Infrared limit of Horava’s gravity with the global Hamiltonian constraint

Archil Kobakhidze*

School of Physics, The University of Melbourne, Victoria 3010, Australia(Received 29 June 2009; revised manuscript received 20 June 2010; published 7 September 2010)

We show that Horava’s theory of gravitation with the global Hamiltonian constraint does not reproduce

general relativity in the infrared domain. There is one extra propagating degree of freedom, besides those

two associated with the massless graviton, which does not decouple.

DOI: 10.1103/PhysRevD.82.064011 PACS numbers: 04.50.Kd, 04.60.Bc, 11.30.Cp

I. INTRODUCTION

Recently, Horava proposed a power-counting renorma-lizable higher-derivative theory of gravitation wherethe full diffeomorphism invariance is broken down to thefoliation-preserving diffeomorphism [1]. Because of thereduced symmetry, the ghost states usually associatedwith the higher time derivatives in general relativity(GR), are removed, and thus the theory is unitary. A vitalquestion is whether Horava’s theory of gravitation has aninfrared limit consistent with observations. Since the ob-servational success of GR is largely based on its fulldiffeomorphism invariance, it is clear that any theorywith reduced diffeomorphism invariance will deviatefrom GR in both the ultraviolet and infrared regimes.From the purely phenomenological point of view, it isimportant to understand whether this deviation in the in-frared regime can be made consistent with observations. InRef. [2], it has been pointed out that the scalar polarizationof the graviton does not decouple in Horava’s theory. Also,it has been shown in Ref. [3] that in Horava’s theory with alocal Hamiltonian constraint the Poisson algebra is notclosed. From this perspective it seems vital to retain the‘‘projectability condition’’ which generates a less restric-tive global Hamiltonian constraint [4]. In this paper, weconsider the infrared limit of Horava’s theory with theglobal Hamiltonian constraint. By applying Dirac’s con-straint analysis, we show that there is one extra propagatingdegree of freedom, besides those two associated with themassless graviton. Therefore, Horava’s theory of gravita-tion does not reproduce general relativity in the infraredregime.

II. DIRAC’S CONSTRAINTANALYSIS

In the infrared limit, Horava’s theory is described by thefollowing action1:

SHorava ¼Z

dtZ�t

d3xð�ij _hij � NH 0 � NiH iÞ; (1)

where

H 0 ¼� ffiffiffih

p ð3ÞRþh�1=2�ij�ij � �

ð3�� 1Þh�1=2�2; (2)

H i ¼ �2rið�ijÞ; (3)

and �ij is a canonical momentum for the spatial metric hijdefined on a spatial hypersurface �t [h � detðhijÞ]. ð3ÞRdenotes the Ricci scalar buildup from the metric hij, andri

is the covariant derivative associated with hij. It has been

argued (but has not been explicitly shown) in Ref. [1] thatthe effective running parameter � has an infrared fixedpoint at � ¼ 1. In what follows, we also assume that� ¼ 1. We also set to 0 the cosmological constant in theoriginal theory [1]. This seems, in principle, possible if the‘‘detailed balance condition’’ of the original theory isabandoned [5]. Anyway, the cosmological constant is notimportant for our analyses. With these simplifying assump-tions, the action (1) looks identical to the GR action withone important exception: The lapse function depends onlyon time variable N ¼ NðtÞ. Therefore, the action (1) isinvariant under the reduced diffeomorphism symmetry,the foliation-preserving diffeomorphism transformations.Since Horava’s theory maintains a smaller diffeomor-

phism group, an important question is, how many degreesof freedom does the theory describe? This question can beanswered by applying Dirac’s constraint analysis. We define

canonical momenta: �N � @LHorava

@ _N� 0, �i

N � @LHorava

@ _Ni� 0,

and �ij � @LHorava

@ _hij. The basic Poisson commutators read

fN;�NgPB ¼ 1; (4)

fNiðxÞ; �jNðyÞgPB ¼ �j

i�3ðx� yÞ; (5)

and

fhijðxÞ; �mnðyÞgPB ¼ 12ð�m

i �nj þ �n

i �mj Þ�3ðx� yÞ: (6)

Note that, because the lapse function depends only on thetime coordinate, its canonical momentum �N is alsox-independent. Consequently, the corresponding secondaryHamiltonian constraint is also satisfied only for the zeromode of H 0. Indeed, taking the Hamiltonian

*archilk@unimelb.edu.au1We adopt units where 16�GN ¼ 1.

