The Existence and Distribution of Photon Spheres Near Spherically Symmetric
Black Holes — A Geometric Analysis

The photon spheres / circular photon orbits are significantly important topics in black hole physics. On one hand, these photon spheres are closely connected with a number of astrophysical observations of black holes in galaxies, such as the gravitational lensing, black hole’s accretion processes and black hole shadows. Particularly, since the observation of high-resolution images of supermassive black holes captured by Event Horizon Telescope (EHT) [1, 2], there has been large interests in studying the photon spheres and black hole shadows. Numerous studies have emerged to extensively investigate the photon spheres and shadow images for various types of black holes [3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 12, 16, 21, 17, 18, 20, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 29, 33, 32, 34, 35, 36]. On the other hand, photon spheres may also provide valuable insights into the physical properties and characteristics of black holes. They can reveal non-trivial aspects related to the spacetime singularities, event horizons, causal structures and asymptotic behaviors associated with black holes and ultracompact objects. For instance, a number of recent researches have pointed out that the number and distribution of photon spheres (or light rings) in black hole spacetimes, horizon-less spacetimes produced by ultracompact objects, and naked singularity spacetimes could exhibit entirely different features [37, 38, 39, 40, 41, 42, 43].

Conventionally, the photon spheres in the vicinity of black holes are obtained from the effective potential of photons moving in black hole spacetimes [44, 45, 46, 47, 27, 48, 49]. While this approach proves highly valuable for analytical and numerical investigations for a specific spacetime, it is advantageous to incorporate geometric and topological approaches, especially when seeking the universal properties associated with photon spheres. The geometric and topological techniques could provide us powerful tools on the studies of photon spheres, without having to solve effective potentials and particle orbits for any particular black hole spacetimes.

Under these circumstances, new approaches using topological invariant (such as topological charge, index of vector fields, and Brouwer’s degree of mapping) and other mathematical concepts in the studying of photon spheres (or light rings) have already been stimulated [50, 42, 55, 43, 56, 51, 52, 53, 54]. Notably, in 2017, P. V. P. Cunha et al. introduced a topological approach to photon spheres / light rings, assigning a topological charge to each photon sphere (or light ring) in spacetime using the topological index of auxiliary vector fields [42]. Within this framework, they successfully demonstrated the existence of photon spheres in black hole spacetimes, and they also constrained the number of photon spheres (the number of stable and unstable photon spheres satisfies $n_{\text{stable}}-n_{\text{unstable}}=-1$ in black hole spacetimes) [43]. Subsequently, this topological approach has been further developed to a wider range of gravitational systems, extending its utility to both photon spheres / light rings and timelike circular orbits [56, 58, 59, 60, 61, 57, 62, 63, 64, 65, 66, 67]. Additionally, S. -W. Wei et al. pointed out that the topological charge can also be used to investigate the topology of black hole spacetimes as well as the black hole thermodynamic properties [69, 68]. These new approaches could provide us new insights on black hole physics and spacetime geometry, potentially providing answers to some important questions about photon spheres, spacetime properties and gravitational fields.

Stimulated from these approaches and modern geometric techniques, we proposed a geometric approach to study photon spheres and black hole shadows for spherically symmetric black holes in recent works [70, 71]. In our geometric approach, the photon spheres / circular photon orbits are analyzed through the optical geometry of black hole spacetime, which can be constructed from a generalization of the Fermat’s principle in curved spacetime [72, 73, 74, 75, 102]. The photon orbits (traveling along null geodesics in spacetime geometry) retain geodesic when they are transformed into optical geometry. Two intrinsic curvatures in the optical geometry — Gaussian curvature and geodesic curvature — can completely determine the stable and unstable photon spheres. Notably, for any photon spheres, the geodesic curvature is precisely zero, namely $\kappa_{g}(r=r_{ph})=0$. The sign of Gaussian curvature identifies the stability of photon spheres. The negative Gaussian curvature signifies the corresponding photon spheres are unstable, while the positive Gaussian curvature indicates the photon spheres are stable. In addition, it has been proven that our geometric approach is completely equivalent to the conventional approach based on effective potential of test particles [70, 71]. Furthermore, this geometric approach can be applied to study the orbits of other particles. Particularly, P. V. P. Cunha et al. extended our approach to study the timelike circular geodesics using the Jocabi geometry of black hole spacetimes [76]. The distinguishing features of the conventional approach (employing the effective potential of test particles moving in spacetimes), topological approach (utilizing the topological charge of photon spheres / topological index of given auxiliary vector fields) and our geometric approach (relying on the Gaussian curvature and geodesic curvature in the optical / Jacobi geometry of black hole spacetimes) are summarized in table 1.

