Ho\V {R} Ava Gravity at a Lifshitz Point: a Progress Report

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 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 Hoˇrava gravity. In particular, we first present four most-studied versions of Hoˇrava 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.

CONTENTS The last particle predicted by SM, the Higgs boson, was finally observed by Large Hadron Collider in 2012 I. Introduction1 [1,2], after 40 years search. On the other hand, general relativity (GR) describes the fourth force, gravity, and II. Hoˇrava Theory of Quantum Gravity3 predicts the existence of cosmic microwave background A. The Minimal Theory6 radiation (CMB), black holes, and gravitational waves B. With Projectability & U(1) Symmetry7 (GWs), among other things. CMB was first observed ac- C. The Healthy Extension 10 cidentally in 1964 [3], and since then various experiments D. With Non-projectability & U(1) Symmetry 11 have remeasured it each time with unprecedented preci- E. Covariantization of Hoˇrava Gravity 13 sions [4–8]. Black holes have attracted a great deal of attention both theoretically and experimentally [9], and III. Black Holes & Thermodynamics 15 various evidences of their existence were found [10]. In particular, on Sept. 14, 2015, the LIGO gravitational IV. Non-relativistic Gauge/Gravity Duality 19 wave observatory made the first-ever successful observation of GWs [11]. The signal was consistent with theo- V. Quantization of Hoˇrava Gravity 20 retical predictions for the GWs produced by the merger of two binary black holes, which marks the beginning of VI. Concluding Remarks 21 a new era: gravitational wave astronomy. Acknowledgements 23 Despite these spectacular successes, we have also been facing serious challenges. First, observations found that Appendix A: Lifshitz Scalar Theory 23 our universe consists of about 25% dark matter (DM) [8]. It is generally believed that such matter should not be arXiv:1701.06087v3 [gr-qc] 4 Apr 2017 References 25 made of particles from SM with a very simple argument: otherwise we should have already observed them directly. 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 with two triumphs, quantum mechanics and general rela- hope that quantum gravitational effects will step in and tivity. On one hand, based on quantum mechanics (QM), resolve the singularity problem. the Standard Model (SM) of Particle Physics was devel- However, when applying the well-understood quantum oped, which describes three of the four interactions: 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] −2 coupling constant GN has the dimension of (mass) (in 2 units where the Planck constant ~ and the speed of light Now the non-degeneracy means that ∂2L/∂x¨2 6= 0, which c are one), whereas it should be larger than or equal to implies that Eq.(1.5) can be cast in the form [19], zero in order for the theory to be renormalizable pertur- d4x(t) ... batively [16]. In fact, the expansion of a given physical = 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 X 2n F = an GN E , (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 2 X ≡ x, P ≡ − , that the combination GN E is dimensionless. Clearly, 1 1 ∂x˙ dt ∂x¨ 2 −1 when E & GN , such expansions diverge. Therefore, ∂L X ≡ x,˙ P ≡ . (1.7) it is expected that perturbative effective QFT is broken 2 2 ∂x¨ down at such energies. It is in this sense that GR is often said not perturbatively renormalizable. The assumption of non-degeneracy guarantees that An improved ultraviolet (UV) behavior can be ob- Eq.(1.7) has the inverse solutionx ¨ = A(X1,X2,P2), so tained by including high-order derivative corrections to that the Einstein-Hilbert action, ∂L = P2. (1.8) Z ∂x¨ x=X , x˙ =X , x¨=A 4 √ 1 2 SEH = d x −gR, (1.2) Then, the corresponding Hamiltonian is given by such as a quadratic term, R Rµν [17]. Then, the gravi- 2 µν X dix tational propagator will be changed from 1/k2 to H = P − L = P X + P A − L, (1.9) i dti 1 2 2 i=1 1 1 4 1 1 4 1 4 1 + GN k + GN k GN k + .... k2 k2 k2 k2 k2 k2 which is linear in the canonical momentum P1 and im- 1 plies that there are no barriers to prevent the system = . (1.3) 2 4 from decay, so the system is not stable generically. k − GN k It is remarkable to note how powerful and general that Thus, at high energy the propagator is dominated by 4 the theorem is: It applies to any Lagrangian of the form the term 1/k , 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, 2 1 1 1 ∂ L(x, x,˙ x¨) = − , (1.4) 2 6= 0, (1.10) 2 4 2 2 −1 ∂x¨ k − GN k k k − GN so the inverse x¨ = A(X ,X ,P ) of Eq.(1.7) exists. The and the first one (1/k2) describes a massless spin-2 gravi- 1 2 2 above considerations can be easily generalized to systems 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- netic energy). It is the existence of this ghost that makes 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- The existence of the ghost is closely related to the fact 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- time derivative terms to the second order. This is exactly 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 onx ¨, i.e., servations. In fact, all the experiments carried out so L = L(x, x,˙ x¨), wherex ˙ = dx(t)/dt, etc. Then, the Euler- 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) ∂x dt ∂x˙ dt2 ∂x¨ tor are much weaker than those in the matter sector 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 of universal horizons and black holes, and discuss the cor- proposed a mechanism of SUSY breaking by coupling a responding thermodynamics, while in Sec. IV, we discuss Lorentz-invariant supersymmetric matter sector to non- the non-relativistic gauge/gravity duality, by paying par- supersymmetric gravitational interactions with Lifshitz ticular attention on spacetimes with Lifshitz symmetry. scaling, and shown that it can lead to a consistent Hoˇrava 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 turbative realm, so that strong interactions will take over the literature. These studies can be easily generalized to at an intermediate scale (which is in between the Lorentz other gravitational theories with broken LI. The review violation and the infrared (IR) scales) and accelerate the is ended in Sec. VI, in which we list some open questions 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 With the above in mind, in this article we shall give an 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], Loop Quantum Gravity (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 and Asymptotic Safety [16, 53], to name only few of them. is not an easy task, considering the fact that the field For more details, see [54]. However, our understanding has been extensively developed in the past few years and on each of them is still highly limited. In particular we do there have been various extensions of Hoˇrava’s original not know the relations among them (if there exists any), minimal theory (to be defined soon below)2. So, one way and more importantly, if any of them is the theory we have or another one has to make a choice on which subjects been looking for 5. One of the main reasons is the lack that should be included in a brief review, like the current 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, which currently is well above the range of any man-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 changing recently with the arrival of precision cosmology 3. With all these reasons, I would like first to offer my [56–63]. In particular, the inconsistency of the theoretical sincere thanks and apologies to whom his/her work is not predictions with current observations, obtained by using mentioned in this review. In addition, there have already the deformed algebra approach in the framework of LQC 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– tered an era in which quantum theories of gravity can be 40], including the one by Hoˇrava himself [41]. There- tested directly by observations. fore, 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: II. HORAVAˇ THEORY OF QUANTUM In the next section (Sec. II), we first give a brief in- GRAVITY troduction to the gauge symmetry that Hoˇrava gravity adopts and the general form of the action that can be con- According to our current understanding, space and structed under such a symmetry. Then, we state clearly time are quantized in the deep Planck regime, and a con- 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, which have been most intensively studied so far. At the end of this section (Sec. II.E), we consider the co-

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- and a large number of cosmological models have been considered cessfully constructed, yet [28–30]. [50]. In all of these models big bang singularity is resolved by 2 Up to the moment of writing this review, Hoˇrava’s seminal paper quantum gravitational effects in the deep Planck regime. Similar [23] has been already cited about 1400 times, see, for example, conclusions are also obtained for black holes [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. so far. 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 i unreasonable to assume that LI is broken in the UV but N = 0, N = 2, [gij] = 0. (2.7) recovered later in the IR. Once LI is broken, one can include only high-order spatial derivative operators into Under the Diff(M, F), on the other hand, they transform the Lagrangian, so the UV behavior can be improved, as, while the time derivative operators are still kept to the δN = ξk∇ N + Nξ˙ + Nξ˙ , second-order, in order to evade Ostrogradsky’s ghosts. k 0 0 k k ˙k ˙ ˙ This was precisely what Hoˇrava did [23]. δNi = Nk∇iξ + ξ ∇kNi + gikξ + Niξ0 + Niξ0, Of course, there are many ways to break LI. But, δgij = ∇iξj + ∇jξi + ξ0g˙ij, (2.8) Hoˇrava chose to break it by considering anisotropic scal- j ing between time and space, where Ni ≡ gijN , and in writing the above we had as- k i sumed that ξ0(t) and ξ t, x are small, so that only t → b−zt, xi → b−1x0i, (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 requires z = 1, while power-counting renomalizibality re- 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- Rij,Kij, ai, ∇i, (2.9) sider spacetimes with d = 3 and take the minimal value z = d, except for particular considerations. Whenever where ai ≡ N,i/N, and ∇i denotes the covariant deriva- this happens, we shall make specific notice. Eq.(2.1) is a tive with respect to gij, while Rij is the 3-dimensional ij reminiscent of Lifshitz’s scalar fields in condensed mat- Ricci tensor constructed from the 3-metric gij and g ik k ter physics [70, 71], hence in the literature Hoˇrava gravity where gijg = δj . Kij denotes the extrinsic curvature is also called the Hoˇrava-Lifshitz (HL) theory. With the tensor of the leaves t= constant, defined as scaling of Eq.(2.1), the time and space have, respectively, 6 1 the dimensions , K ≡ (−g˙ + ∇ N + ∇ N ) , (2.10) ij 2N ij i j j i [t] = −z, xi = −1. (2.2) withg ˙ij ≡ ∂gij/∂t. It can be easily shown that these Clearly, such a scaling breaks explicitly the LI and hence basic quantities are vectors/tensors under the coordinate 4-dimensional diffeomorphism invariance. Hoˇrava as- transformations (2.3), and have the dimensions, sumed that it is broken only down to the level

i i k [Rij] = 2, [Kij] = 3, [ai] = 1, [∇i] = 1. (2.11) t → ξ0(t), x → ξ t, x , (2.3) so the spatial diffeomorphism still remains. The above With the basic blocks of Eq.(2.9) and their dimensions, symmetry is often referred as to the foliation-preserving we can build scalar operators order by order, so the total diffeomorphism, denoted by Diff(M, F). To see how Lagrangian will finally take the form, gravitational fields transform under the above diffeomor- 2z phism, let us first introduce the Arnowitt-Deser-Misner X L = L(n) N,N i, g , (2.12) (ADM) variables [72], g g ij n=0 N,N i, g , (2.4) ij (n) where Lg denotes the part of the Lagrangian that con- i where N,N and gij, denote, respectively, the lapse func- tains operators of the nth-order only. In particular, to tion, shift vector, and 3-dimensional metric of the leaves each order of [k], we have the following independent 7 i t = constant . Under the rescaling (2.1) N,N and gij terms that are all scalars under the transformations of are assumed to scale, respectively, as [23], the foliation-preserving diffeomorphisms (2.3)[73], i 2 i N → N,N → b N , gij → gij, (2.6) 6 ij 2 3 ij i j k 2 [k] : KijK ,K ,R ,RRijR ,RjRkRi , (∇R) ,

