# Gravity’s Rainbow: a bridge towards Hořava–Lifshitz gravity

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## Abstract

We investigate the connection between Gravity’s Rainbow and Hořava–Lifshitz gravity, since both theories incorporate a modification in the ultraviolet regime which improves their quantum behavior at the cost of the Lorentz invariance loss. In particular, extracting the Wheeler–De Witt equations of the two theories in the case of Friedmann–Lemaître–Robertson–Walker and spherically symmetric geometries, we establish a correspondence that bridges them.

## Keywords

Scalar Curvature Planck Scale Potential Term Hamiltonian Constraint Lapse Function## 1 Introduction

The idea that general relativity (GR) is not the fundamental gravitational theory and that it needs to be modified or extended is quite old. On the one hand, the idea of a small-scale, ultraviolet (UV) modification of GR arises from the non-renormalizability of the theory and the difficulties towards its quantization [1]. In particular, since the usual loop-expansion procedure gives rise to UV-divergent Feynman diagrams, the requirement for a UV-complete gravitational theory, which has GR as a low-energy limit, becomes necessary. On the other hand, we know that the large-scale, infrared (IR) modifications of GR might be the explanation of the observed late-time universe acceleration (see [2] and references therein) and/or of the inflationary stage [3]. Due to their significance, both directions led to a huge amount of research.

Concerning the modification of the UV behavior, it was realized that the insertion of higher-order derivative terms in the Lagrangian establishes renormalizability, since these terms modify the graviton propagator at high energies [1]. However, this leads to an obvious problem, namely that the equations of motion involve higher-order time derivatives and thus the application of the theory leads to ghosts. Nevertheless, based on the observation that it is the higher spatial derivatives that improve renormalizability, while it is the higher time derivatives that lead to ghosts, some years ago Hořava had the idea to construct a theory that allows for the inclusion of higher spatial derivatives only. In order to achieve this, and motivated by the Lifshitz theory of solid state physics [4], he broke the “democratic treating” of space and time in the UV regime, introducing an anisotropic, Lifshitz scaling between them [5, 6, 7, 8]. Hence, higher spatial derivatives are not accompanied by higher time ones (definitely this corresponds to Lorentz violation), and thus in the UV the theory exhibits power-counting renormalizability but still without ghosts. Finally, the theory presents GR as an IR fixed point, as required, where Lorentz invariance is restored and space and time are handled on equal footing.

On the other hand, in [9] the authors followed a different approach. In particular, instead of modifying the action, they constructed an UV modification of the metric itself, in a construction named Gravity’s Rainbow (GRw) [9]. Hence, the deformed metric in principle exhibits a different treatment between space and time in the UV, namely on scales near the Planck scale, depending on the energy of the particle probing the space-time, while at low energies one recovers the standard metric, and General Relativity is restored. Physically, one can think of it as a deformation of the metric by the Planck-scale graviton. This deformation has been shown to cure divergences (at least to one loop) avoiding any regularization/renormalization scheme [10, 11]. Hence, due to this advantage, a large amount of research has been devoted to GRw [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

In the present work we are interested in examining whether there is a correspondence between Hořava–Lifshitz gravity and GRw, since both directions result in a modification of the equations in the UV regime, while they both present GR as their low-energy limit. In particular, since GR provides a natural scheme for quantization of the gravitational field, namely the Wheeler–De Witt (WDW) equation [34], which is a quantum version of the Hamiltonian constraint obtained from the Arnowitt–Deser–Misner decomposition of space-time, we will impose the requirement that the WDW equation must be satisfied by GRw and Hořava–Lifshitz gravity, respectively. We will examine this correspondence on the Friedmann–Lemaître–Robertson–Walker (FLRW) metric at the mini-superspace level, where the problem with the scalar graviton is absent, as well as in spherically symmetric geometries.

