Higher order gravity theories and their black hole solutions

TL;DR

This work surveys Lovelock gravity, a natural higher-dimensional extension of GR defined by a hierarchy of curvature invariants ${\cal L}_{(k)}$ that preserve second-order field equations and energy conservation. It develops the differential-form formalism, presents exact static black-hole and soliton solutions, and derives generalized matching (junction) conditions for distributional sources, including codimension-1 and codimension-2 braneworlds with induced gravity. The discussion covers braneworld cosmology, extended Kaluza–Klein reductions, and the resulting 4D effective theories, highlighting how higher-order Lovelock terms modify horizon thermodynamics, induce gravity on branes, and yield nonlinear electrodynamics upon KK reduction. The paper also outlines open problems in stationary solutions, higher-codimension braneworlds, and perturbative spectra, emphasizing the rich phenomenology and technical challenges of Lovelock gravity. Overall, Lovelock theory provides a mathematically consistent, physically rich framework for exploring gravity in higher dimensions with clear routes to 4D phenomenology through branes and KK reduction.

Abstract

We discuss a particular higher order gravity theory, Lovelock theory, that generalises in higher dimensions, 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 of even dimensional manifolds. We will then discuss a generic staticity theorem, which will give us the charged static black hole solutions. We will examine their asymptotic behavior, analyse their horizon structure and briefly their thermodynamics. We will then examine the distributional matching conditions for Lovelock theory. We will see how induced 4 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 1 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 4 dimensional vector-scalar-tensor theory which has the unique property of retaining second order field equations. We will conclude by listing some open problems and common difficulties.

Figures (1)

• Figure 1: Any 2 dimensional compact surface can be continuisly deformed to one of its Euler classes parametrised by $\chi$.