Hořava–Lifshitz holography

Tatsuma Nishioka

Department of Physics, Kyoto University, Kyoto 606-8502, Japan
² Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8582, Japan

Received 23 September 2009, in final form 26 October 2009
Published 16 November 2009

A renormalizable theory of gravity, so-called the Hořava–Lifshitz gravity, has been attracting great interest since the advent of the seminal works of Hořava [1, 2]. This theory does not have the full diffeomorphism invariance, but the following anisotropic scaling with dynamical critical exponent z larger than the spatial dimensions d exists

This theory has a remarkable property such that it describes the non-relativistic theory of gravity in the UV regime, but becomes the Einstein gravity in the IR region. It was extensively applied to a resolution of the cosmological problem including inflation and non-Gaussianity in [3–21], and new solutions were constructed in [22–28]. For the other recent progress, refer the reader to [29–53].

It becomes convenient in the following discussion to take the Wick rotation to the Euclidean space. The ADM decomposition of the (d  +  1)-dimensional Euclidean space is

and the extrinsic curvature for the spacelike slice is given by

The action of the Hořava–Lifshitz gravity is defined as [1, 2]

where we set a future boundary at t  =  t₀ and is the inverse of the DeWitt metric G^ijkl of the space of metrics

The first and second terms in (4) are the kinetic and potential terms, respectively, and E^ij does not contain the time derivative of g_ij. When we require E^ij to be a gradient of some function W[g] with respect to the metric g_ij

it is called a `detailed balance condition' [1, 2]. In general, in the context of condensed matter physics, it is known that theories which satisfy the detailed balance condition have simpler quantum properties than a generic theory, and that the renormalization properties in (d  +  1) dimensions are often inherited from the simpler renormalization of the theory in d dimensions with the action W. It still remains to be understood what the detailed balance condition means in the context of the Hořava–Lifshitz gravity, however. We would like to get a deep insight into the role of it.

In this paper, we derive the detailed balance condition as a solution to the Hamilton–Jacobi equation in the Hořava–Lifshitz gravity. This result leads us to propose the existence of the d-dimensional quantum field theory with the effective action W on the future boundary of the (d  +  1)-dimensional Hořava–Lifshitz gravity from the viewpoint of the holographic renormalization group [54]. This proposal reminds us of the dS/CFT correspondence [55], while the detailed balance condition forces the cosmological constant to be always negative^Note1. In addition, we obtain a Ricci flow equation of the boundary theory as the holographic RG flow, which is the Hamilton equation in the bulk gravity.

Although this proposal is the most salient feature of our work, we should emphasize that we can derive the detailed balance condition and the Ricci flow equation based on the Hamiltonian formulation of the Hořava–Lifshitz gravity without the holography.

The Hamiltonian formulation in the bulk gravity affords us the renormalization group flow of the field theory on the future boundary t  =  t₀ along the time direction as the Hamilton equation. Such a flow is termed a holographic renormalization group flow initiated by [54] (for a comprehensive review see [56] and references therein).

The ADM decomposition (2) is suitable for the Hamiltonian formulation. The conjugate momentum associated with g_ij in the Hořava–Lifshitz action (4) is ( )

and the momenta conjugate to N and N_i are identically zero. The Hamiltonian is

with

The last term in the second line of (9) vanishes when Σ_t is compact space; this is the case we focus on here.

In the Hamiltonian formulation, the conjugate momentum π^ij becomes the independent variable instead of . Using the momentum, we can take the action to the one in the first-order form

Varying this action, we obtain the Hamilton equation

with the Hamiltonian and momentum constraints

Substituting the classical solution g_c into the action (12) and integrating it along the time direction, one can express S[g] as a surface integral with respect to g_c(x, t₀)^Note2

It follows from this relation that S_bdy is the effective action of the d-dimensional quantum field theory on the future boundary using the bulk/boundary relation Z_gravity  =  Z_QFT and Z_gravity  =  exp(–S) in a manner similar to the AdS/CFT correspondence [57–59].

In this case, the momentum is expressed in terms of the boundary action (see [56] for a careful derivation)

We use the same notation g_ij to denote g_c(x, t₀) for simplicity hereafter. Inserting these relations into the constraints (14), we obtain the momentum constraint

which indicates the conservation law of the energy momentum tensor in the d-dimensional QFT, and the Hamilton–Jacobi equation from the Hamiltonian constraint

This equation is easily solved^Note3

where W stands for the rescaled effective action of the d-dimensional QFT

The solution (19) to the Hamiltonian constraint results in the detailed balance condition (7) and we find that W is the (rescaled) effective action of QFT on the future boundary of the Hořava–Lifshitz gravity.