The manuscript is organized as follows: in Sect. 2 we review the basic elements of Hořava–Lifshitz theory, while in Sect. 3 we extract the corresponding WDW equation in the case of FLRW space-time. In Sect. 4 we extract the WDW equation for GRw in the case of FLRW space-time. In Sect. 5 we establish the correspondence between the two theories, while in Sect. 6 we obtain this relation for spherically symmetric space-times. Finally, we summarize our results in Sect. 7.

Throughout this manuscript we use units in which \(\hbar =c=k=1\).

## 2 Hořava–Lifshitz gravity

*N*and shift \(N_{i}\) functions, and the spatial metric \(g_{ij}\) (Latin indices denote spatial coordinates). The coordinate scaling transformations are written as

*g*is the determinant of the spatial metric \(g_{ij}\). The constant \(\lambda \) is a dimensionless running coupling, which takes the value \(\lambda =1\) in the IR limit. The potential part \(\mathcal {L}_{P}\) can in principle contain many terms. However, one can make additional assumptions in order to reduce the possible terms, thus resulting to various versions of the theory. In the following we review the basic ones.

### 2.1 Detailed-balance version

*D*-dimensional one. Physically, it corresponds to the requirement that the potential term should arise from a superpotential. This condition reduces significantly the potential part of the action, resulting in

*w*, \(\mu \) and \(\Lambda \). We mention that the detailed-balance condition, apart from reducing the possible terms in the potential part of the action, additionally correlates their coefficients, and thus the total number of coefficients is smaller than the total number of terms.

### 2.2 Projectable version

*N*. In this case, and neglecting parity-violating terms, the potential part of the action becomes [35, 36]

### 2.3 Non-projectable version

*R*, and \(\mathcal {L}_{4}\) and \(\mathcal {L}_{6}\), respectively, contain all possible fourth and sixth order invariants that can be constructed by \(a_{i}\) and \(g_{ij}\) and their combinations and contractions. Clearly, the above potential term contains much more terms than the projectable or the detailed-balance versions. Lastly, in order to recover GR in the IR limit, apart from the running of \(\lambda \) to 1, \(\eta \) should run to zero too, while \(\xi \) can be set to 1.

We close this section by mentioning that in all versions of Hořava–Lifshitz gravity, Lorentz invariance is violated due to both the kinetic term (since \(\lambda \) is in general not equal to 1) and the terms in the potential. It is approximately and asymptotically restored in the IR, where \(\lambda \) runs to 1 and the potential terms will be significantly suppressed. Thus, one can apply Hořava–Lifshitz gravity in order to investigate its implications, which indeed are found to be rich and interesting at both cosmological [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84] and black hole applications [85, 86, 87, 88, 89, 90, 91].

## 3 The WDW equation in Hořava–Lifshitz gravity

In this section we examine the Wheeler–De Witt (WDW) equation in the framework of Hořava–Lifshitz gravity. For convenience, and in order to simplify the calculations, we focus on the projectable version of the theory, without the detailed-balanced condition, although an extension to the full, non-projectable theory is straightforward.

*a*(

*t*) denotes the scale factor. In this background, the three-dimensional Ricci curvature tensor and the scalar curvature read

*R*the three-dimensional scalar curvature [11]. Indeed, if one starts from the Lagrangian

*b*and

*c*given by (3.9), and extracts the corresponding field equations in the case of FLRW geometry, one will obtain the same equations as those extracted from \(\mathcal {L}_{P}\) in (2.7). Lastly, note that we have used the definitions (3.6), while we have furthermore set \(R_{0}\equiv 6/G=6/l_{p}^{2}\).

## 4 The WDW equation in Gravity’s Rainbow

*E*quantifies the energy scale at which quantum gravity effects become apparent. For instance, one of these effects would be that the graviton distorts the background metric as we approach the Planck scale.

