Higher Order Gravity Theories and Their Black Hole Solutions

In this chapter, we will discuss a particular higher order gravity theory, Lovelock theory, that generalises in higher dimensions than 4, general relativity. After briefly motivating modifications of gravity, we will introduce the theory in question and we will argue that it is a unique, mathematically sensible, and physically interesting extension of general relativity. We will see, by using the formalism of differential forms, the relation of Lovelock gravity to differential geometry and topology or even-dimensional manifolds. We will then discuss a generic staticity theorem, quite similar to Birkhoff's theorem in general relativity, which will give us the charged static black hole solutions. We will examine their asymptotic behaviour, analyse their horizon structure and briefly their thermodynamics. For the thermodynamics we will give a geometric justification of why the usual entropy-area relation is broken. We will then examine the distributional matching conditions for Lovelock theory. We will see how induced four-dimensional Einstein-Hilbert terms result on the brane geometry from the higher order Lovelock terms. With the junction conditions at hand, we will go back to the black hole solutions and give applications for braneworlds: perturbations of codimension I braneworlds and the exact solution for braneworld cosmology as well as the determination of maximally symmetric codimension 2 braneworlds. In both cases, the staticity theorem evoked beforehand will give us the general solution for braneworld cosmology in codimension 1 and maximal symmetry warped branes of codimension 2. We will then end with a discussion of the simplest Kaluza-Klein reduction of Lovelock theory to a four-dimensional vector-scalar-tensor theory which has the unique property of retaining second-order field equations. We will comment briefly the non-linear generalisation of Maxwell's theory and scalar-tensor theory. We will conclude by listing some open problems and common difficulties.