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Static hairy black hole in 4D General Relativity

Marco Astorino

TL;DR

The work demonstrates a static hairy black hole in four-dimensional vacuum GR by introducing a hair parameter $B$ that induces a Petrov type I geometry and an external gravitational field, avoiding asymptotic flatness and spherical symmetry. The horizon becomes oblate with a smaller area, while the temperature, mass, and entropy satisfy a consistent thermodynamic framework after normalisation, and the spacetime remains regular outside the horizon. This solution provides a continuous deformation of Schwarzschild and opens avenues for exploring non-M Minkowskian asymptotics, potential phenomenological signatures (shadows, lensing), and generalisations to electrovacuum, rotation, and cosmological contexts. The background structure connects to a hyperbolic bubble-like spacetime, enriching the landscape of exact solutions and challenging traditional no-hair theorems by relaxing key assumptions.

Abstract

In four-dimensional vacuum general relativity the only known static, exact and analytical black hole solution is given by the Schwarzschild spacetime. In this paper this renowned metric is generalised by adding another integrating constant, a hair that switches the metric from the Petrov type D to the type I. This new parameter represents the intensity of an external gravitational field, which can be considered the hyperbolic generalisation of the Witten's bubble of nothing. No curvature or conical singularities are present outside the event horizon. The no hair arguments are circumvented because the metric is not asymptotically flat, and neither the black hole is spherical. The gravitational hair continuously deforms the Schwarzschild geometry: the horizon becomes oblate, while its area is reduced. Conserved charges and thermodynamic properties of the black hole are studied.

Static hairy black hole in 4D General Relativity

TL;DR

The work demonstrates a static hairy black hole in four-dimensional vacuum GR by introducing a hair parameter that induces a Petrov type I geometry and an external gravitational field, avoiding asymptotic flatness and spherical symmetry. The horizon becomes oblate with a smaller area, while the temperature, mass, and entropy satisfy a consistent thermodynamic framework after normalisation, and the spacetime remains regular outside the horizon. This solution provides a continuous deformation of Schwarzschild and opens avenues for exploring non-M Minkowskian asymptotics, potential phenomenological signatures (shadows, lensing), and generalisations to electrovacuum, rotation, and cosmological contexts. The background structure connects to a hyperbolic bubble-like spacetime, enriching the landscape of exact solutions and challenging traditional no-hair theorems by relaxing key assumptions.

Abstract

In four-dimensional vacuum general relativity the only known static, exact and analytical black hole solution is given by the Schwarzschild spacetime. In this paper this renowned metric is generalised by adding another integrating constant, a hair that switches the metric from the Petrov type D to the type I. This new parameter represents the intensity of an external gravitational field, which can be considered the hyperbolic generalisation of the Witten's bubble of nothing. No curvature or conical singularities are present outside the event horizon. The no hair arguments are circumvented because the metric is not asymptotically flat, and neither the black hole is spherical. The gravitational hair continuously deforms the Schwarzschild geometry: the horizon becomes oblate, while its area is reduced. Conserved charges and thermodynamic properties of the black hole are studied.
Paper Structure (9 sections, 42 equations, 1 figure)

This paper contains 9 sections, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Static, type I, hairy black hole solution in vacuum
  3. Geometrical and physical properties
  4. Thermodynamics
  5. Background
  6. Geodesics
  7. A new look at the black hole inside the expanding bubble of nothing
  8. Conclusions
  9. Acknowledgements

Figures (1)

  • Figure 1: Embedding in Euclidean three-dimensional space $\mathbb{E}^3$ of the black hole's event horizon distorted by the presence of an external gravitational field, for different values of the parameter $B$, while keeping $m=1$. When $B=0$ we have the usual spherical Schwarzschild horizon, as in picture (\ref{['fig:picture-horizons']}.a).