aa r X i v : . [ h e p - t h ] M a y Strong coupling in Hoˇrava gravity.
Christos Charmousis,
1, 2, ∗ Gustavo Niz, † Antonio Padilla, ‡ and Paul M. Saffin § Laboratoire de Physique Theoretique, Universtite Paris Sud-11,91405 Orsay Cedex, France. LMPT UFR Sciences et Techniques, Universite Francois Rabelais, Parc de Grandmont 37200 Tours, France. School of Physics & Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom. (Dated: October 29, 2018)By studying perturbations about the vacuum, we show that Hoˇrava gravity suffers from twodifferent strong coupling problems, extending all the way into the deep infra-red. The first of theseis associated with the principle of detailed balance and explains why solutions to General Relativityare typically not recovered in models that preserve this structure. The second of these occurs evenwithout detailed balance and is associated with the breaking of diffeomorphism invariance, requiredfor anisotropic scaling in the UV. Since there is a reduced symmetry group there are additionaldegrees of freedom, which need not decouple in the infra-red. Indeed, we use the Stuckelberg trickto show that one of these extra modes become strongly coupled as the parameters approach theirdesired infra-red fixed point. Whilst we can evade the first strong coupling problem by breakingdetailed balance, we cannot avoid the second, whatever the form of the potential. Therefore theoriginal Hoˇrava model, and its ”phenomenologically viable” extensions do not have a perturbativeGeneral Relativity limit at any scale . Experiments which confirm the perturbative gravitationalwave prediction of General Relativity, such as the cumulative shift of the periastron time of binarypulsars, will presumably rule out the theory.
I. INTRODUCTION
Hoˇrava has recently proposed an interesting toy model of quantum gravity [1, 2, 3], generating a whole slew ofpublications that examine various aspects of the theory (see, for example [4, 5, 6, 7, 8, 9]). At short distances thetheory describes interacting nonrelativistic gravitons, and is argued to be power counting renormalisable in 3 + 1dimensions. Relativistic physics is supposed to emerge in the infra-red via relevant deformations, such that GeneralRelativity is recovered at large distances. Since Lorentz symmetry is manifestly broken in this theory, there are,in general, a huge number of possible relevant deformations one could include. To restrict the number of possibleparameters in the model, Hoˇrava made use of the principle of ”detailed balance”, as developed in studies of non-equilibrium critical phenomena and quantum critical systems. Whilst this organising principle is elegant, it wouldappear to be an obstacle to recovering GR in the infra-red. This was first illustrated in a study of static sphericallysymmetric solutions that did not recover the Schwarzschild geometry at large distances, unless detailed balance wasbroken [7, 8]. This has led to so called ”phenomenologically viable” extensions of the model that break detailedbalance explicitly [9].In this paper we will show that Hoˇrava gravity suffers from strong coupling problems, with and without detailedbalance, and is therefore unable to reproduce General Relativity in the infra-red. We consider the perturbative theoryabout the vacuum, yielding two important results. The first considers the role of detailed balance in these models. Asthe breaking terms go zero, we find that the linearised gravitational Hamiltonian constraint vanishes off -shell. Thismeans that linearised theory breaks down in this limit, just as it does for the Chern-Simons limit of Gauss-Bonnetgravity [10] (for a review on these gravity theories see [11], [12]). By comparing our equations to their counterpartsin General Relativity, we can see that the ”emergent” Planck length actually diverges in the limit of detailed balance,in contrast to the original claims [2]. This strong coupling behaviour means that the theory with detailed balancedoes not have a perturbative infra-red limit of any sort, explaining the results of [7]. Indeed, from the point of viewof spherically symmetric solutions one sees that the putative higher order terms in the IR are just as important asthe ”lower” order terms. In summary, with detailed balance, we can never hope to recover GR in the infra-red forthe following reason: General Relativity admits an effective linearised description beyond the Schwarzschild radius ofa source, but in Hoˇrava gravity with detailed balance, strong coupling prevents an effective linearised description on any scale. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: paul.saffi[email protected]
