Gravity in the 3+1-Split Formalism I: Holography as an Initial Value Problem

We present a detailed analysis of the 3+1-split formalism of gravity in the presence of a cosmological constant. The formalism helps revealing the intimate connection between holography and the initial value formulation of gravity. We show that the various methods of holographic subtraction of divergences correspond just to different transformations of the canonical variables, such that the initial value problem is properly set up at the boundary. The renormalized boundary energy momentum tensor is a component of the Weyl tensor.Comment: 28 pages; v2: minor improvements, references adde

The Cotton tensor in Riemannian spacetimes

Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components for the first time. Subsequently, we exhibit its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw, and Templeton. For each class examples are given. Finally we investigate the relation between the Cotton tensor and the energy-momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions.Comment: 27 pages, revtex

Modified F(R) Horava-Lifshitz gravity: a way to accelerating FRW cosmology

We propose a general approach for the construction of modified gravity which is invariant under foliation-preserving diffeomorphisms. Special attention is paid to the formulation of modified $F(R)$ Ho\v{r}ava-Lifshitz gravity (FRHL), whose Hamiltonian structure is studied. It is demonstrated that the spatially-flat FRW equations of FRHL are consistent with the constraint equations. The analysis of de Sitter solutions for several versions of FRHL indicates that the unification of the early-time inflation with the late-time acceleration is possible. It is shown that a special choice of parameters for FRHL leads to the same spatially-flat FRW equations as in the case of traditional $F(R)$-gravity. Finally, an essentially most general modified Ho\v{r}ava-Lifshitz gravity is proposed, motivated by its fully diffeomorphism-invariant counterpart, with the restriction that the action does not contain derivatives higher than the second order with respect to the time coordinate.Comment: LaTeX 11 pages. v4: Some errors have been correcte

Aspects of Quantum Gravity in de Sitter Spaces

In these lectures we give a review of recent attempts to understand quantum gravity on de Sitter spaces. In particular, we discuss the holographic correspondence between de Sitter gravity and conformal field theories proposed by Hull and by Strominger, and how this may be reconciled with the finite-dimensional Hilbert space proposal by Banks and Fischler. Furthermore we review the no-go theorems that forbid an embedding of de Sitter spaces in string theory, and discuss how they can be circumvented. Finally, some curious issues concerning the thermal nature of de Sitter space are elucidated.Comment: 36+1 pages, 5 Postscript figures, introduction and section 6 extended, further references, final version to appear in JCA