Hořava Gravity at a Lifshitz Point: A Progress Report

# Hořava 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
July 14, 2019
###### Abstract

Hořava 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, Hořava 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 observational constraints and self-consistency, it turns out that this is not an easy task, and various 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 Hořava gravity. In particular, we first present four most-studied versions of Hořava gravity, by focusing first on their self-consistency and then their consistency with experiments, including the solar system tests and cosmological observations. Then, we provide a general review on the recent developments of the theory in three different but also related areas: (i) universal horizons, black holes and their thermodynamics; (ii) 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.

## I Introduction

In the beginning of the last century, physics started with two triumphs, quantum mechanics and general relativity. On one hand, based on quantum mechanics (QM), the Standard Model (SM) of Particle Physics was developed, which describes three of the four interactions: electromagnetism, and the weak and strong nuclear forces. The last particle predicted by SM, the Higgs boson, was finally observed by Large Hadron Collider in 2012 LHCa (); LHCb (), after 40 years search. On the other hand, general relativity (GR) describes the fourth force, gravity, and predicts the existence of cosmic microwave background radiation (CMB), black holes, and gravitational waves (GWs), among other things. CMB was first observed accidentally in 1964 PW (), and since then various experiments have remeasured it each time with unprecedented precisions cosmoAa (); cosmoAb (); cosmoBa (); cosmoBb (); cosmoC (). Black holes have attracted a great deal of attention both theoretically and experimentally BHs1 (), and various evidences of their existence were found BHs2 (). In particular, on Sept. 14, 2015, the LIGO gravitational wave observatory made the first-ever successful observation of GWs GW (). The signal was consistent with theoretical predictions for the GWs produced by the merger of two binary black holes, which marks the beginning of a new era: gravitational wave astronomy.

Despite these spectacular successes, we have also been facing serious challenges. First, observations found that our universe consists of about dark matter (DM) cosmoC (). It is generally believed that such matter should not be made of particles from SM with a very simple argument: otherwise we should have already observed them directly. Second, spacetime singularities exist generically HE73 (), including those of black hole and the big bang cosmology. At the singularities, GR as well as any of other physics laws are all broken down, and it has been a cherished hope that quantum gravitational effects will step in and resolve the singularity problem.

However, when applying the well-understood quantum field theories (QFTs) to GR to obtain a theory of quantum gravity (QG), we have been facing a tremendous resistance QGa (); QGb (); QGc (): GR is not (perturbatively) renormalizable. Power-counting analysis shows that this happens because in four-dimensional spacetimes the gravitational coupling constant has the dimension of (in units where the Planck constant and the speed of light are one), whereas it should be larger than or equal to zero in order for the theory to be renormalizable perturbatively AS (). In fact, the expansion of a given physical quantity in terms of the small gravitational coupling constant must be in the form

 F=∞∑n=0an(GNE2)n, (1.1)

where denotes the energy of the system involved, so that the combination is dimensionless. Clearly, when , such expansions diverge. Therefore, it is expected that perturbative effective QFT is broken down at such energies. It is in this sense that GR is often said not perturbatively renormalizable.

An improved ultraviolet (UV) behavior can be obtained by including high-order derivative corrections to the Einstein-Hilbert action,

 SEH=∫d4x√−gR, (1.2)

such as a quadratic term, Stelle (). Then, the gravitational propagator will be changed from to

 1k2+1k2GNk41k2+1k2GNk41k2GNk41k2+.... =1k2−GNk4. (1.3)

Thus, at high energy the propagator is dominated by the term , and as a result, the UV divergence can be cured. Unfortunately, this simultaneously makes the modified theory not unitary, as now we have two poles,

 1k2−GNk4=1k2−1k2−G−1N, (1.4)

and the first one () describes a massless spin-2 graviton, while the second one describes a massive one but with a wrong sign in front of it, which implies that the massive graviton is actually a ghost (with a negative kinetic energy). It is the existence of this ghost that makes the theory not unitary, and has been there since its discovery Stelle ().

The existence of the ghost is closely related to the fact that the modified theory has orders of time-derivatives higher than two. In the quadratic case, for example, the field equations are fourth-orders. As a matter of fact, there exists a powerful theorem due to Mikhail Vasilevich Ostrogradsky, who established it in 1850 Ostrogradsky (). The theorem basically states that a system is not (kinematically) stable if it is described by a non-degenerate higher time-derivative Lagrangian. To be more specific, let us consider a system whose Lagrangian depends on , i.e., , where , etc. Then, the Euler-Lagrange equation reads,

 ∂L∂x−ddt∂L∂˙x+d2dt2∂L∂¨x=0. (1.5)

Now the non-degeneracy means that , which implies that Eq.(1.5) can be cast in the form Woodard15 (),

 d4x(t)dt4=F(x,˙x,¨x,...x;t). (1.6)

Clearly, in order to determine a solution uniquely four initial conditions are needed. This in turn implies that there must be four canonical coordinates, which can be chosen as,

 X1 ≡ x,P1≡∂L∂˙x−ddt∂L∂¨x, X2 ≡ ˙x,P2≡∂L∂¨x. (1.7)

The assumption of non-degeneracy guarantees that Eq.(I) has the inverse solution , so that

 ∂L∂¨x∣∣∣x=X1,˙x=X2,¨x=A=P2. (1.8)

Then, the corresponding Hamiltonian is given by

 H=2∑i=1Pidixdti−L=P1X2+P2A−L, (1.9)

which is linear in the canonical momentum and implies that there are no barriers to prevent the system from decay, so the system is not stable generically.

It is remarkable to note how powerful and general that the theorem is: It applies to any Lagrangian of the form . The only assumption is the non-degeneracy of the system,

 ∂2L(x,˙x,¨x)∂¨x2≠0, (1.10)

so the inverse of Eq.(I) exists. The above considerations can be easily generalized to systems with even higher order time derivatives Woodard15 ().