Our second result also applies to those models that have been dubbed ”phenomenologically viable”, and breakdetailed balance explicitly. In some sense it is clear that breaking detailed balance cannot possibly be enough torecover GR in these models. The point is that General Relativity contains full diffeomorphism invariance, so thatthe theory has just two propagating degrees of freedom. Because Lorentz symmetry is necessarily broken in the UV,Hoˇrava gravity contains a reduced set of diffeomorphisms, and must therefore contain more propagating degrees offreedom. If GR is to be recovered in the infra-red, these extra degrees of freedom should decouple from the system.This is not what happens. By restoring the full set of diffeomorphisms using the Stuckelberg trick, we are able to showthat one of the additional degrees of freedom actually becomes strongly coupled as the parameters in the theory flowtowards their desired infra-red fixed points. The scenario is highly reminiscent of Pauli-Fierz massive gravity [13] inwhich the longitudinal scalar becomes strongly coupled as m → II. ANISOTROPIC SCALING AND HO ˇRAVA GRAVITY
We begin by reviewing the basic ideas behind Hoˇrava gravity in scalar field theory, using Lifshitz’s model for a scalarfield that explicitly breaks Lorentz invariance [16] (see also [17, 18]). This provides a different way to regulate theUV divergences of loop diagrams, avoiding violations of unitarity associated with Pauli-Villars and higher derivativeLorentz invariant regulators, and without the need to introduce ficticious non-integer dimensions as in dimensionalregularisation. The hope then is that while Lorentz symmetry is explicity broken at high energy scales, it may berecovered in the IR regime at low energies. Consider, for example, the action [18] S free = Z dt d D x (cid:20)
12 ˙ φ − φ ( −∇ ) z φ ) (cid:21) , (1)This describes a free field fixed point with anisotropic scaling between space and time, x i → lx i , t → l z t, (2)characterised by the ’dynamical critical exponent’ z , so that the scaling dimensions are [ x ] = − t ] = − z . Theaction (1) leads to internal propagators in the UV of the form G ( ω, k ) → | k | − z . (3)For large enough z one sees that the fall-off of the propagator is fast enough to render Feynman diagrams convergent.In fact, the superficial degree of divergence, δ , satisfies δ ≤ ( D − z ) L, (4)for L loops [18]. As it stands, this model is not acceptable because it has no Lorentz symmetry in the IR. This maybe remedied by including a relevant operator of the form S rel = Z dt d D x (cid:20) − c φ ) ∂ i φ∂ i φ (cid:21) , (5)leading to a model that flows to a theory with Lorentz symmetry emergent at low energies, with a light-cone definedby the parameter c ( φ ) . It is interesting to note that if we have a number of matter fields, they can each have theirown Lorentz symmetry. This is not something that is observed experimentally and leads to a fine tuning of themodel. There are further issues that appear once Lorentz symmetry is broken, such as the possibility of a black-holeperpetuum mobile machine [19, 20]. Furthermore, although the action S free + S rel breaks Lorentz invariance in theUV, it does not introduce extra degrees of freedom in the infra-red as the emergent symmetry is not dynamical. InGeneral Relativity, however, diffeomorphism invariance is a dynamical symmetry, so breaking it in the UV could alterthe number of degrees of freedom that propagate in the infra-red. This leads directly to the second strong couplingproblem alluded to earlier.With Lorentz symmetry no longer being used as a guiding principle, there is a great proliferation in the number ofterms that may appear in the action. To ameliorate this, Hoˇrava proposed an organising principle based on detailedbalance [2], which also allows one to put forward a quantum inheritance principle such that the theory in D + 1dimensions acquires the renormalisation properties of the D -dimensional theory [21]. Detailed balance is a statementthat the potential of a D + 1-dimensional theory is obtained from a D -dimensional ”superpotential” by functionaldifferentiation. For example, the scalar field action is given by S = Z dt d D x "
