Where and why does Einstein-Scalar-Gauss-Bonnet theory break down?

TL;DR

This paper investigates where and why Einstein-Scalar-Gauss-Bonnet gravity theories lose predictivity, showing that strong geodesic focusing causes hyperbolicity breakdown, which limits the physical relevance of scalarized black hole solutions.

Contribution

The authors develop a gauge covariant method to analyze hyperbolicity loss in ESGB gravity and numerically study its behavior during gravitational collapse for specific models.

Findings

Hyperbolicity breaks down due to strong geodesic focusing.

Predictivity is maintained only for small GB couplings.

Scalarized black hole solutions require fine-tuned initial conditions.

Abstract

We present a systematic exploration of the loss of predictivity in Einstein-scalar-Gauss-Bonnet (ESGB) gravity. We first formulate a gauge covariant method of characterizing the breakdown of the hyperbolicity of the equations of motion in the theory. With this formalism, we show that strong geodesic focusing leads to the breakdown of hyperbolicity, and the latter is unrelated to the violation of the null convergence condition. We then numerically study the hyperbolicity of the equations during gravitational collapse for two specific ESGB gravity theories: "shift symmetric Gauss-Bonnet gravity" and a version of the theory that admits "spontaneously scalarized" black holes. We devise a "phase space" model to describe the end states for a given class of initial data. Using our phase space picture, we demonstrate that the two theories we consider remain predictive (hyperbolic) for a range of GB couplings. The range of couplings, however, is small, and thus, the presence of "spontaneously scalarized" solutions requires fine-tuning of initial data. Our results, therefore, cast doubt as to whether scalarized black hole solutions can be realistically realized in Nature even if ESGB gravity happened to be the correct gravitational description.