Clearly, with the above theorem one can see that any higher derivative theory of gravity with the Lorentz invariance (LI) and the non-degeneracy condition is not stable. Taking the above point of view into account, recently extensions of scalar-tensor theories were investigated by evading the Ostrogradsky instability LN16 (); RM16 (); Achour ().

Another way to evade Ostrogradsky’s theorem is to break LI in the UV and include only high-order spatial derivative terms in the Lagrangian, while still keep the time derivative terms to the second order. This is exactly what Hořava did recently Horava ().

It must be emphasized that this has to be done with great care. First, LI is one of the fundamental principles of modern physics and strongly supported by observations. In fact, all the experiments carried out so far are consistent with it Liberati13 (), and no evidence to show that such a symmetry must be broken at certain energy scales, although the constraints in the gravitational sector are much weaker than those in the matter sector LZbreaking (). Second, the breaking of LI can have significant effects on the low-energy physics through the interactions between gravity and matter, no matter how high the scale of symmetry breaking is Collin04 (). Recently, it was proposed a mechanism of SUSY breaking by coupling a Lorentz-invariant supersymmetric matter sector to non-supersymmetric gravitational interactions with Lifshitz scaling, and shown that it can lead to a consistent Hořava gravity PT14 () 111 A supersymmetric version of Hořava gravity has not been successfully constructed, yet Xue (); Redigolo (); PS ().. Another scenario is to go beyond the perturbative realm, so that strong interactions will take over at an intermediate scale (which is in between the Lorentz violation and the infrared (IR) scales) and accelerate the renormalization group (RG) flow quickly to the LI fixed point in the IR BPSd (); KS (); Afshordi ().

With the above in mind, in this article we shall give an updated review of Hořava gravity. Our emphases will be on: (i) the self-consistency of the theory, such as free of ghosts and instability; (ii) consistency with experiments, mainly the solar system tests and cosmological observations; and (iii) predictions. One must confess that this is not an easy task, considering the fact that the field has been extensively developed in the past few years and there have been various extensions of Hořava’s original minimal theory (to be defined soon below)222Up to the moment of writing this review, Hořava’s seminal paper Horava () has been already cited about 1400 times, see, for example, https://inspirehep.net/search?p=find+eprint+0901.3775.. So, one way or another one has to make a choice on which subjects that should be included in a brief review, like the current one. Such a choice clearly contains the reviewer’s bias. In addition, in this review we do not intend to exhaust all the relevant articles even within the chosen subjects, as in the information era, one can simply find them, for example, from the list of the citations of Hořava’s paper 333A more complete list of articles concerning Hořava gravity can be found from the citation list of Hořava’s seminal paper Horava (). With all these reasons, I would like first to offer my sincere thanks and apologies to whom his/her work is not mentioned in this review. In addition, there have already existed a couple of excellent reviews on Hořava gravity and its applications to cosmology and astrophysics Muk (); BC (); TPS (); WSV (); Visser11 (); Padilla (); Clifton (), including the one by Hořava himself Hreview (). Therefore, for the works prior to these reviews, the readers are strongly recommended to them for details.

The rest part of the review is organized as follows: In the next section (Sec. II), we first give a brief introduction to the gauge symmetry that Hořava gravity adopts and the general form of the action that can be constructed under such a symmetry. Then, we state clearly the problems with this incarnation. To solve these problems, various modifications have been proposed. In this review, we introduce four of them, respectively, in Sec. II.A - D, which have been most intensively studied so far. At the end of this section (Sec. II.E), we consider the covariantization of these models, which can be considered as the IR limits of the corresponding versions of Hořava gravity. In Sec. III, we present the recent developments of universal horizons and black holes, and discuss the corresponding thermodynamics, while in Sec. IV, we discuss the non-relativistic gauge/gravity duality, by paying particular attention on spacetimes with Lifshitz symmetry. In Sec. V, we consider the quantization of Hořava gravity, and summarize the main results obtained so far in the literature. These studies can be easily generalized to other gravitational theories with broken LI. The review is ended in Sec. VI, in which we list some open questions of Hořava’s quantum gravity and present some concluding remarks. An appendix is also included, in which we give a brief introduction to Lifshitz scalar theory.

Before proceeding to the next section, let us mention some (well studied) theories of QG. These include string/M-Theory stringa (); stringb (); stringc (), Loop Quantum Gravity (LQG) LQGa (); LQGb (); LQGc (); LQGd () 444It is interesting to note that big bang singularities have been intensively studied in Loop Quantum Cosmology (LQC) LQCc (), and a large number of cosmological models have been considered LQCSb (). In all of these models big bang singularity is resolved by quantum gravitational effects in the deep Planck regime. Similar conclusions are also obtained for black holes LQGBHd ()., Causal Dynamical Triangulation (CDT) CDTs (), and Asymptotic Safety AS (); ASa (), to name only few of them. For more details, see Bojowald15 (). However, our understanding on each of them is still highly limited. In particular we do not know the relations among them (if there exists any), and more importantly, if any of them is the theory we have been looking for 555One may never be able to prove truly that a theory is correct, but rather disprove or more accurately constrain a hypothesis Popper (). The history of science tells us that this has been the case so far.. One of the main reasons is the lack of experimental evidences. This is understandable, considering the fact that quantum gravitational effects are expected to become important only at the Planck scale, which currently is well above the range of any man-made terrestrial experiments. However, the situation has been changing recently with the arrival of precision cosmology KW14a (); KW14b (); KW14c (); KW14d (); KW14e (); KW14f (); stringCa (); stringCb (). In particular, the inconsistency of the theoretical predictions with current observations, obtained by using the deformed algebra approach in the framework of LQC deformed2 (), has shown that cosmology has indeed already entered an era in which quantum theories of gravity can be tested directly by observations.

## Ii Hořava Theory of Quantum Gravity

According to our current understanding, space and time are quantized in the deep Planck regime, and a continuous spacetime only emerges later as a classical limit of QG from some discrete substratum. Then, since the LI is a continuous symmetry of spacetime, it may not exist quantum mechanically, and instead emerges at the low energy physics. Along this line of arguing, it is not unreasonable to assume that LI is broken in the UV but recovered later in the IR. Once LI is broken, one can include only high-order spatial derivative operators into the Lagrangian, so the UV behavior can be improved, while the time derivative operators are still kept to the second-order, in order to evade Ostrogradsky’s ghosts. This was precisely what Hořava did Horava ().