6 In this review we will measure canonical dimensions of all objects in the unities of spatial momenta k. But, in Hoˇrava gravity the line element ds is not necessarily given 7 In the ADM decomposition, the line element ds is given by ds2 = by this relation [For example, see Eqs.(2.41) and (2.42)]. Instead, µ ν µν i gµν dx dx with the 4-dimensional metrics gµν and g being one can simply consider N,N , gij as the fundamental quan- given by [72], tities that describe the quantum gravitational ﬁeld of Hoˇrava i ! gravity, and their relations to the macroscopic quantities, such 2 i − 1 N −N + NiN Ni µν N2 N2 as ds, will emerge in the IR limit. gµν = , g = i i j . 8 µ µ k Ni gij N gij − N N Note that with the general diﬀeomorphism, x → ξ t, x , the N2 N2 µ (2.5) fundamental quantity is the Riemann tensor R ναβ . 5

i jk i2 i ij (∇iRjk) ∇ R , aia R, aia aiajR , not reduce the total number of coupling constants signif- i3 i 2 i icantly. aia , a ∆ ai, a i ∆R, ..., To further reduce the number of independent coupling 5 ij ijk l ijk l [k] : KijR , Ril∇jRk, aial∇jRk, constants, Hoˇrava introduced two additional conditions, the projectability and detailed balance [23]. The former a a Kij,Kija , ai K, i j ij i requires that the lapse function N be a function of t only, 4 2 ij i2 i 2 i j [k] : R ,RijR , aia , a i , aia a j, ij i ij i N = N(t), (2.17) a aij, aia R, aiajR , Ra i, 3 [k] : 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 ai has dimension of one only. 0 [k] : γ0, (2.13) Thus, to build an operator out of ai to the sixth-order, there will be many independent combinations of ai and where ω3(Γ) denotes the gravitational Chern-Simons ij its derivatives. However, once the condition (2.17) is im- term, γ0 is a dimensionless constant, ∆ ≡ g ∇i∇j, and posed, all such terms vanish identically, and the total number of the sixth-order terms immediately reduces to ai1i2...in ≡ ∇i1 ∇i2 ...∇in ln(N), 2 seven, given exactly by the first seven terms in Eq.(2.13). ω (Γ) ≡ Tr Γ ∧ dΓ + Γ ∧ Γ ∧ Γ So, the totally number of the independently coupling con- 3 3 stants of the theory now reduce to N = 14, even with the eijk 2 = √ Γm∂ Γl + Γn Γl Γm . (2.14) three parity-violated terms, g il j km 3 il jm kn K Rij, ijkR ∇ Rl , ω (Γ). (2.18) ijk ijk √ 123 ij il j k 3 Here g = det (gij) and ≡ e / g with e = 1, 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) is mathematically elegant and appealing. [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]. time derivative terms only contain in Kij, we can see that Inspired by condensed matter systems [77], in addition the kinetic part LK is the linear combination of the sixth to the projectable condition, Hoˇrava also assumed that order derivative terms, the potential part, LV , can be obtained from a superpotential Wg via the relations [23], 1 ij 2 LK = KijK − λK , (2.15) ζ2 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 3 generalized DeWitt metric, defined as ζ2 = [t] · xi + [K]2 = −z − 3 + 2z = z − 3. (2.16) 1 Therefore, for z = 3, it is dimensionless, and the Gijkl = gikgjl + gilgjk − λgijgkl, (2.20) power-counting analysis given between Eqs.(1.1) and 2 (1.2) shows that the theory now becomes power-counting where λ is the same coupling constant, as introduced in renormalizable. The parameter λ is another dimension- Eq.(2.15), and the superpotential Wg is given by less coupling constant, and LI guarantees it to be one even after radiative corrections are taken into account. Z 1 Z √ W = ω (Γ) + d3x g(R − 2Λ), (2.21) But, in Hoˇrava gravity it becomes a running constant g 3 2 Σ κW Σ due to the breaking of LI. The rest of the Lagrangian (also called the potential) where Σ denotes the leaves of t = constant, Λ the cosmo- will be the linear combination of all the rest terms of logical constant, and κW is another coupling constant of Eq.(2.13), from which we can see that, without protec- the theory. Then, the total Lagrangian Lg = LK − LV , tion of further symmetries, the total Lagrangian of the contains only five coupling constants, ζ, λ, w, κw and Λ. gravitational sector is about 100 terms, which is normally Note that the above detailed balance condition has a considered very large and could potentially diminish the couple of remarkable features [41]: First, it is in the same prediction power of the theory. Note that the odd terms spirit of the AdS/CFT correspondence [78–82], where the 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-equilibrium thermodynamics, the counterpart of the 6 superpotential Wg plays the role of entropy, while the Making the coordinate transformation, term Eij the entropic forces [83, 84]. This might shed √ 2mr light on the nature of the gravitational forces [85]. dtPG = dt + dr, (2.24) However, despite of these desired features, this condi- r − 2m tion leads to several problems, including that the Newto- the above metric takes the Painleve-Gullstrand (PG) nian limit does not exist [86], and the six-order derivative form [101], operators are eliminated, so the theory is still not power- r !2 counting renormalizable [23]. In addition, it is not clear if 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), eral diffeomorphism. As a result, a spin-0 mode of gravi- 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) To solve these problems, various modifications have is not allowed by the foliation-preserving diffeomorphism been proposed [34–41]. In the following we shall briefly (2.3), and as a result, they describe two different space- introduce only four of them, as they have been most ex- times in Hoˇrava theory. In particular, metric (2.25) sat- tensively studied in the literature so far. These are the isfies the projectability condition, while metric (2.23) ones: (i) with the projectability condition - the mini- does not. So, in Hoˇrava theory they belong to the two mal theory [23, 91, 92]; (ii) with the projectability condi- completely different branches, with or without the pro- tion and an extra U(1) symmetry [93, 94]; (iii) without jectability condition. Moreover, in GR the metric the projectability condition but including all the possi- 2m 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) Before considering each of these models in detail, some and (2.25), but it does not belong to any of the two comments on singularities in Hoˇrava gravity are in order, branches of Hoˇrava theory, because v = constant hyper- as they will appear in all of these models and shall be surfaces do not define a (3 + 1)-dimensional foliation, a faced when we consider applications of Hoˇrava gravity to fundamental requirement of Hoˇrava gravity. Third, be- black hole physics and cosmology [99]. First, the nature cause of the difference between the two kinds of coor- of singularities of a given spacetime in Hoˇrava theory dinate transformations, the global structure of a given could be quite different from that in GR, which has the spacetime is also different in GR and Hoˇrava grav- general diffeomorphism, ity [102]. For example, the maximal extension of the Schwarzschild solution was achieved when it is written in the Kruskal coordinates [12], δxµ = ξµ t, xk , (µ = 0, 1, 2, 3). (2.22) 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 Z is the Schwarzschild solution given in the Schwarzschild 2 3 √ Sg = ζ dtd xN g (LK − LV ) , (2.28) coordinates,

where LK is given by Eq.(2.15), while the potential part 2m 2m−1 takes the form, ds2 = − 1 − dt2 + 1 − dr2 + r2dΩ2. r r 1 L = 2Λ − R + g R2 + g R Rij (2.23) V ζ2 2 3 ij 7

1 3 ij i j k by properly choosing the coupling constants of the high- + 4 g4R + g5RRijR + g6RjRkRi ζ order operators gn(n = 2, 3, 7, 8). Note that each of these 1 terms is subjected to radiative corrections. It would be + g R∇2R + g (∇ R ) ∇iRjk . (2.29) ζ4 7 8 i jk very interesting to show that the scalar mode is still stable, even after such corrections are taken into account. 2 Here ζ = 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 analysis invalid, one is λ = 1 and the other is λ = 1/3. A front of R was set to −1, which can be realized by rescal- more careful analysis of these two cases shows that the ing the time and space coordinates [92]. As mentioned equations for ψ and B degenerate into elliptic differential above, Hoˇrava referred to this model as the minimal the- equations, so the scalar mode is no longer dynamical. 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 In addition, this minimal theory still suffers the strong 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: if the theory is renormalizable, all coefficients with experimental constraints. of infinite number of nonlinear terms can be written in To study the stability of the theory, let us consider the 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) perturbative expansion, we need to employ nonperturba- N = 1,N = ∂ B, g = e−2ψδ . (2.30) i i ij ij tive methods to analyze the fate of the scalar graviton After integrating out the B field, the action upto the in the limit. Such an analysis was performed in [34] for quadratic terms of ψ takes the form [104], spherically symmetric, static, vacuum configurations and was shown that the limit is continuously connected to Z ˙ 2 ! GR. A similar consideration for cosmology was given in (2) 2 3 ψ 2 4 2 sg = ζ dtdx − 2 − ψ 1 + α1∂ + α2∂ ∂ ψ , [111, 112], where a fully nonlinear analysis of superhori- cψ zon cosmological perturbations was carried out, and was (2.31) shown that the limit λ → 1 is continuous and that GR is 2 ij 2 where ∂ ≡ δ ∂i∂j, cψ ≡ −(λ − 1)/(3 λ − 1), α1 ≡ recovered. This may be considered as an analogue of the 2 2 (8g2 + 3g3)/ζ and α2 ≡ −(8g7 − 3g8)/ζ . Clearly, to Vainshtein effect first found in massive gravity [113]. 2 avoid ghosts we must assume that cψ < 0, that is, With the projectability condition, the Hamiltonian constraint becomes global, from which it was shown that 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 9 Cosmological perturbations in this version of theory in the IR [92, 104] . This can be seen easily from the has been extensively studied [104, 108, 115–120], and are 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 1 − α k2 + α k4 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. In the UV and intermediate regimes, it can be stable B. With Projectability & U(1) Symmetry

9 Stability of the scalar mode with the projectability condition As mentioned above, the problems plagued in Hoˇrava was also considered in [105]. But, it was found that it exists for gravity are closely related to the existence of the spin- all the value of λ. This is because the detailed balance condition 0 graviton. Therefore, if it is eliminated, all the prob- was also imposed in [105]. lems should be cured. This can be done, for example, 8

by imposing extra symmetries, which was precisely what A LA ≡ 2Λg − R , Hoˇrava and Melby-Thompson (HMT) did in [93]. HMT N introduced an extra local U(1) symmetry, so that the h 2 2 2 i Lλ ≡ 1 − λ ∇ ϕ + 2K∇ ϕ , total symmetry of the theory now is enlarged to, 1 G ≡ R − g R + Λ g . (2.38) U(1) n Diff(M, F). (2.34) ij ij 2 ij g ij 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 behavior of the theory can be considerably improved. On Eq.(2.35). So, the most general form of LV is still given the other hand, under the local U(1) symmetry, the fields by Eq.(2.29). transform as Note that the strong coupling problem no longer exists in the gravitational sector, as the spin-0 graviton now i δαA =α ˙ − N ∇iα, δαϕ = −α, is eliminated. However, when coupled with matter, it δαNi = N∇iα, δαgij = 0 = δαN, (2.35) will appear again for processes with energy higher than [125, 127], where α t, xk is the generator of the local U(1) gauge symmetry. Under the Diff(M, F), the auxiliary fields A 3/2 Mpl 5/4 and ϕ transform as, Λω ≡ Mpl |λ − 1| , (2.39) C δA = ζi∇ A + fA˙ + fA,˙ i where M denotes the Planck mass, and generically i pl δϕ = fϕ˙ + ζ ∇iϕ, (2.36) C Mpl. To solve this problem, one way is to intro- i duce a new energy scale M so that M < Λ , as Blas, while N,N and gij still transform as those given by ∗ ∗ ω Eq.(2.8). With this enlarged symmetry, the spin-0 gravi- Pujolas and Sibiryakov first introduced in the nonpro- ton is indeed eliminated [93, 124]. jectable case [128]. This is reminiscent of the case in At the initial, it was believed that the U(1) symmetry string theory where the string scale is introduced just can be realized only when the coupling constant λ takes below the Planck scale, in order to avoid strong coupling its relativistic value λ = 1. This was very encouraging, [42–44]. In the rest of this review, it will be referred because it is the deviation of λ from one that causes to as the BPS mechanism. The main ideas are the fol- all the problems, including ghost, instability and strong lowing: before the strong coupling energy Λω is reached coupling 11. However, this claim was soon challenged, [cf. Fig. 1], the sixth order derivative operators become and shown that the introduction of the Newtonian pre- dominant, so the scaling law of a physical quantity for potential is so strong that action with λ 6= 1 also has the process with E > M∗ will follow Eq.(2.1) instead of the local U(1) symmetry [94]. It is remarkable that the spin-0 relativistic one (z = 1). Then, with such anisotropic graviton is still eliminated even with an arbitrary value of scalings, it can be shown that all the nonrenornalizable λ first by considering linear perturbations in Minkowski terms (with z = 1) now become either strictly renormal- and de Sitter spacetimes [94, 125], and then by analyzing izable or supperrenormalizable [110], whereby the strong the Hamiltonian structure of the theory [126]. coupling problem is resolved. For more details, we refer The general action for the gravitational sector now readers to [127, 128]. takes the form [125], It should be noted that, in order for the mechanism to work, the price to pay is that now λ cannot be ex- Z 2 3 √ actly equal to one, as one can see from Eq.(2.39). In Sg = ζ dtd xN g LK − LV + Lϕ + LA + Lλ , other words, the theory cannot reduce precisely to GR in (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. ij Lϕ ≡ ϕG 2Kij + ∇i∇jϕ , In addition, the BPS mechanism cannot be applied to the minimal theory presented in the last subsection, because the condition M∗ < Λω, together with the one that instability cannot occur within the age of the universe, 10 In the original paper of HMT, the Newtonian pre-potential was 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 reserve ν for other use. 11 Recall that in the relativistic case, λ = 1 is protected by the Minkowski spacetime is stable, so such a fine-tuning does diffeomorphism,x ˜µ =x ˜µ(t, xk), (µ = 0, 1, 2, 3). With this sym- not exist. metry, λ remains this value even after the radiative corrections Static and spherically symmetric spacetimes were con- are taken into account. sidered in [93, 98, 102, 129–133], and solar system tests 9

