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Exact analytic rotating black-hole solutions with primary hair

Pedro G. S. Fernandes

TL;DR

This work uses a Kerr-Schild ansatz within a restricted Generalized Proca theory motivated by four-dimensional Lovelock reductions to construct exact, analytic rotating black-hole solutions that possess primary hair and are non-circular. The Proca field is aligned with the null direction $l^\mu$ and configurations with $X=0$, allowing a reduced set of equations whose solutions yield a single mass function $M(r,\chi)$ and a Proca profile $\mathcal{A}(r,\chi)$. Exact solutions are found in special parameter regimes (e.g., $c_{4,2}= -c_{4,1}^2/2$, $c_{3,1}=0$, or $c_{4,1}=0$), and further enriched by disformal transformations that generate rotating Boulware-Deser-type geometries; these spacetimes are generically non-circular and exhibit horizon shapes that depend on latitude. The results expand the landscape of analytic rotating black holes beyond Kerr, with potential observable implications for geodesic motion, shadows, and gravitational-wave signals, and open directions for numerical studies and broader Lovelock-like constructions.

Abstract

Exact, analytic, asymptotically flat rotating black-hole solutions are exceedingly rare, with only a handful of examples known. Using a Kerr-Schild ansatz, we derive a multitude of exact, analytic, asymptotically flat rotating black-hole solutions within a broad class of Generalized Proca theories. These black holes differ significantly from Kerr black holes, as they possess primary hair and are non-circular, thus breaking a symmetry that vacuum black holes exhibit in General Relativity.

Exact analytic rotating black-hole solutions with primary hair

TL;DR

This work uses a Kerr-Schild ansatz within a restricted Generalized Proca theory motivated by four-dimensional Lovelock reductions to construct exact, analytic rotating black-hole solutions that possess primary hair and are non-circular. The Proca field is aligned with the null direction and configurations with , allowing a reduced set of equations whose solutions yield a single mass function and a Proca profile . Exact solutions are found in special parameter regimes (e.g., , , or ), and further enriched by disformal transformations that generate rotating Boulware-Deser-type geometries; these spacetimes are generically non-circular and exhibit horizon shapes that depend on latitude. The results expand the landscape of analytic rotating black holes beyond Kerr, with potential observable implications for geodesic motion, shadows, and gravitational-wave signals, and open directions for numerical studies and broader Lovelock-like constructions.

Abstract

Exact, analytic, asymptotically flat rotating black-hole solutions are exceedingly rare, with only a handful of examples known. Using a Kerr-Schild ansatz, we derive a multitude of exact, analytic, asymptotically flat rotating black-hole solutions within a broad class of Generalized Proca theories. These black holes differ significantly from Kerr black holes, as they possess primary hair and are non-circular, thus breaking a symmetry that vacuum black holes exhibit in General Relativity.
Paper Structure (12 sections, 45 equations, 3 figures)

This paper contains 12 sections, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generalized Proca theory and four-dimensional Proca theories of Lovelock gravity
  3. Equations of motion for a rotating Kerr-Schild metric
  4. Exact analytic rotating solutions
  5. Solutions with $c_{4,2}=-c_{4,1}^2/2$
  6. Solutions with $c_{3,1}=0$
  7. Solutions with $c_{4,1}=0$: stealth Kerr
  8. Generating new rotating solutions via disformal transformations
  9. An example: a rotating Boulware-Deser solution
  10. On the breaking of circularity
  11. Conclusions and future work
  12. Linearly independent field equations

Figures (3)

  • Figure 1: Radial coordinate location of the event horizon as a function of the polar angle for the metric described by Eq. \ref{['eq:Solution1']} with $q(\chi)=\mu \chi$, $c_{4,1}=-4$, $c_{3,1}=-16$, $a/\mu=0.2$, $\alpha/\mu^2=0.1$, and $\mu=1$.
  • Figure 2: Radial coordinate location of the event horizon as a function of the polar angle for the metric described by Eq. \ref{['eq:Solution_c31_0']} with $\mu_{\rm ADM}=1$, $q/\mu_{\rm ADM}=1$, $c_{4,2}=-c_{4,1}^2$, $c_{4,1}=1$, $a/\mu_{\rm ADM}=0.5$, and $\alpha/\mu_{\rm ADM}^2=0.4$.
  • Figure 3: Radial coordinate location of the event horizon as a function of the polar angle for the metric described by Eq. \ref{['eq:rotatingBD']} with $q(\chi)=\mu$, $a/\mu=0.5$ and $\alpha/\mu^2=0.4$.