Causality and Hyperbolicity of Lovelock Theories

TL;DR

This paper investigates the causal structure and hyperbolicity of Lovelock gravity theories, showing that gravitational signals are confined within Killing horizons and analyzing stability issues for small black holes.

Contribution

It generalizes previous results to prove Killing horizons are characteristic hypersurfaces for all gravitational modes in Lovelock theories and examines hyperbolicity in various black hole backgrounds.

Findings

Killing horizons are characteristic hypersurfaces for all gravitational degrees of freedom.

Lovelock theories are hyperbolic in Ricci flat type N spacetimes.

Hyperbolicity can be violated near small black hole horizons.

Abstract

In Lovelock theories, gravity can travel faster or slower than light. The causal structure is determined by the characteristic hypersurfaces. We generalise a recent result of Izumi to prove that any Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom of a Lovelock theory. Hence gravitational signals cannot escape from the region inside such a horizon. We investigate the hyperbolicity of Lovelock theories by determining the characteristic hypersurfaces for various backgrounds. First we consider Ricci flat type N spacetimes. We show that characteristic hypersurfaces are generically all non-null and that Lovelock theories are hyperbolic in any such spacetime. Next we consider static, maximally symmetric black hole solutions of Lovelock theories. Again, characteristic surfaces are generically non-null. For some small black holes, hyperbolicity is violated near the horizon. This implies that the stability of such black holes is not a well-posed problem.