EE where

N ≡ (1 − a1σ) N, Ni = Ni + N∇iϕ, A − A Λω γ ≡ (1 − a σ)2 g , σ ≡ . (2.42) M* ij 2 ij N Here a1(≡ υ) and a2 are two coupling constants. It can be shown that the eﬀective metric (2.41) is invariant under the enlarged symmetry (2.34). The matter is mini- II mally coupled with respect to the eﬀective metric γµν via the relation, Z Λ 3 √ i ω M* SM = dtd xN γLM N , N , γjk; ψn , (2.43)

where ψn denotes collectively matter fields. With such a coupling, in [98] the authors calculated explicitly all the parameterized post-Newtonian (PPN) parameters in I terms of the coupling constants of the theory, and showed that the theory satisfies the constraints [134] 12, γ = 1 + (2.1 ± 2.3) × 10−5, β = 1 + (−4.1 ± 7.8) × 10−5, (a) (b) −4 −7 −20 α1 < 10 , α2 < 4 × 10 , α3 < 4 × 10 , −3 −3 −2 FIG. 1. The energy scales: (a) Λω < M∗ and (b) Λω > M∗. ξ < 10 , Γ < 1.5 × 10 , ζ1 < 2 × 10 , In Case (a), the theory becomes strong coupling once the en- −5 −8 −3 ζ2 < 4 × 10 , ζ3 < 10 , ζ4 < 6 × 10 , (2.44) ergy of a system reaches the strong coupling energy Λω. In Case (b), the high-order operators suppressed by M∗ become obtained by all the current solar system tests, where important before the system reaches the strong coupling en- 10 2 2 1 ergy Λω, whereby the (relativistic) IR scaling of the system 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], were in turn investigated. In particular, HMT found that GR can be recovered in the IR if the lapse function N γ = β = 1, α1 = α2 = α3 = ξ = 0, of GR is related to the one N of Hoˇrava gravity and the ζ1 = ζ2 = ζ3 = ζ4 = ζB = 0, (2.46) gauge field A via the relation, N = N − A [93]. This can be further justified by the considerations of geometric in- for terpretations of the gauge field A, in which the local U(1) symmetry was found in the first place. By requiring that (a1, a2) = (1, 0). (2.47) 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] found that the lapse function N and the shift vec- A remarkable feature is that the solar system tests im- tor N i of GR should be given by N = N − υ(A − A)/c2 pose no constraint on the parameter λ. As a result, when and N i = N i + 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- i 1 2 A ≡ −ϕ˙ + N ∇iϕ + N(∇iϕ) . (2.40) presses higher dimensional operators in the theory. This 2 is in sharp contrast to other versions of Hoˇrava gravity without the U(1) symmetry. It should be noted that the Then, it was found that the theory is consistent with the solar system tests for both λ = 1 and λ 6= 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 0 matter and the HMT theory via the effective metric γµν , R ρ 0 dv0 3 0 B ≡ |x−x0| (x − x ) · dt d x appearing in the component htt can be always eliminated by the coordinate transformation δt = 2 µ ν 0 k ds ≡ γµν dx dx ξ (t, x ). However, in Hoˇrava gravity, this symmetry is missing, 2 2 2 i i j j and the term ζB B must be included into htt. So, instead of = −c N dt + γij dx + N dt dx + N dt , the ten PPN parameters introduced in [134], here we have an (2.41) additonal one ζB . For more details, we refer readers to [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- and found that, among other things, the FLRW universe dition [95, 96], is necessarily flat. In the sub-horizon regions, the met- k ric and inflaton are tightly coupled and have the same N = N t, x . (2.48) oscillating frequencies. In the super-horizon regions, the perturbations become adiabatic, and the comoving cur- Although again we are facing the problem of a large num- vature perturbation is constant. Both scalar and tensor ber of coupling constants, BPS showed that the spin-0 graviton can be stabilized even in the IR. This is re- perturbations are almost scale-invariant, and the spec- i trum indices are the same as those given in GR, but alized by including the quadratic term aia into the the ratio of the scalar and tensor power spectra depends Lagrangian, on the high-order spatial derivative terms, and can be 6 different from that of GR significantly. Primordial non- i X (n) LV = 2Λ − R − β0aia + LV , (2.49) Gaussianities of scalar and tensor modes were also stud- n=3 ied in [137, 138] 13. (n) Note that gravitational collapse of a spherically sym- where β0 is a dimensionless coupling constant, and LV 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 perturbations can be still described by Eq.(2.33), but now (minimal) assumption that the junctions must be math- with [95, 96] ematically meaningful in terms of distribution theory. 4 Lately, gravitational collapse in this setup was investi- 2 λ − 1 2 2 k ωk = − 1 k + O 2 , (2.50) gated and various solutions were constructed [141–143]. 3λ − 1 β0 M∗ 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 i because of the presence of the aia term, the scalar 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 < β0 < 2. (2.51) balance condition softly breaking, the number of independent coupling constants can be still significantly reduced. It is remarkable to note that the stability requires β0 6= 0 This is particularly the case when we consider Hoˇrava strictly 14. This will lead to significantly difference from gravity without the projectability condition but with the 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 linear perturbations of the Minkowski spacetime [74, 96], explicitly in [73]. In particular, in the healthy extension (n≥3) −2 2 ij i i to be discussed in the next subsection the detailed bal- LV = ζ γ1R + γ2RijR + γ3R∇ia + γ4ai∆a ance condition cannot be imposed even allowing it to be −4 h 2 2 i + ζ γ5 (∇iR) + γ6 (∇iRjk) + γ7 (∆R) ∇ia broken softly. Otherwise, it can be shown that the sixth- i 2 order operators are eliminated by this condition, and the + γ8a ∆ ai , (2.52) 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 that this version of Hoˇrava gravity has one-to-one corre- It is interesting to note that in the case λ = 1/3 two extra second- class constraints appear [146, 147]. As a result, in this case the spondence to dynamical Newton-Carton geometry with- spin-0 graviton is eliminated even when β0 = 0. However, since out torsion, and a precise dictionary was built [145]. β0 in general is subjected to radiative corrections, it is not clear which symmetry preserves this particular value. In [23], Hoˇrava 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 13 It should be noted that such studies were carried out when power-countering, as the sixth-order operators are eliminated by 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, so the theory is not consistent with observations. For fur- preliminary study indicates that a more general coupling might ther discussions of the running of the coupling constants in terms be needed [139]. of RG flow, see [103]. 11

where γn’s are all dimensionless coupling constants. carried out before the realization of the importance of i As first noticed by BPS, the most stringent constraints the term aia that could play in the IR stability [95, 96], come from the preferred frame effects due to Lorentz vi- so they all unfortunately belong to the branch of the olation, which require [96, 134], non-projectable Hoˇrava gravity that is plagued with the 16 instability problem . Lately, the case with β0 6= 0 were −7 15 |λ − 1| . 4 × 10 ,M∗ . 10 GeV. (2.53) studied in [167–170]. In particular, it was claimed that slowly rotating black holes do not exist in this extension In addition, the timing of active galactic nuclei [148] and [169]. It was shown that this is incorrect [171, 172], and gamma ray bursts [149] require slowly rotating black holes and stars indeed exist in all the four versions of Hoˇrava gravity introduced in this re- M 1010 ∼ 1011 GeV. (2.54) ∗ & view [173] 17. To obtain the constraint (2.54), BPS used the results Cosmological perturbations were studied in [174–176] from the Einstein-aether theory, as these two theories and found that they are consistent with observations. coincide in the IR [150, 151]. Cosmological constraints of Lorentz violation from dark Limits from binary pulsars were also studied recently, matter and dark energy were also investigated in [177– and the most stringent constraints of the theory were 180]. obtained [152, 153]. However, when the allowed range of In addition, the odd terms given in Eq.(2.13) violate the preferred frame effects given by the solar system tests the parity, which can polarize primordial gravitational is saturated, the limit for λ is still given by Eq.(2.53), so waves and lead to direct observations [181, 182]. In particular, it was shown that, because of both parity vio- the upper bound M∗ remains the same. It is also interesting to note that observations of syn- lation and the nonadiabatic evolution of the modes due chrotron radiation from the Crab Nebula seems to require to a modified dispersion relationship, a large polarization that the scale of Lorentz violation in the matter sector of primordial gravitational waves becomes possible, and could be within the range of detection of the BB, TB and must be Mpl, not M∗ [154]. In addition, the consistency of the theory with the current observations of gravita- EB power spectra of the forthcoming CMB observations tional waves from the events GW150914 and GW151226 [182]. Of course, this conclusion is not restricted to this were studied recently, and moderate constraints were ob- version of Hoˇrava gravity, and in principle it is true in all tained [155]. of these four versions presented in this review. Since the spin-0 graviton generically appears in this version of Hoˇrava gravity [156, 157], strong coupling problem is inevistable [158, 159]. However, as mentioned D. With Non-projectability & U(1) Symmetry in the last subsection, this can be solved by introducing an energy scale M∗ that suppresses the high-order opera- A non-trivial generalization of the enlarged symmetry tors [128]. When the energy of the system is higher than (2.34) to the nonprojectable case N = N(t, x) was real- M∗, these high-order operators become dominant, and ized in [73, 97, 98], and has been recently embedded into the scaling law will be changed from z = 1 to with z = d string theory by using the non-relativistic AdS/CFT cor- given by Eq.(2.1). With such anisotropic scalings, all the respondence [183, 184]. In addition, it was also found nonrenornalizable terms (in the case with z = 1) now that it corresponds to the dynamical Newton-Cartan become either strictly renormalizable or supperrenormal- geometry with twistless torsion (hypersurface orthogo- izable, whereby the strong coupling problem is resolved. nal foliation) [145]. A precise dictionary was then con- For more details, we refer readers to [127, 128] 15. Note structed, which connect all fields, their transformations that in this case the strong coupling energy is given by and other properties of the two corresponding theories. [96, 128], Further, it was shown that the U(1) symmetry comes from the Bargmann extension of the local Galilean al- 1/2 Λω = |λ − 1| Mpl, (2.55) gebra that acts on the tangent space to the torsional Newton-Cartan geometries. instead of that given by Eq.(2.39) for the HMT extension The realization of the enlarged symmetry (2.34) in the with a local U(1) symmetry. nonprojectable case can be carried out by first noting With the non-projectability condition, static spacetimes with β0 = 0 have been extensively studied, see for example, [86, 160–166]. Sine all of these works were 16 In [39] 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. 17 It should be noted that “black holes” here are defined by the 15 The mechanism requires the coupling constants of the six-order existence of Killing/sound horizons. But, particles can have ve- derivative terms, represented by γn (√n = 5, 6, 7, 8) in the action locities higher than that of light, once the Lorentz symmetry is 7 (2.52), must be very large γn ' 1/ λ − 1 ≥ 10 . This may broken [25]. Then, Killing/sound horizons are no longer barriers introduce a new hierarchy [74], although it was argued that this to these particles, so such defined “black holes” are not really is technically natural [96, 128]. black to such particles. 12 the fact that Ni and σ defined in Eq.(2.42) are gauge- [74, 97]. To reduce the number, one way is to generalize invariant under the local U(1) transformations and are a the Hoˇrava detailed balance condition (2.19) to [97], vector and scalar under Diff(M,F), respectively. In addi- ijkl tion, they have dimensions two and four, respectively, G 0 Ekl L(V,D) = Eij Ai ij , (2.63) 0 g Aj δαNi = 0 = δασ, [Ni] = 2, [σ] = 4. (2.56) where Eij and Ai are given by Then, the quantity K˜ij defined by 1 δW 1 δW 1 Eij ≡ g ,Ai ≡ a . (2.64) K˜ij ≡ (−g˙ij + ∇iNj + ∇jNi) √ √ 2N g δgij g δai = K + ∇ ∇ ϕ + a ∇ ϕ, (2.57) ij i j (i j) 18 The super-potentials Wg and Wa are constructed as , is also gauge-invariant under the U(1) transformations. ˜ 1 Z Z √ In addition, Kij has the same dimension as Kij, i.e., W = ω (Γ) + µ d3x g R − 2Λ , g w2 3 [K˜ij] = 3, from which one can see that the quantity L˜K , Σ Z √ 1 ij 2 3 X i n L˜K ≡ K˜ K˜ij − λK˜ , (2.58) Wa = d x g Bna ∆ ai, (2.65) n=0 has dimension 6. In addition, the quantity where Bn are coupling constants. However, to have a healthy infrared limit, as mentioned previously, the de- L˜S ≡ σ (ZA + 2Λg − R) . (2.59) tailed balance condition needs to be broken softly, by has also dimension 6 and is gauge-invariant under the adding all the low dimensional operators, so that the po- i i U(1) transformations, where ZA ≡ σ1a ai + σ2a i, with tential finally takes the form [97], σ and σ being two dimensionless coupling constants. 1 2 1 Then, the action L = 2Λ − β a ai − γ R + γ R2 + γ R Rij V 0 i 1 ζ2 2 3 ij Z 2 3 √ ˜ " Sg = ζ dtd xN g LK − γ1R − 2Λ 1 i2 i 2 i j + 2 β1 aia + β2 a i + β3 aia a j i ζ +βa ai + σ (ZA + 2Λg − R) − Lz>1 , (2.60) # ij i ij i is gauge-invariant under the enlarged symmetry (2.34), +β4a aij + β5 aia R + β6aiajR + β7Ra i where Lz>1 denotes the potential part that contains spatial derivative operators higher than second-orders. In- " # 1 2 serting Eqs.(2.57)-(2.59) into the above action, and then ij i + 4 γ5CijC + β8 ∆a , (2.66) after integrating it partially, the total action takes the ζ form [98], where all the coefficients, βn and γn, are dimension- Z 2 3 √ less and arbitrary, except for the ones of the sixth-order Sg = ζ dtd x gN LK − LV + LA + Lϕ + LS , derivative terms, γ5 and β8, which are positive, (2.61) γ5 > 0, β8 > 0, (2.67) where LK and LA are given, respectively, by Eqs.(2.15) and (2.38), but now with as can be seen from Eqs.(2.63)-(2.65). The Cotton tensor Cij is defined as ij Lϕ ≡ ϕG 2Kij + ∇i∇jϕ + ai∇jϕ ijk h i 2 i i ij e j 1 j +(1 − λ) ∆ϕ + ai∇ ϕ + 2 ∆ϕ + ai∇ ϕ K C = √ ∇ R − Rδ . (2.68) g k l 4 l 1 h + Gîjlk 4 (∇ ∇ ϕ) a ∇ ϕ 3 i j (k l) + 5 a(i∇j)ϕ a(k∇l)ϕ + 2 ∇(iϕ aj)(k∇l)ϕ i 18 Note that in [185] a different generalization was proposed, + 6Kija(l∇k)ϕ , in which L(V,D) takes the same form in terms of Eij and ijkl i i G , as that given by Eq.(2.19), but with the superpo- i 1 R LS ≡ σ(σ1a ai + σ2a i), (2.62) tential including a term aia , i.e., Wg = ω3(Γ) + √ w2 Σ R d3x g µ(R − 2Λ) + βa ai. However, in this generalization, where Gîjlk ≡ gilgjk − gijgkl, and L denotes the po- i V the six-order derivative term ∆ai2 does not exist in the potential part of the theory made of a , ∇ and R only. i i ij tential LV , and is a particular case of Eq.(2.66) with β8 = 0. However, as mentioned previously, once the projectabil- Without this term, the sixth-order derivative terms are absent ity condition is abandoned, it gives rise to a prolifera- for scalar perturbations. As a result, the corresponding theory tion of a large number of independent coupling constants is not power-counting renormalizable [97]. 13