Of course, there are many ways to break LI. But, Hořava chose to break it by considering anisotropic scaling between time and space,

 t→b−zt,xi→b−1x′i,(i=1,2,...,d) (2.1)

where denotes the dynamical critical exponent, and LI requires , while power-counting renomalizibality requires , where denotes the spatial dimension of the spacetime Horava (); Visser (); VisserA (); AH07 (); FIIKa (); FIIKb (). In this review we mainly consider spacetimes with and take the minimal value , except for particular considerations. Whenever this happens, we shall make specific notice. Eq.(2.1) is a reminiscent of Lifshitz’s scalar fields in condensed matter physics Lifshitz (); Lifshitz2 (), hence in the literature Hořava gravity is also called the Hořava-Lifshitz (HL) theory. With the scaling of Eq.(2.1), the time and space have, respectively, the dimensions 666In this review we will measure canonical dimensions of all objects in the unities of spatial momenta .,

 [t]=−z,[xi]=−1. (2.2)

Clearly, such a scaling breaks explicitly the LI and hence -dimensional diffeomorphism invariance. Hořava assumed that it is broken only down to the level

 t→ξ0(t),xi→ξi(t,xk), (2.3)

so the spatial diffeomorphism still remains. The above symmetry is often referred as to the foliation-preserving diffeomorphism, denoted by Diff(M, ). To see how gravitational fields transform under the above diffeomorphism, let us first introduce the Arnowitt-Deser-Misner (ADM) variables ADM (),

 (N,Ni,gij), (2.4)

where and , denote, respectively, the lapse function, shift vector, and 3-dimensional metric of the leaves constant 777In the ADM decomposition, the line element is given by with the 4-dimensional metrics and being given by ADM (), (2.5) But, in Hořava gravity the line element is not necessarily given by this relation [For example, see Eqs.(II.2) and (II.2)]. Instead, one can simply consider as the fundamental quantities that describe the quantum gravitational field of Hořava gravity, and their relations to the macroscopic quantities, such as , will emerge in the IR limit.. Under the rescaling (2.1) and are assumed to scale, respectively, as Horava (),

 N→N,Ni→b2Ni,gij→gij, (2.6)

so that their dimensions are

 N=0,[Ni]=2,[gij]=0. (2.7)

Under the Diff(), on the other hand, they transform as,

 δN = ξk∇kN+˙Nξ0+N˙ξ0, δNi = Nk∇iξk+ξk∇kNi+gik˙ξk+˙Niξ0+Ni˙ξ0, δgij = ∇iξj+∇jξi+ξ0˙gij, (2.8)

where , and in writing the above we had assumed that and are small, so that only their linear terms appear. Once we know the transforms (II), we can construct the basic operators of the fundamental variables (2.4) and their derivatives, which turn out to be 888Note that with the general diffeomorphism, , the fundamental quantity is the Riemann tensor .,

 Rij,Kij,ai,∇i, (2.9)

where , and denotes the covariant derivative with respect to , while is the 3-dimensional Ricci tensor constructed from the 3-metric and where . denotes the extrinsic curvature tensor of the leaves = constant, defined as

 Kij≡12N(−˙gij+∇iNj+∇jNi), (2.10)

with . It can be easily shown that these basic quantities are vectors/tensors under the coordinate transformations (2.3), and have the dimensions,

 [Rij]=2,[Kij]=3,[ai]=1,[∇i]=1. (2.11)

With the basic blocks of Eq.(2.9) and their dimensions, we can build scalar operators order by order, so the total Lagrangian will finally take the form,

 Lg=2z∑n=0L(n)g(N,Ni,gij), (2.12)

where denotes the part of the Lagrangian that contains operators of the nth-order only. In particular, to each order of , we have the following independent terms that are all scalars under the transformations of the foliation-preserving diffeomorphisms (2.3) ZSWW (),

 [k]6:KijKij,K2,R3,RRijRij,RijRjkRki,(∇R)2, (∇iRjk)(∇iRjk),(aiai)2R,(aiai)(aiajRij), (aiai)3,aiΔ2ai,(aii)ΔR,..., [k]5:KijRij,ϵijkRil∇jRlk,ϵijkaial∇jRlk, aiajKij,Kijaij,(aii)K, [k]4:R2,RijRij,(aiai)2,(aii)2,(aiai)ajj, aijaij,(aiai)R,aiajRij,Raii, [k]3:ω3(Γ), [k]2:R,aiai, [k]1:None, [k]0:γ0, (2.13)

where denotes the gravitational Chern-Simons term, is a dimensionless constant, , and

 ai1i2...in ≡ ∇i1∇i2...∇inln(N), ω3(Γ) ≡ Tr(Γ∧dΓ+23Γ∧Γ∧Γ) (2.14) = eijk√g(Γmil∂jΓlkm+23ΓnilΓljmΓmkn).

Here and with , etc. Note that in writing Eq.(II), we had not written down all the sixth order terms, as they are numerous and a complete set of it has not been given explicitly KP13 (); CES (). Then, the general action of the gravitational part (2.6) will be the summary of all these terms. Since time derivative terms only contain in , we can see that the kinetic part is the linear combination of the sixth order derivative terms,

 LK=1ζ2(KijKij−λK2), (2.15)

where is the gravitational coupling constant with the dimension

 (2.16)

Therefore, for , it is dimensionless, and the power-counting analysis given between Eqs.(1.1) and (1.2) shows that the theory now becomes power-counting renormalizable. The parameter is another dimensionless coupling constant, and LI guarantees it to be one even after radiative corrections are taken into account. But, in Hořava gravity it becomes a running constant due to the breaking of LI.

