Structure of Kerr black hole spacetimes in Weyl conformal gravity

TL;DR

This paper analyzes the CG Kerr metric within Weyl conformal gravity, detailing the causal and ergoregion structure across the $\gamma-\kappa-a$ parameter space for positive mass and comparing the results to GR Kerr, Kerr-dS, and Kerr-AdS spacetimes. It derives the horizon and ergosurface conditions from quartic and cubic polynomials, identifies three principal extremal limits (extremal spin, horizon cosmological, and the Empty case), and shows that horizon surface gravities and Hawking temperatures vanish at extremality. The analysis reveals a rich landscape of 13 spacetime configurations, governed by the background sign set by $k = \kappa + \frac{\gamma^2(1-\beta\gamma)}{(2-3\beta\gamma)^2}$ and strongly influenced by the CG parameter $\gamma$, with implications for tests of CG versus GR and for near-horizon/quantum gravity studies. The work also demonstrates a conformal equivalence between CG Kerr and CG Schwarzschild in the $a\to0$ limit, and discusses the role of $\gamma$ in shaping cosmological features and the potential observational signatures in extremal regimes.

Abstract

We analyze the stationary, uncharged, rotating, vacuum solution to Weyl conformal gravity. We elucidate the causal and ergoregion structure of the spacetimes found in the parameter space of the metric for positive mass. These are then compared to the analogous Kerr, Kerr-de Sitter, and Kerr-Anti-de Sitter solutions to general relativity. Additionally, we investigate and derive the extremal limits for both the horizons and ergosurfaces of the spacetimes. The horizon surface gravities and Hawking temperatures at the extremal horizon limits are then calculated to show that they vanish.

Figures (8)

• Figure 1: Plots of $V_+(r)$ and $-V_+'(r)$, for $(\gamma, \kappa, a, L_z) = (0.1, 0.01, 0.25, 0)$, within a timelike $\mathrm{T}$ region. We have an $\mathrm{S}^-$ region to the left of the event horizon at $r^{\mathrm{H}}_\mathrm{O}$, and an $\mathrm{S}^+$ region to the right of the cosmological horizon at $r^{\mathrm{H}}_\mathrm{C}$. The points (red) mark the values of these functions at the turning point of $V(r)$ at $r = r^{\mathrm{TP}}$.
• Figure 2: Schematic diagram of a parametric plot $\beta \gamma$ vs $r/\beta$ showing the transitions between the domains in table \ref{['tab:domains']}. Here, the horizons $\mathrm{H}$ are in blue and ergosurfaces $\mathcal{E}$ in dashed red. The extremal horizon limits are marked by stars, and the extremal ergosurface cosmological limit by a diamond. Note that distances between features have been exaggerated for clarity.
• Figure 3: Causal structure, ergoregion structure, and plot of horizons (blue) and ergosurfaces (dashed red) of the CG Kerr spacetime for $(\beta^2\kappa, a/\beta) = (0, 0.25)$. The solid black line represents the Empty case at $\beta \gamma = \frac{2}{3}$.
• Figure 4: Map of the parameter space of $\beta^2 \kappa$ against $\beta \gamma$, for spin $a/\beta = 0.25$, showing the relevant spacetime domains defined in tables \ref{['tab:domains']} and \ref{['tab:domains neg']}.
• Figure 5: Causal structure, ergoregion structure, and plot of horizons (blue) and ergosurfaces (dashed red) of the CG Kerr spacetime for $(\beta^2\kappa, a/\beta) = (0.01, 0.25)$. The solid black line represents the Empty case at $\beta \gamma = \frac{2}{3}$.