It is worth mentioning that the Hamilton equation (13) gives us the holographic renormalization group flow after substituting (16) into it

This is a simpler equation in first order as opposed to the original equation of motion which is second order in time derivatives. The Hamilton equation for π^ij is automatically satisfied and it is enough to solve (13)^Note4. As was mentioned in [1, 2], we can easily find classical solutions in the Hořava–Lifshitz gravity by just solving this equation.

Moreover, this equation has a remarkable property as follows. When we take the d-dimensional effective action W for the Einstein–Hilbert action

the (d  +  1)-dimensional theory becomes the Hořava–Lifshitz gravity with dynamical critical exponent z  =  2. The holographic RG flow (21) is given by [1]

If we take λ  =  1/2, κ_W  =  κ/2, N  =  1 and N_i  =  0, this becomes the Ricci flow equation^Note5. From this viewpoint one may say that the Ricci flow in d dimensions is the holographic RG flow to the (d  +  1)-dimensional Hořava–Lifshitz gravity with z  =  2 and λ  =  1/2.

One simple but interesting application of our results is that static solutions in the Hořava–Lifshitz gravity are obtainable by solving^Note6

In the case of the four-dimensional Hořava–Lifshitz gravity with z  =  3, W is the Einstein–Hilbert action with the gravitational Chern–Simons term in three dimensions (i.e. the topologically massive gravity). Equation (24) is just the Einstein equation in TMG and the interesting solutions were constructed in [60–62]. For example, we can construct the four-dimensional solitonic solution by the use of the Euclidean warped AdS₃ black hole. It would be of wide interest to investigate such a solution in higher dimensions with different dynamical exponents z.

Acknowledgments

We are grateful to K Izumi S Minakami S Mukohyama K Murata T Kobayashi for valuable discussions and S Horiuchi for careful reading of this manuscript. This work is supported by JSPS Grant-in-Aid for Scientific Research no. 19·3589 and by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.

[1] 

Hořava P 2009 Membranes at quantum criticality J. High Energy Phys. JHEP03(2009)020 (arXiv:0812.4287 [hep-th]) 
IOPsciencePreprint

[2] 

Hořava P 2009 Quantum gravity at a Lifshitz point Phys. Rev. D 79 084008 (arXiv:0901.3775 [hep-th]) 
CrossRefPreprint

[3] 

Takahashi T and Soda J 2009  Chiral primordial gravitational waves from a Lifshitz point arXiv:0904.0554 [hep-th]
Preprint

[4] 

Calcagni G 2009  Cosmology of the Lifshitz universe arXiv:0904.0829 [hep-th]
Preprint

[5] 

Kiritsis E and Kofinas G 2009  Hořava–Lifshitz cosmology arXiv:0904.1334 [hep-th]
Preprint

[6] 

Mukohyama S 2009  Scale-invariant cosmological perturbations from Hořava–Lifshitz gravity without inflation arXiv:0904.2190 [hep-th]
Preprint

[7] 

Brandenberger R 2009  Matter bounce in Hořava–Lifshitz cosmology arXiv:0904.2835 [hep-th]
Preprint

[8] 

Piao Y S 2009  Primordial perturbation in Hořava–Lifshitz cosmology arXiv:0904.4117 [hep-th]
Preprint

[9] 

Gao X 2009  Cosmological perturbations and non-Gaussianities in Hořava–Lifshitz gravity arXiv:0904.4187 [hep-th]
Preprint

[10] 

Mukohyama S, Nakayama K, Takahashi F and Yokoyama S 2009  Phenomenological aspects of Hořava–Lifshitz cosmology arXiv:0905.0055 [hep-th]
Preprint

[11] 

Chen B, Pi S and Tang J Z 2009  Scale invariant power spectrum in Hořava–Lifshitz cosmology without matter arXiv:0905.2300 [hep-th]
Preprint

[12] 

Saridakis E N 2009  Horava–Lifshitz dark energy arXiv:0905.3532 [hep-th]
Preprint

[13] 

Mukohyama S 2009 Dark matter as integration constant in Horava–Lifshitz gravity Phys. Rev. D 80 064005 (arXiv:0905.3563 [hep-th]) 
CrossRefPreprint

[14] 

Gao X, Wang Y, Brandenberger R and Riotto A 2009  Cosmological perturbations in Hořava–Lifshitz gravity arXiv:0905.3821 [hep-th]
Preprint

[15] 

Minamitsuji M 2009  Classification of cosmology with arbitrary matter in the Hořava–Lifshitz theory arXiv:0905.3892 [astro-ph.CO]
Preprint

[16] 

Wang A and Wu Y 2009 Thermodynamics and classification of cosmological models in the Horava–Lifshitz theory of gravity J. Cosmol. Astropart. Phys JCAP07(2009)012 (arXiv:0905.4117 [hep-th]) 
IOPsciencePreprint