In the current study, we utilize a novel geometric analysis on the existence of photon spheres as well as the distributions of stable and unstable photon spheres in the vicinity of black holes. Because of the astrophysical observations of black hole shadow images, it is widely accepted that the existence of photon sphere must be a general feature in black hole spacetimes. By assuming the proper asymptotic behaviors of black hole spacetime, this conjecture has been proven from several different schemes [9, 10, 37, 43, 77, 78]. In this work, we provide a new proof on the existence of photon spheres in asymptotically flat, asymptotically de-Sitter and anti de-Sitter black hole spacetimes, following our geometric approach proposed in references [70, 71]. Subsequently, we employ the Gauss-Bonnet theorem to show that the stable and unstable photon spheres near black holes must be one-to-one alternatively separated from each other, such that each unstable photon sphere is sandwiched between two stable photon spheres (and each stable photon sphere is sandwiched between two unstable photon spheres). This photon sphere distribution characteristic enables us to give a new demonstration of the important relation $n_{\text{stable}}-n_{\text{unstable}}=-1$ in black hole spacetimes from our geometric analysis. In the presented work, our analysis is confined to spherically symmetric black holes (with the spacetime metric to be $ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$).

This work is organized in the following way. Section I provides the backgrounds and motivations of our work. Section II offers a concise review of our geometric approach for analyzing photon spheres. Section III presents the proof on the existence of photon spheres in the vicinity of black holes (assuming the proper asymptotic behaviors of black holes spacetimes). Section IV discusses in detail the distribution of stable and unstable photon spheres near black holes. In this section, we provide a new proof on the number of photon spheres $n_{\text{stable}}-n_{\text{unstable}}=-1$. The summary and perspectives are outlined in section V. In addition, some mathematical preliminaries of our work are described in the Appendices, including the optical geometry of black hole spacetime, two intrinsic curvatures in optical geometry (Gaussian curvature and geodesic curvature), the Gauss-Bonnet theorem in differential geometry, the connection between Gaussian curvature and the stability of photon spheres.

In this section, we briefly describe the geometric approach to photon spheres, which was proposed in our recent works [70, 71]. In this approach, a low-dimensional analog of the black hole spacetime — the optical geometry — is implemented. Intrinsic curvatures in optical geometry are capable to determine the locations of photon spheres and their stability in the vicinity of black holes. This approach is applicable to any spherically symmetric spacetime, and it could give completely equivalent results with the conventional geodesic approach based on effective potential of massless photons.

The optical geometry serves as a low-dimensional analog of the spacetime when describing the massless particle’s orbits, providing powerful tools to study the motions of photons (or other massless particles traveling along null geodesics) in the gravitational field [72, 73, 74, 75]. The underlying physical interpretation of the optical geometry can be viewed as a generalization of the Fermat’s principle in curved spacetime. In recent years, the optical geometry has attracted significant interests, and various aspects of black hole physics can be explored using the geometric and topological properties of optical geometry [72, 73, 74]. For instance, G. W. Gibbons and M. C. Werner developed an approach to calculate the gravitational deflection angle using the Gauss-Bonnet theorem in optical geometry [73]. The Gibbons-Werner approach has been widely applied in a variety of gravitational systems and several modified versions of the Gibbons-Werner approach have also been developed [79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94].

