i
[k]0 : γ0, (2.13) Thus, to build an operator out of ai to the sixth-order,
there will be many independent combinations of a and
where ω (Γ) denotes the gravitational Chern-Simons i
3 its derivatives. However, once the condition (2.17) is im-
term, γ is a dimensionless constant, ∆≡gij∇ ∇ , and
0 i j posed, all such terms vanish identically, and the total
a ≡∇ ∇ ...∇ ln(N), number of the sixth-order terms immediately reduces to
i1i2...in i1 i2 in seven,givenexactlybythefirstseventermsinEq.(2.13).
(cid:16) 2 (cid:17)
ω (Γ)≡Tr Γ∧dΓ+ Γ∧Γ∧Γ So,thetotallynumberoftheindependentlycouplingcon-
3 3
stantsofthetheorynowreducetoN =14,evenwiththe
eijk(cid:16) 2 (cid:17)
= √ Γm∂ Γl + ΓnΓl Γm . (2.14) three parity-violated terms,
g il j km 3 il jm kn
Here g = det (g ) and (cid:15)ijk ≡ eijk/√g with e123 = 1, KijRij, (cid:15)ijkRil∇jRkl, ω3(Γ). (2.18)
ij
etc. Note that in writing Eq.(2.13), we had not written It is this version of the HL theory that Hoˇrava referred
down all the sixth order terms, as they are numerous to as the minimal theory [41]. Note that the projectable
and a complete set of it has not been given explicitly condition(2.17)ismathematicallyelegantandappealing.
[74, 75]. Then, the general action of the gravitational It is preserved by the Diff(M, F) (2.3), and forms an
part (2.6) will be the summary of all these terms. Since independent branch of differential geometry [76].
timederivativetermsonlycontaininKij,wecanseethat Inspiredbycondensedmattersystems[77],inaddition
thekineticpartLK isthelinearcombinationofthesixth to the projectable condition, Hoˇrava also assumed that
order derivative terms, the potential part, L , can be obtained from a superpo-
V
tential W via the relations [23],
L = 1 (cid:0)K Kij −λK2(cid:1), (2.15) g
K ζ2 ij 1 δW
L ≡w2E GijklE , Eij ≡ √ g, (2.19)
V ij kl g δg
where ζ2 is the gravitational coupling constant with the ij
dimension where w is a coupling constant, and Gijkl denotes the
(cid:2)ζ2(cid:3)=[t]·(cid:2)xi(cid:3)3+[K]2 =−z−3+2z =z−3. (2.16) generalized DeWitt metric, defined as
Therefore, for z = 3, it is dimensionless, and the Gijkl = 1(cid:0)gikgjl+gilgjk(cid:1)−λgijgkl, (2.20)
2
power-counting analysis given between Eqs.(1.1) and
(1.2)showsthatthetheorynowbecomespower-counting where λ is the same coupling constant, as introduced in
renormalizable. The parameter λ is another dimension- Eq.(2.15), and the superpotential W is given by
g
less coupling constant, and LI guarantees it to be one
even after radiative corrections are taken into account. (cid:90) 1 (cid:90) √
W = ω (Γ)+ d3x g(R−2Λ), (2.21)
But, in Hoˇrava gravity it becomes a running constant g 3 κ2
Σ W Σ
due to the breaking of LI.
The rest of the Lagrangian (also called the potential) whereΣdenotestheleavesoft=constant,Λthecosmo-
will be the linear combination of all the rest terms of logical constant, and κ is another coupling constant of
W
Eq.(2.13), from which we can see that, without protec- the theory. Then, the total Lagrangian L = L −L ,
g K V
tion of further symmetries, the total Lagrangian of the contains only five coupling constants, ζ,λ,w,κ and Λ.
