Aspects of general relativity with negative cosmological constant

Abstract

siehe engl.

Solutions to general relativity with negative cosmological constant are ubiquitous incontemporary physics. They have received a surge of attention due to the Anti-deSitter/conformal field theory correspondence — a conjecture which asserts that quantumgravity on asymptotically Anti-de Sitter spacetimes is equivalent to a quantum fieldtheory with conformal invariance in one dimension lower. In this thesis, we discuss the causal structure and energy of solutions to Einstein gravity with negative cosmological constant in three and four spacetime dimensions. We show that projection diagramscan be used to visualize the causal structure of BTZ black holes and so-called warpedflat spacetimes. The latter correspond to solutions to topologically massive gravitywith positive cosmological constant in (2 + 1) dimensions. It is shown that warped flatspacetimes do not describe black hole solutions. Next, we discuss the energy of solutions to Einstein gravity with negative cosmological constant. To this day, the question of boundedness of energy from below for general relativity with negative cosmological constant remains wide open. For time-symmetric initial data sets, this is the question of whether the energy of asymptotically locally hyperbolic spaces is bounded from below.We show that three-dimensional asymptotically locally hyperbolic spaces with constant negative scalar curvature, arbitrary high genus, and negative total mass exist and explain how this result relates to positive energy theorems in general relativity.