Generalized Lense--Thirring metrics: higher-curvature corrections and solutions with matter

TL;DR

This paper generalizes Lense--Thirring spacetimes to higher dimensions and curvatures, exploring solutions with matter and their properties in various gravity theories, including Lovelock gravity.

Contribution

It introduces a new ansatz for higher-dimensional, multiply-spinning Lense--Thirring solutions applicable to higher curvature gravities and matter, revealing Einstein gravity's uniqueness in four dimensions.

Findings

Constructed slowly rotating solutions in Lovelock gravity.

Showed Einstein gravity is unique in 4D for single metric function solutions.

Recast solutions in Painlevé--Gullstrand form for regularity on horizons.

Abstract

The Lense--Thirring spacetime describes a 4-dimensional slowly rotating approximate solution of vacuum Einstein equations valid to a linear order in rotation parameter. It is fully characterized by a single metric function of the corresponding static (Schwarzschild) solution. In this paper, we introduce a generalization of the Lense--Thirring spacetimes to the higher-dimensional multiply-spinning case, with an ansatz that is not necessarily fully characterized by a single (static) metric function. This generalization lets us study slowly rotating spacetimes in various higher curvature gravities as well as in the presence of non-trivial matter. Moreover, the ansatz can be recast in Painlev{\'e}--Gullstrand form (and thence is manifestly regular on the horizon) and admits a tower of exact rank-2 and higher rank Killing tensors that rapidly grows with the number of dimensions. In particular, we construct slowly multiply-spinning solutions in Lovelock gravity and notably show that in four dimensions Einstein gravity is the only non-trivial theory amongst all up to quartic curvature gravities that admits a Lense--Thirring solution characterized by a single metric function.