PHYSICAL REVIEW D 82, 064011 (2010)

1550-7998=2010=82(6)=064011(4) 064011-1 � 2010 The American Physical Society

H ¼ NZ�t

d3xH 0 þZ�t

d3xNiH i; (7)

one easily finds

_�N ¼ f�N;HgPB ¼Z�t

d3yH 0ðyÞ: (8)

That is, to preserve the primary constraint �N � 0 in time,one must assume

H0 �Z�t

d3xH 0 � 0: (9)

The secondary momentum constraints, on the other hand,are satisfied locally, i.e. at each given point x:

H iðxÞ � 0; (10)

just like in GR [6]. Next, we compute the algebra ofsecondary constraints:

fH0; H0gPB ¼ZZ

d3xd3yð2H iðxÞ@ðxÞi �3ðx� yÞ

þ @ðxÞi H iðxÞ�3ðx� yÞÞ¼

Zd3x@ðxÞi H iðxÞ ¼ 0; (11)

where we have usedRd3x@ðxÞi �3ðxÞ ¼ 0 and H i!jxj!10;

fH iðxÞ; H0gPB ¼ H 0ðxÞ@ðxÞi

Zd3y�3ðx� yÞ ¼ 0; (12)

and the last commutator

fH iðxÞ;H jðyÞgPB ¼ H iðyÞ@ðyÞj �3ðx� yÞ�H jðxÞ@ðxÞi �3ðy� xÞ

� 0 (13)

is the same as the one in GR [6]. Using the above relationswe verify that no further constraints emerge:

_H 0 ¼ fH0; HgPB ¼ 0; _H i ¼ fH i; HgPB � 0: (14)

The set of constraints f�N;�iN;H0;H ig for zero modes

and the set of constraints f�iN;H

ig for propagating modesrepresent nonsingular systems of the first-class constraints.Therefore, we can directly apply Dirac’s method of count-ing the physical degrees of freedom:

½number of physical degrees of freedom�¼ 1

2½number of canonical variables�� ½number of first-class constraints�: (15)

Applying the above counting to nonpropagating(x-independent) zero modes, we obtain 10� 8 ¼ 2, simi-lar to GR. For propagating modes, however, we have oneextra physical degree of freedom: 9� 6 ¼ 3. The differ-ence in global and local degrees of freedom seems to be

related with the nonlocal nature of the Horava gravitywhere the lapse function strictly depends only on thetime coordinate.It is instructive to compare the counting of degrees of

freedom in Horava’s theory with the counting in othertheories with reduced diffeomorphism invariance.Namely, in the unimodular theory of gravitation onlycoordinate transformations with unit Jacobian are admis-

sible. In that case, ðh1=2NÞ is fixed, and, hence, there is noHamiltonian constraint. Instead, there are first-class terti-

ary constraints, @iðh�1=2H 0Þ ¼ 0 [7]. These tertiary con-straints can be solved by defining a new Hamiltonian

constraint, H 00 � H 0 þ h1=2� � 0, where � is a zero-

mode field, @i� ¼ 0. The conservation of this Hamiltonianconstraint then implies that� is actually a ‘‘vacuum’’ field,� ¼ const. Thus, besides the 2 degrees of freedom asso-ciated with massless graviton, one accounts in additionliterally one (not per each point x) global degree of free-dom. This global degree of freedom turns out to be acosmological constant, and the ‘‘cosmic’’ time is itscanonically conjugated variable. Consequently, one canrewrite the unimodular theory as an equivalent fully cova-riant theory by reparameterizing the time coordinate [7]without introducing new propagating degrees of freedom.A similar trick seems impossible in Horava’s theory,because the extra degree of freedom is a propagatingmode. That is to say, to covariantize Horava’s gravity onenecessarily needs to invoke a new dynamical field.2

III. A COVARIANTACTION

In this section wewould like to write down an equivalentto (1) action where the lapse function is promoted to a fullspace-time-dependent field, N ¼ Nðx; tÞ. The action (1)turns then into a Einstein-Hilbert action. The projectabilitycondition on the lapse function can be enforced through theequation of motion @iN ¼ 0. In order to achieve this in acovariant way, we introduce a spatial two-form field Aij,

whose field strength is F ijk ¼ @½iAjk�. The dual field

strength then represents spatial density:

~F � h1=21

3!�ijkF ijk ¼ @i

~Ai; (16)

where ~Ai ¼ h1=2�ijkAjk is a 3-vector density. The diffeo-

morphism invariant action equivalent to (1) then takes theform

Sequiv ¼ SGR þZ

d4xN@i~Ai; (17)

2Such a covariant theory has been proposed recently inRef. [8]. However, it is assumed there that the lapse functionis a full-fledged field. It has been claimed also that for � ¼ 1 theextra degree of freedom ‘‘freezes out.’’ We note here that the setof constraints introduced in Ref. [8] seems to be singular, andthus the counting of degrees of freedom must be taken with care.

