Can Rotating Black Holes Have Short Hairs? (2501.08767v2)

Abstract: Despite the no-hair theorem, several notable hairy black hole (BH) solutions exist in both General Relativity and modified gravity theories. For such hairs to be detectable, they must extend sufficiently beyond the event horizon. This idea has been rigorously formalized by the no-short hair theorem, which dictates that all existing hairs of a static spherically symmetric BH must extend at least to the innermost light ring (LR). However, the theorem's applicability to the astrophysically relevant rotating BHs remains elusive as yet. To address this gap, we examine its validity for rotating BHs in the Konoplya-Rezzolla-Zhidenko-Stuchl\'ik and Johannsen classes. Interestingly, for Klein-Gordon separable BHs in these classes that are solutions of non-vacuum GR, the no-short hair property continues to hold. However, unlike in static cases, this result may not apply in a theory-agnostic fashion due to the rotation-induced repulsive effects. Consequently, we identify a minimal set of additional criteria on the metric and matter content needed for such a generalization in other theories. Our study marks an important first step toward establishing general results on the extent of rotating BH hairs, reinforcing their observational detections. Further extension of this novel result for rotating horizonless objects is also discussed.

• The paper confirms that for KG separable rotating black holes in GR, the no-short hair theorem holds when specific energy and metric conditions are met.
• It employs the KRZS and Johannsen metric classes to simplify the Klein-Gordon equation and geodesic analysis around the black hole.
• Findings imply that detecting hair in rotating black holes requires examining poles and additional criteria beyond the innermost light ring.

An Examination of Rotating Black Holes and the No-Short Hair Theorem

The paper "Can Rotating Black Holes Have Short Hairs?" by Rajes Ghosh and Chiranjeeb Singha addresses the applicability of the no-short hair theorem to rotating black holes (BHs). This inquiry extends previous work on static, spherically symmetric BHs in General Relativity (GR) and modified gravity theories. The no-short hair theorem states that any hair on a static BH must extend at least to the innermost light ring (LR). However, its extension to the more astrophysically relevant rotating BHs remained unexplored until now.

Theoretical Background and Methodology

The paper explores hairy BH solutions in GR and modified gravity. Traditional views, epitomized by the no-hair theorem, restrict BH characterization to mass, charge, and angular momentum, dismissing additional attributes or "hairs." Yet, hairy solutions defying this principle have been discovered, particularly in connection with scalar fields and more complex matter content.

The current research specifically investigates the validity of the no-short hair theorem for rotating BHs in the Konoplya-Rezzolla-Zhidenko-Stuchlík (KRZS) and Johannsen classes. These classes represent metrics where Klein-Gordon (KG) equation separability and geodesic equation simplification are feasible. This separability is instrumental for tractable analysis on the space-time geometry around BHs.

Key Results and Insights

The authors confirm that for KG separable BHs in these metric classes and adhering to GR, the no-short hair property holds. Importantly, they identify a minimal set of additional criteria related to the metric and matter distribution that would allow this theorem to be broader in its applications:

• Energy Conditions: The stress-energy tensor should satisfy the weak energy condition (WEC), possess a non-positive trace, and have energy density decaying faster than $r^{-4}$ at asymptotically flat regions.
• Polar Considerations: Rotation-induced effects, minimal at poles, play a crucial role in preserving no-short hair attributes.

A critical distinction arises in that the extent of hair in rotating configurations is not inherently linked to the LR as it is in static cases. The findings expose that rotating BHs lack a direct geometric criterion like the LR radius in static instances, reflecting the composite effects of rotation on matter fields.

Implications and Future Directions

The implications of this work are substantial, suggesting that no-short hair properties in rotating BHs depend on more nuanced conditions than previously assumed, influenced by the choice of gravitational theory and matter configurations.

Practically, this suggests detection strategies must adapt, potentially looking beyond innermost LRs to auxiliary constructs or specific poles, where theoretical limitations show more promise for observation.

From a theoretical standpoint, it raises the need for further exploration into the nature of "hair" in horizonless rotating objects and across varying asymptotic behaviors (such as de Sitter/anti-de Sitter spacetimes). The possibility of extending these results without relying on KG separability of metrics could significantly alter the landscape of BH physics and their observational signatures.

Conclusion

The paper by Ghosh and Singha marks an important progression in understanding BH characteristics, especially under rotation. Their approach highlights not only the mathematical intricacies required to address rotating BHs but also broadens the theoretical framework under which BH hairs are understood to exist. This paper sets the stage for a deeper exploration of the no-short hair theorem's application to a variety of gravitational and matter field theories, revealing new facets of BH dynamics and offering a refined lens for future observational tests.