: The final report covering the research performed under the Air Force Office of Scientific Research by the Dallas relativity group, summarizes information on personnel involved, on the scientific work done, and on the results obtained in the four years of the grant's duration. Topics investigated were in areas of General relativity, cosmology, and electrodynamics.

Exact Solutions of Einstein's Field Equations.

We construct a new gravitational action which includes cubic curvature interactions and which provides a useful toy model for the holographic study of a three parameter family of four- and higher-dimensional CFT’s We also investigate the black hole solutions of this new gravity theory Further we examine the equations of motion of quasi-topological gravity While the full equations in a general background are fourth-order in derivatives, we show that the linearized equations describing gravitons propagating in the AdS vacua match precisely the second-order equations of Einstein gravity

Black holes in quasi-topological gravity

We present a new cubic theory of gravity in five dimensions which has second-order traced field equations, analogous to BHT new massive gravity in three dimensions. Moreover, for static spherically symmetric spacetimes all the field equations are of second order, and the theory admits a new asymptotically locally flat black hole. Furthermore, we prove the uniqueness of this solution, study its thermodynamical properties and show the existence of a C-function for the theory following the arguments of Anber and Kastor (2008 J. High Energy Phys. JHEP05(2008)061 (arXiv:0802.1290 [hep-th])) in pure Lovelock theories. Finally, we include the Einstein–Gauss–Bonnet and cosmological terms and find new asymptotically AdS black holes at the point where the three maximally symmetric solutions of the theory coincide. These black holes may also possess a Cauchy horizon.

A new cubic theory of gravity in five dimensions: black hole, Birkhoff's theorem and C-function

Lovelock theory is a natural extension of Einstein theory of gravity to higher dimensions, and it is of great interest in theoretical physics as it describes a wide class of models. In particular, it describes string theory inspired ultraviolet corrections to Einstein–Hilbert action, while admits the Einstein general relativity and the so-called Chern–Simons theories of gravity as particular cases. Here, we give an introduction to the black hole solutions of Lovelock theory and analyze their most important properties. These solutions can be regarded as generalizations of the Boulware–Deser solution of Einstein–Gauss–Bonnet gravity, which we discuss in detail here. We briefly discuss some recent progress in understanding these and other solutions, like topological black holes that represent black branes of the theory, and vacuum thin-shell wormhole-like geometries that connect two different asymptotically de Sitter spaces. We also make some comments on solutions with time-like naked singularities.

The Lovelock Black Holes

We classify all the six-derivative Lagrangians of gravity, whose traced field equations are of second or third order, in arbitrary dimensions. In the former case, the Lagrangian in dimensions greater than six reduces to an arbitrary linear combination of the six-dimensional Euler density and the two linearly independent cubic Weyl invariants. In five dimensions, besides the independent cubic Weyl invariant, we obtain an interesting cubic combination, whose field equations for static spherically symmetric spacetimes are of second order. In the latter case, in arbitrary dimensions we obtain two combinations, which in dimension three, are equivalent to the complete contraction of two Cotton tensors. Moreover, we also recover all the conformal anomalies in six dimensions. Finally, we present the general static, spherically symmetric solution for some of these Lagrangians.

Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions

We construct all independent local scalar monomials in the Riemann tensor at an arbitrary dimension, for the special regime of static spherically symmetric geometries. Compared to general spaces, their number is significantly reduced: the extreme example is the collapse of all invariants ~Weyl^k, to a single term at each k. The latter is equivalent to the Lovelock invariant L_k . Depopulation is less extreme for invariants involving rising numbers of Ricci tensors, and also depends on the dimension. The corresponding local gravitational actions and their solution spaces are discussed.

/pdf/curvature-invariants-of-static-spherically-symmetric-11oh2n0ggu.pdf

Curvature invariants of static spherically symmetric geometries

We construct all independent local scalar monomials in the Riemann tensor at arbitrary dimension, for the special regime of static, spherically symmetric geometries. Compared to general spaces, their number is significantly reduced: the extreme example is the collapse of all invariants ~ Weyl^k, to a single term at each k. The latter is equivalent to the Lovelock invariant L_k. Depopulation is less extreme for invariants involving rising numbers of Ricci tensors, and also depends on the dimension. The corresponding local gravitational actions and their solution spaces are discussed.

/pdf/curvature-invariants-of-static-spherically-symmetric-zm0p55m2mz.pdf

A. V. Ryzhov

Papers

Curvature invariants of static spherically symmetric geometries

Curvature invariants of static spherically symmetric geometries