Charged black holes in Weyl conformal gravity

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.

💡 Research Summary

The paper presents a comprehensive study of charged, non‑rotating black‑hole solutions in Weyl conformal gravity (CG), focusing on the dyonic Reissner‑Nordström analogue (CGRN). Starting from the fourth‑order Bach field equations, the authors derive the general static, spherically symmetric metric function (B(r)) as a linear combination of four terms: a constant, a Newtonian (1/r) term, a linear (\gamma r) term, and a quadratic (-\kappa r^{2}) term. The presence of electric and magnetic charges introduces a dimensionless “dyonic” parameter (D_{g}^{2}= -3(Q^{2}+P^{2})/(8\alpha_{g})), which modifies the constant term through the constraint (w^{2}=1+3uv-D_{g}^{2}). Because the CG action is conformally invariant, the stress‑energy tensor is traceless, and the charge contribution appears as a (1/r^{4}) term in the energy density rather than the familiar (1/r^{2}) term of the GR Reissner‑Nordström solution. Consequently, the metric lacks the inverse‑quadratic term entirely, leading to qualitatively new causal structures.

Two families of solutions are examined in detail. When the linear conformal term (\gamma\neq0), the metric reads
(B_{\gamma\neq0}(r)=(1-3\beta\gamma)-\beta(2-3\beta\gamma)r^{-1}+D_{g}^{2}\gamma r+\gamma r-\kappa r^{2}).
When (\gamma=0) the metric simplifies to
(B_{\gamma=0}(r)=w_{0}-2\beta r-\kappa r^{2}) with (w_{0}=\sqrt{1-D_{g}^{2}}).
The condition (D_{g}^{2}\le1) ensures a real constant term; saturation (D_{g}^{2}=1) drives (w_{0}) to zero and eliminates the constant contribution altogether.

Because (B(r)) is a cubic polynomial in (r), it can have up to three real roots, each corresponding to a distinct horizon: an event horizon (H_{E}), a Cauchy horizon (H_{C}), and a cosmological horizon (H_{\Lambda}). By scanning the four‑dimensional parameter space ((\beta,\gamma,\kappa,D_{g})) the authors map out a rich “phase diagram” of horizon configurations. Notable exotic configurations include:

• Nested black holes – a Cauchy horizon trapped between two event horizons, a structure impossible in standard GR.
• Critical photon‑sphere merger – the stable and unstable photon‑sphere radii coalesce at a specific charge magnitude, implying a sudden change in the black‑hole shadow size.
• Extremal limits – several distinct limits where horizons merge: (i) charge‑extremal (H_{CE}) (Cauchy and event horizons coincide), (ii) nested‑extremal (H_{EC}) (event horizon inside a Cauchy horizon), (iii) Nariai‑type (H_{E\Lambda}) (event and cosmological horizons merge), and (iv) triple‑extremal (H_{TL}) where all three horizons coincide. In the triple‑extremal case the surface gravity vanishes, giving a Hawking temperature (T_{H}=0).

Photon‑sphere radii are obtained from the effective potential condition (V_{\text{eff}}’(r)=0), yielding two solutions (r_{\text{ph}}^{\pm}). The analysis shows that for (D_{g}) exceeding a critical value the two radii merge, eliminating the inner stable photon sphere. This predicts a sharp transition in observable shadow features for sufficiently charged CG black holes.

The Hawking temperature is computed via the standard surface‑gravity formula (T_{H}=B’(r_{h})/(4\pi)). For non‑extremal event horizons the temperature depends on all four parameters; increasing charge (larger (D_{g})) generally lowers (T_{H}). At the extremal limits (B’(r_{h})\to0), confirming the zero‑temperature nature of the corresponding configurations.

The authors discuss observational implications. The photon‑sphere merger could be probed by very‑long‑baseline interferometry (e.g., the Event Horizon Telescope) through abrupt changes in shadow diameter. Nested black holes would modify the quasi‑normal mode spectrum of ringdown gravitational‑wave signals, potentially offering a new diagnostic of higher‑order gravity. The zero‑temperature extremal states may be relevant for quantum‑gravity considerations of black‑hole entropy and the information paradox.

In summary, the paper demonstrates that Weyl conformal gravity’s fourth‑order dynamics, combined with electric and magnetic charges, generate a spectrum of black‑hole spacetimes far richer than that of General Relativity. The absence of the (1/r^{2}) charge term, the presence of a linear conformal potential, and the resulting exotic horizon structures (nested horizons, triple‑horizon collisions, photon‑sphere coalescence) open new avenues for both theoretical exploration and astrophysical testing of conformal gravity.