Circular photon orbits and black hole shadows are significantly important issues in physics and astronomy and a number of breakthroughs have been witnessed in recent years. Conventionally the stable and unstable circular photon orbits are obtained using the effective potential of test particles moving in black hole spacetime. In this work a pure geometric approach is developed to calculate these circular photon orbits and black hole shadow radius. Furthermore it can be proved that our geometric approach is completely equivalent to the conventional approach based on effective potentials of test particles.
['General Relativity and Quantum Cosmology', 'Mathematical Physics']
Geometric approach to circular photon orbits and black hole shadows
{'Geometric Approach to Circular Photon Orbits and Black Hole Shadows': 'Chen-Kai Qiao ∗  \nCollege of Science, Chongqing University of Technology, Banan, Chongqing, 400054, China', 'Ming Li †': 'Mathematical Science Research Center, Chongqing University of Technology, Banan, Chongqing, 400054, China (Dated: September 12, 2024)  \nThis supplemental material gives a detailed description on the optical geometry of black hole spacetime, as well as the Gauss curvature and geodesic curvature used in our geometric approach. The first part discusses the optical geometry of spherically symmetric and axi-symmetric black holes. The second part introduces the Gauss curvature and geodesic curvature in surface theory and differential geometry.', 'Unstable Circular Photon Orbit': 'FIG. 1. The schematic plot of the stable and unstable circular photon orbits near black holes. In this figure, we choose the spherically symmetric black hole as a typical example, and the region covered by blue is the interior of black hole.  \n<!-- image -->  \nGLYPH<d835>  \n(GLYPH<d835>)', 'Stable Circular Photon Orbit': "<!-- image -->  \nGLYPH<d835>  \n(GLYPH<d835>)  \ngeometry dτ 2 = g µν dx µ dx ν maintains geodesic in the optical geometry dt 2 = g OP ij dx i dx j [25]. Actually, it can be viewed as the generalization of Fermat's principle in curved stationary spacetime [25, 26, 38].  \nThe following question is how to distinguish stable photon orbits from unstable photon orbits. The following Hadamard theorem in differential geometry would answer this question appropriately.  \nHadamard Theorem: For a two dimensional complete Riemannian manifold with nonpositive Gauss curvature, there is only one geodesic curve from p to q belong to the same homotopy class, and this geodesic curve minimizes the length in this homotopy class [34, 35].  \nFor spherically symmetric black hole, if we restrict the optical geometry in two dimensional equatorial plane, the Gauss curvature in the Hadamard theorem can be calculated through [32, 33]  \nK = -1 √ ˜ g OP [ ∂ ∂φ ( 1 √ ˜ g OP φφ ∂ √ ˜ g OP rr ∂φ ) + ∂ ∂r ( 1 √ ˜ g OP rr ∂ √ ˜ g OP φφ ∂r )] = 1 2 f ( r ) · d 2 f ( r ) dr 2 -[ 1 2 · df ( r ) dr ] 2 (6)  \nFIG. 2. This figure shows several possible bound photon orbits near the stable circular photon orbits. These bound photon orbits could be obtained by perturbation of circular photon orbit at point p , and they may have different shapes. The bound photon orbit 1 labeled in blue is a closed orbit, and the bound photon orbit 2 labeled in green is an unclosed orbit (just like the Mercury procession around the Sun). In the equatorial plane of optical geometry, we can find another point q (we denote the q i for bound orbit i = 1 , 2). When we considering photon orbits starting from p ending with q , there are at least two different photon orbits (one is the stable circular photon orbit, and the other is a bound photon orbit) that can be continuously deformed to each other. Therefore, in the equatorial plane of optical geometry, there are two different geodesic curves from p to q belong to the same homotopy class.  \n<!