Singularities in Horava–Lifshitz theory

Abstract

Singularities in $(3+1)$-dimensional Horava–Lifshitz (HL) theory of gravity are studied. These singularities can be divided into scalar, non-scalar curvature, and coordinate singularities. Because of the foliation-preserving diffeomorphisms of the theory, the number of scalars that can be constructed from the extrinsic curvature tensor $K_{ij}$, the 3-dimensional Riemann tensor and their derivatives is much large than that constructed from the 4-dimensional Riemann tensor and its derivatives in general relativity (GR). As a result, even for the same spacetime, it may be singular in the HL theory but not in GR. Two representative families of solutions with projectability condition are studied, one is the (anti-)de Sitter Schwarzschild solutions, and the other is the Lü–Mei–Pope (LMP) solutions written in a form satisfying the projectability condition – the generalized LMP solutions. The (anti-)de Sitter Schwarzschild solutions are vacuum solutions of both HL theory and GR, while the LMP solutions with projectability condition satisfy the HL equations coupled with an anisotropic fluid with heat flow. It is found that the scalars K and $K_{ij}{K}^{ij}$ are singular only at the center for the de Sitter Schwarzschild solution, but singular at both the center and $r={(3M/|\Lambda |)}^{1/3}$ for the anti-de Sitter Schwarzschild solution. The singularity at $r={(3M/|\Lambda |)}^{1/3}$ is absent in GR. In addition, all the generalized LMP solutions have two scalar curvature singularities, located at either $r=0$ and $r=r_s>0$, or $r=r_1$ and $r=r_2$ with $r_2>r_1>0$, or $r=r_s>0$ and $r=\infty$, depending on the choice of the free parameter λ.

Introduction

There has been considerable interest recently on a theory of quantum gravity proposed by Horava [1], motivated by Lifshitz theory in solid state physics [2], for which the theory is usually referred to as the Horava–Lifshitz (HL) theory. The HL theory is based on the perspective that Lorentz symmetry should appear as an emergent symmetry at long distances, but can be fundamentally absent at high energies [3]. With such a perspective, Horava considered systems whose scaling at short distances exhibits a strong anisotropy between space and time,

$$\mathbf{x}\to \ell \mathbf{x},t\to {\ell }^{z}t.$$

In $(3+1)$-dimensional spacetimes, in order for the theory to be power-counting renormalizable, it needs $z⩾3$. At low energies, the theory is expected to flow to $z=1$, whereby the Lorentz invariance is “accidentally restored”. So, the HL theory is non-relativistic, ultra-violet (UV) complete, explicitly breaks Lorentz invariance at short distances, but is expected to reduce to general relativity (GR) in the infrared (IR) limit.

The effective speed of light in this theory diverges in the UV regime, which could potentially resolve the horizon problem without invoking inflation [4]. The spatial curvature is enhanced by higher-order curvature terms [5], [6], [7], and this opens a new approach to the flatness problem and to a bouncing universe [5], [8], [9]. In addition, in the super-horizon region scale-invariant curvature perturbations can be produced without inflation [10], [11], and the perturbations become adiabatic during slow-roll inflation driven by a single scalar field and the comoving curvature perturbation is constant [11]. Due to all these remarkable features, the theory has attracted lot of attention lately [12], [13], [14].

To formulate the theory, Horava assumed two conditions – detailed balance and projectability (He also considered the case where the detailed balance condition was softly broken) [1]. The detailed balance condition restricts the form of a general potential in a ($D+1$)-dimensional Lorentz action to a specific form that can be expressed in terms of a D-dimensional action of a relativistic theory with Euclidean signature, whereby the number of independent-couplings is considerably limited. The projectability condition, on the other hand, originates from the fundamental symmetry of the theory – the foliation-preserving diffeomorphisms of the Arnowitt–Deser–Misner (ADM) form,

$$d{s}^2=-N^2{c}^{2}d{t}^2+g_{ij}(d{x}^i+N^{i}dt)(d{x}^j+N^{j}dt)(i,j=1,2,3),$$

which require coordinate transformations be only of the types,

$$t\to f(t),x^i\to {\zeta }^i(t,\mathbf{x}),$$

that is, space-dependent time reparameterizations are no longer allowed, although spatial diffeomorphisms are still a symmetry. Then, it is natural, but not necessary, to restrict the lapse function N to be space-independent, while the shift vector $N^i$ and the 3-dimensional metric $g_{ij}$ in general depend on both time and space,

$$N=N(t),N^i=N^i(t,x),g_{ij}=g_{ij}(t,x).$$

This is the projectability condition, and clearly is preserved by the foliation-preserving diffeomorphisms (1.3). However, due to these restricted diffeomorphisms, one more degree of freedom appears in the gravitational sector – a spin-0 graviton. This is potentially dangerous, and needs to be highly suppressed in the IR regime, in order to be consistent with observations. Similar problems also raise in other modified theories, such as massive gravity [15].

Under the rescaling (1.1), the dynamical variables N, $N^i$ and $g_{ij}$ scale as,

$$N\to N,N^i\to {\ell }^{-2}{N}^i,g_{ij}\to g_{ij}.$$

Note that in [7], [11], the constant c in the metric (1.2) was absorbed into N, so that there the lapse function scaled as ${\ell }^{-2}$.