12 ˙ φ − (cid:18) δWδφ (cid:19) , (6)with a superpotential W [ φ ] = Z d D x (cid:20) ∂ i φ∂ i φ + 12 mφ (cid:21) . (7)In this case, one obtains a z = 2 theory in the UV, with a Lorentz invariant Klein Gordon theory in the IR, withan ”emergent” speed of light c φ ) = 2 m . In the first of our results, we shall find that the gravity model constructedusing detailed balance does not have a well defined perturbation limit about its vacuum. However, we ought to notethat this is not an artefact of detailed balance in general. For example, the scalar model above has well defined wavesolutions in the vacuum φ ( t, x ) = e i ( ωt + k.x ) , ω = ± ( | k | + m ) . (8)Since the theory is linear, the perturbations are also waves. The situation in gravity is rather more subtle owing tothe fact that there are constraints, specifically the Hamiltonian constraint, as we shall see in the next section. Theconstraint equations lead to strong coupling unless detailed balance is broken.The gravitational theory based on the violation of Lorentz symmetry has been clearly presented in Hoˇrava’s paper [2]and we refer the reader to that work for more detail. One of Hoˇrava’s key assumptions is the explicit breaking of fullfour dimensional diffeomorphism invariance to a subgroup that preserves a foliation structure of space-like slices. Thisenables him to make use of anisotropic scaling in the UV as in the Lifshitz model we have just discussed. Followingfrom the ADM decomposition of the metric, and the Einstein equations [22], the fundamental objects of interest arethe fields N ( t, x ), N i ( t, x ), g ij ( t, x ), corresponding to the lapse, shift and spatial metric of the ADM decomposition, ds = ˆ g µν dx µ dx ν = − N c dt + g ij ( dx i + N i dt )( dx j + N j dt ) . (9)Under the new, restricted, set of diffeomorphisms x i → x i − ζ i ( t, x ) , t → t − f ( t ) (10)the fields transform as follows δg ij → δg ij + 2 ∇ ( i ζ j ) + f ˙ g ij , (11) δN i → δN i + ∂ i ( ζ j N j ) − ζ j ∇ [ i N j ] + ˙ ζ j g ij + ˙ f N i + f ˙ N i , (12) δN → δN + ζ j ∂ j N + ˙ f N + f ˙ N. (13)where indices are raised/lowered using g ij , and ∇ i is the covariant derivative on the space-like slices.The transformation laws represent an important deviation from standard General Relativity, where full 4 D diffeo-morphism invariance is present. Indeed, note that the last of these transformations shows that if N is restricted tobe ”projectable” [2], i.e. N = N ( t ), then this condition is maintained under the restricted diffeomorphism group.Projectable solutions to Hoˇrava’s theory cannot, therefore, be transformed into non-projectable solutions, in contrastto General Relativity. This explicitly illustrates the fact that solutions to Hoˇrava gravity cannot be specified usingthe 4 D metric alone–one must always specify the foliation. Furthermore, although one is free to impose projectabilityat the level of solutions in Hoˇrava gravity, doing so prevents us from finding the full set of solutions. Again, this is notthe case in GR where one can always use the full set of diffeomorphisms to render any solution locally projectable.In this paper, we shall consider the general case, as Hoˇrava does, where N is a function of both x i and t . We notethat imposing projectability at the level of theory, as advocated in [9], alters the theory explicitly, since the equationsof motion for the lapse can only then be expressed as integrals over space. Such a modification of Hoˇrava gravitywould appear to be inherently non-local, so we will not consider it here.The action for Hoˇrava gravity is made up of a kinetic term, and a potential term satisfying ”detailed balance”, S H = Z dtd x √ gN ( T − V ) . (14)The kinetic term is constructed out of the extrinsic curvature of the foliations, as this is covariant under the remnantdiffeomorphism symmetry, K ij = 12 N ( ˙ g ij − ∇ i N j − ∇ j N i ) . (15)Requiring the kinetic term to be at most quadratic in K yields T = 2 κ ( K ij K ij − λK ) = 2 κ K ij G ijkl K kl (16)where we have introduced the the de Witt metric, G ijkl = 12 ( g ik g jl + g il g jk ) − λg ij g kl , (17)whose inverse is given by G ijkl = 12 ( g ik g jl + g il g jk ) − ˜ λg ij g kl , ˜ λ = λ λ − . (18)The dimensionless parameter λ is taken to run with scale. In order to have any hope of recovering General Relativityin the IR, one must assume that λ = 1 corresponds to the infra-red fixed point. The potential term is constructed outof the spatial metric and its derivatives. Inspired by methods used in quantum critical systems and non-equilibriumcritical phenomena, Hoˇrava restricts the large class of possible potentials using the principle of detailed balanceoutlined above. This requires that the potential takes the form V = κ √ g δWδg ij G ijkl √ g δWδg kl (19)= κ E ij G ijkl E kl . (20)Note that by constructing E ij as a functional derivative it automatically becomes transverse from within the foliationslices, ∇ i E ij = 0. We can derive the field equations by varying the action (14) with respect to each of the fields [4],1 √ g δS H δN = − ( T + V ) , (21)1 √ g δS H δN i = 4 κ ∇ i π ij , (22)1 √ g δS H δg ij = − κ h ˙ π ij + N Kπ ij + 2 ∇ k ( π k ( i N j ) ) − N k ∇ k π ij + 2 N K ki π jk i + 12 N ( T − V ) g ij − κ (cid:2) ∆( N χ ij ) + N E ik χ jk (cid:3) , (23)where π ij = K ij − λKg ij , χ ij = E ij − ˜ λEg ij , (24)and the operator ∆ is defined as ∆ h ij = lim ǫ → ǫ (cid:0) E ij [ g + ǫh ] − E ij [ g ] (cid:1) . (25)Having constructed the gravitational theory following the same principles as those for the scalar field, it remains topick the superpotential W [ g ij ]. In 3 + 1 dimensions, this must be chosen such that we have anisotropic scaling with adynamical critical exponent z ≥
