Petrov Type, Principal Null Directions, and Killing Tensors of Slowly-Rotating Black Holes in Quadratic Gravity

TL;DR

This study analyzes the symmetries of slowly-rotating black holes in quadratic gravity theories, revealing they are Petrov type I and lack higher-rank Killing tensors, implying no additional conserved quantities.

Contribution

It computes principal null directions and Killing tensors for black holes in quadratic gravity, showing the absence of a fourth constant of motion in these solutions.

Findings

Both spacetimes are Petrov type I.

No nontrivial Killing tensors of rank 6 or 2 found.

Suggests unknown solutions lack a fourth conserved quantity.

Abstract

The ability to test general relativity in extreme gravity regimes using gravitational wave observations from current ground-based or future space-based detectors motivates the mathematical study of the symmetries of black holes in modified theories of gravity. In this paper we focus on spinning black hole solutions in two quadratic gravity theories: dynamical Chern-Simons and scalar Gauss-Bonnet gravity. We compute the principal null directions, Weyl scalars, and complex null tetrad in the small-coupling, slow rotation approximation for both theories, confirming that both spacetimes are Petrov type I. Additionally, we solve the Killing equation through rank 6 in dynamical Chern-Simons gravity and rank 2 in scalar Gauss-Bonnet gravity, showing that there is no nontrivial Killing tensor through those ranks for each theory. We therefore conjecture that the still-unknown, exact, quadratic-gravity, black-hole solutions do not possess a fourth constant of motion.