The rest of the Lagrangian (also called the potential) will be the linear combination of all the rest terms of Eq.(II), from which we can see that, without protection of further symmetries, the total Lagrangian of the gravitational sector is about 100 terms, which is normally considered very large and could potentially diminish the prediction power of the theory. Note that the odd terms given in Eq.(II) violate the parity. So, to eliminate them, we can simply require that parity be conserved. However, since there are only six such terms, this will not reduce the total number of coupling constants significantly.

To further reduce the number of independent coupling constants, Hořava introduced two additional conditions, the projectability and detailed balance Horava (). The former requires that the lapse function be a function of only,

 N=N(t), (2.17)

so that all the terms proportional to and its derivatives will be dropped out. This will reduce considerably the total number of the independent terms in Eq.(2.12), considering the fact that has dimension of one only. Thus, to build an operator out of to the sixth-order, there will be many independent combinations of and its derivatives. However, once the condition (2.17) is imposed, all such terms vanish identically, and the total number of the sixth-order terms immediately reduces to seven, given exactly by the first seven terms in Eq.(II). So, the totally number of the independently coupling constants of the theory now reduce to , even with the three parity-violated terms,

 KijRij,ϵijkRil∇jRlk,ω3(Γ). (2.18)

It is this version of the HL theory that Hořava referred to as the minimal theory Hreview (). Note that the projectable condition (2.17) is mathematically elegant and appealing. It is preserved by the Diff() (2.3), and forms an independent branch of differential geometry MM03 ().

Inspired by condensed matter systems Cardy (), in addition to the projectable condition, Hořava also assumed that the potential part, , can be obtained from a superpotential via the relations Horava (),

 LV≡w2EijGijklEkl,Eij≡1√gδWgδgij, (2.19)

where is a coupling constant, and denotes the generalized DeWitt metric, defined as

 Gijkl=12(gikgjl+gilgjk)−λgijgkl, (2.20)

where is the same coupling constant, as introduced in Eq.(2.15), and the superpotential is given by

 Wg=∫Σω3(Γ)+1κ2W∫Σd3x√g(R−2Λ), (2.21)

where denotes the leaves of constant, the cosmological constant, and is another coupling constant of the theory. Then, the total Lagrangian , contains only five coupling constants, and .

Note that the above detailed balance condition has a couple of remarkable features Hreview (): First, it is in the same spirit of the AdS/CFT correspondence AdSCFTa (); AdSCFTb (); AdSCFTc (); AdSCFTd (); AdSCFTe (), where the superpotential is defined on the 3-dimensional leaves, , while the gravity is (3+1)-dimensional. Second, in the non-equilibrium thermodynamics, the counterpart of the superpotential plays the role of entropy, while the term the entropic forces OM (); MO (). This might shed light on the nature of the gravitational forces Verlinde ().

However, despite of these desired features, this condition leads to several problems, including that the Newtonian limit does not exist LMP (), and the six-order derivative operators are eliminated, so the theory is still not power-counting renormalizable Horava (). In addition, it is not clear if this symmetry is still respected by radiative corrections.

Even more fundamentally, the foliation-preserving diffeomorphism (2.3) allows one more degree of freedom in the gravitational sector, in comparing with that of general diffeomorphism. As a result, a spin-0 mode of gravitons appears. This mode is potentially dangerous and may cause ghosts and instability problems, which lead the constraint algebra dynamically inconsistent CNPS (); LP (); BPS (); HKLG ().

To solve these problems, various modifications have been proposed Muk (); BC (); TPS (); WSV (); Visser11 (); Padilla (); Clifton (); Hreview (). In the following we shall briefly introduce only four of them, as they have been most extensively studied in the literature so far. These are the ones: (i) with the projectability condition - the minimal theory Horava (); SVWa (); SVWb (); (ii) with the projectability condition and an extra U(1) symmetry HMT (); daSilva (); (iii) without the projectability condition but including all the possible terms - the healthy extension BPSa (); BPSb (); and (iv) with an extra U(1) symmetry but without the projectability condition ZWWS (); ZSWW (); LMWZ ().

Before considering each of these models in detail, some comments on singularities in Hořava gravity are in order, as they will appear in all of these models and shall be faced when we consider applications of Hořava gravity to black hole physics and cosmology CW10 (). First, the nature of singularities of a given spacetime in Hořava theory could be quite different from that in GR, which has the general diffeomorphism,

 δxμ=ξμ(t,xk),(μ=0,1,2,3). (2.22)

In GR, singularities are divided into two different kinds: spacetime and coordinate singularities ES77 (). The former is real and cannot be removed by any coordinate transformations of the type given by Eq.(2.22). The latter is coordinate-dependent, and can be removed by proper coordinate transformations of the kind (2.22). Since the laws of coordinate transformations in GR and Hořava theory are different, it is clear that the nature of singularities are also different. In GR it may be a coordinate singularity but in Hořava gravity it becomes a spacetime singularity. Second, two different metrics may represent the same spacetime in GR but in general it is no longer true in Hořava theory. A concrete example is the Schwarzschild solution given in the Schwarzschild coordinates,

 ds2=−(1−2mr)dt2+(1−2mr)−1dr2+r2dΩ2. (2.23)

Making the coordinate transformation,

 dtPG=dt+√2mrr−2mdr, (2.24)

the above metric takes the Painleve-Gullstrand (PG) form PG (),

 ds2=−dt2PG+(dr+√2mrdtPG)2+r2dΩ2. (2.25)

In GR we consider metrics (2.23) and (2.25) as describing the same spacetime (at least in the region ), as they are connected by the coordinate transformation (2.24), which is allowed by the symmetry (2.22) of GR. But this is no longer the case when we consider them in Hořava gravity. The coordinate transformation (2.24) is not allowed by the foliation-preserving diffeomorphism (2.3), and as a result, they describe two different spacetimes in Hořava theory. In particular, metric (2.25) satisfies the projectability condition, while metric (2.23) does not. So, in Hořava theory they belong to the two completely different branches, with or without the projectability condition. Moreover, in GR the metric

 ds2=−(1−2mr)dv2+2dvdr+r2dΩ2, (2.26)

describes the same spacetime as those of metrics (2.23) and (2.25), but it does not belong to any of the two branches of Hořava theory, because constant hypersurfaces do not define a ()-dimensional foliation, a fundamental requirement of Hořava gravity. Third, because of the difference between the two kinds of coordinate transformations, the global structure of a given spacetime is also different in GR and Hořava gravity GLLSW (). For example, the maximal extension of the Schwarzschild solution was achieved when it is written in the Kruskal coordinates HE73 (),

 ds2=−e−r/2mrdUdV+r2(U,V)dΩ2. (2.27)

But the coordinate transformations that bring metric (2.23) or (2.25) into this form are not allowed by the foliation-preserving diffeomorphism (2.3). For more details, we refer readers to CW10 (); GLLSW ().