Mathematically, the optical geometry can be constructed in several equivalent ways. A straightforward method to construct the optical geometry is from the spacetime geometry through a continuous transformation with the null constraint $d\tau^{2}=-ds^{2}=0$

Furthermore, for spherically symmetric black holes, the photon sphere can always be constricted in the equatorial plane without loss of generality. In this way, a 2-dimensional optical geometry can be constructed.

The detailed construction and explicit metric form of the optical geometry for spherically symmetric black hole spacetimes and rotational symmetric black spacetimes are presented in Appendix A. Notably, in the context of spherically symmetric black hole spacetimes, the corresponding optical geometry $dt^{2}=\tilde{g}^{\text{OP-2d}}_{ij}dx^{i}dx^{j}$ is a 2-dimensional Riemannian geometry [73, 74].

The optical geometry of black hole spacetime has some desirable properties. The photon orbits, which follow null geodesic curves in spacetime geometry (characterized by a 4-dimensional Lorentz geometry), are converted to spatial geodesic curves when transforming into optical geometry. In other words, the null geodesic curve $\gamma=\gamma(\tau)$ in spacetime geometry $ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ retains geodesic in the optical geometry $dt^{2}=g^{\text{OP}}_{ij}dx^{i}dx^{j}$. Additionally, the optical geometry possesses simpler intrinsic geometric and topological properties than the 4-dimensional spacetime geometry (because the dimension is reduced). In this low-dimensional geometry, a number of elegant theorems in classical surface theory, modern differential geometry and topology could offer powerful tools to study the particle motions in the vicinity of black holes.

In our geometric approach developed in references [70, 71], the analysis of photon spheres / circular photon orbits and their stability is implemented in the 2-dimensional equatorial plane of optical geometry (obtained from equation (2)). The most important intrinsic curvatures in this 2-dimensional optical geometry are Gaussian curvature and geodesic curvature, which are explained in Appendix B. The Gaussian curvature is the curvature of a 2-dimensional surface $S$, quantifying how much this surface deviates from being intrinsically flat. The geodesic curvature, on the other hand, describes the curvature of a continuous curve $\gamma(s)$, which measures how far this curve departs from being a geodesic curve on this surface [100, 98, 99]. Using the Gaussian curvature and geodesic curvature in optical geometry, the locations of photon spheres near black holes and their stability can be completely determined. Firstly, the photon sphere $r=r_{ph}$ near the black hole, which is a geodesic curve in spacetime geometry, becomes a spatial geodesic curve in the 2-dimensional optical geometry, and its geodesic curvature vanishes naturally [70, 71]

The solution of this equation gives the exact position of photon spheres in the vicinity of black holes. Further, it is also demonstrated that the above geodesic curvature criterion in our geometric approach is equivalent to the effective potential criterion within the conventional approach [70, 71].

Secondly, the stability of photon spheres near black holes can be constrained by the topological properties of optical geometry, especially the existence of conjugate points. For stable and unstable photon spheres, the behavior of photon orbits under a perturbation from photon spheres are completely different. When photons are perturbed at a given point on the unstable photon sphere, they would either escape to infinity or fall into black holes. Conversely, when photons are perturbed from a stable photon sphere, apart from going to infinity or falling into black holes, they could also travel along another bound photon orbit near the original photon sphere. Mathematically, these distinct characteristics of stable and unstable photon sphere are reflected by conjugate points in the manifold. There are conjugate points in the stable photon sphere, while no conjugate points could exist in the unstable photon sphere. Thus, the existence or absence of conjugate points offer us a novel means to distinguish the stable and unstable photon spheres. A classical theorem in differential geometry and topology — Cartan-Hadamard theorem — strongly constrain the Gaussian curvature and the existence of conjugate points. By applying the Cartan-Hadamard theorem in the equatorial plane of optical geometry, we derive the following criterion to determine the stability of photon spheres / circular photon orbits [70, 71].