3, in order that the theory be power counting renormalisable. This follows from thefact that in D+1-dimensions, the scaling dimension of κ is given by[ κ ] = z − D . (26)Fully relativistic theories such as general relativity must always have z = 1. In the next section we will focus on thecase of z = 3, so that κ is dimensionless in 3 + 1 dimensions. To illustrate what we mean by this definition, note that we can define the Lichnerowicz operator in a similar way, − ∆ L h ij =lim ǫ → ǫ ( R ij [ g + ǫh ] − R ij [ g ]). III. Z=3 HO ˇRAVA GRAVITY WITH AND WITHOUT DETAILED BALANCE
Given the guiding principle of detailed balance, the unique z = 3 theory in 3+1 dimensions, with additional relevantdeformations in the IR may be obtained from the following superpotential [2], W [ g ij ] = 1 w Z ω (Γ) + µ Z d x √ g ( R − W ) . (27)The z = 3 contribution comes from gravitational Chern-Simons action in 3-dimensions, where ω (Γ) = T r (cid:18) Γ ∧ d Γ + 23 Γ ∧ Γ ∧ Γ (cid:19) . (28)Again, all the couplings are taken to run with scale, with scaling dimensions [ w ] = 0 , [ µ ] = 1 , [Λ W ] = 2. Variationof this action yields E ij = 1 w C ij − µ (cid:0) G ij + Λ W g ij (cid:1) , (29)where G ij is the Einstein tensor on the spatial slices, and C ij is the Cotton tensor C ij = ǫ kl ( i ∇ k R j ) l . (30)Hoˇrava originally argued that this theory flowed from λ = 1 / λ = 1 in the IR, thereby recoveringGeneral Relativity at low energies, with an emergent speed of light, c , Newton’s constant, G N , and cosmologicalconstant, Λ, given by c = κ µ r Λ W − λ , G N = κ πc , Λ = 32 Λ W . (31)However, a study of spherically symmetric solutions in this theory [7] seems to indicate that this is not the case.One has to break detailed balance in order to recover the corresponding solutions in General Relativity. We willnow show that this is because detailed balance leads to strong coupling on all scales, so that one cannot consistentlytruncate the higher derivative operators in the infra-red. To elucidate the specific role played by detailed balance letus break it explicitly. Clearly there are a number of ways in which one can do this. A set of relevant breaking termswas proposed in [9], although we note here that their list did not include terms like R dtd x √ gN C ij R ij , which seemperfectly reasonable at first glance. For simplicity, we will perform a minimal breaking of detailed balance by addinga term to the action of the form S br = − κ (cid:18) ǫ − λ (cid:19) Z dtd x √ gN ( R − β ) (32)where, from the point of view of the z = 3 theory at short distances, the new parameters have scaling dimension[ ǫ ] = 4, [ β ] = 2. We will also include a generic matter contribution, S m , so that the full action is now given by S = S H + S br + S m . (33)Of course, it is not exactly clear how we should couple matter in this theory, as we no longer have the guiding handof Lorentz invariance to assist us. We will not worry about those issues here, merely assuming that it can be donein some consistent way, so that the matter fields act as sources in our equations of motion. The field equations nowtake the form 1 √ g δS H δN − κ (cid:18) ǫ − λ (cid:19) ( R − β ) = − √ g δS m δN = ρ, (34)1 √ g δS H δN i = − √ g δS m δN i = v i , (35)1 √ g δS H δg ij − κ (cid:18) ǫ − λ (cid:19) (cid:20) ∇ i ∇ j − ( g ij ∇ + G ij + 3 β g ij ) (cid:21) N = − √ g δS m δg ij = τ ij . (36)The energy-momentum fields of the matter contribution ( ρ, v i , τ ij ) satisfy the following conservation laws Z d x √ g (cid:20) ˙ g ij τ ij − N ( ρ √ g )˙ √ g − N i ( v i √ g )˙ √ g (cid:21) = 0 , (37)2 ∇ i τ ij − ρ∂ j N + ( v i √ g )˙ √ g + N j ∇ i v i + 2 v i ∇ [ i N j ] = 0 . (38)These deviate slightly from the usual conservation of energy-momentum, ˆ ∇ µ T µν = 0, because we only have thereduced set of diffeomorphisms outlined in the previous section.We now wish to define vacua in this theory, in the absence of these matter fields. Owing to the fact that we have areduced set of diffeomorphisms, it is not enough to impose, say, maximal symmetry in 3 + 1 dimensions. We must alsodefine the foliation. To this end we note that the momentum conjugate to g ij is given by p ij = √ gπ ij , and require itto vanish on the vacuum, so that ¯ K ij = 0. Further, we choose the gauge ¯ N i = 0, and require that the spatial metric,¯ g ij is a homogeneous Einstein space ¯ g ij dx i dx j = dr − γ r + r d Ω (39)with constant Ricci curvature ¯ R ij = γg ij . In geometric terms