ARCHIL KOBAKHIDZE PHYSICAL REVIEW D 82, 064011 (2010)

064011-2

where SGR is the standard Einstein-Hilbert action. Varying

the above action with respect to ~Ai we obtain the desiredconstraint equation

@iN ¼ 0: (18)

Also, the action (18) implies the local Hamiltonianconstraint

H 0 þ @i~Ai � 0: (19)

Integrating the above equation over the constant time

hypersurface and assuming ~Ai!jxj!10, we reproducethe global Hamiltonian constraint (9).

Observe now that the modified Hamiltonian constraint(19) together with the momentum constraint equations andthe dynamical equations of motion form the following setof Einstein’s equations:

ð4ÞG�� ¼ �Fn�n�; (20)

where n� ¼ ð1; 0; 0; 0; Þ and ð4ÞG�� is the Einstein tensor

built from 4D metric tensor ð4Þg��. The additional term in

(17) can be viewed as the action functional of a pressure-

less dust, where F ¼ 2 ~F =h1=2 is the energy density of dustparticles as seen by observers who are at rest in theconstant time hypersurfaces. This observation has beenfirst made in Ref. [4]. However, this is not the ordinarydust fluid as its total energy is zero:

Zd3x ~F ¼ 0: (21)

Note that the ‘‘free’’ (vacuum) limit of Eqs. (20) cannot beachieved because, according to Eq. (19), FðxÞ ¼ 0 wouldcorrespond to the local Hamiltonian constraint for H 0

which is not admissible in Horava’s gravity. In fact, takingthe trace of (20) one can solve for the nondynamicalauxiliary field F:

F ¼ 1

g00R: (22)

Substituting (22) back into (20) we obtain the followingequations:

R�� ��1

2g�� � 1

g00n�n�

�R ¼ 0: (23)

These equations can be viewed as the vacuum equations ofthe theory. They must be supplemented by the condition

(18), where N ¼ ð�g00Þ�1=2. Applying the contractedBianchi identities to (23) one finds @0R ¼ 0, which inturn implies R ¼ �ð ~xÞ, where �ð ~xÞ is a function of spatialcoordinates only. Therefore, the spatial dependence of F(23) is entirely determined by this function F ¼ 1

g00�,

and the global Hamiltonian constraint reads asRd3xh1=2� ¼ 0. Finally, the coupling to a matter energy-

momentum tensor T�� is described by adding 12 �

ðT�� � T��

n�n�g00

Þ to the right-hand side of Eq. (23).

IV. THE WEAK-FIELD LIMIT

Let us now consider the linearized version of (23), byexpanding the metric around flat Minkowski background��:

ð4Þg�� � �� þ ��: (24)

This expansion is justified, providing one considers F in(22) as a small perturbation: F � ���OðhÞ.We choose to work in the temporal gauge �0 ¼ 0. In

this gauge the constraint equation (18) is automaticallysatisfied. The above gauge fixing conditions are preservedby the residual diffeomorphism transformations with �0 ��ð ~xÞ, �i � �ið ~xÞ þ t�0ð ~xÞ. In Ref. [1] this residual invari-ance has been used to impose further conditions: @iij �@j

ii ¼ 0. However, these conditions imply immediately

H 0 � @jð@iij � @jiiÞ þOð2Þ ¼ 0, and therefore they

are not admissible in the theory without a localHamiltonian constraint. That is to say, the absence ofthe local Hamiltonian constraint prevents us from remov-ing an extra (compared to GR) propagating degree offreedom, in full accordance with our generic Dirac con-straint analysis.To see how this extra degree of freedom affects the

gravitational interactions between matter sources, let usmore closely inspect the linearized equations amendedby the conserved matter energy-momentum tensorT��ð@�T�� ¼ 0Þ:

h�� � @�@��� � @�@

��� þ @��

þ ð�� þ 2n�n�Þð@�@� �hÞ¼ �2ðT�� þ n�n�TÞ; (25)

where h � @�@�, � ��, and T � T�

� . The solution to(25) can be written as

�� ¼ ~�� þ GR��; (26)

where GR�� is the solution of the corresponding Einstein’s

equation:

GR�� ¼

ZdyGGR

��� ðx� yÞT� ðyÞ: (27)

GGR��� in the above equation is the GR causal graviton

propagator in the temporal gauge, n�GGR��� ¼

n�GGR��� ¼ 0. ~�� in (26) satisfies the homogeneous

equation

h~�� � @�@� ~�� � @�@

� ~�� þ @�� ~

þ ð�� þ 2n�n�Þð@�@� �h~Þ ¼ 0 (28)

and, thus, contributes to the on-shell part of the totalpropagator, modifying normal analytic properties whichcharacterize the standard causal propagators. Indeed,the standard causal propagator is determined (up to atensorial part) by the pole structure in the momentum