In terms of Rij and R, we have [97], the high-order spatial derivative operators of the theory. Thus, the next leading-order contributions come ij 1 3 5 ij i j k 3 CijC = R − RRijR + 3RjRkRi + R∆R from the 3-dimensional gravitational Chern-Simons term. 2 2 8 With some reasonable arguments, it is shown that this i jk k + (∇iRjk) ∇ R + ∇kG , (2.69) 3-dimensional operator is the only one that violates the parity and in the meantime has non-vanishing contribu- where 19 tions to non-Gaussianities. k 1 jk j ik 3 k Finally we note that, instead of introducing the U(1) G ≡ R ∇jR − Rij∇ R − R∇ R. (2.70) 2 8 gauge field A and the pre-Newtonian potential ϕ, the au- When σ1 = σ2 = 0, it was shown that the spin-0 gravi- thors in [190] introduced two Lagrange multipliers ACKO ton is eliminated [73, 97, 98]. This was further confirmed and ΛCKO (Don’t confuse with the U(1) gauge field A by Hamiltonian analysis [186]. For the universal coupling and the cosmological constant Λ introduced above), and with matter given by Eqs.(2.41)-(2.43), all the PPN pa- found that in the non-projectable case the spin-0 gravi- rameters were calculated and given explicitly in terms of ton can be eliminated for certain choices of the coupling the coupling constants [98, 187]. In particular, it was constants of the theory, while for other choices the spin-0 found that with the choice of the parameters a1 and a2 graviton is still present. However, it is not clear which given by Eq.(2.47), the relativistic results (2.46) are also symmetry, if there is one, will preserve these particular obtained. choices, so that the spin-0 graviton is always absent. Oth- When σ1σ2 6= 0 the spin-0 graviton appears [98, 186]. erwise, it will appear generically once radiative correc- It was shown that the theory is ghost-free for the choice tions are taken into account. It is interesting to note that, of λ given by Eq.(2.32), and the spin-0 graviton is stable in contrast to the non-projectable case, in the projectable in both of the IR and UV, provided that the following case the spin-0 graviton is always eliminated [190]. holds [98], In addition, Hoˇrava gravity with mix spatial and time derivatives has been also studied recently in order to − + β8 > 0, σ2 < σ2 < σ2 , (2.71) avoid unacceptable violations of Lorentz invariance in the matter sector [191, 192]. Unfortunately, it was found ± p 2 2 where σ2 ≡ 4 −γ1 ± γ1 − β0/2 and γ1 − β0/2 ≥ 0. that the theory contains four propagating degrees of free- In this case, the analysis of the PPN parameters shows dom, as opposed to three in the standard Hoˇrava gravity, that the corresponding theory is consistent with all the and the new degree of freedom is another scalar graviton, solar system tests in a large region of the phase space which is unstable at low energies [193]. [98, 187]. In particular, when the two coupling constants σ2 and β0 satisfy the relations [98] E. Covariantization of Hoˇrava Gravity σ2 = 4(1 − a1), β0 = −2(γ1 + 1), (2.72) the relativistic values of the PPN parameters given by As GR is general covariant and consistent with all the Eq.(2.46) can be still achieved. observations, both cosmological and astrophysical, it is The strong coupling problem appearing in the gravita- desired to write Hoˇrava Gravity also in a general covari- tional and/or matter sectors can be resolved [73], by the ant form 20, in order to see more clearly the difference BPS mechanism introduced previously [128]. between the two theories. One may take one step further The consistency of the theory with cosmology was and consider such a process as a mechanism to obtain studied in [188] when matter is minimally coupled to the the IR limit of the UV complete theory of Hoˇrava grav- theory. For the universal coupling of Eqs.(2.41) -(2.43), ity with the symmetry of general covariance. such studies have not been worked out, yet [139]. To restore the full general covariance, one way is to The effects of parity violation on non-Gaussianities of make the foliation dynamical by using the Stückelberg primordial gravitational waves were also studied recently trick [194], in which the space-like hypersurfaces of t = [189]. By calculating the three point function, it was constant in the (3+1) ADM decompositions are replaced found that the leading-order contributions to the non- by a scalar field φ = constant, which is always time-like Gaussianities come from the usual second-order deriva- [88, 195], tive terms, which produce the same bispectrum as that found in GR. The contributions from high-order spatial µ φ,µ µν uµu = −1, uµ ≡ √ ,X ≡ −γ φ,µφ,ν > 0. (2.73) n-th derivative terms are always suppressed by a fac- X n−2 tor (H/M∗) (n ≥ 3), where H denotes the inflationary energy and M∗ the suppression mass scale of

20 This process in general introduces new physics, for example, the appearance of the instantaneous mode in the khronometric the- 19 Note that, from Gauss’s theorem, the integral of ∇ Gk yields, √ √ k ory, which is absent in both Hoˇrava and Einstein-aether theories, R dtdx3N g∇ Gk = − R dtdx3N ga Gk + R dtGkdS . k k Σt k as shown below. 14

Clearly, choosing φ = t, the original foliations are ob- ai → aλ, tained. This gauge choice is often referred to as the uni- ν µ ∂⊥ϕ → u hν Dµϕ, tary gauge [88]. Then, the gauge symmetry t → ξ (t) 0 R → R , translates into the symmetry of the Stückelberg field, ij µν Kij → Kµν , ˜ φ → φ = f(φ), (2.74) Cij → Cµν , (2.82) where f(φ) is an arbitrary monotonic function of φ only, where so that 1 i ˜ ∂⊥ ≡ −∂t + N ∇i , φ,µ f φ,µ N u˜µ ≡ = √ = f uµ, (2.75) p ˜ 1 X X Cµν ≡ ηµαβ∇ Rν − δν R , (2.83) α β 4 β where f ≡ Sign[df(φ)/dφ]. Once uµ is introduced, all µαβ other geometric quantities can be constructed from γµν where η is the three-dimensional volume element de- µαβ µαβδ µαβδ and uµ by following Israel’s method for space-like hyper- fined as η = η uδ, and η denotes the four- µνβ surfaces [196, 197]. For example, the extrinsic curvature dimensional volume element with ∇αη = 0 [195]. Kµν is given by, In the case with an extra U(1) symmetry, we also replace the gauge field A and the pre-Newton potential ϕ, λ 1 α β Kµν = hλµD uν = √ h h DαDβφ, (2.76) respectively, by [198], X µ ν 1 1 λ ν µ 2 where D denotes the covariant derivative with respect A → √ σ + u h Dµϕ + (∇λϕ) , X ν 2 to γµν , and hµν is the projection operator, defined as ϕ → ϕ, (2.84) hµν ≡ γµν + uµuν . (2.77) where σ defined by Eq.(2.42) and ϕ are scalars un-

Defining the compatible covariant derivative ∇µ on the der both the U(1) and the Diff(M, F) transformations. hypersurfaces φ = constant as, Therefore, the resulting theories from the ones with the local U(1) symmetry will contain two Lagrangian multi- ∇ T α1...αk ≡ hα1 ...hαk hσ1 ...hσl hν D T δ1...δk , (2.78) pliers σ and ϕ, each of them acts like a scalar field, but µ β1...βl δ1 δk β1 βl µ ν σ1...σl none of them is dynamical, in contrast to the Stückelberg we find that field φ, introduced above. This is similar to what was γ done in [190], and it would be very interesting to find R = hαhβh hδ (4)R + K K − K K , µνλρ µ ν λ ρ αβγδ νλ µρ νρ µλ out if they are related one to the other. α β γ ∇µRνλ = hµhν hλDαRβγ , (2.79) Applying the above Stückelberg trick to the healthy extension [95, 96], it can be shown that the resulting ac- (4) where Rαβγδ denotes the 4D Riemann tensor made tion in the IR is identical to the hypersurface-orthogonal out of the metric γµν , and Rµνλρ the intrinsic Riemann Einstein-aether theory [199, 200], as shown explicitly in tensor with R ≡ Rλ . Thus, we find that µν µλν [150, 151], provided that the aether four-vector uµ satis- α β (4) α β (4) fies the hypersurface-orthogonal conditions, Rµν = hµhν Rαβ + u u Rαµβν λ u D u = 0, (2.85) + KµλK ν − KKµν , [α β λ] R ≡ hµν R = (4)R + K2 − K Kµν + 2D Gα, µν µν α which are necessary and sufficient conditions for uµ to be α λ α α λ G ≡ u Dλu − u Dλu . (2.80) given by Eq.(2.73)[201]. From Eq.(2.85) it can be shown that Note that in writing the above expressions, we had used 2 µ α β β α the relation [196, 197], ω ≡ a aµ + Dαuβ D u − Dαuβ D u , (2.86)

