Loading the stable photon sphere in Weyl conformal gravity metrics

TL;DR

This paper studies the accumulation of null matter on the stable photon sphere of the Mannheim–Kazanas metric in Weyl conformal gravity. By modeling a zero-width thin shell via a delta‑like source in the Bach equations, it shows that a radial‑pressure jump occurs only when the shell radius differs from the photon-sphere radii, and that the stable photon sphere’s area remains invariant under loading. A critical loading amplitude produces an extremal horizon exactly at the stable photon sphere, with a near‑horizon AdS$_2\times$S$^2$ geometry whose AdS$_2$ radius equals the S$^2$ radius and is independent of the cosmological curvature. This decoupling from the cosmological term in CG contrasts with GR results and suggests novel γ–κ dynamics and potential extensions to finite-width shells and time‑dependent CG solutions.

Abstract

We investigate the accumulation of null matter at the stable photon sphere in the Mannheim-Kazanas metric, the analogue to the Schwarzschild solution in Weyl's conformal theory of gravity. In our toy problem in which we consider an infinitely-thin shell, we find that a jump in radial pressure ${T^r}_r$ is induced across the shell unless the shell has a radius of either the unstable or stable photon sphere radii. We then find that upon loading the stable photon sphere, its area remains invariant. Furthermore, at a critical threshold loading limit for this zero-width null matter shell, we are able to produce a metric containing an extremal horizon with an AdS$_2\times$S$^2$ geometry completely independent of the cosmological curvature. This hitherto unencountered and therefore unexpected result is a phenomenon unseen in standard nonconformal second-order metrics with nonzero cosmological constants.

Figures (6)

• Figure 1: The effective potential $V_\text{eff}(r)$\ref{['eq:V_eff']} experienced by null particles in the MK metric \ref{['eq:MK_metric']} against $\beta/r$ for $-2\leq\beta\gamma\leq2$ in steps of $1/3$. $\beta/r$ as the radial parameter maps $r/\beta\rightarrow+\infty$ to $\beta/r=0$ and $r/\beta=0$ to $\beta/r\rightarrow+\infty$; as a result, the $V_\mathrm{eff}$ intercept corresponds naturally to the cosmological curvature $\kappa$. Stars/rhombuses denote stable/unstable photon spheres, and solid/hollow markers show whether they exist in $\mathrm{T}$/$\mathrm{S}$ regions. The dotted/dashed grey lines denote stable/unstable photon sphere radii at arbitrary $\beta\gamma$.
• Figure 2: Energy levels of resonant null orbits for $V_\mathrm{eff}(r;\,\beta\gamma=1,\,\beta^2\kappa=-0.5)$ with their corresponding closed rosette orbits visualised to the right. The grey dashed lines correspond to $V_\mathrm{eff}(r=r_\mathrm{ust})$, and the black dashed lines correspond to $V_\mathrm{eff}(r=r_\mathrm{st})$. $R_n$ corresponds to the ratio of azimuthal oscillations to radial oscillations.
• Figure 3: A dimensionless parameter map spanning $(-\infty, +\infty)$ in both $\beta\gamma$ and $\beta^2\kappa$, showing whether $r_\mathrm{st}$ exists in a $\mathrm{T}$ (yellow) or $\mathrm{S}$ (pink) region. The blue boundary corresponds to \ref{['eq:ps_hor']}. The darkened region corresponds to $r_\mathrm{st}<0$.
• Figure 4: The jump in pressure \ref{['eq:pressure_jump']} for different values of $\beta\gamma$, arising as a result of the infinitely thin shell of an arbitrary radius $r_\mathrm{sh}/\beta$. Predictably, \ref{['eq:pressure_jump']} has two roots: one at the unstable photon sphere $r_\mathrm{ust}$ (hollow black circle), and the stable photon sphere $r_\mathrm{st}$ (coloured filled circles).
• Figure 5: Plots of $B(r)$ for different configurations of initial MK parameter values and loading amplitudes. Dashed horizontal grey line: $B(r)=0$, dashed vertical black line: $r=r_\mathrm{st}$.