The detailed interpretation on the relationship between Gaussian curvature, stability of photon spheres and the Cartan-Hadamard theorem is presented in Appendix D. This conclusion, derived purely from the geometric and topological properties of optical geometry, is independent of the specific metric form of black hole spacetimes (only constraining the sign of Gaussian curvature). Consequently, the analysis can be universally applicable to any spherically spherical black hole. Furthermore, it has been demonstrated that the Gaussian curvature criterion for stable and unstable photon spheres is equivalent to the effective potential criterion within the conventional approach [70, 71]

The distinguishing features of our geometric approach, the conventional effective potential approach, the topological approach and their equivalence have been summarized in table 1.

In the following, we present the detailed expression of the geodesic curvature and Gaussian curvature in the optical geometry of spherically symmetric black holes. The spacetime metric for general static spherically symmetric black holes can be expressed as

The corresponding optical geometry restricted in the equatorial plane $\theta=\pi/2$ is given by

where $\overline{g}_{\phi\phi}(r)=g_{\phi\phi}(r,\theta=\pi/2)$. The geodesic curvature of a circle with constant radius $r$ (such as the photon spheres) in the 2-dimensional optical geometry can be calculated via

and the Gaussian curvature in the 2-dimensional optical geometry can be calculated through

with $\tilde{g}^{\text{OP-2d}}=\text{det}\big{(}\tilde{g}^{\text{OP-2d}}_{ij}\big{)}$ to be the determinant of the 2-dimensional optical geometry metric.

In this section, we prove the existence of photon spheres in the vicinity of spherically symmetric black holes. The proof is based on our geometric approach and the asymptotic behaviors of black holes, irrespective of any specific spacetime metric functions $g_{tt}(r)$ and $g_{rr}(r)$. In this study, we give the detailed proof for black holes with most commonly asymptotic behaviors — the asymptotically flat, asymptotically de-Sitter and asymptotically anti de-Sitter black holes. However, as will be shown through the subsequent analysis, a similar analysis can be applied to a wide range class of black holes with other asymptotic behaviors.

In our geometric analysis, the criterion for the presence of photon spheres is the vanishing of geodesic curvature, namely $\kappa_{g}(r=r_{ph})=0$. Therefore, the existence of photon spheres is equivalent to that the equation $\kappa_{g}(r)=0$ admit at least one solution [103]. To prove the existence of photon spheres near black holes, the crucial point is analyzing the behavior of geodesic curvature for a circular curve in the 2-dimensional optical geometry, especially in the near horizon region and the infinite distance region. Notably, for the spacetimes generated by black holes, a recent study have shown that the geodesic curvature $\kappa_{g}(r)$ in the optical geometry in near horizon limit is related to the surface gravity of black holes via $\lim_{r\to r_{H}}\kappa_{g}(r)=-\kappa_{\text{surface}}<0$ [76], which can be viewed as a general feature of optical geometry. Under these circumstances, if the geodesic curvature satisfies $\lim_{r\to\infty}\kappa_{g}(r)>0$ in the infinite distance limit, as illustrated in figure 1, then the equation $\kappa_{g}(r)=0$ must have at least one solution, thereby proving the existence of photon spheres in black hole spacetimes. In the following, we derive the infinite distance limit $\lim_{r\to\infty}\kappa_{g}(r)>0$ for the asymptotically flat black holes, asymptotically de-Sitter and asymptotically anti de-Sitter black holes, respectively.

Asymptotically Flat Black Holes: The asymptotically flat black holes has the following analytical metric expansions in the $r\to\infty$ limit

with $a_{i}$ and $b_{j}$ to be the characteristic expansion coefficients. Furthermore, for any spherically symmetric black holes, the metric components $g_{\theta\theta}(r)=r^{2}$ and $g_{\phi\phi}(r)=r^{2}\sin^{2}\theta$ can be achieved through a coordinate transformation. Using these metric expansions, we eventually obtain the geodesic curvature for a circular curve in the infinite distance limit

The notation $\lim_{r\to\infty}\kappa_{g}(r)=0^{+}$ in this expression indicates that the geodesic curvature satisfies $\kappa_{g}(r)\to 0$ and $\kappa_{g}(r)>0$ in the $r\to\infty$ limit. This is exactly the case illustrated in figure 1, suggesting that the equation $\kappa_{g}(r)=0$ has at least one solution. Consequently, the existence of photon spheres is demonstrated for asymptotically flat black hole spacetimes.