w
gravitationalsectorisabout100terms,whichisnormally Note that the above detailed balance condition has a
considered very large and could potentially diminish the coupleofremarkablefeatures[41]: First,itisinthesame
prediction power of the theory. Note that the odd terms spiritoftheAdS/CFTcorrespondence[78–82],wherethe
given in Eq.(2.13) violate the parity. So, to eliminate superpotential is defined on the 3-dimensional leaves, Σ,
them, we can simply require that parity be conserved. while the gravity is (3+1)-dimensional. Second, in the
However, since there are only six such terms, this will non-equilibriumthermodynamics, thecounterpartofthe
6
superpotential W plays the role of entropy, while the Making the coordinate transformation,
g
term Eij the entropic forces [83, 84]. This might shed √
2mr
light on the nature of the gravitational forces [85]. dt =dt+ dr, (2.24)
PG r−2m
However, despite of these desired features, this condi-
tionleadstoseveralproblems,includingthattheNewto- the above metric takes the Painleve-Gullstrand (PG)
nianlimitdoesnotexist[86],andthesix-orderderivative form [101],
operatorsareeliminated,sothetheoryisstillnotpower-
(cid:32) (cid:114) (cid:33)2
countingrenormalizable[23]. Inaddition,itisnotclearif 2m
ds2 =−dt2 + dr+ dt +r2dΩ2. (2.25)
this symmetry is still respected by radiative corrections. PG r PG
Even more fundamentally, the foliation-preserving dif-
feomorphism (2.3) allows one more degree of freedom in In GR we consider metrics (2.23) and (2.25) as describ-
the gravitational sector, in comparing with that of gen- ing the same spacetime (at least in the region r > 2m),
eraldiffeomorphism. Asaresult, aspin-0modeofgravi- as they are connected by the coordinate transformation
tons appears. This mode is potentially dangerous and (2.24), which is allowed by the symmetry (2.22) of GR.
may cause ghosts and instability problems, which lead But this is no longer the case when we consider them
the constraint algebra dynamically inconsistent [87–90]. in Hoˇrava gravity. The coordinate transformation (2.24)
isnotallowedbythefoliation-preservingdiffeomorphism
To solve these problems, various modifications have
(2.3), and as a result, they describe two different space-
been proposed [34–41]. In the following we shall briefly
times in Hoˇrava theory. In particular, metric (2.25) sat-
introduce only four of them, as they have been most ex-
isfies the projectability condition, while metric (2.23)
tensively studied in the literature so far. These are the
does not. So, in Hoˇrava theory they belong to the two
ones: (i) with the projectability condition - the mini-
completely different branches, with or without the pro-
mal theory [23, 91, 92]; (ii) with the projectability condi-
jectability condition. Moreover, in GR the metric
tion and an extra U(1) symmetry [93, 94]; (iii) without
the projectability condition but including all the possi- (cid:18) 2m(cid:19)
ds2 =− 1− dv2+2dvdr+r2dΩ2, (2.26)
ble terms - the healthy extension [95, 96]; and (iv) with r
an extra U(1) symmetry but without the projectability
condition [73, 97, 98]. describes the same spacetime as those of metrics (2.23)
and (2.25), but it does not belong to any of the two
Beforeconsideringeachofthesemodelsindetail,some
branches of Hoˇrava theory, because v = constant hyper-
commentsonsingularitiesinHoˇravagravityareinorder,
surfaces do not define a (3+1)-dimensional foliation, a
as they will appear in all of these models and shall be
fundamental requirement of Hoˇrava gravity. Third, be-
facedwhenweconsiderapplicationsofHoˇravagravityto
cause of the difference between the two kinds of coor-
black hole physics and cosmology [99]. First, the nature
dinate transformations, the global structure of a given
of singularities of a given spacetime in Hoˇrava theory
spacetime is also different in GR and Hoˇrava grav-
could be quite different from that in GR, which has the
ity [102]. For example, the maximal extension of the
general diffeomorphism,
Schwarzschildsolutionwasachievedwhenitiswrittenin
δxµ =ξµ(cid:0)t,xk(cid:1), (µ=0,1,2,3). (2.22) the Kruskal coordinates [12],
e−r/2m
ds2 =− dUdV +r2(U,V)dΩ2. (2.27)
In GR, singularities are divided into two different kinds: r
spacetime and coordinate singularities [100]. The former But the coordinate transformations that bring metric
is real and cannot be removed by any coordinate trans- (2.23) or (2.25) into this form are not allowed by the
formations of the type given by Eq.(2.22). The latter foliation-preserving diffeomorphism (2.3). For more de-
is coordinate-dependent, and can be removed by proper tails, we refer readers to [99, 102].
