Abstract
In this paper, we present two new families of spatially homogeneous black hole solution for z=4 Hořava–Lifshitz Gravity equations in (4+1) dimensions with general coupling constant lambda and the especial case lambda =1, considering beta =-1/3. The three-dimensional horizons are considered to have Bianchi types II and III symmetries, and hence the horizons are modeled on two types of Thurston 3-geometries, namely the Nil geometry and H^2times R. Being foliated by compact 3-manifolds, the horizons are neither spherical, hyperbolic, nor toroidal, and therefore are not of the previously studied topological black hole solutions in Hořava–Lifshitz gravity. Using the Hamiltonian formalism, we establish the conventional thermodynamics of the solutions defining the mass and entropy of the black hole solutions for several classes of solutions. It turned out that for both horizon geometries the area term in the entropy receives two non-logarithmic negative corrections proportional to Hořava–Lifshitz parameters. Also, we show that choosing some proper set of parameters the solutions can exhibit locally stable or unstable behavior.
Highlights
1.1 General considerationsThe non-relativistic power counting renormalizable theory of Horava–Lifshitz gravity was proposed by Horava at the Lifshitz point aimed at resolving the problems concerning the ultraviolet behavior of Einstein gravity [1,2,3]
We are interested in spatially homogeneous black hole solutions for z = 4 Horava–Lifshitz gravity on (4+1) dimensional spacetimes, where the three-dimensional horizons are assumed to be homogeneous spaces corresponding to Bianchi types I I and I I I with closed geometries of Nil and H 2 × R, respectively
These negatively curved homogeneous geometries are non-trivial in the sense that they are not constant scalar curvature type geometries that have been extensively studied in previous topological black hole solutions in Horava–Lifshitz gravity
Summary
The non-relativistic power counting renormalizable theory of Horava–Lifshitz gravity was proposed by Horava at the Lifshitz point aimed at resolving the problems concerning the ultraviolet behavior of Einstein gravity [1,2,3]. Considering Horava–Lifshitz gravity as a candidate quantum gravity theory and the importance of investigating AdS/CFT correspondence in the framework of this theory [66,67], it is interesting to find black hole solutions with special Thurston type horizon geometries for (4 + 1) dimensional Horava–Lifshitz gravity, for which the power counting super renormalizability requires z = 4. We are interested in spatially homogeneous black hole solutions for z = 4 Horava–Lifshitz gravity on (4+1) dimensional spacetimes, where the three-dimensional horizons are assumed to be homogeneous spaces corresponding to Bianchi types I I and I I I with closed geometries of Nil and H 2 × R, respectively These negatively curved homogeneous geometries are non-trivial in the sense that they are not constant scalar curvature type geometries that have been extensively studied in previous topological black hole solutions in Horava–Lifshitz gravity.
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