*E*to evolve depending on

*t*, one finds that the extrinsic curvature of the metric (4.1) reads

## 5 Correspondence of Gravity’s Rainbow with Hořava–Lifshitz gravity

*ad hoc*, it can be supported by invoking the dispersion relation of a massless graviton, which, as we show in Appendix B, for a FLRW background acquires the form

*k*the constant dimensionless radial wavenumber, and thus in the present case of GRw it is modified to

*D*scalar curvature. Moreover, note that the energy-momentum tensor has dimensions of energy density. Thus, and in order to take the comparison on general grounds, one can assume that \(g_{2}\left( E\left( a\left( t\right) \right) /E_{P}\right) \) can be represented by a formal expansion in powers of \(E/E_{P}\), identifying the coefficients order by order. However, since in the present work we are comparing GRw with the Hořava–Lifshitz gravity with \(z=3\), the formal Taylor expansion is truncated at the second order.

## 6 Correspondence in spherically symmetric backgrounds

*N*(

*r*) and

*b*(

*r*) are arbitrary functions of the radial coordinate

*r*, denoted as the lapse function and the form function respectively. In this case, the energies now depend on the shape function \(b\left( r\right) \) and the radial coordinate

*r*, namely

*R*is given by

*r*, and we have used the mixed Ricci tensor \(R_{j\text { }}^{a}\) with components

*E*on \(b\left( r\right) \), that is, we assume \(\text {d}E\left( b\left( r\right) \right) /\text {d}b=0\). In this case the scalar curvature simplifies to

## 7 Conclusions

In this work we explored the connection between two Lorentz-violating theories, namely GRw and Hořava–Lifshitz gravity. In GRw, it is the metric that incorporates all the distortion of the space-time when one approaches the Planck scale, while in Hořava–Lifshitz gravity, it is the potential part of the action (or the Hamiltonian) that acquires higher-order curvature terms. Usually GRw is switched on because a Planckian particle distorts the gravitational metric tensor \(g_{\mu \nu }\). However, since in the present application we have neglected any matter fields, the only particle appearing is the graviton. Since the graviton is the quantum particle associated with the quantum fluctuations of the space-time, we conclude that it is the gravitational field itself that is responsible for such a distortion. This is also enforced by the dispersion relation relating the graviton energy and the scale factor, namely the scalar curvature, in the case where an FLRW background is imposed, or the graviton energy and the shape function in the case where a spherically symmetric background is imposed.

As we have shown, one can indeed establish a correspondence between the two theories, through the examination of their Wheeler–De Witt equations. However, although we have explicitly shown this in the case of two physically interesting space-times, namely the FLRW and the spherically symmetric ones, and thus we have a strong indication that this correspondence is not an artifact of the space-time symmetries but rather it arises from the features of the two theories, a general proof (or disproof) in the case of arbitrary metrics is still needed. In order to handle this issue, one might use the well-known relation between Hořava–Lifshitz gravity and Einstein-aether theory [93, 94, 95].

It is interesting to mention that GRw, in the FLRW background, generates Hořava–Lifshitz gravity under a specific form of \(f\left( R\right) \) theory, with *R* the three-dimensional scalar curvature. A similar result was pointed out in [92], where a connection between the rainbow’s functions and a specific \(f\left( R\right) \) form seems to be evident. In our analysis we saw that the obtained correspondence includes information even for the terms of the type \(R^{ij}R_{ij}\), \(RR^{ij}R_{ij}\) and \(R_{j}^{i}R_{k}^{j}R_{i}^{k}\), which were not explicitly included. Hence, we deduce that in order to incorporate higher-curvature terms, it is likely that the rainbow’s functions must include terms of the form \(R^{ij}R_{ij}\) etc., a possibility that could be encoded in the Kretschmann scalar. These issues reveal that the bridge between GRw and Hořava–Lifshitz gravity could be much richer, and it deserves further investigation.

We close this work by mentioning that in the above analysis we have remained at the background level, as a first step towards bridging the two theories. However, it is required and it is interesting to examine their relation at the perturbation level too, since there are many examples of theories that coincide at the background level, while being distinguishable or different when one incorporates the perturbations. Furthermore, relating the perturbations between GRw and Hořava–Lifshitz gravity becomes necessary having in mind the problems of the extra mode propagation that appears in the simple versions of the latter [96, 97, 98, 99]. Since such a detailed analysis lies beyond the scope of the present manuscript it is left for a future investigation.

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