coordinate transformations of the kind (2.22). Since the
laws of coordinate transformations in GR and Hoˇrava
theory are different, it is clear that the nature of sin- A. The Minimal Theory
gularities are also different. In GR it may be a coor-
dinate singularity but in Hoˇrava gravity it becomes a
If we only impose the parity and projectability condi-
spacetime singularity. Second, two different metrics may
tion (2.17), the total action for the gravitational sector
represent the same spacetime in GR but in general it is
can be cast in the form [91, 92],
no longer true in Hoˇrava theory. A concrete example
(cid:90) √
is the Schwarzschild solution given in the Schwarzschild S =ζ2 dtd3xN g(L −L ), (2.28)
g K V
coordinates,
where L is given by Eq.(2.15), while the potential part
K
(cid:18) 2m(cid:19) (cid:18) 2m(cid:19)−1 takes the form,
ds2 =− 1− dt2+ 1− dr2+r2dΩ2.
r r L =2Λ−R+ 1 (cid:0)g R2+g R Rij(cid:1)
(2.23) V ζ2 2 3 ij
7
1 (cid:16) (cid:17)
+ g R3+g RR Rij +g RiRjRk by properly choosing the coupling constants of the high-
ζ4 4 5 ij 6 j k i orderoperatorsg (n=2,3,7,8). Notethateachofthese
n
+ 1 (cid:2)g R∇2R+g (∇ R )(cid:0)∇iRjk(cid:1)(cid:3).(2.29) terms is subjected to radiative corrections. It would be
ζ4 7 8 i jk very interesting to show that the scalar mode is still sta-
ble, even after such corrections are taken into account.
Here ζ2 = 1/16πG, and gn(n = 2,...8) are all dimen- On the other hand, from Eq.(2.31) it can be seen that
sionless coupling constants. Note that, without loss of there are two particular values of λ that make the above
the generality, in writing Eq.(2.29) the coefficient in the analysisinvalid,oneisλ=1andtheotherisλ=1/3. A
frontofRwassetto−1,whichcanberealizedbyrescal- more careful analysis of these two cases shows that the
ing the time and space coordinates [92]. As mentioned equationsforψ andB degenerateintoellipticdifferential
above, Hoˇravareferredtothismodelastheminimal the- equations,sothescalarmodeisnolongerdynamical. As
ory [41]. a result, the Minkowski spacetime in these two cases are
In the IR, all the high-order curvature terms (with co- stable.
efficients gn’s) drop out, and the total action reduces to It is also interesting to note that the de Sitter space-
the Einstein-Hilbert action, provided that the coupling time in this minimal theory is stable [106, 107]. See also
constant λ flows to its relativistic limit λGR = 1 in the a recent study of the issue in a closed FLRW universe
IR. This has not been shown in the general case. But, [108].
with only the three coupling constants (ζ, Λ,λ), it was Inaddition,thisminimaltheorystillsuffersthestrong
found that the Einstein-Hilbert action with Λ = 0 is an coupling problem [106, 109], so the power-counting anal-
attractor in the phase space of RG flow [103]. In addi- ysis presented above becomes invalid. It must be noted
tion, RG trajectories with a tiny positive cosmological that this does not necessarily imply the loss of pre-
constant also come with a value of λ that is compatible dictability: ifthetheoryisrenormalizable,allcoefficients
with experimental constraints. of infinite number of nonlinear terms can be written in
Tostudythestabilityofthetheory,letusconsiderthe terms of finite parameters in the action, as several well-
linear perturbations of the Minkowski background (with known theories with strong coupling (e.g., [110]) indi-
Λ=0), cate. However, because of the breakdown of the (naive)
perturbativeexpansion,weneedtoemploynonperturba-
N =1, N =∂ B, g =e−2ψδ . (2.30)
i i ij ij tive methods to analyze the fate of the scalar graviton
in the limit. Such an analysis was performed in [34] for
After integrating out the B field, the action upto the
sphericallysymmetric,static,vacuumconfigurationsand
quadratic terms of ψ takes the form [104],
was shown that the limit is continuously connected to
s(2) =ζ2(cid:90) dtdx3(cid:32)−ψ˙2 −ψ(cid:0)1+α ∂2+α ∂4(cid:1)∂2ψ(cid:33), G[11R1.,A11s2i]m, iwlahrerceonasifduellryatnioonnlifnoreacroasmnaollyosgiys owfassupgievrehnoriin-
g c2 1 2
ψ zon cosmological perturbations was carried out, and was
(2.31) shownthatthelimitλ→1iscontinuousandthatGRis
where ∂2 ≡ δij∂i∂j, c2ψ ≡ −(λ − 1)/(3λ − 1), α1 ≡ recovered. This may be considered as an analogue of the
(8g +3g )/ζ2 and α ≡ −(8g −3g )/ζ2. Clearly, to Vainshtein effect first found in massive gravity [113].