(4)R uαuβ = K2 − K Kαβ + D Gα, (2.81) λ αβ αβ α vanishes identically, where aµ ≡ u Dλuµ. Then, one can λ add the term, where K ≡ Kλ. Then, to obtain the relativistic version of Hoˇrava gravity, one can simply identify the quantities Z 4√ 2 appearing in Hoˇrava gravity in the ADM decompositions ∆Sæ ≡ c0 dx −γ ω , (2.87) with those obtained in the unitary gauge, which leads to the following replacements in the action, where c0 is an arbitrary constant and γ ≡ det(γµν ), into 1 the action of the Einstein-aether theory [199, 200], N → √ , Z X 4 √ h 2 µ 2 Sæ = d x −γ c1 (Dµuν ) + c2 (Dµu ) ∇i → ∇λ, 15

µ ν µ i + c3 (D u )(Dν uµ) − c4a aµ , (2.88) Future t to eliminate one of the four independent coupling con- Future stants ci (i = 1, 2, 3, 4) of the Einstein-aether theory, so p p that there are only three independent coupling constants in terms of the Stückelberg field φ, which will be referred Simultaneous to as the khronon field, and the corresponding theory the Past the khronometric theory [96, 202], which was also referred Past to as the “T-theory” in [150, 151]. (a) (b) Some comments are in order. First, the action of Eq.(2.88) is the most general action of the Einstein- FIG. 2. (a) The light cone of the event p in special relativity. aether theory in which the field equations for γµν and (b) The causal structure of the point p in theories with broken uµ are the second-order differential equations [199, 200]. LI. This figure is adopted from [102]. In this theory, in addition to the spin-2 graviton encountered in GR, there are two extra modes, the spin-1 and spin-0 gravitons, each of them moves with a different ve- and firewalls [208, 209]. ˇ locity vs [199, 200]. To avoid the gravitational Cerenkov Lately, such studies have gained further momenta in effects [203], each of them must be not less than the speed the framework of gravitational theories with broken LI, of light, vs ≥ c. Second, the khronometric theory is including Hoˇrava gravity. In such theories, due to the fundamentally different from the Einstein-aether theory breaking of LI, the dispersion relation of a massive parti- even in the domain where the hypersurface-orthogonal cle contains generically high-order momentum terms [34– conditions (2.85) hold. This can be seen clearly by the 41], fact that there is an additional mode, the instantaneous propagation of signals, in the khronometric theory [202],  2(z−1) n X p which is absent in the Einstein-aether gravity. This is E2 = m2 + c2p2 1 + a , (3.1) p  n M  closely related to the fact that in terms of the khronon n=1 ∗ field φ, the action (2.88) are in general fourth-order differential equations. It is the existence of this mode that from which we can see that both of the group and phase may provide a mechanism to restore the second law of velocities, vg ≡ E/p and vp ≡ dE/dp, become unbounded thermodynamics [202]. It is also interesting to note that as p → ∞, where E and p are the energy and momen- this instantaneous mode does not exist in the healthy tum of the particle considered, and cp and an’s are coeffi- extension either [95, 96]. It is introduced by the covari- cients, depending on the species of the particle, while M∗ antization process with the use of the Stückelberg trick. is the suppression energy scale of the higher-dimensional We also note that in [88, 96], the covariantization of the operators. As an immediate result, the causal structure projectable Hoˇrava gravity was also studied, and showed of the spacetimes in such theories is quite different from that this minimal theory is always fine-tuning due to the that given in GR, where the light cone at a given point p instability of the spin-0 graviton in this version of Hoˇrava plays a fundamental role in determining the causal rela- gravity. tionship of p to other events [cf. Fig.2]. However, once LI is broken, the causal structure will be dramatically changed. For example, in the Newtonian theory, time is III. BLACK HOLES & THERMODYNAMICS absolute and the speeds of signals are not limited. Then, the causal structure of a given point p is uniquely deter- Black holes have been objects of intense study both mined by the time difference, ∆t ≡ tp − tq, between the theoretically and observationally over half a century now two events. In particular, if ∆t > 0, the event q is to the [9, 10], and so far there are many solid observational evi- past of p; if ∆t < 0, it is to the future; and if ∆t = 0, the dences for their existence in our universe, especially after two events are simultaneous. In Hoˇrava gravity, a similar the first detection of gravitational waves from two binary situation occurs. This immediately makes the definitions black holes on September 14, 2015 by aLIGO [11]. The- of black holes given in GR [12] invalid. oretically, such investigations have been playing a funda- To provide a proper definition of black holes, mental role in the understanding of the nature of grav- anisotropic conformal boundaries [210] and kinematics of ity in general, and QG in particular. They started with particles [102, 167, 211–214] have been studied within the the discovery of the laws of black hole mechanics [204] framework of Hoˇrava gravity. Lately, a potential break- and Hawking radiation [205], which led to the profound through was the discovery that there still exist absolute recognition of the thermodynamic interpretation of the causal boundaries, the so-called universal horizons, in four laws [206] and the reconstruction of GR as the ther- theories with broken LI [202, 215]. Particles even with in- modynamic limit of a more fundamental theory of gravity finitely large velocities would just move around on these [207]. More recently, they play the essential role in the boundaries and cannot escape to infinity. The main idea understanding of the AdS/CFT correspondence [78–82] is as follows. In a given spacetime, a globally timelike 16

scalar field φ might exist [216] 21. Then, similar to the t Newtonian theory, this field defines a global absolute time, and all particles are assumed to move along the increasing direction of the timelike scalar field, so the causality is well defined, similar to the Newtonian case µ [Cf. Fig.2]. In such a spacetime, there may exist a sur- ζ face as shown in Fig.3, denoted by the vertical solid line, located at r = rUH . Given that all particles move along the increasing direction of the timelike scalar field, from d φ Fig.3 it is clear that a particle must cross this surface and move inward, once it arrives at it, no matter how large of its velocity is. This is an one-way membrane, φ = φ0 and particles even with infinitely large speed cannot escape from it, once they are inside it. So, it acts as an Black Hole absolute horizon to all particles (with any speed), which 0 r r is often called the universal horizon [202, 215, 216]. At UH KH r the universal horizon, we have dt·dφ = 0, or equivalently,

ζ · u = 0, (3.2) FIG. 3. Illustration of the bending of the φ = constant surfaces, and the existence of the universal horizon in the Schwarzschild spacetime [216], where φ denotes the globally where ζ(≡ ∂t) denotes the asymptotically timelike Killing λ timelike scalar field, and t and r are the Painlevé-Gullstrand vector, and u(≡ uλdx ) is defined in terms of the globally timelike scalar field φ via Eq.(2.73). It is because coordinates. Particles move always along the increasing direction of φ. The Killing vector ζµ = δµ always points upward at of this that the globally timelike scalar field φ is also t each point of the plane. The vertical dashed line is the loca- called the khronon field. But, there is a fundamental dif- tion of the Killing horizon, r = rKH . The universal horizon, ference between the scalar field introduced here and the denoted by the vertical solid line, is located at r = rUH , which one introduced in Section II.E. In particular, the one in- is always inside the Killing horizon. This figure is adopted troduced in Section II.E is part of the gravitational field from [217]. and represents the extra degrees of the gravitational sector, while the one introduced here is a test field, which plays the same role as a Killing vector field does, and no 1 Dβuα = θhαβ + σαβ + ωαβ − aαuβ, (3.4) effects on the spacetime, but simply describes the prop- 3

erties of such a given spacetime. So, whether such a field with c13 ≡ c1 + c3, c14 ≡ c1 + c4, and exists or not only indicates whether a globally timelike λ scalar φ can be defined or not in such a given spacetime. θ ≡ Dλu , hαβ = γαβ + uαuβ, A Killing vector in a given spacetime background sat- 1 σαβ ≡ D(βuα) + a(αuβ) − θhαβ, isfies the Killing equations, D(αζβ) = 0. To define the 3 globally timelike scalar field φ, one way is to adopt the ωαβ ≡ D[βuα] + a[αuβ], (3.5) equations for the khronon field given by the general action (2.88), which can be written in the form [151] 22, where (A, B) ≡ (AB +BA)/2. When uµ is hypersurface- orthogonal, we have ω2 = 0, so the last term in the above " # Z √ 1 action vanishes identically, as mentioned in Section II.E. S = d4x −γ c θ2 + c σ2 − c a2 + c ω2 , (3.3) φ 3 θ σ a ω Even so, the equation for φ still contains three free parameters, (cθ, cσ, ca), and their physical interpretations where are not clear. Hence, the variation of Sφ with respect to φ yields the cθ ≡ c13 + 3c2, cσ ≡ c13, khronon equation,

cω ≡ c1 − c3, ca ≡ c14, µ DµA = 0, (3.6) where [171] 23, (δµ + uµu ) 21 In [202, 215] the globally timelike scalar field is identified to the Aµ ≡ ν √ ν Æν , hypersurface-orthogonal aether field uµ, in which the aether is X part of the gravitational field, the existence of which violates Æν ≡ D J γν + c a Dν uγ , LI. However, to apply such a concept to other theories (with- γ 4 γ out an aether field), a generalization is needed. In [216], the hypersurface-orthogonal aether field was promoted to a field that plays the same role as a Killing vector does in GR. 22 The quantity σ introduced here must not be confused with one 23 Notice the difference between the signatures of the metric chosen introduced in Eq.(2.42). in this paper and the ones in [171]. 17

α αβ α β α β J µ ≡ c1g gµν + c2δµ δν + c3δν δµ three well-known black hole solutions: Schwarzschild, Schwarzschild anti-de Sitter, and Reissner-Nordström − c uαuβg D uν . (3.7) 4 µν β [238], which are also solutions of Hoˇrava gravity [102]. To solve the above equations for φ, one can divide it In addition, universal horizons exist not only in the into two steps: First solve Eq.(3.6) in terms of uµ. Once low energy limit but also in the UV regime [217]. It is in- uµ is known, one can solve φ from the definition, teresting to note that in [239], the effects of higher-order derivative terms on the existence of universal horizons q φ = −γαβφ φ u . (3.8) were studied, and found that, if a three-Ricci curvature ,µ ,α ,β µ squared term is joined in the ultraviolet modification, the universal horizon appearing in the low energy limit was Eq.(3.6) is a second-order differential equation for u , µ turned into a spacelike singularity. This is possible, as and to uniquely determine it, two boundary conditions the universal horizons might not be stable against non- are needed. These two conditions can be chosen as fol- linear perturbations [202]. lows [202]: (i) It will be aligned asymptotically with the timelike Killing vector, At the universal horizon, the first law of black hole mechanics exists for the neutral Einstein-aether black uµ ∝ ζµ. (3.9) holes [240], provided that the surface gravity is defined by [241], (ii) The second condition can be that the khronon has a 1 α λ regular future sound horizon, which is a null surface of κUH ≡ u Dα uλζ , (3.11) the effective metric, 2