INFRARED LIMIT OF HORAVA’s GRAVITY WITH THE . . . PHYSICAL REVIEW D 82, 064011 (2010)

064011-3

space, GðpÞ / ðp2 � i�Þ�1 ¼ P:V:p�2 � i��ðp2Þ. Thispropagator is obtained by prescribing appropriateasymptotic boundary conditions at t ¼ �1. Assumingthat GGR

��� in (27) is such a causal propagator, it is

easy to see that the total propagator in the Horava gravitywill contain an additional, Lorentz noninvariant, on-shellpiece due to the contribution from ~�� obeying (28):

GGR��� ðpÞ / P:V:p�2 � i�ð1þ fðpÞÞ�ðp2Þ, where

fðp0; pi ¼ 0Þ ¼ 0. Since the pole structure of the propa-gator is ultimately related to the unitarity and causality, thegraviton exchange amplitudes in Horava’s theory are likelyto fail to satisfy unitarity and causality conditions. This isthe consequence of the modified asymptotic boundaryconditions in the Horava gravity due to the nondecouplingof the extra scalar mode which has been established rigor-ously, without recourse to a background metric and thelinearized approximation, in the previous sections.

The discontinuity of an apparent GR limit is typical intheories with broken diffeomorphism invariance. The well-known example is the Pauli-Fierz theory of massive grav-ity. The massless limit of massive gravity is known to bediscontinuous [9,10] and does not coincide with GR, atleast at perturbative level. The key reason is, of course,breaking of the diffeomorphism invariance by the gravitonmass, so massless and massive theories describe a differentnumber of degrees of freedom. In particular, the scalargraviton does not decouple in the massless limit.Similarly, in Horava’s theory with a global Hamiltonian

constraint, breaking of the full diffeomorphism invarianceresults in propagating a massless scalar graviton.Moreover, contrary to the massive gravity [11,12], theproblem of an extra unwanted degree of freedom seemsto persist in Horava’s theory with the cosmological con-stant, because the dust acts not like a mass but as a sourcewhich cannot be switched off.In conclusion, Horava’s theory of gravitation with the

global Hamiltonian constraint in the infrared regime con-tains an extra propagating massless degree of freedomwhich cannot be removed. This, together with other nega-tive results reported in the literature, puts serious doubt onthe validity of the theory.

ACKNOWLEDGMENTS

I am indebted to Ray Volkas for discussions on variousaspects of Horava’s gravity. Thework was supported by theAustralian Research Council.Note added.—During the preparation of this paper,

Ref. [13] appeared on the hep-th archive, where someimportant issues related with the extra degree of freedomin Horava’s theory with a local Hamiltonian constraint hasbeen clarified. They have also briefly discussed the versionof Horava’s theory with a global Hamiltonian constraint,by associating it with a covariant theory admitting a ghostcondensate. This treatment is different from the one dis-cussed in the present paper.

[1] P. Horava, Phys. Rev. D 79, 084008 (2009); J. HighEnergy Phys. 03 (2009) 020.

[2] C. Charmousis, G. Niz, A. Padilla, and P.M. Saffin, J.High Energy Phys. 08 (2009) 070.

[3] M. Li and Y. Pang, J. High Energy Phys. 08 (2009) 015.[4] S. Mukohyama, Phys. Rev. D 80, 064005 (2009).[5] T. Sotiriou, M. Visser, and S. Weinfurtner, Phys. Rev. Lett.

102, 251601 (2009).[6] B. S. DeWitt, Phys. Rev. 160, 1113 (1967).[7] M. Henneaux and C. Teitelboim, Phys. Lett. B 222, 195

(1989).

[8] C. Germani, A. Kehagias, and K. Sfetsos, J. High EnergyPhys. 09 (2009) 060.

[9] H. van Dam and M. J. G. Veltman, Nucl. Phys. B22, 397(1970).

[10] V. I. Zakharov, Pis’ma Zh. Eksp. Teor. Fiz. 12, 447 (1970)[JETP Lett. 12, 312 (1970)].

[11] I. I. Kogan, S. Mouslopoulos, and A. Papazoglou, Phys.Lett. B 503, 173 (2001).

[12] M. Porrati, Phys. Lett. B 498, 92 (2001).[13] D. Blas, O. Pujolas, and S. Sibiryakov, J. High Energy

Phys. 10 (2009) 029.

ARCHIL KOBAKHIDZE PHYSICAL REVIEW D 82, 064011 (2010)

064011-4