Asymptotically de Sitter and Asymptotically Anti de-Sitter Black Holes: In the infinite distance limit, the asymptotically de Sitter / asymptotically anti de-Sitter black holes have the asymptotic analytical metric expansions

with $a_{i}$ and $b_{j}$ to be the characteristic expansion coefficients, and $\Lambda$ is the cosmological constant. The asymptotically de Sitter black holes admit a positive cosmological constant $\Lambda>0$, while the asymptotically anti de Sitter black holes have a negative cosmological constant $\Lambda<0$. Similarly, since we are focused on the spherically symmetric black holes, the metric components $g_{\theta\theta}(r)=r^{2}$ and $g_{\phi\phi}(r)=r^{2}\sin^{2}\theta$ can always be achieved through a coordinate transformation. Using the above metric expansions, we obtain the geodesic curvature in the infinite distance limit

The notation $\lim_{r\to\infty}\kappa_{g}(r)=0^{+}$ indicates that the geodesic curvature satisfies $\kappa_{g}(r)\to 0$ and $\kappa_{g}(r)>0$ in the $r\to\infty$ limit, which is precisely the same with those for asymptotically flat black holes. This successfully reproduces the case illustrated in figure 1, thus the equation $\kappa_{g}(r)=0$ must have at least one solution. The existence of photon spheres is demonstrated for both asymptotically de Sitter and asymptotically anti de-Sitter black holes.

In conclusion, for asymptotically flat, asymptotically de-Sitter and asymptotically anti de-Sitter black holes, the geodesic curvatures all satisfy $\lim_{r\to r_{H}}\kappa_{g}(r)<0$ in the near horizon region and $\lim_{r\to\infty}\kappa_{g}(r)=0^{+}$ in the infinite distance region, which are all consistent with the behavior shown in figure 1. The indicates that the equation $\kappa_{g}(r)=0$ must have at least one solution, explicitly proving the existence of photon spheres near such black holes. Furthermore, by assuming the continuity of geodesic curvature $\kappa_{g}(r)$ [104], it can also be easily derived that the total number of photon spheres in the vicinity of black holes must be odd, namely $n_{\text{stable}}+n_{\text{unstable}}=2k+1$ (which is extremely helpful for providing a new proof of the relation $n_{\text{stable}}-n_{\text{unstable}}=-1$ in the next section). In the present study, we haven’t presented the detailed calculation of geodesic curvature in the infinite distance limit for black holes with other asymptotic behaviors. However, the similar analysis can be easily extended to black holes with other asymptotic behaviors.

In this section, we present a discussion on the distribution of photon spheres near spherically symmetric black holes. Based on our geometric analysis, the following property of photon sphere distribution can be demonstrated:

The stable and unstable photon spheres near black holes are one-to-one alternatively separated from each other, such that each unstable photon sphere is sandwiched between two stable photon spheres (and visa versa, each stable photon sphere is sandwiched between two unstable photon spheres), as illustrated in figure 2.

Furthermore, this distribution characteristic could lead to a new proof of a theorem on photon spheres proposed by Cunha et al. recently [43], which suggested the numbers of stable photon spheres and unstable photon spheres satisfy $n_{\text{stable}}-n_{\text{unstable}}=-1$. The $w=n_{\text{stable}}-n_{\text{unstable}}=-1$ can be viewed as topological invariant / topological charge for black hole spacetimes.

