A no-hair theorem for the galileon (1202.1296v1)

Abstract: We consider a galileon field coupled to gravity. The standard no-hair theorems do not apply, because of the galileon's peculiar derivative interactions. We prove that, nonetheless, static spherically symmetric black holes cannot sustain non-trivial galileon profiles. Our theorem holds regardless of whether there are non-minimal couplings between the galileon and gravity of the covariant galileon type.

• The paper establishes that static, spherically symmetric black holes cannot support non-trivial Galileon profiles.
• Researchers prove the result by leveraging a conserved Galileon current that vanishes at the event horizon and throughout spacetime.
• The findings imply that black holes in modified gravity lack additional Galileon charges, confirming their predictable simplicity.

A No-Hair Theorem for the Galileon

In the paper "A No-Hair Theorem for the Galileon" by Lam Hui and Alberto Nicolis, a pivotal analysis is presented regarding the properties of black holes within the framework of modified theories of gravity, specifically those involving Galileon fields. The paper establishes a comprehensive no-hair theorem for static, spherically symmetric black holes in the presence of Galileon derivatives, addressing whether these cosmic objects can support non-trivial Galileon profiles.

Galileon theories have emerged as significant due to their potential to modify gravity at cosmological scales and satisfy the null energy condition in a ghost-free manner. Given their relevance in scenarios including cosmic acceleration and massive gravity, it is essential to evaluate whether existing no-hair theorems can apply to Galileon fields. Traditional no-hair theorems, such as those conjectured by Bekenstein, impose constraints on the behavior of scalar fields around black holes, suggesting that such fields cannot sustain long-lived "hair" or additional charges independent of primary black hole features like mass or angular momentum.

This paper delineates a proof in four primary steps, notably:

1. Conservation of the Galileon equation of motion (EOM): The Galileon EOM, in absence of external sources, manifests as a current conservation equation. This equation is invariant under Galileon shift symmetries, implying conservation even in complex, non-minimally coupled scenarios involving gravity and Galileon fields.
2. Vanishing of the Noether current at the horizon: Utilizing the symmetries intrinsic to static and spherically symmetric configurations, it is derived that the Noether current exhibits a vanishing behavior at the event horizon of the black hole.
3. Propagation of the current's vanishing property: The authors extend this vanishing property across the entire spacetime by leveraging the conservation of the current, ultimately deducing that the Galileon field must be trivial throughout the black hole's exterior.
4. Uniqueness of the trivial solution: By integrating the conservation equations and analyzing the boundary conditions at infinity, it is argued that the only physical solution fulfilling the required criteria is a constant, trivial field.

The implication of these results is profound, as it implies that black holes in such setups cannot harbor additional "Galileon charges" beyond conventional mass parameters, ensuring characterizable simplicity akin to that of classical no-hair theorems. The proofs constructed in this paper do not rely on the specific form of Einstein’s equations, reflecting their broad applicability across various gravitational contexts.

The implications extend notably to astrophysical observations and potential tests of modified gravity theories. The assertion that the Galileon charge vanishes stipulates an enhancement of the predictability of black hole properties in universes governed by beyond-standard theories of gravity. Furthermore, this framework may influence ongoing research in massive gravity theories, especially concerning the stability and boundary conditions in Galileon-modified spacetimes.

Future research could focus on extending the analysis to scenarios involving non-static, rotating black holes or those influenced by external cosmological fields, factors which often provide richer phenomenology. Additionally, exploring the interactions between Galileon fields and other exotic scalar fields might yield new insights into the symmetry principles governing our universe.

In summary, this paper serves as a crucial touchpoint for understanding the interactions between scalar fields with derivative couplings and black holes, reaffirming their fundamental behavior in a modified gravitational context and illuminating avenues for future exploration in theoretical physics and observational astrophysics.