[17] 

Nojiri S and Odintsov S D 2009  Covariant Horava-like renormalizable gravity and its FRW cosmology arXiv:0905.4213 [hep-th]
Preprint

[18] 

Park M i 2009  A Test of Horava gravity: the dark energy arXiv:0906.4275 [hep-th]
Preprint

[19] 

Yamamoto K, Kobayashi T and Nakamura G 2009 Breaking the scale invariance of the primordial power spectrum in Horava–Lifshitz cosmology Phys. Rev. D 80 063514 (arXiv:0907.1549 [astro-ph.CO]) 
CrossRefPreprint

[20] 

Wang A and Maartens R 2009  Linear perturbations of cosmological models in the Horava–Lifshitz theory of gravity without detailed balance arXiv:0907.1748 [hep-th]
Preprint

[21] 

Kobayashi T, Urakawa Y and Yamaguchi M 2009  Large-scale evolution of the curvature perturbation in Horava–Lifshitz cosmology arXiv:0908.1005 [astro-ph.CO]
Preprint

[22] 

Lu H, Mei J and Pope C N 2009  Solutions to Hořava gravity arXiv:0904.1595 [hep-th]
Preprint

[23] 

Nastase H 2009  On IR solutions in Hořava gravity theories arXiv:0904.3604 [hep-th]
Preprint

[24] 

Colgain E O and Yavartanoo H 2009  Dyonic solution of Hořava–Lifshitz gravity arXiv:0904.4357 [hep-th]
Preprint

[25] 

Cai R G, Cao L M and Ohta N 2009  Topological black holes in Hořava–Lifshitz gravity arXiv:0904.3670 [hep-th]
Preprint

[26] 

Cai R G, Liu Y and Sun Y W 2009  On the z  =  4 Hořava–Lifshitz gravity arXiv:0904.4104 [hep-th]
Preprint

[27] 

Ghodsi A and Hatefi E 2009  Extremal rotating solutions in Horava gravity arXiv:0906.1237 [hep-th]
Preprint

[28] 

Lee H W, Kim Y W and Myung Y S 2009  Extremal black holes in the Hořava–Lifshitz gravity arXiv:0907.3568 [hep-th]
Preprint

[29] 

Visser M 2009  Lorentz symmetry breaking as a quantum field theory regulator arXiv:0902.0590 [hep-th]
Preprint

[30] 

Maccione L, Taylor A M, Mattingly D M and Liberati S 2009 Planck-scale Lorentz violation constrained by ultra-high-energy cosmic rays J. Cosmol. Astropart. Phys JCAP04(2009)022 arXiv:0902.1756 [astro-ph.HE]) 
IOPsciencePreprint

[31] 

Carvalho P R S and Leite M M 2009  Callan–Symanzik–Lifshitz approach to generic competing systems arXiv:0902.1972 [hep-th]
Preprint

[32] 

Hořava P 2009  Spectral dimension of the universe in quantum gravity at a Lifshitz point arXiv:0902.3657 [hep-th]
Preprint

[33] 

Volovich A and Wen C 2009  Correlation functions in non-relativistic holography arXiv:0903.2455 [hep-th]
Preprint

[34] 

Jenkins A 2009  Constraints on emergent gravity arXiv:0904.0453 [gr-qc]
Preprint

[35] 

Kluson J 2009  Branes at quantum criticality arXiv:0904.1343 [hep-th]
Preprint

[36] 

Pal S S 2009  Non-relativistic supersymmetric Dp branes arXiv:0904.3620 [hep-th]
Preprint

[37] 

Chen B and Huang Q G 2009  Field theory at a Lifshitz point arXiv:0904.4565 [hep-th]
Preprint

[38] 

Myung Y S and Kim Y W 2009  Thermodynamics of Hořava–Lifshitz black holes arXiv:0905.0179 [hep-th]
Preprint

[39] 

Cai R G, Hu B and Zhang H B 2009  Dynamical scalar degree of freedom in Horava–Lifshitz gravity arXiv:0905.0255 [hep-th]
Preprint

[40] 

Orlando D and Reffert S 2009  On the renormalizability of Horava–Lifshitz-type gravities arXiv:0905.0301 [hep-th]
Preprint

[41] 

Chen S and Jing J 2009  Quasinormal modes of a black hole in the deformed Horava–Lifshitz gravity arXiv:0905.1409 [gr-qc]
Preprint

[42] 

Konoplya R A 2009 Towards constraining of the Horava–Lifshitz gravities Phys. Lett. B 679 499 (arXiv:0905.1523 [hep-th]) 
CrossRefPreprint

[43] 

Chen S b and Jing J l 2009 Strong field gravitational lensing in the deformed Horava–Lifshitz black hole Phys. Rev. D 80 024036 (arXiv:0905.2055 [gr-qc]) 
CrossRefPreprint