(φ) 2 which was obtained by considering the peeling behavior γµν = γµν − cφ − 1 uµuν , (3.10) of ray trajectories of constant khronon field φ. However, for the charged Einstein-aether black holes, such a first where c (≡ (c + c )/c ) denotes the speed of the φ 13 2 14 law is still absent [242]. khronon 24. Using the tunneling method, Hawking radiation at the It is interesting to note that when ca = c14 = 0, the khronon has an infinitely large velocity, and its sound universal horizon for a scalar field that violates the lo- horizon now coincides with the universal horizon. Then, cal LI was studied, and found that the universal horizon the second condition becomes to require that the univer- radiates as a blackbody at a fixed temperature [243]. A sal horizon be regular. This is an interesting choice, and different approach was taken in [241], in which ray tra- it also reduces the free parameters in the action (3.3) jectories in such black hole backgrounds were studied, from three, (c , c , c ), to two, (c , c ). Out of these two and evidence was found, which shows that Hawking ra- θ σ a θ σ diation is associated with the universal horizon, while the only the ratio cθ/cσ appears in the globally timelike field equation (3.6). “lingering” of low-energy ray trajectories near the Killing horizon hints a reprocessing there. However, the study In addition, since uµ is always timelike, Eq.(3.2) is possible only inside a Killing horizon, in which ζ becomes of a collapsing null shell showed that the mode passing spacelike. That is, the radius of a universal horizon is across the shell is adiabatic at late time [244]. This im- always smaller than that of a Killing horizon, which is plies that large black holes emit a thermal flux with a physically expected, as the universal horizon represents temperature fixed by the surface gravity of the Killing the one-way membrane for particles with any large ve- horizon. This, in turn, suggests that the universal hori- locities, including the infinitely large one. zon should play no role in the thermodynamic properties Since they were first discovered in [202, 215], uni- of these black holes, although it must be noted that in versal horizons have been studied intensively, and al- such a setting, the khronon field is not continuous across ready attracted lots of attention. These include uni- the collapsing null shell. As mentioned above, a globally- versal horizons in static spacetimes [123, 218–234], and defined khronon plays an essential role in the causality of the ones with rotation [217, 235, 236] 25. In particu- the theory, so it is not clear how the results presented in lar, it was showed that universal horizons exist in the [244] will be changed once the continuity of the khronon field is imposed. On the other hand, using the Hamilton-Jacobi method, quantum tunneling of both relativistic and non- 24 To avoid the Cerenkovˇ effects, in the khronometric theory, or relativistic particles at Killing as well as universal hori- more general, in the Einstein-aether theory, cφ is required to be zons of Einstein-Maxwell-aether black holes were studied no less than the speed of light cφ ≥ c [203]. However, since in [245], after higher-order curvature corrections are taken the current case, the khronon is treated as a probe field, such into account. It was found that only relativistic particles requirement is not needed. are created at the Killing horizon, and the corresponding 25 When in the process of submitting this review for publication, an interesting paper on Smarr Formula for Lorentz-breaking grav- radiation is thermal with a temperature exactly the same ity [237] appears. The formula, derived by using the Noether as that found in GR. In contrary, only non-relativistic charge analysis, is applicable not only to static cases but also particles are created at the universal horizon and are ra- with rotations. diated out to infinity with a thermal spectrum. However, 18

different species of particles, in general, experience differ- In addition, gravitational collapse of a scalar field in ent temperatures, the Einstein-aether theory was studied in [252], prior to the discovery of universal horizons. Lately, it was revis- z≥2 2(z − 1) κUH TUH = , (3.12) ited [253]. However, due to the specially slicing of the z 2π spacetime carried out in [252], the numerical simulations where κUH is the surface gravity calculated from cannot penetrate inside universal horizons, so it is not Eq.(3.11) and z is the exponent of the dominant term clear whether or not universal horizons have been formed in the UV [cf. Eq.(3.1)]. When z = 2 we have the stan- in such simulations. dard result, In any case, the universal horizons defined by Eq.(3.2) κUH are in terms of the timelike Killing vector, which exists T z=2 = , (3.13) UH 2π only in stationary spacetimes. To study the dynam- which was first obtained in [241, 243]. ical formation of universal horizons from gravitational Recently, more careful studies of ray trajectories collapse, a generalization of Eq.(3.2) to non-stationary showed that the surface gravity for particles with a non- spacetimes is needed. One way, as first proposed in [254], relativistic dispersion relation (3.1) is given by [246], is to replace the timelike Killing vector by a Kodama-like vector [255], which reduces to the timelike Killing vector 2(z − 1) κz≥2 = κ , (3.14) in the stationary limit. To be more specific, let us con- UH z UH sider spacetimes described by the metric, so that Eq.(3.11) is true only for particles with z = 2. ds2 = g dxadxb + R2 x0, x1 dΣ2 , (a, b = 0, 1), (3.16) The same results were also obtained in [247]. It is re- ab k z≥2 z≥2 markable to note that in terms of κUH and TUH , the in the coordinates, xµ = x0, x1, θ, $ , (µ = 0, 1, 2, 3), standard relationship between the temperature and sur- where k = 0, ±1, and face gravity of a black hole still holds here,  2 z≥2 dθ2 + sin θd$2, k = 1, z≥2 κ  T = UH . (3.15) 2 2 2 UH 2π dΣk = dθ + d$ , k = 0, (3.17)  2 2 2 Is this a coincidence? Without a deeper understanding dθ + sinh θd$ , k = -1. of thermodynamics of universal horizons, it is difficult to The normal vector n to the hypersurface R = C is say. But, whenever case like this raises, it is worthwhile µ 0 given by, of paying some special attention on it. In particular, does entropy of a universal horizon also depend on the ∂(R − C0) 0 1 dispersion relations of particles? nµ ≡ = δ R0 + δ R1, (3.18) ∂xµ µ µ To understand the problem better, another important a issue is: Can universal horizons be formed from grav- where C0 is a constant and Ra ≡ ∂R/∂x . Setting itational collapse? so they can naturally exist in our µ µ µ universe [140]. To answer this question, the collapse of ζ ≡ δ0 R1 − δ1 R0, (3.19) a spherically symmetric thin-shell in a flat background µ that finally forms a Schwarzschild black hole was studied we can see that ζ is always orthogonal to nµ, ζ · [248], in which the globally timelike scalar field is taken n = 0. The vector ζµ is often called Kodama vector as the Cuscuton field [249], a scalar field with infinitely and plays important roles in black hole thermodaynam- large sound speed. It was shown that an observer inside ics [256, 257]. For spacetimes that are asymptotically flat the universal horizon of the Schwarzschild radius cannot there always exists a region, say, R > R∞, in which nµ send a signal outside, after a stage in collapse, even us- and ζµ are, respectively, space- and time-like, that is, ing signals that propagate infinitely fast in the preferred (n · n)| > 0, (ζ · ζ)| < 0. (3.20) frame. Then, it was argued that this universal horizon R>R∞ R>R∞ should be considered as the future boundary of the classical space-time. Lately, such studies were generalized to An apparent horizon may form at RAH , at which nµ the Reissner-Nordstrom and Kerr black holes [250] 26. becomes null, (n · n)| = 0, (3.21) R=RAH 26 It should be noted that in these studies the Schwarzschild-like coordinates were used, in which the metric outside of the col- where RAH < R∞. Then, in the internal region R < lapsing thin shell is valid only for radius of the shell greater than RAH , the normal vector nµ becomes timelike. Therefore, the radii of the Killing horizons. But, universal horizons occur we have always inside Killing horizons. In addition, the Cuscuton theory in general does not have the symmetry (2.74) that reflects the  > 0, R > RAH , gauge symmetry t → ξ0(t) of Hoˇrva gravity. For other related as-  pects of Cuscuton theory to Hoˇrava gravity and Einstein-aether (n · n) = = 0,R = RAH , (3.22) theory, see [251].  < 0, R < RAH . 19

Since Eq.(3.20) always holds, we must have crystals may be dual to certain nonrelativistic gravitational theories in the Lifshitz space-time background,  < 0, R > RAH , 2z 2 2  2 r 2 r i i ` 2 (ζ · ζ) = = 0,R = RAH , (3.23) ds = − dt + dx dx + dr , (4.1)  ` ` r > 0, R < RAH , where z is a dynamical critical exponent, and ` a dimen- that is, ζµ becomes null on the apparent horizon, and sional constant. Clearly, the above metric is invariant spacelike inside it. under the anisotropic scaling, We define an apparent universal horizon as the hyper- t → bzt, xi → bxi, r → b−1r. (4.2) surface at which Thus, for z 6= 1 the relativistic scaling is broken in the sector (t, xi), and to have the above Lifshitz space-time as (u · ζ)|R=R = 0. (3.24) UH a solution of GR, it is necessary to introduce gauge fields to create a preferred direction, so that the anisotropic Since uµ is globally timelike, Eq.(2.19) is possible only scaling (4.2) becomes possible. In [269], this was realized when ζµ is spacelike. Clearly, this is possible only inside by two p-form gauge fields with p = 1, 2, and was soon the apparent horizon, that is, RUH < RAH . In the static case, the apparent horizons defined above generalized to different cases [270]. reduce to the Killing horizons, and the apparent universal It should be noted that the Lifshitz space-time is sin- horizons defined by Eq.(3.24) are identical to those given gular at r = 0 [269], and this singularity is generic in by Eq.(3.2). the sense that it cannot be eliminated by simply em- bedding it to high-dimensional spacetimes, and that test The above definition is only for spacetimes in which the particles/strings become infinitely excited when passing metric can be cast in the form (3.16). However, the gener- through the singularity [271, 272]. To resolve this issue, alization to other cases is straightforward. In particular, various scenarios have been proposed. There have been it was generalized in terms of an optical scalar built with also attempts to cover the singularity by horizons, and the preferred flow defined by the preferred spacetime fo- replace it by Lifshitz solitons [270]. liation [258]. On the other hand, since the anisotropic scaling is built in by construction in Hoˇrava gravity, it is natural to expect that the Hoˇrava theory should provide a minimal IV. NON-RELATIVISTIC GAUGE/GRAVITY holographic dual for non-relativistic Lifshitz-type field DUALITY theories with the anisotropic scaling and dynamical exponent z. Indeed, recently it was showed that the Lifshitz Anisotropic scaling plays a fundamental role in quan- spacetime (4.2) is a vacuum solution of Hoˇrava gravity in tum phase transitions in condensed matter and ultra- (2+1) dimensions, and that the full structure of the z = 2 cold atomic gases [259, 260]. Recently, such studies anisotropic Weyl anomaly can be reproduced in dual field have gained considerable momenta from the community theories [273], while its minimal relativistic gravity coun- of string theory in the content of gauge/gravity dual- terpart yields only one of two independent central charges ity [261, 263, 316]. This is a duality between a QFT in in the anomaly. D-dimensions and a QG, such as string theory, in (D+1)- Note that in [273], only the IR limit of the (2+1)- dimensions. An initial example was found between the dimensional Hoˇrava theory was considered. Later, the supersymmetric Yang-Mills gauge theory with maximal effects of high-order operators on the non-relativistic Lif- supersymmetry in four-dimensions and a string theory shitz holography were studied [274], and found that the on a five-dimensional anti-de Sitter space-time in the low Lifshitz space-time is still a solution of the full theory of energy limit [80]. Soon, it was discovered that such a du- the Hoˇrava gravity. The effects of the high-oder operators ality is not restricted to the above systems, and can be on the space-time itself is simply to shift the Lifshitz dy- valid among various theories and in different spacetime namical exponent z. However, the asymptotic behavior backgrounds [261, 263, 316]. of a (probe) scalar field near the boundary gets dramat- One of the remarkable features of the duality is that it ically modified in the UV limit, because of the presence relates a strong coupling QFT to a weak coupling gravita- of the high-order operators in this regime. Then, accord- tional theory, or vice versa. This is particularly attractive ing to the gauge/gravity duality, this in turn affects the to condensed matter physicists, as it may provide hopes two-point correlation functions. to understand strong coupling systems encountered in The above studies in the framework of Hoˇrava grav- quantum phase transitions, by simply studying the dual ity were soon generalized to various cases, in which vari- weakly coupling gravitational theory [264–268]. Other- ous Lifshitz soliton, Lifshitz spacetimes with hyperscaling wise, it has been found extremely difficult to study those violations, and Lifshitz charged black hole solutions in systems. Such studies were initiated in [269], in which it terms of universal horizons [216, 221–226, 230, 232, 234] was shown that nonrelativistic QFTs that describe mul- as well as in terms of Killing horizons [275–277] were ob- ticritical points in certain magnetic materials and liquid tained and studied. 20

