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

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.

💡 Research Summary

In this paper the authors rigorously develop a generating technique that produces exact solutions of Einstein‑bumblebee gravity from any known vacuum solution of General Relativity. The key assumptions are that the bumblebee vector field Bμ is frozen at its vacuum expectation value bμ, that its field strength vanishes (∂μBν−∂νBμ=0), and that the potential V and its derivative vanish, i.e. the field sits at the minimum of the potential. Under these conditions the bumblebee field equation reduces to bμRμν=0, while the modified Einstein equation simplifies to Rμν=ξ∇α∇(μ(bαbν)). By interpreting bμ as a closed one‑form, the authors invoke the Poincaré lemma to write b=b dρ, where ρ is a scalar function. Introducing Gaussian normal coordinates (ρ, xi) the metric takes the form ds²=ε dρ²+γij(ρ,xi)dxi dxj, and the extrinsic curvature Kij=½∂ργij appears naturally. The field equations then become a set of equations for Kij and γij that are identical to the vacuum Einstein equations after a simple rescaling of the ρ‑coordinate: ρ→ρ/(1+εξb²). Consequently, any vacuum metric can be transformed into a bumblebee solution simply by (i) bringing it to Gaussian normal form, (ii) performing the rescaling, and (iii) adding the term ξ/(1+εξb²) bμbν to the metric. The bumblebee vector itself is given by bμ=b/(√(1+εξb²))∂μρ, i.e. it is proportional to the tangent of the geodesic (timelike or spacelike) whose Hamilton‑Jacobi function is ρ.

A crucial observation is that the scalar ρ must satisfy the Hamilton‑Jacobi equation ĝμν∂μρ∂νρ=ε in the original vacuum background ĝμν. Therefore, once the Hamilton‑Jacobi separability of the background is known, the required ρ can be constructed directly from the known action‑angle variables. This links the generation of the bumblebee field to the integrability properties of the seed spacetime.

The authors extend the technique to include a non‑zero cosmological constant Λ and an electromagnetic field. The presence of Λ merely modifies the Hamilton‑Jacobi equation, while the electromagnetic field enters through the condition bμFμν=0, which can be satisfied by aligning bμ with a Killing direction orthogonal to the field strength. Thus the algorithm works for Einstein‑Maxwell‑bumblebee systems as well.

Applying the method to the most general separable type‑D solution—Kerr‑Newman‑Taub‑NUT‑(A)dS—they obtain a family of bumblebee‑deformed metrics. The deformation depends on the choice of the geodesic used to define ρ; consequently the resulting spacetime is not unique. By examining several sub‑cases (static Reissner‑Nordström, Kerr, Kerr‑(A)dS, Taub‑NUT, etc.) they show how the bumblebee vector can be aligned with the timelike Killing vector (∂t), the azimuthal Killing vector (∂φ), or a linear combination thereof. The requirement that the bumblebee field be globally real (b²>0) imposes constraints on the constants of motion (energy E and angular momentum L) of the chosen geodesic. In rotating spacetimes, for instance, only geodesics with a specific ratio L/E lead to a real bumblebee field, effectively restricting the admissible families of solutions.

The paper concludes that the “add a bμbν term” prescription is not only unique but also algorithmic: given any vacuum solution with separable Hamilton‑Jacobi structure, one can systematically construct its bumblebee counterpart, even in the presence of Λ and electromagnetic fields. The non‑uniqueness arising from the choice of geodesic and the reality conditions provide new physical insight into how Lorentz‑violating vector fields can be embedded in realistic black‑hole geometries. This work thus opens a systematic pathway to explore a broad class of exact Lorentz‑violating spacetimes, including rotating, charged, and NUT‑type black holes with (anti‑)de Sitter asymptotics.