Black hole hair in generalized scalar-tensor gravity: An explicit example (1408.1698v2)

Abstract: In a recent Letter we have shown that in shift-symmetric Horndeski theory the scalar field is forced to obtain a nontrivial configuration in black hole spacetimes, unless a linear coupling with the Gauss-Bonnet invariant is tuned away. As a result, black holes generically have hair in this theory. In this companion paper, we first review our argument and discuss it in more detail. We then present actual black hole solutions in the simplest case of a theory with the linear scalar-Gauss-Bonnet coupling. We generate exact solutions numerically for a wide range of values of the coupling and also construct analytic solutions perturbatively in the small coupling limit. Comparison of the two types of solutions indicates that non-linear effects that are not captured by the perturbative solution lead to a finite area, as opposed to a central, singularity. Remarkably, black holes have a minimum size, controlled by the length scale associated with the scalar-Gauss-Bonnet coupling. We also compute some phenomenological observables for the numerical solution for a wide range of values of the scalar-Gauss-Bonnet coupling. Deviations from the Schwarzschild geometry are generically very small.

• The paper demonstrates that incorporating a scalar-Gauss–Bonnet coupling leads to non-trivial scalar hair in black holes.
• The authors use numerical simulations and perturbative analysis to show a threshold black hole size and finite area singularity formation.
• The findings suggest potential observable deviations from general relativity, offering tests for scalar-tensor gravity in astrophysics.

Black Hole Hair in Generalized Scalar-Tensor Gravity: An Analytical and Numerical Inquiry

The paper "Black hole hair in generalized scalar-tensor gravity: An explicit example" by Sotiriou and Zhou explores the intricate domain of black hole physics within the framework of shift-symmetric Horndeski theories. The fundamental focus of their investigation is on the configurations of scalar fields in black hole spacetimes and whether such black holes retain the so-called 'hair' in these theories, exploring specifically the association with the scalar-Gauss–Bonnet coupling term.

Theoretical Background and Framework

Black holes in classical general relativity are often described as exhibiting “no hair,” meaning they can be completely described by just three external parameters: mass, charge, and angular momentum. However, this simplified characterisation is potentially disrupted when considering scalar-tensor theories, where scalar fields couple non-minimally to gravity.

The paper points out shifts in perspective when the Shift-Symmetric Generalized Galileon (SSGG) — a broader extension of Horndeski's theory — is integrated with black hole environments. Previous conjectures have suggested that no-hair theorems may not hold universally in these contexts, especially when the scalar field configuration is not trivial.

Methodological Innovations and Analysis

Sotiriou and Zhou challenge previous no-hair assertions by showing that if a certain linear coupling between the scalar field and the Gauss-Bonnet invariant is not precisely set aside, black holes indeed exhibit 'hair'. The authors provide two pathways to solutions incorporating such couplings:

1. Numerical Solutions: By circumventing traditional difficulties with explicit solutions, the authors employed numerical techniques to generate solutions that demonstrate deviations from Schwarzschild geometry. These solutions reveal a critical insight: a threshold exists for the minimum black hole size governed by the coupling length scale. Moreover, non-linear effects become pivotal, leading to a finite area singularity rather than a central one.
2. Analytic Solutions: Perturbative solutions in the small-coupling limit further buttress their claims, demonstrating subtle departures from expected configurations when these couplings are incorporated. Interestingly, the perturbative approach shows that finite area singularities are not captured to first order but become apparent at second-order perturbations.

Implications and Theoretical Consequences

The paper advances the theoretical understanding by demonstrating that the presence of the scalar-Gauss-Bonnet term inevitably leads to non-trivial scalar configurations around black holes, even within the confines of second-order derivative theories. This presence denotes potential avenues for observational exploration as it implies deviations from general relativity’s predictions even with stationary solutions.

These findings have broader implications, hinting at potential observational evidence of scalar charges in cases where black holes are solitary and not overly perturbed by substantial external fields. However, the paper rightly tempers expectations regarding the magnitude of such deviations, estimating them to be minor under most realistic astrophysical conditions.

Future Prospects

The authors also suggest an intriguing frontier for future exploration: the role of axially symmetric, i.e., rotating, black holes in these theories. They succinctly argue that if spherically symmetric black holes do not have scalar hair, it might be inferred that the slowly rotating counterparts likely exhibit similar attributes.

In conclusion, Sotiriou and Zhou’s exploration has convincingly demonstrated the nuanced landscape of scalar-tensor theories in black hole environments, specifically within the confines of shift-symmetric Horndeski theories. Their contributions outline a pivotal shift from traditional general relativity, establishing a bedrock for subsequent theoretical and observational studies to probe the nature of scalar fields in our universe further. As new astronomical technologies seek more precise measurements of black holes, these insights could become vital in deciphering whether modifications from general relativity are discernible amid these cosmic phenomena.