Exact Kerr-Newman-(A)dS and other spacetimes in bumblebee gravity: employing a novel generating technique

TL;DR

This work establishes a unique generating technique for exact solutions in Einstein–bumblebee gravity under the strong assumptions $B_{\mu}=b_{\mu}$ and $B_{\mu\nu}=0$, yielding the modified metric $g_{\mu\nu} = \tilde{g}_{\mu\nu} + \dfrac{\xi}{1+\epsilon\xi b^2} b_{\mu} b_{\nu}$ with the bumblebee field aligned to a background geodesic via the Hamilton–Jacobi equation. It then generalizes the approach to cases with nonzero cosmological constant and with electromagnetic fields, showing that the field equations reduce to vacuum or Einstein–Maxwell equations in a tilde frame, enabling a broad generation of bumblebee spacetimes from known seeds. The technique is applied to the Kerr–Newman–Taub–NUT–(A)dS family, where the bumblebee field depends on geodesic data $(E,L,C)$ and is not unique, and the global reality of the field imposes strong constraints on admissible geodesics, especially near horizons. Overall, the paper provides a constructive, geometrically grounded method for exploring Lorentz-violating spacetimes in extended gravity and clarifies how reality conditions and charge–rotation parameters shape the landscape of viable solutions.

Abstract

In this work, we show that if the bumblebee field in the Einstein-bumblebee theory is given by its vacuum expectation value ($B_μ=b_μ$) and it is not dynamical ($\partial_μB_ν-\partial_νB_μ=0$), then these conditions uniquely provide a generating technique, allowing us to construct exact solutions to bumblebee gravity from the vacuum solutions by adding a term $\sim b_μb_ν$ to the metric tensor. Also, we show that the bumblebee field within this technique is proportional to the tangential vector of the (timelike or spacelike) geodesic curve in the background vacuum spacetime, and can be easily found knowing the solution to the Hamilton-Jacobi equation. Moreover, we prove that this technique can be extended to the case of any non-zero cosmological constant and the presence of the electromagnetic field. We apply this generating technique and obtain the bumblebee extension of the Kerr-Newman-Taub-NUT-(anti-)de Sitter spacetime. We show that this extension is not unique, as it depends on the exact geodesic curve one chooses to associate a bumblebee field with. Then, by considering various special cases of this generic solution, we demonstrate that the condition of the global reality of the bumblebee field limits the set of geodesics with which we can associate it.

Figures (5)

• Figure 1: The ranges of energy $E$ and the Carter constant $C$, where the bumblebee field (\ref{['68']}) is either globally real (green ranges) or becomes imaginary at some range (red ranges). Black curves represent the border between such ranges. Left panel is the plot for the spacelike bumblebee field $\epsilon=+1$, while the right panel is the plot for the timelike bumblebee field $\epsilon=-1$. The green dot represents the special value $C=4m^2$ at which the bumblebee field becomes globally real even for $E=0$ (but only for the case $\epsilon=+1$).
• Figure 2: The ranges of energy $E$ and the rotation parameter $a/m$, where the bumblebee field (\ref{['f_pr']})-(\ref{['h_pr']}) is either globally real (green ranges) or becomes imaginary at some range (red ranges). Black curves represent the border between such ranges. The left panel is the plot for the spacelike bumblebee field $\epsilon=+1$, while the right panel is the plot for the timelike bumblebee field $\epsilon=-1$. The dashed line represents the value of $a/m=1$, where the black hole becomes extremal.
• Figure 3: The ranges of energy $E$ and the Carter constant $C$, where the bumblebee field (\ref{['68']}) for the Schwarzschild--(A)dS black hole with $f=1-\dfrac{2m}{r}-\dfrac{\Lambda}{3}r^2$ is either globally real (green ranges) or becomes imaginary at some range (red ranges). Various plots represent various values of the cosmological constant for spacelike $\epsilon=+1$ and timelike $\epsilon=-1$ bumblebee fields.
• Figure 4: The ranges of energy $E$ and the rotation parameter $a/m$, where the bumblebee field for Kerr--(A)dS spacetime is either globally real (green ranges) or becomes imaginary (red ranges) for different $\lambda=\Lambda m^2$. Black curves represent the border between such ranges. Plots represent only the cases with $\epsilon=+1$, because for the case $\epsilon=-1$ the corresponding pictures are the same as for the case without cosmological constant, see Fig. \ref{['fig2']}. The dashed line represents the value of $a/m$, where the black hole becomes extremal.
• Figure 5: The ranges of energy $E$ and the charge parameter $q/m$, where the bumblebee field for the Reissner--Nordström--(A)dS spacetime is either globally real (green ranges) or becomes imaginary (red ranges) for different $\lambda=\Lambda m^2$. Black curves represent the border between such ranges. The dashed line represents the value of $q/m$, where the black hole becomes extremal.