Kerr black holes with Proca hair (1603.02687v1)

Abstract: Bekenstein proved that in Einstein's gravity minimally coupled to one (or many) real, Abelian, Proca field, stationary black holes (BHs) cannot have Proca hair. Dropping Bekenstein's assumption that matter inherits spacetime symmetries, we show this model admits asymptotically flat, stationary, axi-symmetric, regular on and outside an event horizon BHs with Proca hair, for an even number of real (or an arbitrary number of complex) Proca fields. To establish it, we start by showing that a test, complex Proca field can form bound states, with real frequency, around Kerr BHs: stationary Proca clouds. These states exist at the threshold of superradiance. It was conjectured in arXiv:1403.2757, that the existence of such clouds at the linear level implies the existence of a new family of BH solutions at the non-linear level. We confirm this expectation and explicitly construct examples of such Kerr black holes with Proca hair (KBHsPH). For a single complex Proca field, these BHs form a countable number of families with three continuous parameters (ADM mass, ADM angular momentum and Noether charge). They branch off from the Kerr solutions that can support stationary Proca clouds and reduce to Proca stars when the horizon size vanishes. We present the domain of existence of one family of KBHsPH, as well as its phase space in terms of ADM quantities. Some physical properties of the solutions are discussed; in particular, and in contrast with Kerr BHs with scalar hair, some spacetime regions can be counter-rotating with respect to the horizon. We further establish a no-Proca-hair theorem for static, spherically symmetric BHs but allowing the complex Proca field to have a harmonic time dependence, which shows BHs with Proca hair in this model require rotation and have no static limit. KBHsPH are also disconnected from Kerr-Newman BHs with a real, massless vector field.

• The paper demonstrates that Kerr black holes can possess Proca hair by relaxing traditional symmetry restrictions.
• It employs harmonic time dependence and an axially symmetric metric ansatz to construct stationary solutions that satisfy all energy conditions.
• Numerical analysis maps a domain of existence for these solutions, revealing new insights into black hole physics and superradiance.

Kerr Black Holes with Proca Hair: An Analytical Overview

The paper "Kerr Black Holes with Proca Hair" provides a detailed examination of black holes (BHs) coupled with Proca fields in the framework of general relativity. Historically, the no-hair theorems have posited that black holes are characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum. By integrating the Einstein's theory of gravity with Abelian Proca fields, the authors challenge this hypothesis by demonstrating the existence of Kerr black holes that can possess "hair," or additional parameters in the form of Proca fields.

Key Methodologies

This research focuses on the implications of coupling complex Proca fields with rotating black holes. Kerr black holes, which possess both mass and angular momentum, serve as the foundational solution for this paper:

• The analysis begins with a foundational review of Bekenstein’s theorem, which traditionally forbids the existence of Proca hair in static black holes due to the symmetry conditions imposed on spacetime.
• By examining scenarios where symmetry inheritance is dropped, the authors construct stationary black hole solutions.
• The introduction of harmonic time dependencies in Proca field dynamics allows for the existence of hair, contradicting traditional theorems assuming symmetry inheritance.

Numerical Results and Critical Observations

• Proca Hair: The authors introduce stationary Proca clouds around Kerr BHs and demonstrate how these clouds can lead to a new family of solutions, termed as Kerr Black Holes with Proca Hair (KBHsPH). The critical innovation illustrated here is in permitting a complex Proca field, which allows radial solutions that decay exponentially, due to the massiveness of the field itself.
• Existence Line and Domain: By utilizing axially symmetric metric ansatz and solving through numerical adjustments, the paper identifies the conditions under which these solutions exist. They map out a domain of existence, illustrating the dependency of these domains upon the frequency and horizon size of solutions, while contrasting it with known scalar field results.
• Energy Conditions: Scenarios depicted show compliance with all energy conditions -- a notable achievement in constructing counter-examples to the no-hair theorem.
• Angular Quantities: The complexities of angular relationships introduce conditions under which Noether charges exist without accompanying Gauss laws.

Implications and Theoretical Perspectives

• Stationary Solutions and Superradiance: The paper's understanding of stationary Proca clouds underlines a subtlety in contemporary black hole physics: the viability of superradiant regimes that can support new hair solutions.
• Theoretical Speculation: Based on the successful construction of these solutions, the authors speculate on the richness of the parameter space possible for black holes, suggesting further studies on self-interactions and combined scalar and Proca fields.
• No-Hair Conjecture: This work not only refines underlying conditions under which the no-hair theorem could be bypassed but also illustrates that real astrophysical processes may reflect these novel dynamics.

Directions for Future Development

The work hints at many exciting paths forward, including:

• Extending these results to self-interacting Proca models, which could further illustrate the complexity of potentials around black holes.
• Investigating further combinations with scalar fields, possibly revealing even more exotic black hole solutions.

This paper marks an important step toward redefining our understanding of black holes, suggesting they can harbor much richer structures than classical models predict. As research continues in this direction, the interplay of field theory, relativity, and astrophysical phenomena promises new insights into the nature of gravitational interactions and spacetime.