-- image -->  \nGLYPH<d835>  \n2  \nwhere ˜ g OP = det (˜ g OP ij ) is determinant of optical metric. For stable circular photon orbit, there are bound photon orbits nearby, which may have different shapes. In the equatorial plane of optical geometry, we can find two points p and q such that there are at least two geodesic curves (one is the stable photon orbit, the other is a bound photon orbit) belong to the same homotopy class. Figure 2 illustrates two possible bound photon orbits and the corresponding choice of points p and q . In such cases, the Gauss curvature of optical geometry should be positive, otherwise it would violate the Hadamard theorem. On the contrary, for unstable circular photon orbit, no bound photon orbits homotopic to this circular photon orbit exist. Then unstable circular photon orbit itself forms the whole homotopy class, which correspond to the negative Gauss curvature in the Hadamard theorem (we assume the Gauss curvature of optical geometry is nonzero, otherwise the black hole spacetime would by flat). Based on the above discussions, we obtain the following criterion to determine the stable and unstable circular photon orbits [39].  \nK < 0 ⇒ The circular photon orbit r = r ph is unstable > 0 The circular photon orbit r = r is stable  \nK ⇒ ph  \nFIG. 3. The black hole shadow and unstable circular photon orbit. The light beam emitted from infinity with impact parameter b = b critical would reach the unstable circular photon orbit exactly. The light beam emitted with impact parameter b < b critical would fall into the black hole.  \n<!-- image -->  \nBlack Hole Shadow: Black hole shadow is the dark silhouette of black hole image in a bright background. The size and shape of black hole shadow depends not only on black hole parameters, but also on the position of observers [17]. In this work, we concentrate on the idealized situation where observer is located at infinity, and there are no light sources between the observer and black hole. In this case, the radius of black hole shadow detected by observer is just the critical value of impact parameter b critical . In the gravitational field, light beams emitted from infinity with impact parameter b = b critical would reach the unstable circular photon orbit exactly, as illustrated in figure 3  \nFollowing the conventional definition, the impact parameter can be expressed as [15]  \nb ≡ ∣ ∣ ∣ L E ∣ ∣ ∣ = ∣ ∣ ∣ r 2 sin 2 θ · dφ/dλ f ( r ) · dt/dλ ∣ ∣ ∣ (7)  \n∣ ∣ ∣ ∣ where E and L is the conserved energy and angular momentum per unit mass  \nE = f ( r ) · dt dλ (8a)  \nL = r 2 sin 2 θ · dφ dλ (8b)  \nSince the impact parameter b ≡ L/E is a conserved quantity along the geodesics, we can calculate the critical impact parameter b critical at any specific point along the photon trajectory. A simple and convenient choice are the points in the unstable circular photon orbit r = r unstable  \nb critical = ∣ ∣ ∣ ∣ L E ∣ ∣ ∣ ∣ r = r unstable = ∣ ∣ ∣ ∣ r 2 f ( r ) · dφ/dλ dt/dλ ∣ ∣ ∣ ∣ r = r unstable (9)  \nTABLE I. The circular photon orbit and black hole shadow radius detected by observer at infinity. This table summarize the results for Schwarzschild black hole and Reissner-Nordstrom black hole. All quantities calculated using our new geometric approach agree with the results obtained using conventional approach based on effective potential. a  \nwhere θ = π/ 2 has been used in the equatorial plane.  \nThe radius of black hole shadow detected by observer at infinity can be calculated in the optical geometry. In the spacetime geometry, photon orbits are along lightlike / null geodesics, and their tangent vector satisfies T · T = g µν dx µ dλ dx ν dλ = 0, with λ to be any affine parameter of null geodesics. However, when transformed into the optical geometry, the photon orbits become spatial geodesics with vanishing geodesic curvature κ g = 0, as we have emphasized previously [40]. The stationary time coordinate t in spacetime geometry exactly reduces to the arc-length parameter / spatial distance parameter in optical geometry. In the optical geometry, the tangent vector of arbitrary photon orbits T OP = d/dt with respect to arc-length parameter t becomes a unit vector ( | T OP | = 1). Here, we restrict