Toward regular black holes in sixth-derivative gravity

TL;DR

This paper investigates static, spherically symmetric solutions in sixth-derivative gravity, demonstrating conditions for regular black holes and highlighting differences from lower-derivative theories.

Contribution

It proves that only regular solutions admit Frobenius expansions around the origin in the general theory, and explores solution behaviors under specific coupling constraints.

Findings

Regular solutions are the only Frobenius-expandable solutions at the origin.

Black hole solutions can be identified by expansions around non-zero radii.

Conditions like $R=0$ and $g_{tt}g_{rr}=-1$ are too restrictive in sixth-derivative gravity.

Abstract

We study spherically symmetric static solutions of the most general sixth-derivative gravity using series expansions. Specifically, we prove that the only solutions of the complete theory (i.e., with generic coupling constants) that possess a Frobenius expansion around the origin, $r=0$, are necessarily regular. When restricted to specific branches of theories (i.e., imposing particular constraints on the coupling constants), families of potentially singular solutions emerge. By expanding around $r=r_0 \neq 0$, we identify solutions with black hole horizons. Finally, we argue that, unlike in fourth-derivative gravity, the conditions $R=0$ and $g_{tt}g_{rr}=-1$ are too restrictive for sixth-derivative gravity solutions.