we are asking for our 3 dimensional foliation to bemaximally symmetric and furthermore that the foliation be trivially embedded (totally geodesic). Given that 3-spaceis conformally flat, ¯ E ij = q ¯ g ij where q = µ ( γ − W ). The N i equation (35) is satisfied automatically. The N equation (34), which is essentially the Hamiltonian constraint, yields q = ǫ ( β − γ ), and so γ = 2Λ W − µ ǫ ± r ǫ + ǫµ β − W ) ! . (40)It remains to impose the g ij equation (36), which constrains the background lapse function ¯ N . The quantity ∆ N ¯ g ij is easily derived by making use of the transformation laws for G ij and C ij under conformal transformations. We findthat − κ (cid:18) µq + 2 ǫ − λ (cid:19) (cid:2) ∇ i ∇ j − g ij ( ∇ + γ ) (cid:3) ¯ N = 0 . (41)For detailed balance, we have ǫ = q = 0, and so ¯ N is unconstrained. This is consistent with the findings of [7]. Awayfrom detailed balance, we find that ¯ N = p − γr /
2, so that the full 3 + 1 dimensional metric corresponds to amaximally symmetric spacetime with curvature γ/
2, written in global coordinates.Let us now reintroduce the matter fields, and consider perturbations about the vacuum δN = n ( t, x ) , δN i = n i ( t, x ) , δg ij = h ij ( t, x ) . (42)It is convenient to introduce E ij = E ij − qg ij , as this vanishes on the background. The unbroken potential now takesthe form V = κ h E ij G ijkl E kl + 2 q (1 − λ ) E + 3 q (1 − λ ) i , (43)and the Hamiltonian constraint (34) may be written − κ K ij G ijkl K kl − κ (cid:20) E ij G ijkl E kl + 12 (cid:18) − λ (cid:19) ( µq + 2 ǫ )( R − γ ) (cid:21) = ρ (44)where we have used the fact that q = ǫ ( β − γ ) and E = µ ( R − γ ). Perturbing this equation to linear order isnow easy, since the first two terms are already second order owing to the fact that both K ij and E ij vanish on thebackground. Lumping all higher order corrections alongside the matter field, the Hamiltonian constraint gives − κ (cid:18) µq + 2 ǫ − λ (cid:19) δR = ρ + non-linear corrections . (45) We will denote vacuum expectation values for all fields with a “bar”.
For detailed balance ( ǫ = q = 0), we immediately see that linearised perturbation theory is not well defined in thepresence of matter. Higher order terms always dominate, and one loses predictive power. This is characteristic of strongcoupling, and is reminiscent of the Chern-Simons limit in Gauss Bonnet gravity [10]. Perturbation theory around thevacuum is strongly coupled on all scales, even in the deep infra-red. We have included a matter component to renderthis explicit, although it ought to be clear that vacuum fluctuations will also be strongly coupled since genericallyone does not expect all non-linear corrections to vanish identically. Of course, one might hope to alleviate thisstrong coupling problem by perturbing about a different background. However, on temporal/spatial scales that aresmall compared to the scale set by the background extrinsic curvature/spatial curvature, our vacuum solution wouldrepresent a good approximation for the background, and one would immediately lose predictability. For example,cosmological perturbations about an FRW background would become strongly coupled on subhorizon scales.Of course, one can avoid this problem by moving away from detailed balance. Indeed, it is instructive to compareequation (45) with the corresponding equation in General Relativity − c πG N δR = ρ + non-linear corrections . (46)This suggests that if General Relativity is indeed recovered in the infra-red, it does so with an emergent Newtonconstant G N = κ / πc and an emergent speed of light c = κ r ǫ + µq/ − λ = κ vuut ∓ (cid:16) ǫ + ǫµ ( β − W ) (cid:17) / − λ . (47)We immediately see that the upper branch of solutions is ruled out, as the emergent speed of light is imaginary. Evenon the lower branch, as one approaches detailed balance c →
0, and so G N → ∞ , which means the effective Plancklength, l pl = p ¯ hG N /c , diverges, as expected due to strong coupling on all scales. Away from detailed balance,strong coupling only kicks in below the emergent Planck length, and it is natural to ask if indeed General Relativitycan be recovered in the infra-red, as is perhaps suggested by the form of equation (45). To establish this properly wemust also look at the linearised N i and g ij equations, and compare them with their GR counterparts. An entirelyequivalent, but more convenient approach, however, is to simply compute the effective action to quadratic order inthe fields propagating on the background. We shall do this presently.Let us rewrite the action as the emergent GR piece, plus corrections S = S GR + S UV + S m , (48)where S GR = 116 πGc Z dtd x √ gN (cid:2) K ij K ij − K − c ( R − γ ) (cid:3) , (49) S UV = Z dtd x √ gN (cid:20) κ − λ ) K − κ E ij G ijkl E kl (cid:21) . (50)It is sufficient to compute S UV and S m to quadratic order. The latter is given by δ S m = − Z dtd x √ ¯ g (cid:2) nρ + n i v i + h ij τ ij (cid:3) . (51)Because K ij and E ij vanish on the background, it is also straightforward to compute δ S UV = Z dtd x √ ¯ g ¯ N (cid:20) κ − λ )( δK ) − κ δ E ij ¯ G ijkl δ E kl (cid:21) (52)where δK = 12 ¯ N h ˙ h − ∇ i n i i , δ E ij = 1 w ǫ kl ( i ∇ k ψ j ) l − µ ψ ij (53)and ψ ij = δ (cid:16) G ij + γ g ij (cid:17) = − ∇ ( h ij − h ¯ g ij ) + ∇ ( i ∇ k h j ) k − ∇ i ∇ j h + ¯ g ij ∇ k ∇ l h kl + γ h ij (54)Assuming that λ flows to 1 in the infra-red, it would appear that δ S UV → − κ Z dtd x √ ¯ g ¯ N δ E ij ¯ G ijkl δ E kl . (55)This piece contains contributions that are higher order in the appropriate derivative operators, and can be ignoredat low energies, compared with δ S GR . This would suggest that provided we break detailed balance, we can indeedrecover General Relativity at low energies. However, such a naive analysis clearly does not tell the full story. Recallthat our original theory was invariant under a reduced set of diffeomorphisms. This means the theory should containmore degrees of freedom than General Relativity. If our theory is to recover GR in the infra-red, where did the extradegrees of freedom go? One faces a similar scenario when studying Pauli-Fierz massive gravity theories [13]. A massivegraviton has 5 propagating degrees of freedom whereas as a massless graviton has just two. As we take the gravitonmass to zero in Pauli Fierz theory, the extra 3 degrees of freedom do not all disappear. In fact, it turns out that thelongitudinal scalar mode becomes strongly coupled [14], and is responsible for the famous vDVZ discontinuity [15].The behaviour of the additional degrees of freedom in Pauli-Fierz gravity is most clearly understood by artificiallyrestoring the full gauge invariance using the Stuckelberg trick [14]. This was first introduced to study massive Abeliangauge theories, although we shall apply it to the case in hand. To begin with, note that under the full set ofdiffeomorphisms present in General Relativity, ( t, x i ) → ( t − f ( t, x ) , x i − ζ i ( t, x )), our ADM variables transform onthe background as follows n → n + ζ k ∇ k ¯ N + ˙ f ¯ N + f ˙¯ N , (56) n i → n i + ˙ ζ j ¯ g ij − ¯ N c ∂ i f , (57) h ij → h ij + 2 ∇ ( i ζ j ) . (58)We now introduce the Stuckelberg fields ξ i ( t, x ), φ ( t, x ), whose scaling dimensions are the same as x and t respectively.If we perform the following field redefinitions in the action n → n + ξ k ∇ k ¯ N + ˙ φ ¯ N + φ ˙¯ N, (59) n i → n i + ˙ ξ j ¯ g ij − ¯ N c ∂ i φ, (60) h ij → h ij + 2 ∇ ( i ξ j ) , (61)we find that δ S → δ S + Z dtd x √ ¯ g ¯ N c (cid:20) κ − λ ) (cid:18) N ∇ i ( ¯ N ∇ i φ ) δK + c ¯ N ( ∇ i ( ¯ N ∇ i φ )) (cid:19) − φ (cid:18) ˙ ρc + ∇ i ( ¯ N v i )¯ N (cid:19)(cid:21) , (62)where we have made use of the energy conservation laws (37) and (38). The action is manifestly invariant under (56)to (58), along with the following shifts in the Stuckelberg fields ξ i → ξ i − ζ i , φ → φ − f. (63)The first Stuckelberg field ξ i clearly plays no role. Not so the other Stuckelberg field, φ . Its equation of motion isgiven by κ (1 − λ ) ∇ i (cid:20) ¯ N ∇ i (cid:18) δK + c ¯ N ∇ j ( ¯ N ∇ j φ ) (cid:19)(cid:21) = ¯ N ˙ ρc + ∇ i ( ¯ N v i ) + non-linear corrections , (64)where we have included contributions from terms in the action beyond quadratic order. Now as λ →
1, we see thatthe Stuckelberg field becomes strongly coupled, in direct analogy with the longitudinal scalar degree of freedom inPauli Fierz gravity. The matter contribution makes this manifest. Indeed, when matter is present, we can even see thestrong coupling of the scalar mode directly from the linearised equations of motion. To see this, consider linearisedperturbations that are scalars with respect to the 3 D diffeomorphisms on spatial slices, δN = n, δN i = ∇ i α, h ij = σ ¯ g ij + ∇ i ∇ j θ (65)It is convenient to make use of the remnant diffeomorphism (13) to gauge away θ . The linearised Hamiltonianconstraint (45) now yields (cid:18) ∇ + 32 γ (cid:19) σ = 8 πG N c ρ (66)where we have expressed everything in terms of the emergent speed of light (47) and Newton constant, G N = κ / πc .Given the linearised form of the N i equation (35), ∇ j δπ ij = 8 πG N cv i (67)we make use of the solution (66) and the equation of motion (41) for ¯ N , to show that¯ N (1 − λ ) (cid:18) ∇ + 32 γ (cid:19) (3 ˙ σ + 2 ∇ α ) = 16 πG N c (cid:0) ¯ N ˙ ρ + c ∇ i ( ¯ N v i ) (cid:1) (68)Now in General Relativity where one has the full set of 4 D diffeomorphisms, the right hand side of the above equationvanishes automatically by energy-conservation, ˆ ∇ µ T µν = 0, and is therefore consistent with λ ≡
