A General Discussion on Photon Spheres in Different Categories of Spacetimes

TL;DR

The paper develops a metric-agnostic geometric framework to study photon spheres across black holes, ultra-compact objects, regular spacetimes, and naked singularities. Photon spheres are characterized by the vanishing geodesic curvature in the 2D optical geometry, $\kappa_g(r_{\text{ph}})=0$, with stability set by the sign of the Gaussian curvature $\mathcal{K}(r)$, linking to the effective-potential picture via $\kappa_g(r_{\text{ph}})=0 \Leftrightarrow \frac{dV_{\text{eff}}}{dr}\big|_{r_{\text{ph}}}=0$. The authors derive universal counting relations $n=n_{\text{stable}}+n_{\text{unstable}}$ and $w=n_{\text{stable}}-n_{\text{unstable}}$, showing that spacetimes with horizons yield $n=2k+1$ and $w=-1$, while horizonless cases yield $n=2k$ and $w=0$, with nondeterministic outcomes for certain naked-singularity cases. They connect these predictions to observations via photon rings in VLBI imaging and discuss extensions to axisymmetric spacetimes and to massive-particle orbits, highlighting potential observational tests with future interferometry and gravitational-wave data.

Abstract

Photon spheres have attracted considerable interest in the studies of black holes and other astrophysical objects. For different categories of spacetimes (or gravitational sources), the existence of photon spheres and their distributions are dramatically influenced by the geometric and topological properties of spacetimes and characteristics of the corresponding gravitational fields. In this work, we carry out a geometric analysis on photon spheres for different categories of spacetime (including black hole spacetime, ultra-compact object's spacetime, regular spacetime, and naked singularity spacetime). Some universal properties and conclusions are obtained for these spacetimes. We mostly focus on the existence of photon spheres, the total number of photon spheres $n = n_{\text{stable}} + n_{\text{unstable}}$, the subtraction of stable photon sphere and unstable photon sphere $w = n_{\text{stable}} - n_{\text{unstable}}$ in different categories of spacetimes. These conclusions are derived solely from geometric properties of optical geometry of spacetimes, irrelevant to the specific spacetime metric forms. Besides, our results successfully recover some important theorems on photon spheres proposed in recent years.

Figures (5)

• Figure 1: The variation of geodesic curvature $\kappa_{g}(r)$ with respect to radial coordinate $r$ in different categories of spacetimes. Case I: For black hole spacetime and regular spacetime (with the presence of horizons) satisfying $\lim_{r \to r_{H}} \kappa_{g}(r) < 0$ and $\lim_{r \to \infty} \kappa_{g}(r) = 0^{+}$, the equation $\kappa_{g}(r)=0$ must have at least one solution outside the event horizon. The dashed line draws the one-solution case, and the solid line draws the three-solution case. Case II: For ultra-compact object’s spacetime, naked singularity spacetime (with finite first-order metric derivative), and regular spacetime (without horizons) satisfying $\lim_{r \to 0} \kappa_{g}(r) > 0$ and $\lim_{r \to \infty} \kappa_{g}(r) = 0^{+}$, the equation $\kappa_{g}(r)=0$ must have either no solution or an even number of solutions. The dashed line illustrates the no-solution case, and the solid line draws the two-solution case. Case III: For naked singularity spacetime (with first-order metric derivative diverging to positive infinity at spacetime singularity $\lim_{r\to 0}\frac{df(r)}{dr} = +\infty$), the geodesic curvature satisfies $\lim_{r \to 0} \kappa_{g}(r) = \text{indefinite}$ and $\lim_{r \to \infty} \kappa_{g}(r) = 0^{+}$. It is impossible to determine whether equation $\kappa_{g}(r)=0$ has solutions or not. The solid line depicts an example where the solution of $\kappa_{g}(r)=0$ exists, while the dashed line illustrates an example where $\kappa_{g}(r)=0$ has no solutions. In this figure, the notation $\lim_{r\to\infty}\kappa_{g}(r)=0^{+}$ represents that the geodesic curvature satisfies $\kappa_{g}(r) \to 0$ and $\kappa_{g}(r)>0$ in the infinite distance limit $r \to \infty$.
• Figure 2: The stable and unstable photon spheres in spacetimes are one-to-one alternatively separated from each other.
• Figure 3: The choice of region $D$ in 2-dimensional optical geometry in the Gauss-Bonnet theorem, which is used to demonstrate that stable and unstable photon spheres in spacetime are one-to-one alternatively separated from each other.
• Figure 4: This figure summarizes the four different cases in which the surface integral of Gaussian curvature vanishes in region $D$ (namely $\int_{D}\mathcal{K}\cdot dS = 0$). Case I: the inner boundary photon sphere is stable and the outer boundary photon sphere is unstable, with $\mathcal{K}(r_{i}) > 0$ and $\mathcal{K}(r_{i+1}) < 0$. Case II: the inner boundary photon sphere is unstable and the outer boundary photon sphere is stable, with $\mathcal{K}(r_{i}) < 0$ and $\mathcal{K}(r_{i+1}) > 0$. Case III: both the inner and outer boundary photon spheres are stable, with $\mathcal{K}(r_{i}) > 0$ and $\mathcal{K}(r_{i+1}) > 0$. Case IV: both the inner and outer boundary photon spheres are unstable, with $\mathcal{K}(r_{i}) < 0$ and $\mathcal{K}(r_{i+1}) < 0$. In the last two cases, a further analysis suggests that the inner and outer boundary photon spheres at $r=r_{i}$ and $r=r_{i+1}$ cannot be adjacent, and an additional photon sphere must exist between $r_{i}$ and $r_{i+1}$. It is worth noting that the $\sqrt{\tilde{g}^{\text{OP-2d}}}$ in surface element is always positive, hence the sign of Gaussian curvature can be easily seen from this figure. In particular, the left panel shows the situations where Gaussian curvature is continuous in region $D$, while the right panel represents the situations where Gaussian curvature admits a finite number of discontinuous points with respect to radial coordinate $r$.
• Figure 5: This figure summarizes the four different cases in which the surface integral of Gaussian curvature satisfies $\int_{D}\mathcal{K}\cdot dS < 0$. Case I: the innermost photon sphere is unstable, with $\mathcal{K}(r_{H}) < 0$ and $\mathcal{K}(r_{\text{innermost}}) < 0$. Case II: the innermost photon sphere is unstable, with $\mathcal{K}(r_{H}) > 0$ and $\mathcal{K}(r_{\text{innermost}}) < 0$. Case III: the innermost photon sphere is stable, with $\mathcal{K}(r_{H}) < 0$ and $\mathcal{K}(r_{\text{innermost}}) > 0$. Case IV: the innermost photon sphere is stable, with $\mathcal{K}(r_{H}) > 0$ and $\mathcal{K}(r_{\text{innermost}}) > 0$. In the last two cases, a further analysis suggests that an additional photon sphere $r=\bar{r}$ must exist between the event horizon $r=r_{H}$ and the faked "innermost" photon sphere $r=r_{\text{innermost}}$. It is worth noting that the $\sqrt{\tilde{g}^{\text{OP-2d}}}$ in surface element is always positive, hence the sign of Gaussian curvature can be easily seen from this figure. In particular, the left panel shows the situations where Gaussian curvature is continuous in region $D$, while the right panel represents the situations where Gaussian curvature admits a finite number of discontinuous points with respect to radial coordinate $r$.