In this study, instead of defining a topological charge $w$ relevant to photon spheres, our analysis is carried out from the Gauss-Bonnet theorem applied to optical geometry. Firstly, we select a region $D$ in the equatorial plane of optical geometry. The corresponding region $D$, which is shown in figure 3, is an annular zone bounded by two adjacent photon spheres without any other boundaries. It is a complexly connected region, such that the Euler characteristic number of this region is $\chi(D)=0$, which can be directly obtained from the triangulation of region $D$. The application of Gauss-Bonnet theorem within this region yields

which means the “averaged” Gaussian curvature between two adjacent photon spheres is zero.

In particular, inspired by recent works, a reduction on the surface integral of Gaussian curvature in the 2-dimensional optical geometry can be carried out [90, 91]. The Gauss-Bonnet theorem for the annular region $D$ eventually leads to

A simple derivation of the reduced formula (15) is presented in Appendix C. Here, $H(r)$ is a function defined in reference [90]

with $\tilde{g}^{\text{OP-2d}}$ to be the determinant of 2-dimensional optical geometry metric. Notably, the last equality in (16) comes from the explicit expression of geodesic curvature for circles given in equation (8). Particularly, both the inner and outer boundaries of region $D$ are photon spheres with vanishing geodesic curvature, so it is necessary to have $\kappa_{g}(r_{i+1})=\kappa_{g}(r_{i})=0\Rightarrow H(r_{i+1})=H(r_{i})=0$ required for the selected annular region $D$.

There are four different cases that could lead to the surface integral of Gaussian curvature vanishes in region $D$. These cases have been summarized in figure 4. The vanishing of surface integral $\int_{D}\mathcal{K}\cdot dS=2\pi\int_{r=r_{i}}^{r=r_{i+1}}\mathcal{K}(r)\sqrt{\tilde{g}^{\text{OP-2d}}}dr=0$ means the shadow areas depicted in figure 4 in the integration are zero. In the first two cases, the inner and outer photon spheres are of different types (one is a stable photon sphere, and the other is an unstable photon sphere), resulting in on-to-one alternating distributions of stable and unstable photon spheres, as depicted in figure 2. However, in the last two cases, the inner and outer photon spheres are of the same types, which violates the one-to-one alternatively separated distribution of stable and unstable photon spheres. Fortunately, a further analysis suggest that there must exist an additional photon sphere between the inner boundary photon sphere $r=r_{i}$ and outer boundary photon sphere $r=r_{i+1}$. Taking the case III as an example to show this conclusion. If the inner and outer photon spheres are both stable photon spheres with $\mathcal{K}(r=r_{i})>0$ and $\mathcal{K}(r=r_{i+1})>0$, we can always find a position $\bar{r}_{i,i+1}$ such that the region $D$ is divided into two parts, demanding that the integration of Gaussian curvature $\int\mathcal{K}(r)\sqrt{\tilde{g}^{\text{OP-2d}}(r)}dr$ over intervals $[r_{i},\bar{r}_{i,i+1}]$ and $[\bar{r}_{i,i+1},r_{i+1}]$ both equal to zero (and the shadow areas in intervals $[r_{i},\bar{r}_{i,i+1}]$ and $[\bar{r}_{i,i+1},r_{i+1}]$ in the lower-left panel of figure 4 are both zero). From the simplified integration formulas in (15), it is evident that this position satisfies $H(\bar{r}_{i,i+1})=H(r_{i})=H(r_{i+1})$ and $\kappa_{g}(\bar{r}_{i,i+1})=\kappa_{g}(r_{i+1})=\kappa_{g}(r_{i})=0$, indicating $r=\bar{r}_{i,i+1}$ must be the location of a new photon sphere. Furthermore, the negative Gaussian curvature in lower-left panel of figure 4 indicates that this new photon sphere at $r=\bar{r}_{i,i+1}$ is an unstable photon sphere, which is opposite to the inner and outer photon spheres at $r=r_{i}$, $r=r_{i+1}$ (both of them are stable photon spheres with positive Gaussian curvature). A similar analysis can be carried out for case IV, where an additional stable photon sphere must exist between the inner and outer unstable photon spheres at $r=r_{i}$ and $r=r_{i+1}$. This analysis remains valid even when there are finite discontinuous points in the Gaussian curvature with respect to radial coordinate $r$ (as presented in figure 5) [105]. In conclusions, among all four cases, we have proven that two stable photon spheres (or two unstable photon spheres) cannot be adjacent. The stable and unstable photon spheres in the vicinity of spherically symmetric black holes must be one-to-one alternatively separated from each other. Each stable photon sphere is sandwiched between two nearby unstable photon spheres, and each unstable photon sphere is sandwiched between two stable photon spheres, as indicated in figure 2.