1. However, inHoˇrava gravity, with a reduced set of diffeomorphisms, the reduced version of energy-conservation (37) merely requires R d x √ g ¯ N ˙ ρ = 0 on this background, and places no constraint on ∇ i ( ¯ N v i ). Therefore, by introducing, say, a non-zerovalue for ∇ i ( ¯ N v i ), the scalar field equation (68) clearly runs into problems with strong coupling as we approach thedesired infra-red fixed point, λ → − λ ) into the Stuckelberg field, defin-ing ˆ φ = φ (1 − λ ). However, the non-linear corrections will generically include terms that schematically go like κ (1 − λ ) c m c ( ∂ t ) m t ( ∇ ) m x ( h ij ) m h ( n i ) (5 − m c − m t − m x +3 m φ ) / ( n ) m n ( φ ) m φ . Upon replacing φ with ˆ φ , such a term con-tains an overall factor of (1 − λ ) − m φ , and will diverge for m φ ≥
2. Of course, this ought to be checked explicitly byintroducing the Stuckelberg fields beyond linear order, and computing the higher order action, but this is beyond thescope of the current paper.Note that unlike the previous case, the strong coupling associated with the Stuckelberg field has nothing to do withdetailed balance. It is merely an artifact of the reduced set of diffeomorphisms present in the theory, and occurs evenwhen detailed balance is broken. As one approaches λ →
1, General Relativity is not recovered because the extradegrees of freedom present in the full theory do not all decouple. On the contrary, one of those degrees of freedombecomes strongly coupled, and one recovers General Relativity with an additional strongly coupled scalar.
IV. DISCUSSION
By considering perturbations about the vacuum we have shown that Hoˇrava gravity generically suffers from strongcoupling problems on all scales, essentially ruling out the theory as a viable model of the Universe. The strong couplingproblems come in two different guises. The first problem is related to the principle of detailed balance, and can bealleviated by adding terms to the action that explicitly break this principle. This radically increases the number ofparameters one can introduce into the model, and although this would be undesirable from an aesthetic perspective,one could take the view that it would be a small price to pay for a viable model of quantum gravity. Unfortunatelybreaking detailed balance is not enough, since it does not save us from the second of our strong coupling problems.This is related to the fact that Lorentz invariance is explicitly broken in the UV and one is forced to give up the fullset of diffeomorphisms present in General Relativity. The result is that there are extra degrees of freedom that canstill propagate in the infra-red, one of which becomes strongly coupled on all scales as the parameters in the theoryapproach their desired infra-red fixed point.Whilst we have explicitly shown these effects for a particular model, we note that they are generic to any modelbased on Hoˇrava’s ideas. Consider first the strong coupling problem associated with detailed balance. Whatever thechoice of superpotential, W [ g ], for detailed balance, the Hamiltonian constraint is given by − κ K ij G ijkl K kl − κ E ij G ijkl E kl = ρ. (69)The vacuum solution ( ρ = 0) is given by ¯ K ij = 0, and so ¯ E ij = 0. Perturbations about the vacuum now yield − κ ¯ K ij ¯ G ijkl δK kl − κ E ij ¯ G ijkl δE kl = ρ + non-linear corrections . (70)Clearly the left-hand side of the above equation vanishes automatically, which is precisely the first strong couplingissue seen in section III.We now turn our attention to the strong coupling associated with broken diffeomorphism invariance. This is presenteven without detailed balance, and regardless of how one breaks it. To see this, note that for the theory to have0any hope of recovering GR in the infra-red, the quadratic action must take the form S = S GR + S UV + S m , where S UV ≪ S GR in the infra-red. Now given the form of the kinetic term in Hoˇrava’s model, S UV = Z dtd x √ ¯ g ¯ N (cid:20) κ − λ )( δK ) + UV corrections coming from the potential (cid:21) . (71)Assuming one breaks detailed balance such that the potential is still just a function of the spatial metric and itsspatial derivatives, then regardless of its precise form, one will find that the UV corrections above will be invariantunder h ij → h ij + 2 ∇ ( i ζ j ) , and as such unaffected by the Stuckelberg fields. The only terms that result in explicitdependence on those fields are S m and the ( δK ) term above. Therefore the Stuckelberg analysis carried out inthe previous section can be extrapolated to apply to any breaking of detailed balance, and one recovers the strongcoupling problem as λ → γ = 0. Of course, one could always foliatea maximally symmetric spacetime along surfaces with