Charged black holes in Weyl conformal gravity

TL;DR

This work analyzes the charged, nonrotating black-hole solutions in Weyl conformal gravity (CGRN), revealing a rich array of spacetime geometries driven by the dyonic charge and the CG parameters. By deriving photon-sphere radii, horizon equations, Hawking temperatures, and extremal-limit conditions for both γ≠0 and γ=0 branches, the study uncovers novel structures such as nested black holes with a Cauchy horizon between event horizons and a critical D_g^2=1 threshold that enables a three-horizon extremal triple limit. The absence of a 1/r^2 term in CGRN (even with charge) destabilizes standard GR intuitions about horizon sequences and singularities, producing diverse causal regions and potential near-horizon geometries not realized in GR. The results provide a comprehensive map of CGRN spacetimes and motivate future work on interior solutions, negative-mass regimes, and extensions to rotating CG Kerr–Newman analogues.

Abstract

We present a parametric study of the spacetime structures obtainable in Weyl conformal gravity's dyonic Reissner-Nordström solution. We derive expressions for photon sphere radii and horizons for this metric in terms of the conformal gravity parameters, from which we then determine analytic formulae for extremal limits and Hawking temperatures. Due to the surprising lack of the inverse quadratic $1/r^2$ term in this fourth-order metric, there is no guarantee for the innermost horizon of a black hole spacetime to be a Cauchy horizon, which is in direct contrast to the corresponding metric in general relativity. For example, for certain parameter values, a ``nested black hole'' is seen to exist; in such a spacetime, we find a Cauchy horizon trapped between two event horizons, which is not a structure known to be obtainable in standard general relativity. In addition to such exotic spacetimes, we also find a critical value for the electric and magnetic charges, at which the stable and unstable photon spheres of the metric merge, and we obtain extremal limits where three horizons collide.

Figures (10)

• Figure 1: Diagram showing the nature of the singularity in the CGRN metric, for given combinations of $D_g^2$ and $\beta\gamma$. Blue regions correspond to spacelike singularities ($B(0)\rightarrow-\infty$), red regions correspond to timelike singularities ($B(0)\rightarrow+\infty$), and the thick black contour that demarcates these regions corresponds to spacetimes with no singularities. $B^{\gamma\neq0}_\mathrm{CGRN}(r)$\ref{['eq:CGRN_metric_1']} is ill-defined at $\beta\gamma=0$, so we notate this discontinuity by a dashed line.
• Figure 2: The effective potential $V_\mathrm{eff}$\ref{['eq:V_eff']} for null particles in the $\gamma\neq0$ CGRN metric \ref{['eq:CGRN_metric_1']} against $r/\beta$, for different values of charge $D_g$. Stars and diamonds correspond to stable and unstable photon spheres $r_\mathrm{st}$ and $r_\mathrm{ust}$\ref{['eq:photonspheres']} respectively, and the circles denote the marginally stable saddle point photon sphere $r_\mathrm{mst}$\ref{['eq:saddle']}. The black dashed and dotted lines show the locations of the stable and unstable photon spheres as charge is increased to its maximum value of $D_g^2=1$.
• Figure 3: Horizon plots (a) for the CG Schwarzschild metric \ref{['eq:MK_metric']} and (b) for the $\gamma\neq0$ CG Reissner-Nordström metric \ref{['eq:CGRN_metric_1']}. In both cases the horizon radii are found as roots of the cubic equation \ref{['eq:CGRN_horizons']} for different values of $\beta\,\gamma$ and $\beta^2\,\kappa$. Solid curves give the horizon radii and dashed ones give photon sphere radii from \ref{['eq:photonspheres']}. We show the full curves for photon sphere solutions, but note that photon spheres cannot exist in spacelike $\mathrm{S}^-$ or $\mathrm{S}^+$ regions, where $B(r)<0$. Thus the photon spheres occur above but not below the extremal horizon limits where 2 horizons merge.
• Figure 4: Horizon curves as in figure \ref{['fig:D = 0.25 Structure']} but for $\beta^2\,\kappa=0$ and $\pm0.01$. Black, cyan, lime, and magenta correspond to $\Delta_\mathrm{H}^{\gamma\neq0}=0$ for $D_g^2=0$ (CGS), $1/6$, $2/6$, and $3/6$, respectively. The discontinuities at $\beta\,\gamma=0$ are noted as thin black horizontal lines on each subplot for clarity.
• Figure 5: Horizon plots for $\gamma=0$ CGRN spacetimes \ref{['eq:CGRN_horizons_gamma0']}, for various values of $D_g^2$.