### ii.1 The Minimal Theory

If we only impose the parity and projectability condition (2.17), the total action for the gravitational sector can be cast in the form SVWa (); SVWb (),

 Sg=ζ2∫dtd3xN√g(LK−LV), (2.28)

where is given by Eq.(2.15), while the potential part takes the form,

 LV = 2Λ−R+1ζ2(g2R2+g3RijRij) (2.29) +1ζ4(g4R3+g5RRijRij+g6RijRjkRki) +1ζ4[g7R∇2R+g8(∇iRjk)(∇iRjk)].

Here , and are all dimensionless coupling constants. Note that, without loss of the generality, in writing Eq.(2.29) the coefficient in the front of was set to , which can be realized by rescaling the time and space coordinates SVWb (). As mentioned above, Hořava referred to this model as the minimal theory Hreview ().

In the IR, all the high-order curvature terms (with coefficients ’s) drop out, and the total action reduces to the Einstein-Hilbert action, provided that the coupling constant flows to its relativistic limit in the IR. This has not been shown in the general case. But, with only the three coupling constants (), it was found that the Einstein-Hilbert action with is an attractor in the phase space of RG flow CRS (). In addition, RG trajectories with a tiny positive cosmological constant also come with a value of that is compatible with experimental constraints.

To study the stability of the theory, let us consider the linear perturbations of the Minkowski background (with ),

 N=1,Ni=∂iB,gij=e−2ψδij. (2.30)

After integrating out the field, the action upto the quadratic terms of takes the form WM10 (),

 s(2)g=ζ2∫dtdx3⎛⎝−˙ψ2c2ψ−ψ(1+α1∂2+α2∂4)∂2ψ⎞⎠, (2.31)

where , and . Clearly, to avoid ghosts we must assume that , that is,

 (i)λ>1or(ii)λ<13. (2.32)

However, in these intervals the scalar mode is not stable in the IR SVWb (); WM10 () 999 Stability of the scalar mode with the projectability condition was also considered in BS (). But, it was found that it exists for all the value of . This is because the detailed balance condition was also imposed in BS ().. This can be seen easily from the equation of motion of in the momentum space,

 ¨ψk+ω2kψk=0, (2.33)

where . In the intervals of Eq.(2.32), we have in the IR, so also becomes negative, that is, the theory suffers tachyonic instability. In the UV and intermediate regimes, it can be stable by properly choosing the coupling constants of the high-order operators . Note that each of these terms is subjected to radiative corrections. It would be very interesting to show that the scalar mode is still stable, even after such corrections are taken into account.

On the other hand, from Eq.(2.31) it can be seen that there are two particular values of that make the above analysis invalid, one is and the other is . A more careful analysis of these two cases shows that the equations for and degenerate into elliptic differential equations, so the scalar mode is no longer dynamical. As a result, the Minkowski spacetime in these two cases are stable.

It is also interesting to note that the de Sitter spacetime in this minimal theory is stable WW10 (); HWW (). See also a recent study of the issue in a closed FLRW universe MFM ().

In addition, this minimal theory still suffers the strong coupling problem KA (); WW10 (), so the power-counting analysis presented above becomes invalid. It must be noted that this does not necessarily imply the loss of predictability: if the theory is renormalizable, all coefficients of infinite number of nonlinear terms can be written in terms of finite parameters in the action, as several well-known theories with strong coupling (e.g., Pol ()) indicate. However, because of the breakdown of the (naive) perturbative expansion, we need to employ nonperturbative methods to analyze the fate of the scalar graviton in the limit. Such an analysis was performed in Muk () for spherically symmetric, static, vacuum configurations and was shown that the limit is continuously connected to GR. A similar consideration for cosmology was given in Izumi:2011eh (); GMW (), where a fully nonlinear analysis of superhorizon cosmological perturbations was carried out, and was shown that the limit is continuous and that GR is recovered. This may be considered as an analogue of the Vainshtein effect first found in massive gravity Vainshtein:1972sx ().

With the projectability condition, the Hamiltonian constraint becomes global, from which it was shown that a component which behaves like dark matter emerges as an “integration constant” of dynamical equations and momentum constraint equations Muk09 ().

Cosmological perturbations in this version of theory has been extensively studied CHZ (); WM10 (); WWM (); Wang10 (); KUY (); GKS (); CB11 (); MFM (), and are found consistent with current observations.

In addition, spherically symmetric spacetimes without/with the presence of matter were also investigated IM09 (); GPW (); Satheesh (), and was found that the solar system tests can be satisfied by properly choosing the coupling constants of the theory.