2 3 2 7 8
avoid ghosts we must assume that c2 <0, that is, With the projectability condition, the Hamiltonian
ψ
constraintbecomesglobal,fromwhichitwasshownthat
1 a component which behaves like dark matter emerges
(i)λ>1 or (ii)λ< . (2.32)
3 as an “integration constant” of dynamical equations and
momentum constraint equations [114].
However, in these intervals the scalar mode is not stable
Cosmological perturbations in this version of theory
in the IR [92, 104] 9. This can be seen easily from the
hasbeenextensivelystudied[104,108,115–120],andare
equation of motion of ψ in the momentum space,
found consistent with current observations.
In addition, spherically symmetric spacetimes with-
ψ¨ +ω2ψ =0, (2.33)
k k k out/with the presence of matter were also investigated
where ω2 ≡ c2 (cid:0)1−α k2+α k4(cid:1)k2. In the intervals of [121–123], and was found that the solar system tests can
k ψ 1 2 be satisfied by properly choosing the coupling constants
Eq.(2.32), we have c2 <0 in the IR, so ω2 also becomes
ψ k of the theory.
negative, that is, the theory suffers tachyonic instabil-
ity. IntheUVandintermediateregimes,itcanbestable
B. With Projectability & U(1) Symmetry
As mentioned above, the problems plagued in Hoˇrava
9 Stability of the scalar mode with the projectability condition
wasalsoconsideredin[105]. But,itwasfoundthatitexistsfor gravity are closely related to the existence of the spin-
allthevalueofλ. Thisisbecausethedetailedbalancecondition 0 graviton. Therefore, if it is eliminated, all the prob-
wasalsoimposedin[105]. lems should be cured. This can be done, for example,
8
A(cid:16) (cid:17)
by imposing extra symmetries, which was precisely what
L ≡ 2Λ −R ,
Hoˇrava and Melby-Thompson (HMT) did in [93]. HMT A N g
introduced an extra local U(1) symmetry, so that the L ≡(cid:0)1−λ(cid:1)(cid:104)(cid:0)∇2ϕ(cid:1)2+2K∇2ϕ(cid:105),
λ
total symmetry of the theory now is enlarged to,
1
U(1)(cid:110)Diff(M, F). (2.34) Gij ≡Rij − 2gijR+Λggij. (2.38)
The extra U(1) symmetry is realized by introducing two Here Λg is another coupling constant, and has the same
auxiliary fields, the U(1) gauge field A and the Newto- dimension of R. Note that the potential LV takes the
nian pre-potential ϕ 10. Under this extended symme- same form as that given in the case without the extra
try, the special status of time maintains, so that the U(1) symmetry. This is because gij does not change
anisotropic scaling (2.1) is still valid, whereby the UV under the local U(1) symmetry, as it can be seen from
behaviorofthetheorycanbeconsiderablyimproved. On Eq.(2.35). So, the most general form of LV is still given
theotherhand,underthelocalU(1)symmetry,thefields by Eq.(2.29).
transform as Notethatthestrongcouplingproblemnolongerexists
in the gravitational sector, as the spin-0 graviton now
δ A=α˙ −Ni∇ α, δ ϕ=−α,
α i α is eliminated. However, when coupled with matter, it
δ N =N∇ α, δ g =0=δ N, (2.35) will appear again for processes with energy higher than
α i i α ij α
where α(cid:0)t,xk(cid:1) is the generator of the local U(1) gauge [125, 127],
symmetry. Under the Diff(M,F), the auxiliary fields A (cid:18)M (cid:19)3/2
and ϕ transform as, Λω ≡Mpl Cpl |λ−1|5/4, (2.39)
δA=ζi∇ A+f˙A+fA˙,
i where M denotes the Planck mass, and generically
pl
δϕ=fϕ˙ +ζi∇iϕ, (2.36) C (cid:28) Mpl. To solve this problem, one way is to intro-
while N,Ni and g still transform as those given by duce a new energy scale M∗ so that M∗ < Λω, as Blas,
ij
Pujolas and Sibiryakov first introduced in the nonpro-
Eq.(2.8). Withthisenlargedsymmetry,thespin-0gravi-
jectable case [128]. This is reminiscent of the case in
ton is indeed eliminated [93, 124].
