Chronology Protection of Rotating Black Holes in a Viable Lorentz-Violating Gravity

TL;DR

The paper analyzes the causal structure of a rotating black-hole solution in the low-energy sector of non-projectable Horava gravity with a Lorentz-violating Maxwell field under a cosmological constant $\Lambda> -3/a^2$. It demonstrates that, although the region containing closed timelike curves (CTCs) matches Kerr-Newman, chronology is protected by a torus-shaped ${\Sigma}^2=0$ singularity that acts as a boundary, rendering the causality-violating region inaccessible to exterior observers. The authors provide the explicit metric in Boyer-Lindquist coordinates with $\rho^2$, $\Delta_r$, $\Delta_\theta$, $\Xi$, and $\Sigma^2$, analyze different mass-rotation regimes, and prove Proposition 1: the maximally extended spacetime is causal for $\Lambda> -3/a^2$ via stable time functions $T$ with $(\nabla_\mu T)(\nabla^\mu T)<0$. This work supports Hawking's chronology protection in a Lorentz-violating gravity context, while leaving open questions about the full Horava theory and the potential for Doran-like horizon-penetrating coordinates.

Abstract

We study causal properties of the recently found rotating black-hole solution in the low-energy sector of Horava gravity as a viable Lorentz-violating (LV) gravity in four dimensions with the LV Maxwell field and a cosmological constant $Λ(>-3/a^2)$ for an arbitrary rotation parameter $a$. The region of non-trivial causality violation containing closed timelike curves is exactly the same as in the Kerr-Newman or the Kerr-Newman-(Anti-)de Sitter solution. Nevertheless, chronology is protected in the new rotating black hole because the causality violating region becomes physically inaccessible by exterior observers due to the new three-curvature singularity at its boundary that is topologically two-torus including the usual ring singularity at $(r,θ)=(0,π/2)$. As a consequence, the physically accessible region outside the torus singularity is causal everywhere.

Figures (6)

• Figure 1: $r$ vs. $\theta{\in[0,\pi]}$ of singularity surfaces determined by ${\Sigma}^2 (r, \theta)=0$ for $m=2$ with $a=2$, $1$, and $0.1$, which correspond to the closed curves from the bottom to the top, respectively. As $a \rightarrow 0$, the singularity surfaces reduce to a point singularity of the Schwarzschild black hole at $r=0$.
• Figure 2: The maximal extension of the rotating black-hole solution for $m^2>a^2$ by identifying the top of the disk $x^2+y^2<a^2$ with $z=0$ in the region $r>0$ (the left chart) with the bottom of the corresponding disk in the region $r<0$ (the right chart) and vice versa. The torus singularity of ${\Sigma}^2=0$ exists in the region $r\leq 0$ that includes the usual ring singularity of $\rho^2=0$ located at $(r,\theta)=(0,\pi/2)$.
• Figure 3: A closed curved in the spacetime, which cannot be non-spacelike everywhere.
• Figure 4: $r$ vs. $\theta{\in[0, \pi]}$ of the singularity surface (\ref{['singular_AdS2']}) with ${\xi=1,}~ a=1$ and $m=2$. In the left panel, we vary ${\Lambda}=-1,0,1$ (from the outer to inner curves) with $q_e=q_m=0$. In the right panel, we vary $q_e=0,0.5,1$ (from the inner to outer curves) with $q_m=0$ and ${\Lambda}=0$.
• Figure 5: The maximal extension of the generalized rotating black-hole solutions with electromagnetic charges and cosmological constant, with the similar identification of the disk regions as in Fig. 2. The torus singularity of ${\Sigma}^2=0$ spreads in both regions $r>0$ and $r<0$ and envelopes the ring singularity of $\rho^2=0$ at $(r,\theta)=(0,\pi/2)$.