[44] 

Li M and Pang Y 2009 A trouble with Hořava–Lifshitz gravity J. High Energy Phys. JHEP08(2009)015 (arXiv:0905.2751 [hep-th]) 
IOPsciencePreprint

[45] 

Kim Y W, Lee H W and Myung Y S 2009  Nonpropagation of scalar in the deformed Hořava–Lifshitz gravity arXiv:0905.3423 [hep-th]
Preprint

[46] 

Calcagni G 2009  Detailed balance in Horava–Lifshitz gravity arXiv:0905.3740 [hep-th]
Preprint

[47] 

Sakamoto M 2009 Strong coupling quantum Einstein gravity at a z  =  2 lifshitz point Phys. Rev. D 79 124038 (arXiv:0905.4326 [hep-th]) 
CrossRefPreprint

[48] 

Park M I 2009 The black hole and cosmological solutions in IR modified Horava gravity J. High Energy Phys. JHEP09(2009)123 (arXiv:0905.4480 [hep-th]) 
IOPsciencePreprint

[49] 

Myung Y S 2009  Propagations of massive graviton in the deformed Hořava–Lifshitz gravity arXiv:0906.0848 [hep-th]
Preprint

[50] 

Germani C, Kehagias A and Sfetsos K 2009 Relativistic quantum gravity at a Lifshitz point J. High Energy Phys. JHEP09(2009)060 (arXiv:0906.1201 [hep-th]) 
IOPsciencePreprint

[51] 

Mukohyama S 2009 Caustic avoidance in Horava–Lifshitz gravity J. Cosmol. Astropart. Phys JCAP09(2009)005 (arXiv:0906.5069 [hep-th]) 
IOPsciencePreprint

[52] 

Harko T, Kovacs Z and Lobo F S N 2009  Solar system tests of Hořava–Lifshitz gravity arXiv:0908.2874 [gr-qc]
Preprint

[53] 

Bogdanos C and Saridakis E N 2009  Perturbative instabilities in Horava gravity arXiv:0907.1636 [hep-th]
Preprint

[54] 

Boer J de, Verlinde E P and Verlinde H L 2000 On the holographic renormalization group J. High Energy Phys. JHEP08(2000)003 (arXiv:hep-th/9912012) 
IOPsciencePreprint

[55] 

Strominger A 2001 The dS/CFT correspondence J. High Energy Phys. JHEP10(2001)034 (arXiv:hep-th/0106113) 
IOPsciencePreprint

[56] 

Fukuma M, Matsuura S and Sakai T 2003 Holographic renormalization group Prog. Theor. Phys. 109 489 (arXiv:hep-th/0212314) 
CrossRefPreprint

[57] 

Maldacena J M 1998 The large N limit of superconformal field theories and supergravity Adv. Theor. Math. Phys. 2 231 

Maldacena J M 1999 Int. J. Theor. Phys. 38 1113 (arXiv:hep-th/9711200) 
CrossRefPreprint

[58] 

Gubser S S, Klebanov I R and Polyakov A M 1998 Gauge theory correlators from non-critical string theory Phys. Lett. B 428 105 (arXiv:hep-th/9802109) 
CrossRefPreprint

[59] 

Witten E 1998 Anti-de Sitter space and holography Adv. Theor. Math. Phys. 2 253 (arXiv:hep-th/9802150) 
Preprint

[60] 

Nutku Y 1993 Exact solutions of topologically massive gravity with a cosmological constant Class. Quantum Grav. 10 2657 
IOPscience

[61] 

Bouchareb A and Clement G 2007 Black hole mass and angular momentum in topologically massive gravity Class. Quantum Grav. 24 5581 (arXiv:0706.0263 [gr-qc]) 
IOPsciencePreprint

[62] 

Anninos D, Li W, Padi M, Song W and Strominger A 2009 Warped AdS₃ black holes J. High Energy Phys. JHEP03(2009)130 (arXiv:0807.3040 [hep-th]) 
IOPsciencePreprint

Notes

Note1

 We are grateful to S Mukohyama for pointing out this feature.

Note2

 We are grateful to K Murata for discussion of this point.

Note3

 We must mention that there might exist other solutions than the one (19) we obtained here. In fact, there is an ambiguity such that we can add a zero norm term to the right (or left)-hand side of (19), though we can remove this ambiguity by adding a zero norm term to E^ij in the action from the beginning.

Note4

 We are grateful to K Izumi for informing us of this point.

Note5

 We can set N_i  =  0 using the d-dimensional diffeomorphism, but we are not sure if we can take N  =  1 by the reparametrization of the time t → f(t).

Note6

 One can find the same discussion in [1] under the assumption of (21).