string theory where the string scale is introduced just
At the initial, it was believed that the U(1) symmetry
belowthePlanckscale, inordertoavoidstrongcoupling
can be realized only when the coupling constant λ takes
[42–44]. In the rest of this review, it will be referred
its relativistic value λ = 1. This was very encouraging,
to as the BPS mechanism. The main ideas are the fol-
because it is the deviation of λ from one that causes
lowing: before the strong coupling energy Λ is reached
all the problems, including ghost, instability and strong ω
coupling 11. However, this claim was soon challenged, [cf. Fig. 1], the sixth order derivative operators become
dominant, so the scaling law of a physical quantity for
and shown that the introduction of the Newtonian pre-
process with E > M will follow Eq.(2.1) instead of the
potential is so strong that action with λ(cid:54)=1 also has the ∗
relativistic one (z = 1). Then, with such anisotropic
localU(1)symmetry[94]. Itisremarkablethatthespin-0
scalings, it can be shown that all the nonrenornalizable
gravitonisstilleliminatedevenwithanarbitraryvalueof
terms (with z =1) now become either strictly renormal-
λ first by considering linear perturbations in Minkowski
izable or supperrenormalizable [110], whereby the strong
anddeSitterspacetimes[94,125],andthenbyanalyzing
coupling problem is resolved. For more details, we refer
the Hamiltonian structure of the theory [126].
readers to [127, 128].
The general action for the gravitational sector now
It should be noted that, in order for the mechanism
takes the form [125],
to work, the price to pay is that now λ cannot be ex-
(cid:90) √ (cid:16) (cid:17)
S =ζ2 dtd3xN g L −L +L +L +L , actly equal to one, as one can see from Eq.(2.39). In
g K V ϕ A λ
otherwords,thetheorycannotreducepreciselytoGRin
(2.37) the IR. However, since GR has achieved great success in
low energies, λ cannot be significantly different from one
where L and L are given by Eqs.(2.15) and (2.29),
K V in the IR, in order for the theory to be consistent with
and
observations.
(cid:16) (cid:17)
L ≡ϕGij 2K +∇ ∇ ϕ , In addition, the BPS mechanism cannot be applied to
ϕ ij i j
the minimal theory presented in the last subsection, be-
causetheconditionM <Λ ,togetherwiththeonethat
∗ ω
instability cannot occur within the age of the universe,
10 IntheoriginalpaperofHMT,theNewtonianpre-potentialwas requires fine-tuning |λ−1| < 10−24, as shown explicitly
denoted by ν [93]. In this review, we shall replace it by ϕ, and in [106]. However, in the current setup (with any λ), the
reserveν forotheruse.
Minkowskispacetimeisstable,sosuchafine-tuningdoes
11 Recall that in the relativistic case, λ = 1 is protected by the
diffeomorphism,x˜µ =x˜µ(t,xk), (µ=0,1,2,3). Withthissym- not exist.
metry, λ remains this value even after the radiative corrections Staticandsphericallysymmetricspacetimeswerecon-
aretakenintoaccount. sidered in [93, 98, 102, 129–133], and solar system tests
9
E E where
N ≡(1−a σ)N, N =N +N∇ ϕ,
1 i i i
Λ A−A
M ω γ ≡(1−a σ)2g , σ ≡ . (2.42)
* ij 2 ij N
Here a (≡ υ) and a are two coupling constants. It can
1 2
be shown that the effective metric (2.41) is invariant un-
der the enlarged symmetry (2.34). The matter is mini-
II
mallycoupledwithrespecttotheeffectivemetricγ via
µν
the relation,
Λω M* SM =(cid:90) dtd3xN√γLM(cid:0)N,Ni,γjk;ψn(cid:1), (2.43)
where ψ denotes collectively matter fields. With such
n
a coupling, in [98] the authors calculated explicitly all
the parameterized post-Newtonian (PPN) parameters in
I
termsofthecouplingconstantsofthetheory,andshowed
that the theory satisfies the constraints [134] 12,
γ =1+(2.1±2.3)×10−5,
(a) (b) β =1+(−4.1±7.8)×10−5,
α <10−4, α <4×10−7, α <4×10−20,
1 2 3
FIG. 1. The energy scales: (a) Λω <M∗ and (b) Λω >M∗. ξ <10−3, Γ<1.5×10−3, ζ1 <2×10−2,
InCase(a),thetheorybecomesstrongcouplingoncetheen- ζ <4×10−5, ζ <10−8, ζ <6×10−3, (2.44)
2 3 4