Recently, another important discovery of the non- that guarantees the counterterms required to absorb the relativistic gauge/gravity duality is the one-to-one cor- loop divergences to be local and marginal or relevant with respondence between the Hoˇrava gravity with the en- respect to the anisotropic scaling. Then, gauge invari- larged symmetry (2.34) and the Newton-Cartan geom- ance of the counterterms is achieved by making use of etry (NCG) [145]. In particular, the projectable Hoˇrava the background-covariant formalism. gravity discussed in Section II.B corresponds to the dy- Along a similar line, Li et al. studied the quantization namical NCG without torsion, while the non-projectable of Hoˇrava gravity both with and without the projectabil- Hoˇrava gravity discussed in Section II.D corresponds to ity condition (2.17) in (1+1)-dimensional (2D) space- the dynamical NCG with twistless torsion. A precise times [286, 287]. In such spacetimes, Einstein’s theory dictionary between these two theories was established. is trivial [288–290]. But, this is not the case of Hoˇrava Restricted to (2+1) dimensions, the effective action for gravity, due to the foliation-preserving diffeomorphism dynamical twistless torsional NCG with 1 < z ≤ 2 was (2.3)[291, 292], although the total degree of freedom constructed by using the NCG invariance, and demon- of the theories is zero [286, 287]. In particular, in the strated that this exactly agrees with the most general projectable case, when only gravity is present, the sys- forms of the Hoˇrava actions constructed in [73, 97, 98]. tem can be quantized by following the canonical Dirac Further, the origin of the local U(1) symmetry was identi- quantization [293], and the corresponding wavefunction fied as coming from the Bargmann extension of the local is normalizable [286]. It is remarkable to note that in this Galilean algebra that acts on the tangent space to the case the corresponding Hamilton can be written in terms torsional NCG. Such studies have already attracted lots of a simple harmonic oscillator, whereby the quantization of attention and generalized to other cases [270]. can be carried out quantum mechanically in the standard In addition, it was shown that the non-projectable way. When minimally coupled to a scalar field, the mo- Hoˇrava gravity with the enlarged symmetry (2.34)[73, mentum constraint can be solved explicitly in the case 97, 98] can be also deduced from non-relativistic QFTs where the fundamental variables are functions of time with conserved particle number, by using the symme- only. In this case, the coupled system can also be quan- tries: time dependent spatial diffeomorphisms acting on tized by following the Dirac process, and the correspond- the background metric and U(1) invariance acting on ing wavefunction is also normalizable. A remarkable fea- the background fields which couple to particle number ture is that orderings of the operators from a classical [183, 184]. As Hoˇrava gravity is presumed to be UV Hamilton to a quantum mechanical one play a funda- complete, in principle this duality allows holography to mental role in order for the Wheeler-DeWitt equation to move beyond the large N limit [79–82]. have nontrivial solutions. In addition, the space-time is Finally, we would like to note that one may flip the well quantized, even when it is classically singular. logics around and study QG by using the gauge/gravity In the non-projectable case, the analysis of the 2D duality from well-known QFTs [278, 279]. Hamiltonian structure shows that there are two first-class and two second-class constraints [287]. Then, following Dirac one can quantize the theory by first requiring that V. QUANTIZATION OF HORAVAˇ GRAVITY the two second-class constraints be strongly equal to zero, which can be carried out by replacing the Poisson bracket Despite numerous efforts and the vast literature on by the Dirac bracket [293]. The two first-class constraints Hoˇrava gravity, so far its quantization has not been give rise to the Wheeler-DeWitt equations. It was found worked out in its general form, yet. Nevertheless, partic- that in this case the characteristics of classical spacetimes ular situations have been investigated, and some (promis- are encoded solely in the phase of the solutions of these ing) results have been obtained. In particular, in (3+1)- equations. dimensional spacetimes with the projectability and de- On the other hand, in CDTs [52] a preferred frame tailed balance conditions (2.17)-(2.21), the renormaliz- is also part of the process of quantization, which is quite ability of Hoˇrava gravity was shown to reduce to the similar to the foliation specified in Hoˇrava gravity. It was one of the corresponding (2+1)-dimensional topologically precisely this similarity that led the authors of [294] to massive gravity [280], by using stochastic quantization show the exact equivalence between the 2D CDT and the [281–283]. Even though the renormalizability of the lat- 2D projectable Hoˇrava gravity. Such studies were further ter has not been rigorously proven, it is often thought generalized to the case coupled with a scalar field [295] that it should be the case [284]. As already pointed out and (2+1)-dimensional spacetimes [296, 297]. In particu- in [280], the equivalence of the renormalizability between lar, in [296] it was shown the existence of known and novel the two theories is closely related by the detailed bal- macroscopic phases of spacetime geometry, and found ev- ance condition (2.19)-(2.21), and it is not clear how to idence for the consistency of these phases with solutions generalize it to the case without this condition. to the equations of motion of classical Hoˇrava gravity. In Lately, the above obstacle is circulated by properly particular, the phase diagram seemingly contains a phase choosing a gauge that ensures the correct anisotropic transition between a time-dependent de Sitter-like phase scaling of the propagators and their uniform falloff at and a time-independent phase. In [297], spacetime con- large frequencies and momenta [285]. It is this choice densation phenomena were considered and shown that a 21

successful condensation in (2+1)-dimensions can be ob- Hoˇrava gravity were derived by using the heat-kernel co- tained from a minisuperspace model of Hoˇrava gravity, efficients for Laplacian operators obeying anisotropic dis- but not from GR. persion relations in curved spacetimes [317, 318], and The close relations between CDTs and Hoˇrava gravity found that the Gaussian fixed point is an infrared at- have been further explored from the point of view of spec- tractor for the renormalization group flow of Newton’s tral dimensions. In particular, after extending the defi- constant, and the high-energy phase of the theory is nition of spectral dimension in fractal and lattice geome- screened by a Landau pole. When coupled to n Lifshitz tries to theories on smooth spacetimes with anisotropic scalars, the Gaussian fixed point ensures that the theory scaling, Hoˇrava showed that a (d+1)-dimensional space- is asymptotically free in the large-N expansion, indicating time with a dynamical critical exponent z has the spec- that the theory is perturbatively renormalizable. Earlier tral dimension [298], studies along the same line were carried out in [74, 319– 325], and various results were obtained. d d = 1 + . (5.1) s z Thus, in the IR (z = 1) the spectral dimension is identical VI. CONCLUDING REMARKS to the macroscopic dimension N(≡ d + 1) of spacetimes, N = ds. But, in the UV (z = d) the spectral dimension is In this review, we have summarized the recent devel- always two, no matter what dimensions of the spacetimes opments in Hoˇrava gravity. In particular, after giving a are in the IR. Therefore, (spectral) dimension is emergent brief introduction of the general ideas of Hoˇrava at the and it flows from two at short distances to (d+1) at large beginning of Section II, we have pointed out some po- ones. This was further confirmed in [299, 300]. Remark- tential issues presented in the original incarnation of the ably, this is also the qualitative behavior found in CDTs theory, and then introduced four most-studied modifi- [301] 27. Even more surprising, the same results were ob- 28 cations, depending on the facts being with or without tained in other theories of gravity , such as asymptotic the projectability condition and a local U(1) symmetry. safety [304, 305], LQG [306, 307], spin foams [308], non- In the presentation for each of these four versions, we commutative theory [309], Wheeler-DeWitt equation of have stated clearly their current status in terms of self- GR [310, 311], and causal set theory [312], to name only consistency of the theory, consistency with experiments, a few of them. For more details, we refer readers to [313]. mainly with solar system tests and cosmological observa- In addition, it was shown recently that holographic tions. renormalization of relativistic gravity in asymptotically In the projectable case without the local U(1) sym- Lifshitz spacetimes naturally reproduces the structure of metry presented in Section II.A, the minimal theory, the Hoˇrava gravity [314]: The holographic counterterms in- main issues are instability [92, 104, 105], and strong cou- duced near anisotropic infinity take the form of the action pling of the spin-0 graviton [106, 109]. The perturbation with the anisotropic scaling (2.1). In the particular case analysis shows that the Minkowski spacetime is not stable of (3 + 1) bulk dimensions and z = 2 asymptotic scal- due to the presence of the spin-0 graviton [92, 104, 105], ing near infinity, a logarithmic counterterm was found, which questions the viability of the theory (although the related to anisotropic Weyl anomaly of the dual confor- de Sitter spacetime is [106, 107]). To solve the strong mal field theory. It is this counterterm that reproduces coupling problem, one way is to borrow the well-known precisely the action of conformal gravity at a z = 2 Lif- Vainshtein mechanism [113]. This has been done in the shitz point in (2 + 1) dimensions, which is of anisotropic static and cosmological settings [34, 111, 112], but for local Weyl invariance and satisfies the detailed balance more general cases it has not been worked out, yet. condition. It was also shown how the detailed balance In the projectable case with the local U(1) symmetry is a consequence of relations among holographic coun- presented in Section II.B, the spin-0 graviton is elimi- terterms. A similar relation also holds in the relativistic nated, so the two issues, instability and strong coupling case of holography in AdS . Upon analytic continuation, 5 mentioned above, are absent in the gravitational sector, similar to the relativistic case [315? , 316], the action of but the strong coupling problem re-appears when coupled anisotropic conformal gravity produces the square of the with matter [125, 127]. To solve the problem, one way is wavefunction of the dual system. to introduce a new energy scale M , as shown in Fig.1, It is also very interesting to note that recently one- ∗ first proposed for the non-projectable case [128]. There- loop effective action and beta functions of the projectable fore, this version is self-consistent. It was shown that it is also consistent with all the solar system tests [98], provided the universal coupling with matter is adopted 29. 27 Scale dependence of spacetime dimension was first considered in anisotropic cosmology [302] and later in string theory where a thermodynamic dimension was also found to be two at high temperature [303]. 28 It should be noted that different definitions of dimension have 29 It should be noted that the consistency of this version of the been adopted in these theories, and in principle they do not theory with solar system tests excludes the possibility of minimal necessarily agree. coupling with matter [98]. 22