### ii.2 With Projectability & U(1) Symmetry

As mentioned above, the problems plagued in Hořava gravity are closely related to the existence of the spin-0 graviton. Therefore, if it is eliminated, all the problems should be cured. This can be done, for example, by imposing extra symmetries, which was precisely what Hořava and Melby-Thompson (HMT) did in HMT (). HMT introduced an extra local U(1) symmetry, so that the total symmetry of the theory now is enlarged to,

 U(1)⋉Diff(M,F). (2.34)

The extra U(1) symmetry is realized by introducing two auxiliary fields, the gauge field and the Newtonian pre-potential 101010In the original paper of HMT, the Newtonian pre-potential was denoted by HMT (). In this review, we shall replace it by , and reserve for other use.. Under this extended symmetry, the special status of time maintains, so that the anisotropic scaling (2.1) is still valid, whereby the UV behavior of the theory can be considerably improved. On the other hand, under the local symmetry, the fields transform as

 δαA = ˙α−Ni∇iα,δαφ=−α, δαNi = N∇iα,δαgij=0=δαN, (2.35)

where is the generator of the local gauge symmetry. Under the Diff(), the auxiliary fields and transform as,

 δA = ζi∇iA+˙fA+f˙A, δφ = f˙φ+ζi∇iφ, (2.36)

while and still transform as those given by Eq.(II). With this enlarged symmetry, the spin-0 graviton is indeed eliminated HMT (); WW11 ().

At the initial, it was believed that the symmetry can be realized only when the coupling constant takes its relativistic value . This was very encouraging, because it is the deviation of from one that causes all the problems, including ghost, instability and strong coupling 111111Recall that in the relativistic case, is protected by the diffeomorphism, . With this symmetry, remains this value even after the radiative corrections are taken into account.. However, this claim was soon challenged, and shown that the introduction of the Newtonian pre-potential is so strong that action with also has the local symmetry daSilva (). It is remarkable that the spin-0 graviton is still eliminated even with an arbitrary value of first by considering linear perturbations in Minkowski and de Sitter spacetimes daSilva (); HW11 (), and then by analyzing the Hamiltonian structure of the theory Kluson11 ().

The general action for the gravitational sector now takes the form HW11 (),

 Sg = ζ2∫dtd3xN√g(LK−LV+Lφ+LA+Lλ),

where and are given by Eqs.(2.15) and (2.29), and

 Lφ ≡ φGij(2Kij+∇i∇jφ), LA ≡ AN(2Λg−R), Lλ ≡ (1−λ)[(∇2φ)2+2K∇2φ], Gij ≡ Rij−12gijR+Λggij. (2.38)

Here is another coupling constant, and has the same dimension of . Note that the potential takes the same form as that given in the case without the extra U(1) symmetry. This is because does not change under the local U(1) symmetry, as it can be seen from Eq.(II.2). So, the most general form of is still given by Eq.(2.29).

Note that the strong coupling problem no longer exists in the gravitational sector, as the spin-0 graviton now is eliminated. However, when coupled with matter, it will appear again for processes with energy higher than HW11 (); LWWZ (),

 Λω≡Mpl(MplC)3/2|λ−1|5/4, (2.39)

where denotes the Planck mass, and generically . To solve this problem, one way is to introduce a new energy scale so that , as Blas, Pujolas and Sibiryakov first introduced in the nonprojectable case BPSc (). This is reminiscent of the case in string theory where the string scale is introduced just below the Planck scale, in order to avoid strong coupling stringa (); stringb (); stringc (). In the rest of this review, it will be referred to as the BPS mechanism. The main ideas are the following: before the strong coupling energy is reached [cf. Fig. 1], the sixth order derivative operators become dominant, so the scaling law of a physical quantity for process with will follow Eq.(2.1) instead of the relativistic one (). Then, with such anisotropic scalings, it can be shown that all the nonrenornalizable terms (with ) now become either strictly renormalizable or supperrenormalizable Pol (), whereby the strong coupling problem is resolved. For more details, we refer readers to LWWZ (); BPSc ().

It should be noted that, in order for the mechanism to work, the price to pay is that now cannot be exactly equal to one, as one can see from Eq.(2.39). In other words, the theory cannot reduce precisely to GR in the IR. However, since GR has achieved great success in low energies, cannot be significantly different from one in the IR, in order for the theory to be consistent with observations.

In addition, the BPS mechanism cannot be applied to the minimal theory presented in the last subsection, because the condition , together with the one that instability cannot occur within the age of the universe, requires fine-tuning , as shown explicitly in WW10 (). However, in the current setup (with any ), the Minkowski spacetime is stable, so such a fine-tuning does not exist.

Static and spherically symmetric spacetimes were considered in HMT (); LMW (); AP10 (); GSW (); GLLSW (); BLW11 (); RLPV (); LMWZ (), and solar system tests were in turn investigated. In particular, HMT found that GR can be recovered in the IR if the lapse function of GR is related to the one of Hořava gravity and the gauge field via the relation, HMT (). This can be further justified by the considerations of geometric interpretations of the gauge field , in which the local U(1) symmetry was found in the first place. By requiring that the line element is invariant not only under the Diff(M, ) but also under the local U(1) symmetry, the authors in LMW () found that the lapse function and the shift vector of GR should be given by and , where is a dimensionless coupling constant, and

 A≡−˙φ+Ni∇iφ+12N(∇iφ)2. (2.40)

Then, it was found that the theory is consistent with the solar system tests for both and , provided that . To couple with matter fields, in LMWZ () the authors considered a universal coupling between matter and the HMT theory via the effective metric ,

 ds2 ≡ γμνdxμdxν = −c2N2dt2+γij(dxi+Nidt)(dxj+Njdt),

where

 N ≡ (1−a1σ)N,Ni=Ni+N∇iφ, γij ≡ (1−a2σ)2gij,σ≡A−AN. (2.42)

Here and are two coupling constants. It can be shown that the effective metric (II.2) is invariant under the enlarged symmetry (2.34). The matter is minimally coupled with respect to the effective metric via the relation,

 SM=∫dtd3xN√γLM(N,Ni,γjk;ψn), (2.43)

where denotes collectively matter fields. With such a coupling, in LMWZ () the authors calculated explicitly all the parameterized post-Newtonian (PPN) parameters in terms of the coupling constants of the theory, and showed that the theory satisfies the constraints Will () 121212Note that in general covariant theory including GR, the term appearing in the component can be always eliminated by the coordinate transformation . However, in Hořava gravity, this symmetry is missing, and the term must be included into . So, instead of the ten PPN parameters introduced in Will (), here we have an additonal one . For more details, we refer readers to LMWZ ().,

 γ=1+(2.1±2.3)×10−5, β=1+(−4.1±7.8)×10−5, α1<10−4,α2<4×10−7,α3<4×10−20, ξ<10−3,Γ<1.5×10−3,ζ1<2×10−2, ζ2<4×10−5,ζ3<10−8,ζ4<6×10−3, (2.44)

obtained by all the current solar system tests, where

 Γ≡4β−γ−3−103ξ−α1+23α2−23ζ1−13ζ2. (2.45)

In particular, one can obtain the same results as those given in GR Will (),

 γ=β=1,α1=α2=α3=ξ=0, ζ1=ζ2=ζ3=ζ4=ζB=0, (2.46)

for

 (a1,a2)=(1,0). (2.47)

It is interesting to note that this is exactly the case first considered in HMT HMT ().