ergy of a system reaches the strong coupling energy Λ . In
ω
Case(b),thehigh-orderoperatorssuppressedbyM become obtained by all the current solar system tests, where
∗
important before the system reaches the strong coupling en-
10 2 2 1
ergy Λ , whereby the (relativistic) IR scaling of the system
ω Γ≡4β−γ−3− ξ−α + α − ζ − ζ . (2.45)
is taken over by the anisotropic scaling (2.1). 3 1 3 2 3 1 3 2
In particular, one can obtain the same results as those
given in GR [134],
wereinturninvestigated. Inparticular,HMTfoundthat
GR can be recovered in the IR if the lapse function N γ =β =1, α =α =α =ξ =0,
1 2 3
of GR is related to the one N of Hoˇrava gravity and the ζ =ζ =ζ =ζ =ζ =0, (2.46)
1 2 3 4 B
gauge field A via the relation, N =N−A [93]. This can
befurtherjustifiedbytheconsiderationsofgeometricin- for
terpretationsofthegaugefieldA,inwhichthelocalU(1)
(a ,a )=(1,0). (2.47)
symmetrywasfoundinthefirstplace. Byrequiringthat 1 2
the line element is invariant not only under the Diff(M,
It is interesting to note that this is exactly the case first
F) but also under the local U(1) symmetry, the authors
considered in HMT [93].
in[129]foundthatthelapsefunctionN andtheshiftvec-
Aremarkablefeatureisthatthesolarsystemtestsim-
tor Ni of GR should be given by N =N −υ(A−A)/c2
posenoconstraintontheparameterλ. Asaresult,when
and Ni = Ni+N∇iϕ, where υ is a dimensionless cou-
combined with the condition for the avoidance of the
pling constant, and
strong coupling problem, these conditions do not lead
to an upper bound on the energy scale M that sup-
1 ∗
A≡−ϕ˙ +Ni∇ ϕ+ N(∇ ϕ)2. (2.40) presses higher dimensional operators in the theory. This
i 2 i
is in sharp contrast to other versions of Hoˇrava gravity
without the U(1) symmetry. It should be noted that the
Then, itwasfoundthatthetheoryisconsistentwiththe
solar system tests for both λ = 1 and λ (cid:54)= 1, provided
that |υ −1| < 10−5. To couple with matter fields, in
[98] the authors considered a universal coupling between
12 Note that in general covariant theory including GR, the term
matterandtheHMTtheoryviatheeffectivemetricγµν, B≡(cid:82) |x−ρ(cid:48)x(cid:48)|(x−x(cid:48))· ddvt(cid:48)d3x(cid:48) appearinginthecomponenthtt
canbealwayseliminatedbythecoordinatetransformationδt=
ds2 ≡γ dxµdxν ξ0(t,xk). However,inHoˇravagravity,thissymmetryismissing,
µν
=−c2N2dt2+γ (cid:0)dxi+Nidt(cid:1)(cid:0)dxj +Njdt(cid:1), and the term ζBB must be included into htt. So, instead of
ij the ten PPN parameters introduced in [134], here we have an
(2.41) additonaloneζB. Formoredetails,wereferreadersto[98].
10
physical meaning of the gauge field A and the Newto- C. The Healthy Extension
nian prepotential ϕ were also studied in [135] but with a
different coupling with matter fields. Instead of eliminating the spin-0 graviton, BPS chose
Inflationary cosmology was studied in detail in [136], to live with it and work with the non-projectability con-
andfoundthat, amongotherthings, theFLRWuniverse dition [95, 96],
is necessarily flat. In the sub-horizon regions, the met-
ric and inflaton are tightly coupled and have the same N =N(cid:0)t, xk(cid:1). (2.48)
oscillating frequencies. In the super-horizon regions, the
Althoughagainwearefacingtheproblemofalargenum-
perturbations become adiabatic, and the comoving cur-
ber of coupling constants, BPS showed that the spin-0
vature perturbation is constant. Both scalar and tensor
graviton can be stabilized even in the IR. This is re-
perturbations are almost scale-invariant, and the spec- alized by including the quadratic term (cid:0)a ai(cid:1) into the
trum indices are the same as those given in GR, but i
Lagrangian,
the ratio of the scalar and tensor power spectra depends
on the high-order spatial derivative terms, and can be
6
different from that of GR significantly. Primordial non- L =2Λ−R−β a ai+(cid:88)L(n), (2.49)
V 0 i V
Gaussianities of scalar and tensor modes were also stud-
n=3
ied in [137, 138] 13.
