[Paper Review] Hairy black holes in the bigravity theory

Mikhail S. Volkov|arXiv (Cornell University)|Feb 29, 2012

Black Holes and Theoretical Physics1 citations

TL;DR

This paper investigates hairy black holes in a ghost-free bigravity theory with one massive and one massless graviton. It presents numerical evidence for multiple solution classes with shared event horizons and Hawking temperatures, differing in horizon structure and asymptotic behavior, including asymptotically AdS and flat solutions, and identifies globally regular 'gravity lumps' and Vainshtein-screened solutions with matter.

ABSTRACT

We study black holes in the recently proposed ghost-free theory with two gravitons, one of which is massive and another is massless. These black holes possess a regular event horizon which is common for both metrics and has the same values of the surface gravity and Hawking temperature with respect to each metric. The ratio of the event horizon radii measured by the two metrics is a free parameter that labels the solutions. We present a numerical evidence for their existence and find that they comprise several classes. Black holes within each class approach the same AdS-type asymptotic at infinity but differ from each other in the event horizon vicinity where the short-range massive modes reside. In addition, there are solutions showing a curvature singularity at a finite proper distance from the horizon. For some special solutions the graviton mass may become effectively imaginary, causing oscillations around the flat metric at infinity. The only asymptotically flat black hole we find -- the Schwarzschild solution obtained by identifying the two metrics -- seems to be exceptional, since changing even slightly its horizon boundary conditions completely changes the asymptotic behavior at infinity. We also construct globally regular solutions describing `lumps of pure gravity' which can be viewed as black hole remnants in the limit where the event horizon shrinks. Finally, adding a matter source we obtain globally regular and asymptotically flat solutions exhibiting the Vainstein mechanism of recovery of General Relativity in a finite region.

Motivation & Objective

• To explore black hole solutions in a ghost-free bigravity theory featuring one massive and one massless graviton.
• To understand the role of the event horizon in a two-metric framework where both metrics share the same horizon and thermodynamic properties.
• To investigate the asymptotic behavior and curvature structure of these black holes, including singularities and AdS-type limits.
• To construct globally regular solutions representing gravity lumps and study their relation to black hole remnants.
• To examine the recovery of General Relativity via the Vainshtein mechanism in the presence of matter sources.

Proposed method

• Numerical construction of black hole solutions in a ghost-free bigravity theory with two metrics—one massive and one massless.
• Implementation of shared event horizon boundary conditions for both metrics, ensuring identical surface gravity and Hawking temperature.
• Analysis of the ratio of horizon radii between the two metrics as a free parameter labeling distinct solution classes.
• Study of asymptotic behavior at infinity, identifying AdS-type and asymptotically flat solutions depending on boundary conditions.
• Investigation of curvature singularities at finite proper distance from the horizon in certain solution branches.
• Introduction of matter sources to probe the Vainshtein mechanism, leading to recovery of General Relativity in finite regions.

Experimental results

Research questions

• RQ1 How do black hole solutions in bigravity theory differ from standard general relativistic black holes in terms of metric structure and horizon properties?
• RQ2 What determines the asymptotic behavior of these black holes, and can they be asymptotically AdS or flat depending on boundary conditions?
• RQ3 Can curvature singularities form at finite distances from the horizon in bigravity black holes, and under what conditions?
• RQ4 How does the graviton mass behave in different solution branches, and can it become effectively imaginary?
• RQ5 To what extent can the Vainshtein mechanism restore General Relativity in the presence of matter in this bigravity framework?

Key findings

• Black hole solutions in the bigravity theory possess a common event horizon with identical surface gravity and Hawking temperature for both the massive and massless metrics.
• The ratio of horizon radii between the two metrics acts as a free parameter, labeling distinct solution classes with different near-horizon structures.
• Solutions exhibit multiple asymptotic behaviors: some approach AdS-type infinity, while others are asymptotically flat, with the latter being highly sensitive to horizon boundary conditions.
• Some solutions feature a curvature singularity at a finite proper distance from the horizon, indicating non-regular behavior despite a regular horizon.
• For certain solutions, the graviton mass becomes effectively imaginary, leading to oscillatory behavior around the flat metric at infinity.
• The only asymptotically flat black hole found—identified as the Schwarzschild solution—appears exceptional, as small changes in horizon conditions drastically alter its asymptotic structure.