However, for such a coupling, the consistency with cos- first discovered that black holes can exist (theoretically) mological observations have not been worked out, yet, al- in gravitational theories with broken Lorentz invariance though a preliminary study indicates that a more general [202, 215], as one would expect that one can explore coupling may be needed [139]. An interesting problem in the physics arbitrarily near the singularity, once particles this version of the theory is the physical interpretations of with arbitrarily large speeds are allowed. Then, a nat- the pre-Newtonian potential ϕ and the U(1) gauge field ural question is whether such a black hole has entropy A. Such understanding may also be able to shed lights or not? Using the same arguments as we did in GR, on the coupling of the gravitational sector with matter. it is not difficult to be convinced that it should have. Along this direction, the non-relativistic gauge/gravity Otherwise, the second law of thermodynamics will be correspondence found recently in [145] may provide some violated [243]. This in turn raises several other inter- hints. esting questions, and one of them is: how the four laws In the non-projectable case without the local U(1) sym- of thermodynamics look like now? The first law in the metry presented in Section II.C, the healthy extension, it neutral (without charges) case can be generalized, after is shown that the theory is self-consistent and also con- a new definition of surface gravity given by Eq.(3.11) is sistent not only with solar system tests and cosmological adopted [241, 243]. But, for charged black holes, such observations, but also with the binary pulsar and gravita- a generalization is still absent [242]. Recent investiga- tional wave observations [152, 153, 155]. The strong cou- tions [245–247] with a more general dispersion relation pling problem [158] can be solved by introducing a new showed that both of the temperature and surface grav- energy scale M∗, as shown explicitly in [128], by properly ity obtained by considering peering behavior of rays near choosing the coupling constants. One of the concerning the universal horizons depend on the parameter z, which of this version is the possibility to introduce a hierarchy characterizes the leading order k2z of the dispersion rela- [74], although it was argued that this is technically quite tion [cf. Eqs.(3.12) and (3.14)]. Then, even in the neutral nature [96, 128]. Another question is the large number of case, it seems that the entropy S of the black hole should the coupling constants (which is about 100). With such be also z-dependent, if the first law, dE = T dS, still a large number, the question of the prediction power of holds. the theory raises again, although in the IR only five of For the zeroth law, it holds automatically in all the them are relevant [95, 96]. examples considered so far, as they are either in static In the non-projectable case with the local U(1) sym- spacetimes or in rotating (2+1)-dimensions, in which the metry presented in Section II.D, the spin-0 graviton is universal horizons always occur on a hypersurface on absent only in particular cases [73, 98, 186]. But, it is which we have r = rUH = constant. Kinematic consid- always stable in a large region of the parameters space, erations prove that this is also the case in general [229], whenever it is present. The strong coupling problem in although concrete examples in more general spacetimes general also exists, but can be solved by also introducing have not been found so far. a new energy scale M∗, as one did in the last two ver- For the second law of thermodynamics, Blas and sions. By softly breaking the detailed balance condition, Sibiryakov considered two possibilities where the miss- the number of the coupling constants can be significantly ing entropy can be found [202]: (i) It is accumulated reduced (to 15) [73, 97], while the power-counting renor- somewhere inside the black hole (BH). BS studied the malizability is still valid. It is interesting to note that stabilities of the universal horizons and found that they this is not the case without the local U(1) symmetry, are linearly stable. But, they argued that after nonlin- the healthy extension, as shown explicitly in [73]. It was ear effects are taken into account, these horizons will be shown that it is also consistent with all the solar sys- turned into singularities with finite areas. One hopes tem tests, provided the universal coupling with matter that in the full Hoˇrava gravity this singularity is resolved is adopted [98]. As in the projectable case, for such a into a high-curvature region of finite width accessible to coupling, the consistency of the theory with cosmologi- the instantaneous and fast high-energy modes. In this cal observations has not been worked out, yet [139]. It is way the BH thermodynamics can be saved. (ii) A BH interesting to note that this version has been embedded has a large amount of static long hairs, which have tails into string theory recently by using the non-relativistic that can be measured outside the horizon [326]. After gauge/gravity correspondence [183, 184]. I has been also measuring them, an outer observer could decode the en- shown that it has one-to-one correspondence to the dy- tropy that had fallen into the BH. However, BS found namical Newton-Cartan geometry [145]. that spherically symmetric hairs do not exist, and to have With all the above in mind, we have considered the this scenario to work, one has to use non-spherically sym- recent developments of Hoˇrava gravity in three differ- metric hairs. Clearly, to have a better understanding of ent but also related areas: (a) black holes and their the second law, much work in this direction needs to be thermodynamics in gravitational theories with broken done. The same is true for the investigations of the third Lorentz invariance, including Hoˇrava theory; (b) non- law. relativistic gauge/gravity correspondence in the frame- The investigations of the non-relativistic gauge/gravity work of Hoˇrava gravity; and (c) quantization of Hoˇrava duality in the framework of Hoˇrava gravity is still in its theory. It was somehow very surprising when it was infancy, and various open issues remain, including the 23 corresponding QFTs for the versions of Hoˇrava gravity sions, comments and suggestions on the subjects pre- without the U(1) symmetry, their dictionaries, and so sented here. The author would like to give his particular on. By flipping the logic, another important question thanks to Dr. Elias C. Vagenas, the Review Editor of is: can we get deeper insights about the quantization of IJMPD, for his kind invitation to write this review arti- Hoˇrava gravity from some well-known systems of QFTs cle and his patience and strong support during the whole through this correspondence? The answer seems very process. This work is supported in part by CiênciaSem positive [278, 279], but systematic investigations along Fronteiras, Grant No. A045/2013 CAPES, Brazil, and this direction are still absent. Chinese NSF, Grant Nos. 11375153 and 11173021. Quantizing gravity is the main motivation for Hoˇrava to propose his theory in 2009 [23], but a rigorous proof of its renormalizability is still absent, although it is power- APPENDIX A: LIFSHITZ SCALAR THEORY counting renormalizable by construction. So far, it has been shown that it is renormalizable only in a few par- To introduce Lifshitz scalar theory, let us begin with a ticular cases. These include (1+1)-dimensional space- free relativistic scalar field in a (3+1)-dimensional spacetimes both with and without the projectability condition time. For the sake of simplicity and without loss of the [286, 287, 294], and spacetimes with the projectable con- generality, we also assume that the spacetime is flat, so dition [285]. It is still an open question how to generalize the action takes the form, such studies to other cases, such as the one without the Z (free) 1 3 ˙2 projectability conditions and the ones with the local U(1) Sφ = dtd x φ + φ∆φ , (A.1) symmetry. In addition, what is the RG flow? So far, we 2 2 have assumed that the theory can arbitrarily approach where ∆(≡ ∂i ) denotes the Laplacian operator. Clearly, GR in the IR. But, without working out the RG flow in it is invariant under the general transformations (2.22). detail, we really do not know if this is indeed the case or Under the isotropic rescaling, not. Currently, only very limited situations were consid- −1 i −1 i ered [103, 280]. t → b t, x → b x , φ → bφ, (A.2) Other important issues include how the theory couples 0 Sφ is also invariant Sφ → Sφ. Denoting the units of with matter [74, 98], what are the effects of the Lorentz length, time and mass, respectively, by L, T and M, we violation [27, 31–33]. To the latter, we need at least to find that the speed c, energy E, and Planck constant h stay below current experimental constraints [24, 25]. As have the dimensions, mentioned in the Introduction, this is not an easy task [∆x] L ML2 at all [26, 27, 31–33]. To have a proper understanding of 2 [c] = = , [E] = mc = 2 , these effects, the coupling of Hoˇrava gravity with matter [∆t] T T will play a crucial role. [E] ML2T −2 ML2 [h] = = = . (A.3) Therefore, although Hoˇrava gravity seems to be a very [ν] T −1 T interesting and promising alternative in quantization of Such, choosing the natural units c = h = 1 implies gravity, for it to be really a viable theory, there is still a long list of questions that need to be addressed properly. L = T,E = M,L = M −1 = E−1. (A.4) In the rest of this section, we shall use the natural units. Then, for a process with ∆φ ' E, we have ACKNOWLEDGEMENTS p ∆t ' E−1, ∆` ≡ ∆x2 + ∆y2 + ∆z2 ' E−1, ∆φ ∆φ ∆φ The author would like to express his gratitude to ' ' ' E2. (A.5) his collaborators in this fascinating field. They include ∆t ∆xi ∆` Elcio Abdalla, Ahmad Borzou, Ronggen Cai, Gerald Hence, we find that Cleaver, Chikun Ding, Otavio Goldoni, Jared Green- Z E ∆φ2 wald, A.Emir Gumrukcuoglu, Yongqing Huang, Jiliang dtd3xφ˙2 ' ∆t (∆`)3 Jing, Jonatan Lenells, Bao-Fei Li, Kai Lin, JianXin 0 ∆t 2 Lu, Roy Maartens, Shinji Mukohyama, Antonios Papa- ' E−1 · E−3 · E2 'O(1), zoglou, V. H. Satheeshkumar, M. F. da Silva, Fu-Wen Z Shu, Miao Tian, David Wands, Xinwen Wang, Qiang (free) 1 3 ˙2 Sφ = dtd x φ + φ∆φ 'O(1). (A.6) Wu, Yumei Wu, Zhong-Chao Wu, Jie Yang, Razieh 2 Yousefi, Wen Zhao, and Tao Zhu. He would also like (free) to thank N. Afshordi, D. Blas, R.H. Brandenberger, B. Thus, the integral Sφ is always finite no matter how Chen, P. Hoˇrava, T. Jacobson, E.B. Kiritsis, J. Kluson, large E will be, which is called power-counting renormal- K. Koyama, S. Liberati, K.-i. Maeda, C. M. Melby- izable. What will happen when self-interaction is taken Thompson, A. Padilla, M. Pospelov, O. Pujolas, E.N. into account? For example, let us consider the term, Saridakis, M. Sasaki, S. Sibiryakov, A.M. da Silva, J. (s.i.) Z S = dtd3x φ2∆φ, (A.7) Soda, T.P. Sotiriou and M. Visser, for valuable discus- φ 24

( which is scaling as < 1, E < M , = ∗ (A.15) > 1, E > M∗. (s.i.) i 2 −1−3+2+2+1 Sφ → [dt] dx φ [∆] [φ] = b Therefore, when E M , we have = b, (A.8) ∗ Z E z 1 3 ˙2 φ∆ φ under the rescaling (A.2). Then, we find that Sφ ≡ dtd x φ + φ∆φ + 2z−2 2 0 M∗ Z E Z E (s.i.) 3 2 1 3 2 Sφ = dtd x φ ∆φ ' E, (A.9) ' dtd x φ˙ + φ∆φ , (A.16) 0 2 0 which becomes unbounded as E → ∞, and is said which is finite and invariant under the relativistic scaling (power-counting) non-renormalizable. On the other (A.2), as shown above. However, when E M∗, we have (s.i.) hand, as E → 0, we have Sφ → 0, which is said Z E z 1 3 ˙2 φ∆ φ irrelevant in low energy limit [110]. In general, for any Sφ ' dtd x φ + , (A.17) 2 M 2z−2 R E 3 0 ∗ given operator Oφ, if 0 dtd x Oφ scales as which is invariant only under the anisotropic rescaling Z E 3 δ between time and space, dtd x Oφ ' b , (A.10) 0 t → b−zt, xi → b−1xi, φ → bαφ φ. (A.18) under the rescaling (A.2), we have [110] Under this new rescaling, we find Z E 3 δ Z E dtd x Oφ ' E dtd3xφ˙2 ' b−z−3+2z+2αφ = bz−3+2αφ , (A.19) 0  0 ∞, δ > 0, Non-renormalizable,  where αφ ≡ [φ]. Thus, the scaling-invariance of this term = finite, δ = 0, Strictly renormalizable, (A.11) requires 0, δ < 0, Super-renormalizable, 1 α = − (z − 3). (A.20) as E → ∞. On the other hand, as E → 0, we find that φ 2 Z E Thus, for z = d = 3 the scalar field becomes dimension- 3 δ dtd x Oφ ' E less, and the interacting term (A.7) now scales as 0  (s.i.) 0, δ > 0, irrelevant, S ' b−z−3+2αφ+2+αφ = b−4, (A.21)  φ = finite, δ = 0, marginal, (A.12)  (s.i.) ∞, δ < 0, relevant. that is, under the new scaling, the interaction term Sφ is scaling with an negative power of b. As a result, we In the rest of this section, we shall use frequently the ter- have minology given above without any further explanations. (s.i.) −4/3 For more details, see [110]. In general, we have Sφ ' E → 0, (A.22) (total) (free) (s.i.) as E → ∞, where we assumed that ∆t ' E−1 ' p−1/3, S = S + S + ..., φ φ φ where ∆` ' p−1. Thus, under the new scaling, the non- = O(1) + E + E2 + ... → ∞, (A.13) (s.i.) renormalizable term Sφ now becomes renormlizable! as E → ∞, that is, the theory is not power-counting Adding all the relevant terms into (A.14), we find that renornalizable. the quadratic action To make the theory renormalizable, Lifshitz [70, 71] (bare) 1 Z added the following non-relativistic term into the free S = dtd3x φ˙2 + φOˆ φ , (A.23) φ 2 φ action (A.1), is power-counting renormalizable, where 1 Z φ∆zφ ∆S ≡ dtd3x , (A.14) 3 2 φ 2z−2 ∆ ∆ 2 2 2 M∗ ˆ Oφ ≡ g3 4 − g2 2 + c ∆ − m , (A.24) M∗ M∗ where z(≥ 2) is an integer, and M is a constant, which ∗ where g , g , c and m are coupling constants. Similarly, represents the energy suppressing the operator φ∆zφ. 3 2 we can add the potential term, V (φ), into the above ac- Thus, for an integration with ∆φ ' E, now we have tion, so finally we have 1 φ∆zφ E 2(z−1) Z (total) 1 3 ˙2 ˆ 2z−2 ' Sφ = dtd x φ + φOφφ − V (φ) , (A.25) M∗ φ∆φ M∗ 2 25 which is clearly also power-counting renormalizable [23, that there are an infinite number of UV divergent terms 65–67]. Note that in the most general case all the coeffi- in one-loop calculations [68, 69], and to eliminate these cients in Eq.(A.24) should be functions of φ [67–69]. In divergences, one needs to introduce an infinite number of addition, all n-point interaction of the forms [68, 69], counterterms, which make the predictability of the theory questionable. To overcome this problem, one way is to Z (zi,n) 3 zi n introduce the shift symmetry [68, 69], Sφ ≡ λ(zi,n) dtd x (∆ φ ), (A.26)

φ → φ + φ0, (A.28) should be also included into the action where zi ≤ z, which contains 2z spatial derivatives acting on n φ’s. i where φ is a constant. Once this symmetry is imposed, it The dimension of λ is 0 (zi,n) can be shown that only finite terms of the forms (A.24) 2(z − z ) satisfy this condition. Hence, in this case only a finite λ = i , (A.27) number of counterterms are needed, and the theory be- (zi,n) z comes predictable. which is always non-negative for zi ≤ z. As a result, this Recently, the theory of Lifshitz scalar fields has been term is power-counting renormalizable. Once all such intensively studied and made applications to various terms are included into the total action, it was shown cases [327–333].

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