A remarkable feature is that the solar system tests impose no constraint on the parameter . As a result, when combined with the condition for the avoidance of the strong coupling problem, these conditions do not lead to an upper bound on the energy scale that suppresses higher dimensional operators in the theory. This is in sharp contrast to other versions of Hořava gravity without the U(1) symmetry. It should be noted that the physical meaning of the gauge field and the Newtonian prepotential were also studied in AdsSilva () but with a different coupling with matter fields.

Inflationary cosmology was studied in detail in HWW12 (), and found that, among other things, the FLRW universe is necessarily flat. In the sub-horizon regions, the metric and inflaton are tightly coupled and have the same oscillating frequencies. In the super-horizon regions, the perturbations become adiabatic, and the comoving curvature perturbation is constant. Both scalar and tensor perturbations are almost scale-invariant, and the spectrum indices are the same as those given in GR, but the ratio of the scalar and tensor power spectra depends on the high-order spatial derivative terms, and can be different from that of GR significantly. Primordial non-Gaussianities of scalar and tensor modes were also studied in HW12 (); HWYZ () 131313 It should be noted that such studies were carried out when matter is minimally coupled to HWW12 (), and for the universal coupling (II.2) such studies have not been worked out, yet. A preliminary study indicates that a more general coupling might be needed MWWZ (). .

Note that gravitational collapse of a spherically symmetric object was studied systematically in GLSW (), by using distribution theory. The junction conditions across the surface of a collapsing star were derived under the (minimal) assumption that the junctions must be mathematically meaningful in terms of distribution theory. Lately, gravitational collapse in this setup was investigated and various solutions were constructed daSilva1 (); daSilva2 (); daSilva3 ().

We also note that with the U(1) symmetry, the detailed balance condition can be imposed BLW (). However, in order to have a healthy IR limit, it is necessary to break it softly. This will allow the existence of the Newtonian limit in the IR and meanwhile be power-counting renormalizable in the UV. Moreover, with the detailed balance condition softly breaking, the number of independent coupling constants can be still significantly reduced. This is particularly the case when we consider Hořava gravity without the projectability condition but with the U(1) symmetry. Note that, even in the latter, the U(1) symmetry is crucial in order not to have the problem of power-counting renormalizability in the UV, as shown explicitly in ZSWW (). In particular, in the healthy extension to be discussed in the next subsection the detailed balance condition cannot be imposed even allowing it to be broken softly. Otherwise, it can be shown that the sixth-order operators are eliminated by this condition, and the resulting theory is not power-counting renormalizable.

It is also interesting to note that, using the non-relativistic gravity/gauge correspondence, it was found that this version of Hořava gravity has one-to-one correspondence to dynamical Newton-Carton geometry without torsion, and a precise dictionary was built HO15 ().

### ii.3 The Healthy Extension

Instead of eliminating the spin-0 graviton, BPS chose to live with it and work with the non-projectability condition BPSa (); BPSb (),

 N=N(t,xk). (2.48)

Although again we are facing the problem of a large number of coupling constants, BPS showed that the spin-0 graviton can be stabilized even in the IR. This is realized by including the quadratic term into the Lagrangian,

 LV=2Λ−R−β0aiai+6∑n=3L(n)V, (2.49)

where is a dimensionless coupling constant, and denotes the Lagrangian built by the nth-order operators only. Then, in the IR it can be shown that the scalar perturbations can be still described by Eq.(2.33), but now with BPSa (); BPSb ()

 ω2k=λ−13λ−1(2β0−1)k2+O(k4M2∗), (2.50)

where is the energy scale of the theory. While the ghost-free condition still leads to the condition (2.32), because of the presence of the term, the scalar mode becomes stable for

 0<β0<2. (2.51)

It is remarkable to note that the stability requires strictly 141414It is interesting to note that in the case two extra second-class constraints appear BRS (); BR16 (). As a result, in this case the spin-0 graviton is eliminated even when . However, since in general is subjected to radiative corrections, it is not clear which symmetry preserves this particular value. In Horava (), Hořava showed that at this fixed point the theory has a conformal symmetry, 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 power-countering, as the sixth-order operators are eliminated by this condition; and (ii) in the IR the Newtonian limit does not exist, so the theory is not consistent with observations. For further discussions of the running of the coupling constants in terms of RG flow, see CRS ().. This will lead to significantly difference from the case . The stability of the spin-2 mode can be shown by realizing the fact that only the following high-order terms have contributions in the quadratic level of linear perturbations of the Minkowski spacetime BPSb (); KP13 (),

 L(n≥3)V = ζ−2(γ1R2+γ2RijRij+γ3R∇iai+γ4aiΔai) (2.52) + ζ−4[γ5(∇iR)2+γ6(∇iRjk)2+γ7(ΔR)∇iai +γ8aiΔ2ai],

where ’s are all dimensionless coupling constants.

As first noticed by BPS, the most stringent constraints come from the preferred frame effects due to Lorentz violation, which require BPSb (); Will (),

 |λ−1|≲4×10−7,M∗≲1015GeV. (2.53)

In addition, the timing of active galactic nuclei Timing () and gamma ray bursts GammaRay () require

 M∗≳1010∼1011GeV. (2.54)

To obtain the constraint (2.54), BPS used the results from the Einstein-aether theory, as these two theories coincide in the IR Jacob (); Jacob13 ().