Proof of the theorem $n_{\text{stable}}-n_{\text{unstable}}=-1$ in reference [43]: To provide a new proof of the theorem $n_{\text{stable}}-n_{\text{unstable}}=-1$ for black hole spacetimes proposed by Cunha et al. in reference [43], we should further demonstrate the innermost photon sphere (or outermost photon sphere) in black hole spacetimes to be an unstable photon sphere. Given the odd total number of photon spheres near black holes $n_{\text{stable}}+n_{\text{unstable}}=2k+1$ (which have been explained at the end of section III), and the one-to-one alternative separation of stable and unstable photon spheres, the numbers of stable and unstable photon spheres are constrained to be $n_{\text{stable}}=k$ and $n_{\text{unstable}}=k+1$. This fulfills the proof on relation $n_{\text{stable}}-n_{\text{unstable}}=-1$.

To demonstrate the innermost photon sphere near a black hole to be unstable, we once again apply the Gauss-Bonnet theorem to a similar annular region depicted in figure 3. However, in this case, modifications are made for the inner and outer boundaries of region $D$: the inner boundary circle approaches to the event horizon $r_{\text{min}}\to r_{H}$ and the outer boundary circle is the innermost photon sphere $r_{\text{max}}=r_{\text{innermost}}$. The application of Gauss-Bonnet theorem in such region yields

The contour integral of geodesic curvature along the outer boundary circle (which is a photon sphere) is vanished, which suggests

Here, $C(r_{H})$ is the circumference of inner boundary $r\to r_{H}$ in 2-dimensional optical geometry. Notably, the relationship between geodesic curvature $\kappa_{g}(r)$ in the near horizon limit and the surface gravity in black hole spacetime $\lim_{r\to r_{H}}\kappa_{g}(r)=\kappa_{\text{surface}}<0$ is used. Besides, the direction of inner boundary of region $D$ in figure 3 is chosen opposite to the increasing of azimuthal angle $\phi$, which also contributes to an additional minus sign.

There are four different cases that could lead to the surface integral of Gaussian curvature satisfies $\int_{D}\mathcal{K}\cdot dS\propto-2\pi r_{H}\kappa_{\text{surface}}<0$. These cases have been summarized in figure 6. The negative value of surface integral $\int_{D}\mathcal{K}\cdot dS=2\pi\int_{r_{H}}^{r_{\text{innermost}}}\mathcal{K}(r)\sqrt{\tilde{g}^{\text{OP-2d}}(r)}dr<0$ indicates that the shadow areas depicted in figure 6 in the integration is negative. In the first two cases, the innermost photon spheres are both unstable photon spheres with negative Gaussian curvature, which is exactly what we need in the demonstration of $n_{\text{stable}}-n_{\text{unstable}}=-1$. However, in the last two cases, it appears that the innermost photon spheres are stable photon spheres, which could hinder the demonstration of $n_{\text{stable}}-n_{\text{unstable}}=-1$. Fortunately, a simple analysis suggests that an additional photon sphere must exist between the event horizon $r=r_{H}$ and the photon sphere $r=r_{\text{innermost}}$. In these cases, the truly innermost photon sphere should locate at a smaller radius $r=\bar{r}$, rather than $r=r_{\text{innermost}}$. As shown in the lower panel of figure 6, in the last two cases, it is always possible to find a new position $r=\bar{r}$ in the interval $[r_{H},r_{\text{innermost}}]$, such that the integral $\int\mathcal{K}(r)\sqrt{\tilde{g}^{\text{OP-2d}}(r)}dr$ vanishes in interval $[\bar{r},r_{\text{innermost}}]$ (and the shadow areas in this interval in the lower panel of figure 6 are zero). By utilizing the simplified integration formulas in equations (15), the vanishing of integral suggests that $H(\bar{r})=H(r_{\text{innermost}})=0$ and $\kappa_{g}(\bar{r})=\kappa_{g}(r_{\text{innermost}})=0$, indicating that an additional photon sphere must exist at position $r=\bar{r}$. Moreover, from the lower panel of figure 6, we observe that the negative Gaussian curvature constrains this new photon sphere at position $r=\bar{r}$ to be an unstable photon sphere. In a summary, in all four cases, we have demonstrated the innermost photon spheres must be unstable, accomplishing the proof of the theorem $n_{\text{stable}}-n_{\text{unstable}}=-1$ in reference [43]. Similar to the analysis of photon sphere distribution on previous pages, our discussion presented here still holds even if the Gaussian curvature admits finite discontinuous points, as presented in figure 7 [105].

