A new exact rotating spacetime in vacuum: The Kerr--Levi-Civita Spacetime

TL;DR

The paper introduces the Kerr–Levi-Civita spacetime, a new exact rotating vacuum solution obtained by applying the inversion symmetry of the Ernst equations to the Kerr seed in the magnetic WLP form. The resulting metric is regular (no curvature singularities or CTCs), has horizons at the Kerr radii, and exhibits a Levi-Civita–like asymptotic structure that strongly influences its ergoregion, making it frame-dependent and extending to infinity. Although it shares horizons with Kerr, the Kerr–LC geometry lacks a Kerr–Schild representation and features a frame-dependent ergoregion shaped by the Levi-Civita background. The work opens avenues for further study of geodesics, hidden symmetries, shadows, lensing, and conserved charges in this novel regular rotating spacetime, and clarifies its distinction from Kerr–Swirling in terms of gauge choices and transformations.

Abstract

We construct a new rotating solution of Einstein's theory in vacuum by exploiting the Lie point symmetries of the field equations in the complex potential formalism of Ernst. In particular, we perform a discrete symmetry transformation, known as inversion, of the gravitational potential associated with the Kerr metric. The resulting metric describes a rotating generalization of the Schwarzschild--Levi-Civita spacetime, and we refer to it as the Kerr--Levi-Civita metric. We study the key geometric features of this novel spacetime, which turns out to be free of curvature singularities, topological defects, and closed timelike curves. These attractive properties are also common to the extremal black hole and the super-spinning case. The solution is algebraically general (Petrov-type I), and its horizons lie at the horizon radii of the Kerr black hole. The ergoregions, however, are strongly influenced by the Levi-Civita-like asymptotic structure, producing an effect akin to the magnetized Kerr--Newman and swirling solutions. Interestingly, while its static counterpart permits a Kerr--Schild representation, the Kerr--Levi-Civita metric does not admit such a formulation.

Figures (2)

• Figure 1: In panels (a), (b), and (c), we show a cross section, taken at $Y=0$, of the embeddings of the outer-horizon surfaces of Kerr (gray) and Kerr--LC (black) black holes for $a=1/4$ (solid) and $a=1/2$ (dashed), $a=a_0$ (solid) and $a=0.75$, and $a=a_{\mathrm{iso}}\approx 0.84$ (solid) and $a=a_{\max}$ (dashed), respectively. The function $Z$ is the embedding function and $X^2+Y^2 = \mathcal{R}^2$, where $\mathcal{R}$ is the proper circumferential radius at $r=r_{\!+}$. In panel (d), we graph the Gaussian curvature ${K}$ of the outer horizon at the equator (black, solid) and the poles (gray, dashed). The solid curve crosses ${K}=0$ exactly at $a=a_0$. The two curves meet at $a=a_{\mathrm{iso}}$ and terminate at $a=a_{\max}$. We have set $m=1$.
• Figure 2: Cross section, taken at $\mathrm{y}=0$, of the ergoregion (gray fill) dressing the event-horizon of a Kerr--LC black hole with $r_+=1$ and $r_-=1/2$ in rotating frames with angular velocity $\alpha=0$ (a), $\alpha=1.64$ (b), and $\alpha=3$ (c). In panel (d), we provide a density map of the absolute difference $\Delta\Omega$ in the above spacetime. We use total white for $\Delta\Omega=0$. The darker the color is, the bigger the difference becomes. The black ellipsoid represents the black hole.