Note that gravitational collapse of a spherically sym- where β0 is a dimensionless coupling constant, and L(Vn)
metric object was studied systematically in [140], by us- denotes the Lagrangian built by the nth-order operators
ing distribution theory. The junction conditions across only. Then, in the IR it can be shown that the scalar
the surface of a collapsing star were derived under the perturbationscanbestilldescribedbyEq.(2.33),butnow
(minimal) assumption that the junctions must be math- with [95, 96]
ematically meaningful in terms of distribution theory.
λ−1 (cid:18) 2 (cid:19) (cid:18) k4 (cid:19)
Lately, gravitational collapse in this setup was investi- ω2 = −1 k2+O , (2.50)
gated and various solutions were constructed [141–143]. k 3λ−1 β0 M∗2
We also note that with the U(1) symmetry, the de-
where M is the energy scale of the theory. While the
∗
tailed balance condition can be imposed [144]. However,
ghost-free condition still leads to the condition (2.32),
in order to have a healthy IR limit, it is necessary to because of the presence of the (cid:0)a ai(cid:1) term, the scalar
i
break it softly. This will allow the existence of the New-
mode becomes stable for
tonian limit in the IR and meanwhile be power-counting
renormalizable in the UV. Moreover, with the detailed 0<β <2. (2.51)
0
balanceconditionsoftlybreaking,thenumberofindepen-
dentcouplingconstantscanbestillsignificantlyreduced. Itisremarkabletonotethatthestabilityrequiresβ0 (cid:54)=0
This is particularly the case when we consider Hoˇrava strictly 14. This will lead to significantly difference from
gravitywithouttheprojectabilityconditionbutwiththe the case β0 =0. The stability of the spin-2 mode can be
U(1) symmetry. Note that, even in the latter, the U(1) shown by realizing the fact that only the following high-
symmetry is crucial in order not to have the problem order terms have contributions in the quadratic level of
of power-counting renormalizability in the UV, as shown linearperturbationsoftheMinkowskispacetime[74,96],
explicitly in [73]. In particular, in the healthy extension
L(n≥3) =ζ−2(cid:0)γ R2+γ R Rij +γ R∇ ai+γ a ∆ai(cid:1)
to be discussed in the next subsection the detailed bal- V 1 2 ij 3 i 4 i
(cid:104)
ance condition cannot be imposed even allowing it to be +ζ−4 γ (∇ R)2+γ (∇ R )2+γ (∆R)∇ ai
5 i 6 i jk 7 i
brokensoftly. Otherwise, itcanbeshownthatthesixth-
orderoperatorsareeliminatedbythiscondition,andthe +γ ai∆2a (cid:3), (2.52)
8 i
resulting theory is not power-counting renormalizable.
It is also interesting to note that, using the non-
relativistic gravity/gauge correspondence, it was found
14 Itisinterestingtonotethatinthecaseλ=1/3twoextrasecond-
that this version of Hoˇrava gravity has one-to-one corre-
class constraints appear [146, 147]. As a result, in this case the
spondence to dynamical Newton-Carton geometry with-
spin-0gravitoniseliminatedevenwhenβ0 =0. However, since
out torsion, and a precise dictionary was built [145]. β0 ingeneralissubjectedtoradiativecorrections,itisnotclear
whichsymmetrypreservesthisparticularvalue. In[23],Hoˇrava
showedthatatthisfixedpointthetheoryhasaconformalsym-
metry, provided that the detailed balance condition is satisfied.
But, there are two issues related to the detailed balance condi-
tion as mentioned before: (i) the resulted theory is no longer
13 It should be noted that such studies were carried out when power-countering,asthesixth-orderoperatorsareeliminatedby
matter is minimally coupled to gµν [136], and for the universal this condition; and (ii) in the IR the Newtonian limit does not
coupling (2.42) such studies have not been worked out, yet. A exist,sothetheoryisnotconsistentwithobservations. Forfur-
preliminary study indicates that a more general coupling might therdiscussionsoftherunningofthecouplingconstantsinterms
beneeded[139]. ofRGflow,see[103].