The No-Short Hair Theorem for Black Holes and Wormholes in Extra-Dimension

TL;DR

This work extends the no-short hair theorem beyond 4D Einstein gravity to extra-dimensional and modified-gravity settings, establishing when hair must lie outside the photon sphere in dD GR, 5D Einstein–Gauss–Bonnet theory, braneworld black holes, and braneworld wormholes. It derives generalized hair bounds via the hair function ε ≡ e^{−δ} r^d T^r_r, analyzes specific solutions (linearized BBH, tidal RN on the brane, DS/Morris–Thorne wormholes), and identifies key conditions (e.g., α>0 in EGB, BD parameter constraints, and RS1 on-brane WEC) that ensure no-short hair persists. An Appendix set offers a theory-agnostic derivation via null circular orbits and detailed BD and scalar-tensor analyses, illustrating how the theorem adapts to scalar couplings and higher-curvature terms. The paper argues that observational signatures—e.g., quasinormal modes and photon rings—could test the no-short hair bounds and hence probe the existence of extra dimensions or string-inspired corrections to gravity. Overall, the results provide a framework linking hair bounds to higher-dimensional physics and suggest pathways for using astrophysical data to constrain or reveal extra-dimensional theories.

Abstract

The no-short hair theorem for static spherically symmetric black holes in general theory of relativity asserts that if a black hole has hair, that hair must extend beyond the lowest photon sphere radius of the black hole. This report generalizes the theorem to various extra-dimensional cases of black holes and wormholes in post-Einstein gravity and shows how the observational satisfaction of the no-short hair theorem may lead to a probe for the existence of extra-dimensions. We also show how we may lead to argue that this no-short hair theorem is satisfied for black holes in theories of quantum gravity that naturally precludes extra-dimensional cases, like string theory.