Singularities in Horava-Lifshitz theory Rong-Gen Cai a ∗ and Anzhong Wang b † a Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China b GCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA (Dated: November 7, 2018) 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-dimesnional 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 Lu-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 = (3 M/ | Λ | ) 1 / 3 for the anti-de Sitter Schwarzschild solution. The singularity at r = (3 M/ | Λ | ) 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 = ∞ , depending on the choice of the free parameter λ . arXiv:1001.0155v2 [hep-th] 21 Feb 2010 PACS numbers: 04.60.-m; 98.80.Cq; 98.80.-k; 98.80.Bp curvature is enhanced by higher-order curvature terms [5–7], and this opens a new approach to the flatness prob- lem and to a bouncing universe [5, 8, 9]. In addition, in the super-horizon region scale-invariant curvature per- turbations can be produced without inflation [10, 11], and the perturbations become adiabatic during slow-roll inflation driven by a single scalar field and the comov- ing curvature perturbation is constant [11]. Due to all these remarkable features, the theory has attracted lot of attention lately [12–14]. There has been considerable interest recently on a the- ory of quantum gravity proposed by Horava [1], moti- vated 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 fun- damentally 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, To formulate the theory, Horava assumed two condi- tions – detailed balance and projectability (He also con- sidered 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 rel- ativistic theory with Euclidean signature, whereby the number of independent-couplings is considerably limited. The projectability condition, on the other hand, orig- inates from the fundamental symmetry of the theory – the foliation-preserving diffeomorphisms of the Arnowitt- Deser-Misner (ADM) form, In (3 + 1)-dimensional spacetimes, in order for the the- ory 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 re- stored.” 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 rel- ativity (GR) in the infrared (IR) limit. The effective speed of light in this theory diverges in the UV regime, which could potentially resolve the hori- zon problem without invoking inflation [4]. The spatial which require coordinate transformations be only of the types, that is, space-dependent time reparameterizations are no longer allowed, although spatial diffeomorphisms are still ds 2 = − N $^{2}$c$^{2}$dt2 + g$_{ij}$ ( dx i + N $^{i}$dt ) ( dx j + N $^{j}$dt ) , ( i, j = 1 , 2 , 3) , (1.2) t → f ( t ) , x i → ζ $^{i}$( t, x ) , (1.3) x → ℓ x , t → ℓ $^{z}$t. (1.1) I. INTRODUCTION $^{∗}$Electronic address: cairg@itp.ac.cn $^{†}$Electronic address: anzhong˙wang@baylor.edu