The Lovelock gravity in the critical spacetime dimension

TL;DR

This paper explores the properties of vacuum solutions in pure Lovelock gravity, revealing that in odd critical dimensions, these solutions are trivial and flat, extending known results from Einstein gravity to higher-order Lovelock theories.

Contribution

It generalizes the triviality of vacuum solutions to all odd critical dimensions in pure Lovelock gravity, and introduces Lovelock analogues of black holes with a cosmological constant.

Findings

Vacuum solutions are trivial and flat in odd critical dimensions for pure Lovelock gravity.

In the case n=1, solutions describe a global monopole field.

Adding Lambda yields Lovelock analogues of BTZ black holes.

Abstract

It is well known that the vacuum in the Einstein gravity, which is linear in the Riemann curvature, is trivial in the critical (2+1=3) dimension because vacuum solution is flat. It turns out that this is true in general for any odd critical $d=2n+1$ dimension where $n$ is the degree of homogeneous polynomial in Riemann defining its higher order analogue whose trace is the nth order Lovelock polynomial. This is the "curvature" for nth order pure Lovelock gravity as the trace of its Bianchi derivative gives the corresponding analogue of the Einstein tensor \cite{bianchi}. Thus the vacuum in the pure Lovelock gravity is always trivial in the odd critical (2n+1) dimension which means it is pure Lovelock flat but it is not Riemann flat unless $n=1$ and then it describes a field of a global monopole. Further by adding Lambda we obtain the Lovelock analogue of the BTZ black hole.