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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) renormalizable. 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 perturbatively [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 scaling between time and space, where Ni ≡ gijNj, and in writing the above we had assumed 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 diffeomorphism (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 coordinate 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 stable, 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 instability. 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 under 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 metric 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 condition 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].