An equivalent way to demonstrate the theorem $n_{\text{stable}}-n_{\text{unstable}}=-1$ is by showing the outermost photon sphere to be an unstable one. This can also be achieved through similar processes using the Gauss-Bonnet theorem for an annular region in figure 3, with the inner boundary circle to be the outermost photon sphere $r_{\text{min}}=r_{\text{outermost}}$ and the outer boundary circle approaching to infinity $r_{\text{max}}\to\infty$. The detailed proof along this direction could be a bit more difficult than what we have presented in this work, as it relies on the detailed asymptotic behavior / asymptotically analytical expansion of the Gaussian curvature in the infinite distance limit.

The photon sphere / circular photon orbit has become a significantly important topic in black hole physics. In this study, we provide a geometric analysis of photon spheres in spherically symmetric black hole spacetimes. The existence and distribution characteristics of photon spheres in the vicinity of black holes are derived from geometric perspective. Our analysis follows the geometric approach in our recent works, in which the location of photon spheres and their stability are completely determined by geodesic curvature and Gaussian curvature in the optical geometry of black hole spacetimes.

In the current work, some basic conclusions on photon spheres in black hole spacetime have been rederived. Firstly, for black holes with most commonly asymptotic properties (the asymptotically flat black holes, asymptotically de-Sitter and asymptotically anti de-Sitter black holes), the existence of photon spheres is successfully demonstrated from the behaviors of geodesic curvature in near horizon limit and infinite distance limit. Secondly, applying the Gauss-Bonnet theorem in optical geometry, we demonstrate that the stable photon spheres and unstable photon spheres must be one-to-one alternatively separated from each other (each stable photon sphere is sandwiched between two nearby unstable photon spheres; each unstable photon sphere is sandwiched between two nearby stable photon spheres). Finally, we provide a new geometric proof of the theorem $n_{\text{stable}}-n_{\text{unstble}}=-1$ for black hole spacetimes. Furthermore, our geometric analysis presented in this work offer a new pathway to explore photon spheres and their distributions in gravitational fields, complementing the existing methods (such as the conventional effective potential approach, the topological approach).

Our analysis in this study is applicable to any spherically symmetric black holes with the general spacetime metric $ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$, regardless of the specific analytical expression for metric components. Consequently, a similar analysis can be easily extended to photon spheres in other kinds of gravitational systems, including spacetimes generated by astrophysical ultracompact objects (such as dwarfs and neutron stars) without the presence of event horizons, regular spacetimes in void of singularities, and the naked singularity spacetimes. The detailed and comprehensive analysis of these cases can be carried out in future studies. Hopefully, the geometric analysis provided in current work could be generalized to other gravitational systems, providing us insights and understanding on photon spheres, gravity theories and mathematical physics.