non-vanishing extrinsic curvature. It is difficult to see how thiswould correspond to a better choice of vacuum since the conjugate momenta no longer vanish and we move away fromtestable regions of GR. In any case, one could always work on temporal scales much larger than the scale set by theextrinsic curvature and reapply our analysis on those scales. This would presumably set the strong coupling scale tobe in the inverse of the extrinsic curvature scale. For example, using a cosmological slicing of de Sitter space wouldresult in strong coupling problems inside the cosmological horizon.The strong coupling problems guarantee that perturbative General Relativity cannot be reproduced in the infra-redin Hoˇrava gravity. This would seem to disagree with the results of [7] that recover the Schwarzschild solution whenone breaks detailed balance. However, there is no disagreement. The symmetries imposed on the solutions in [7]prevent the strongly coupled scalar mode from being excited. Therefore, evidence of this mode may well be absent inclassical local tests of general relativity that implement weak and slowly moving sources. Generically, however, thetroublesome scalar will be excited. If, for example, we allowed for time dependence, while keeping spherical symmetry,one would expect this scalar mode to kick in and be responsible for a breaking of Birkhoff’s theorem. Indeed thepresence of a strong coupled scalar mode in the gravity spectrum casts serious doubts on the validity of this theoremand signals the probable presence of gravitational radiation from spherical sources. Furthermore, the linearised versionof General Relativity is used to study the effects of gravitational radiation emitted by binary pulsars, and containsexcellent agreement with observation [23]. In Hoˇrava gravity we have seen that we have no reliable linearised theoryto work with due to strong coupling of an extra scalar degree of freedom. Even if it were tractable, it seems unlikelythat a non-linear analysis could recover the successes of General Relativity in this instance, since the gravitons willgenerically couple to the strongly coupled mode through higher order interactions. Our conclusion then is that Hoˇravagravity in its current form is almost certainly ruled out. Acknowledgments
We would like to thank Ed Copeland, Nemanja Kaloper and Kazuya Koyama for useful discussions. AP is fundedby a Royal Society University Research Fellowship, and GN by STFC. CC thanks Elias Kiritsis and Olindo Corradinifor early discussions on the subject of Hoˇrava gravity, and for the hospitality shown by Nottingham University duringthe inception of this work. [1] P. Hoˇrava, JHEP , 020 (2009) [arXiv:0812.4287 [hep-th]].[2] P. Hoˇrava, Phys. Rev. D , 084008 (2009) [arXiv:0901.3775 [hep-th]].[3] P. Hoˇrava, arXiv:0902.3657 [hep-th].[4] E. Kiritsis and G. Kofinas, arXiv:0904.1334 [hep-th].[5] G. E. Volovik, arXiv:0904.4113 [gr-qc]. G. Calcagni, arXiv:0904.0829 [hep-th]. H. Nastase, arXiv:0904.3604 [hep-th].R. G. Cai, L. M. Cao and N. Ohta, arXiv:0904.3670 [hep-th]. T. Takahashi and J. Soda, arXiv:0904.0554 [hep-th]. X. Gao,arXiv:0904.4187 [hep-th]. Y. S. Piao, arXiv:0904.4117 [hep-th]. S. Mukohyama, arXiv:0904.2190 [hep-th]. R. G. Cai, Y. Liuand Y. W. Sun, arXiv:0904.4104 [hep-th]. D. Orlando and S. Reffert, arXiv:0905.0301 [hep-th]. [6] R. G. Cai, B. Hu and H. B. Zhang, arXiv:0905.0255 [hep-th]. T. Nishioka, arXiv:0905.0473 [hep-th]. J. Kluson,arXiv:0905.1483 [hep-th]. R. A. Konoplya, arXiv:0905.1523 [hep-th]. H. Nikolic, arXiv:0904.3412 [hep-th].[7] H. Lu, J. Mei and C. N. Pope, arXiv:0904.1595 [hep-th].[8] A. Kehagias and K. Sfetsos, arXiv:0905.0477 [hep-th].[9] T. Sotiriou, M. Visser and S. Weinfurtner, arXiv:0904.4464 [hep-th].[10] C. Charmousis and A. Padilla, JHEP (2008) 038 [arXiv:0807.2864 [hep-th]].[11] J. Zanelli, arXiv:hep-th/0502193.[12] C. Charmousis, Lect. Notes Phys. , 299 (2009) [arXiv:0805.0568 [gr-qc]].[13] M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A (1939) 211.[14] N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Annals Phys. (2003) 96 [arXiv:hep-th/0210184].[15] H. van Dam and M. J. G. Veltman, Nucl. Phys. B (1970) 397. V. I. Zakharov, JETP Lett. (1970) 312 [Pisma Zh.Eksp. Teor. Fiz. (1970) 447].[16] E. . M. Lifshitz, Zh. Eksp. Teor. Fiz. D (1941) 255 & 269.[17] D. Anselmi and M. Halat, Phys. Rev. D , 125011 (2007) [arXiv:0707.2480 [hep-th]].[18] M. Visser, arXiv:0902.0590 [hep-th].[19] T. Jacobson and A. C. Wall, arXiv:0804.2720 [hep-th].[20] S. L. Dubovsky and S. M. Sibiryakov, Phys. Lett. B , 509 (2006) [arXiv:hep-th/0603158].[21] P. Hoˇrava, arXiv:0811.2217 [hep-th].[22] R. L. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. , 1595 (1960).[23] J. H. Taylor and J. M. Weisberg, Astrophys. J.253