Limits from binary pulsars were also studied recently, and the most stringent constraints of the theory were obtained YBYB (); YBBY (). However, when the allowed range of the preferred frame effects given by the solar system tests is saturated, the limit for is still given by Eq.(2.53), so the upper bound remains the same.

It is also interesting to note that observations of synchrotron radiation from the Crab Nebula seems to require that the scale of Lorentz violation in the matter sector must be , not LMS12 (). In addition, the consistency of the theory with the current observations of gravitational waves from the events GW150914 and GW151226 were studied recently, and moderate constraints were obtained YYP ().

Since the spin-0 graviton generically appears in this version of Hořava gravity Kluson10 (); DJ11 (), strong coupling problem is inevistable PS09 (); KP10 (). However, as mentioned in the last subsection, this can be solved by introducing an energy scale that suppresses the high-order operators BPSc (). When the energy of the system is higher than , these high-order operators become dominant, and the scaling law will be changed from to with given by Eq.(2.1). With such anisotropic scalings, all the nonrenornalizable terms (in the case with ) now become either strictly renormalizable or supperrenormalizable, whereby the strong coupling problem is resolved. For more details, we refer readers to LWWZ (); BPSc () 151515 The mechanism requires the coupling constants of the six-order derivative terms, represented by in the action (2.52), must be very large . This may introduce a new hierarchy KP13 (), although it was argued that this is technically natural BPSb (); BPSc ().. Note that in this case the strong coupling energy is given by BPSb (); BPSc (),

 Λω=|λ−1|1/2Mpl, (2.55)

instead of that given by Eq.(2.39) for the HMT extension with a local U(1) symmetry.

With the non-projectability condition, static spacetimes with have been extensively studied, see for example, LMP (); CCOa (); KS09 (); CCOb (); Myung09 (); Mann09 (); CJ09 (); Park09 (). Sine all of these works were carried out before the realization of the importance of the term that could play in the IR stability BPSa (); BPSb (), so they all unfortunately belong to the branch of the non-projectable Hořava gravity that is plagued with the instability problem 161616In Visser11 () attention has been called on the self-consistency of the theory. In particular, one would like first resolve the instability problem before considering any application of the theory.. Lately, the case with were studied in KK09 (); Park12 (); BS12a (); BRS15 (). In particular, it was claimed that slowly rotating black holes do not exist in this extension BS12a (). It was shown that this is incorrect Wang12 (); BS12b (), and slowly rotating black holes and stars indeed exist in all the four versions of Hořava gravity introduced in this review Wang13 () 171717It should be noted that “black holes” here are defined by the existence of Killing/sound horizons. But, particles can have velocities higher than that of light, once the Lorentz symmetry is broken LZbreaking (). Then, Killing/sound horizons are no longer barriers to these particles, so such defined “black holes” are not really black to such particles..

Cosmological perturbations were studied in KUYb (); CHZb (); FB12 () and found that they are consistent with observations. Cosmological constraints of Lorentz violation from dark matter and dark energy were also investigated in BS11b (); BIS (); ABLS (); ABILS ().

In addition, the odd terms given in Eq.(II) violate the parity, which can polarize primordial gravitational waves and lead to direct observations TS09 (); WWZZ (). In particular, it was shown that, because of both parity violation and the nonadiabatic evolution of the modes due to a modified dispersion relationship, a large polarization of primordial gravitational waves becomes possible, and could be within the range of detection of the BB, TB and EB power spectra of the forthcoming CMB observations WWZZ (). Of course, this conclusion is not restricted to this version of Hořava gravity, and in principle it is true in all of these four versions presented in this review.

### ii.4 With Non-projectability & U(1) Symmetry

A non-trivial generalization of the enlarged symmetry (2.34) to the nonprojectable case was realized in ZWWS (); ZSWW (); LMWZ (), and has been recently embedded into string theory by using the non-relativistic AdS/CFT correspondence JK1 (); JK2 (). In addition, it was also found that it corresponds to the dynamical Newton-Cartan geometry with twistless torsion (hypersurface orthogonal foliation) HO15 (). A precise dictionary was then constructed, which connect all fields, their transformations and other properties of the two corresponding theories. Further, it was shown that the U(1) symmetry comes from the Bargmann extension of the local Galilean algebra that acts on the tangent space to the torsional Newton-Cartan geometries.

The realization of the enlarged symmetry (2.34) in the nonprojectable case can be carried out by first noting the fact that and defined in Eq.(II.2) are gauge-invariant under the local U(1) transformations and are a vector and scalar under Diff(M,F), respectively. In addition, they have dimensions two and four, respectively,

 δαNi=0=δασ,[Ni]=2,[σ]=4. (2.56)

Then, the quantity defined by

 ~Kij ≡ 12N(−˙gij+∇iNj+∇jNi) (2.57) = Kij+∇i∇jφ+a(i∇j)φ,

is also gauge-invariant under the U(1) transformations. In addition, has the same dimension as , i.e., , from which one can see that the quantity ,

 ~LK≡~Kij~Kij−λ~K2, (2.58)

has dimension 6. In addition, the quantity

 ~LS≡σ(ZA+2Λg−R). (2.59)

has also dimension 6 and is gauge-invariant under the U(1) transformations, where , with and being two dimensionless coupling constants. Then, the action

 Sg = ζ2∫dtd3xN√g[~LK−γ1R−2Λ (2.60) +βaiai+σ(ZA+2Λg−R)−Lz>1],

is gauge-invariant under the enlarged symmetry (2.34), where denotes the potential part that contains spatial derivative operators higher than second-orders. Inserting Eqs.(2.57)-(2.59) into the above action, and then after integrating it partially, the total action takes the form LMWZ (),

 Sg=ζ2∫dtd3x√gN(LK−LV+LA+Lφ+LS),

where and are given, respectively, by Eqs.(2.15) and (II.2), but now with

 Lφ ≡ φGij(2Kij+∇