So far most of the work on the HL theory has abandoned the projectability condition but kept the detailed balance [4], [6], [12], [13], [14]. One of the main reasons is that the detailed balance condition leads to a very simple action, and the resulted theory is much easier to deal with, while abandoning projectability condition gives rise to local rather than global Hamiltonian constraint and energy conservation. However, with detailed balance a scalar field is not UV stable [5], and gravitational perturbations in the scalar section have ghosts [1] and are not stable for any given value of the dynamical coupling constant λ [16]. In addition, detailed balance also requires a non-zero (negative) cosmological constant, breaks the parity in the purely gravitational sector [17], and makes the perturbations not scale-invariant [18]. Breaking the projectability condition, on the other hand, can cause strong couplings [19] and gives rise to an inconsistency theory [20].

To resolve these problems, various modifications have been proposed [21]. In particular, Blas, Pujolas and Sibiryakov (BPS) [22] showed that the strong coupling problem can be solved without projectability condition (in which the lapse function becomes dependent on both t and $x^i$), when terms constructed from the 3-vector

$$a_i\equiv \frac{{\partial }_{i}N}{N},$$

are included. Contrary claims can be found in [23]. In addition, it is not clear how the inconsistency problem [20] is resolved in such a generalization.

On the other hand, Sotiriou, Visser and Weinfurtner (SVW) formulated the most general HL theory with projectability but without detailed balance conditions [17]. The total action consists of three parts, kinetic, potential and matter,

$$S={\zeta }^2\int dt{d}^{3}xN\sqrt{g}({\mathcal{L}}_K-{\mathcal{L}}_V+{\zeta }^{-2}{\mathcal{L}}_M),$$

where $g=\det g_{ij}$, and

$${\mathcal{L}}_K=\frac{1}{c^2}[K_{ij}{K}^{ij}-(1-\xi )K^2],$$$${\mathcal{L}}_V=2\Lambda -R+\frac{1}{{\zeta }^2}(g_2{R}^2+g_3{R}_{ij}{R}^{ij})+\frac{1}{{\zeta }^4}(g_4{R}^3+g_{5}R{R}_{ij}{R}^{ij}+g_6{R}_j^i{R}_k^j{R}_i^k)+\frac{1}{{\zeta }^4}[g_{7}R{\mathrm{\nabla }}^{2}R+g_8({\mathrm{\nabla }}_i{R}_{jk})\left({\mathrm{\nabla }}^i{R}^{jk}\right)].$$

Here ${\zeta }^2=1/16\pi G$, and c denotes the speed of light. In the “physical” units, one can set $c=1$ [17]. The covariant derivatives and Ricci and Riemann terms are all constructed from the three-metric $g_{ij}$, while $K_{ij}$ is the extrinsic curvature,

$$K_{ij}=\frac{1}{2N}(-{\dot{g}}_{ij}+{\mathrm{\nabla }}_i{N}_j+{\mathrm{\nabla }}_j{N}_i),$$

where $N_i=g_{ij}{N}^j$. The constants ξ, $g_I$ ($I=2,\dots ,8$) are coupling constants, and Λ is the cosmological constant. In the IR limit, all the high-order curvature terms (with coefficients $g_I$) drop out, and the total action reduces when $\xi =0$ to the Einstein–Hilbert action.

The SVW generalization seems to have the potential to solve the above mentioned problems [24], although it was found that gravitational scalar perturbations either have ghosts ($0⩽\xi ⩽2/3$) or are not stable ($\xi <0$) [7], [25]. In order to avoid ghost instability, one needs to assume $\xi ⩽0$. Then, the sound speed $c_{\psi }^2=\xi /(2-3\xi )$ becomes imaginary, which leads to an IR instability. Izumi and Mukohyama showed that this type of instability does not show up if $|c_{\psi }|$ is less than a critical value [26].

It is fair to say, in order to have a viable HL theory, much work needs to be done, and various aspects of the theory ought to be explored, including the renormalization group flows [27], Vainshtein mechanism [15], [28], solar system tests [29], Lorentz violations [30], and its applications to cosmology [7], [11].

In this Letter, we shall study another important issue in the HL theory – the problem of singularities, which is closely related to the issue of black holes in this theory [12]. Although we are initially interested in the case with projectability condition, our conclusions can be equally applied to the HL theory without projectability condition. The extrinsic curvature $K_{ij}$ and the 3-dimensional Riemann tensor $R_{jkl}^i$ are not tensors under the 4-dimensional Lorentz transformations,

$$x^{\mu }\to x^{\prime \mu }={\zeta }^{\mu }(t,x^i).$$

As a result, in GR one usually does not use them to construct gauge-invariant quantities. However, in the HL theory, due to the restricted diffeomorphisms, these quantities become tensors, and can be easily used to construct various scalars. If any of such scalars is singular, such a singularity cannot be limited by the restricted coordinate transformations (1.3). Then, we may say that the spacetime is singular. It is exactly in this vein that we study singularities in the HL theory. In particular, we first generalize the definitions of scalar, non-scalar and coordinate singularities in GR to the HL theory in Section 2, and then in Section 3 we study two representative families of spherical static solutions of the HL theory, and identify scalar curvature singularities using the three quantities K, $K_{ij}{K}^{ij}$ and R. In Section 4, we present our main conclusions and remarks. There is also an Appendix A, in which we show explicitly that the second class of the LMP solutions written in the ADM frame with projectability condition in general satisfy the HL equations coupled with an anisotropic fluid with heat flow.

Before proceeding further, we would like to note that black holes in GR for asymptotically-flat spacetimes are well defined [31]. However, how to generalize such definitions to more general spacetimes is still an open question [32], [33]. The problem in the HL theory becomes more complicated [26], [34], partially because of the fact that particles in the HL theory can have non-standard dispersion relations, and therefore no uniform maximal speed exists. As a result, the notion of a horizon is observer-dependent.