the optical geometry in the equatorial plane, the tangent vector of photon orbits reduce to T OP = ( dr dt , dφ dt ). Furthermore, for unstable circular photon orbit, we have the result  \nr = r unstable = constant ⇒ dr dt ∣ ∣ ∣ ∣ r = r unstable = 0 ⇒| T OP · T OP | = ∣ ∣ ∣ ∣ ˜ g OP φφ · dφ dt · dφ dt ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ r 2 f ( r ) · dφ dt · dφ dt ∣ ∣ ∣ ∣ r = r unstable = 1 ⇒ r sh = b critical = ∣ ∣ ∣ ∣ r 2 f ( r ) · dφ/dλ dt/dλ ∣ ∣ ∣ ∣ r = r unstable = √ r 2 unstable f ( r unstable ) (10)  \nThis is the analytical expression for black hole shadow radius detected by observer located at infinity, and it is consistent with previous results [13, 17]. Based on above procedures, the shadow radius for arbitrary spherically symmetric black hole with metric from in equation (3) can be calculated. In the present work, we choose two typical examples in general relativity -- Schwarzschild black hole and Reissner-Nordstrom black hole. The circu-l  \nr photon orbits and black hole shadow radius for these black holes are summarized in table I.  \nEquivalence between Our Geometric Approach and Conventional Approach: The geometric approach developed in this work is completely equivalent to the conventional approach based on effective potentials of test particles moving in the gravitational field. In conventional approach, the circular photon orbit can be solved by analyzing the extreme points of effective potential V eff ( r ). Particularly, the unstable circular photon orbit r = r unstable corresponds to local maximum of effective potential, and the stable circular photon orbit r = r stable is the local minimum of effective potential.  \nWe now demonstrate the equivalence between these two approaches. In spherically symmetric black hole spacetime, if we restrict test particles moving in the equatorial plane θ = π/ 2, the equation of motion eventually reduces to [14]  \n1 2 · ( dr dλ ) 2 + V eff ( r ) = 1 2 · E 2 (11)  \nwhere effective potential of test particles is defined by  \nV eff ( r ) ≡ f ( r ) 2 · ( L 2 r 2 + /epsilon1 ) (12)  \nHere, for massless particles /epsilon1 = 0, and for massive particles /epsilon1 = 1. The circular photon orbit r = r ph correspond to the extreme point of effective potential [14, 15]  \ndV eff ( r ) dr ∣ ∣ ∣ ∣ r = r ph = 0 ⇒ [ L 2 2 r 2 · df ( r ) dr -L 2 · f ( r ) r 3 ] r = r ph = 0 (13)  \nwhere we have used /epsilon1 = 0 for massless photons. Note that the angular momentum L is a conserved quantity along photon orbit, which is independent of r . Comparing equation (13) with equation (5), it is clearly demonstrated that they are equivalent to each other. When  \nTABLE II. The features of our geometric approach developed in this work and the conventional approach.  \nr = r ph is the extreme point of effective potential V eff ( r ), the geodesic curvature along the circle r = r ph vanishes precisely, which makes it to be the circular photon orbit for spherically symmetric black holes.  \nThen we analyze the stability of circular photon orbits. For unstable circular photon orbit r = r unstable , the effective potential should reach its local maximum [15, 18]  \nd 2 V eff ( r ) dr 2 ∣ ∣ ∣ ∣ r = r unstable < 0 ⇒ [ L 2 2 r 2 · d 2 f ( r ) dr 2 -2 L 2 r 3 · df ( r ) dr + 3 L 2 · f ( r ) r 4 ] r = r unstable < 0 ⇒ [ L 2 2 r 2 · d 2 f ( r ) dr 2 -L 2 · f ( r ) r 4 ] r = r unstable < 0 ⇒ [ f ( r ) 2 · d 2 f ( r ) dr 2 -f ( r ) · f ( r ) r 2 ] r = r unstable < 0 ⇒ [ f ( r ) 2 · d 2 f ( r ) dr 2 -( 1 2 df ( r ) dr ) 2 ] r = r unstable < 0 (14)  \nIn the derivation, we have used the relation df ( r ) /dr = 2 f ( r ) /r in equation (13) for unstable circular photon orbit repeatedly. Recall the expression for Gauss curvature in equation (6), the last line of this inequality implies that Gauss curvature must be negative ( K < 0). On the contrary, the stable photon orbit, which correspond to the local minimum of effective potential, would imply the Gauss curvature to be positive ( K > 0). In this way, the equivalence between our geometric approach developed in this work and the conventional approach is demonstrated. The features of these two approaches and their equivalence are summarized in table II.  \nSummary and Prospects: In this work, a pure geometric approach is developed to calculate the circular photon orbit and black hole shadow radius. This approach is quite simple and general, regardless of particular metric forms of black hole spacetime. The Gauss  \ncurvature, geodesic curvature in optical geometry and Hadamard theorem in differential geometry offer a new pathway to calculate the circular photon orbits and black hole shadows. This approach indicates that the optical geometry may give us profound insights on black hole properties, and it is worthy of extensive investigations. Furthermore, we demonstrate that our approach is completely equivalent to the conventional approach based on effective potentials of test particles.  \nThere are several possible extensions of our geometric approach in the near future. Firstly, the similar algorithm can be apply to rotational black holes, whose optical geometry is a Randers-Finsler manifold. Secondly, the approach developed in this work, which is given for massless photons, can also be generated to the cases of massive particles. For circular orbits of massive particles, the utilization of Gauss curvature, geodesic curvature and Hadamard theorem may take place in the Jacobi geometry of black hole spacetime, rather than the optical geometry. 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For light rays travel in a curved and stationary spacetime with a given choice of time coordinate t , spatial light orbits starting at a fixed light emission event would be the spatial curves that makes the stationary arrival time minimal (namely δ [ ∫ γ dt ] = 0 is satisfied). In the optical geometry of black hole spacetime dt 2 = g OP ij dx i dx j , since the stationary time t becomes arc length (spatial distance), the variation condition δ [ ∫ γ dt ] = 0 indicates that light rays always travel along spatial geodesics in optical geometry. In this way, the observation -- the null geodesic curve in spacetime geometry dτ 2 = g µν dx µ dx ν maintains geodesic in the optical geometry dt 2 = g OP ij dx i dx j -- is the generalization of Fermat's principle in curved stationary spacetime. The readers could find discussion of generalized Fermat's principle in references [25, 26, 29].\n- [39] Although black hole spacetime has singularities, its optical geometry is defied outside the horizons, so it is usually a geodesic complete manifold. Since time coordinate t in spacetime geometry becomes arc length parameter in optical geometry, the endless of time coordinate t outside horizons would imply the geodesic completeness of optical geometry.\n- [40] In a recent work, Z. Li, G. Zhang and A. Ovgun calculated circular orbits using the same geodesic curvature condition κ g = 0 [31]. However, they did not distinguish the stable circular orbits from unstable circular orbits. Their discussion is about the circular orbits for massive particles (whose geodesic curvature κ g is calculated in the Jacobi geometry of black hole spacetime).", 'Geometric approach to circular photon orbits and black hole shadows': 'Chen-Kai Qiao ∗  \nCollege of Science, Chongqing University of Technology, Banan, Chongqing, 400054, China', '1. OPTICAL GEOMETRY OF BLACK HOLE SPACETIME': "This section gives an introduction of the optical geometry of black hole spacetime. The optical geometry can be obtained from spacetime geometry dτ 2 = g µν dx µ dx ν by imposing the null constraint dτ 2 = 0. The optical geometry attracted a number of interests in recent years, and many aspects of black hole physics can be analyzed and studied using optical geometry [1-3]. Particularly, for stationary black hole spacetime, the photon orbits (which are along lightlike / null geodesics in four dimensional spacetime / Lorentz manifold), become spatial geodesics and minimize the stationary time t when transformed into optical geometry. The stationary time coordinate t plays the role of arc length parameter / spatial distance parameter in optical geometry, and δ [ ∫ γ dt ] = 0 is satisfied along the photon orbits. This conclusion can be viewed as the generalization of the renowned Fermat's principle in curved manifold [3, 4], as we have discussed in the main text. In our geometric approach, both the circular photon orbits and black hole shadow radius are analyzed and calculated in the equatorial plane of optical geometry.  \nThe properties of optical geometry strongly depend on symmetries of gravitational field and black hole spacetime. For a spherically symmetric black hole, its optical geometry gives a Riemannian manifold [1-3]. To clarify this point, consider a spherically symmetric black hole spacetime with the metric  \ndτ 2 = g tt dt 2 -g rr dr 2 -g θθ dθ 2 -g φφ dφ 2 (1)  \nits optical geometry is obtained from the null constraint dτ 2 = 0, which eventually gives a three dimensional Riemannian manifold  \ndt 2 = g OP ij dx i dx j = g rr g tt · dr 2 + g θθ g tt · dθ 2 + g φφ g tt · dφ 2 (2)  \nWhen studying the circular photon orbits, one can always restricts this optical geometry to the equatorial plane θ = π/ 2 without loss of generality. Besides, for static and spherically symmetric black holes, another equivalent way to define optical geometry is from the conformal transformation of spacetime and extracting its spatial part [3].  \nFor rotational / axi-symmetric black holes, the correspond optical geometry is a Randers-Finsler manifold [4-6]. In these cases, we should consider the standard rotational black hole metric  \ndτ 2 = g tt dt 2 -2 g tφ dtdφ -g rr dr 2 -g θθ dθ 2 -g φφ dφ 2 (3)  \nthe optical geometry can be obtained in a similar way by imposing the null constraint dτ 2 = 0. The arc length / spatial distance in optical geometry eventually reduces to  \ndt = √ g rr g tt · dr 2 + g θθ g tt · dθ 2 + g 2 tφ + g tt g φφ g 2 tt · dφ 2 + g tφ g tt · dφ (4)  \nThis gives a class of Renders-Finsler manifolds. The Renders-Finsler geometry is the extension of Riemannian geometry [7], in which the arc length / spatial distance can be separated into two parts  \ndt = √ α ij dx i dx j + β i dx i (5)  \nFIG. 1. Illustration of the embedding of surface S in three dimensional Euclidean space. In this figure, S is a two dimensional curved surface, γ = γ ( s ) is a curve lived in surface S , and p is an arbitrary point along this curve. The { p ; e 1 , e 2 , e 3 } are frame fields in three dimensional Euclidean space such that e 1 = T is the unit tangent vector of curve γ = γ ( s ) and e 3 = n is the unit normal vector of surface S . The tangent plane of curved surface S at point p are generated by frames e 1 and e 2 . The curve ˜ γ = ˜ γ ( ˜ s ) is the projection of γ = γ ( s ) on tangent plane of surface S at point p .  \n<!-- image -->  \nGLYPH<d835>  \n= GLYPH<d835>  \nGLYPH<d835>  \nHere, the first part α ij is a Riemannian metric, and the second part β = β i dx i is an one-form which measures the departure of this Renders-Finsler geometry from the Riemannian geometry dt 2 = α ij dx i dx j . The above RendersFinsler geometry reduces to Riemannian geometry if and only if β = 0.  \nIn this work, we only duel with a class of spherically symmetric black holes, whose optical geometry is Riemannian manifold. Therefore, in the present work, both Gauss curvature and geodesic curvature are calculated using formulas in Riemannian geometry. The investigations on circular photon orbits and black hole shadows for rotational black holes, whose optical geometry gives a Renders-Finsler manifold, are left for incoming studies.", '2. GAUSS CURVATURE AND GEODESIC CURVATURE IN TWO DIMENSIONAL SURFACE': "The section gives an introduction of the Gauss curvature and geodesic curvature in surface theory and differential geometry. In our geometric approach, both the circular photon orbits and black hole shadow radius are calculated and analyzed in the optical geometry of black hole spacetime, rather than in four dimensional spacetime (Lorentz manifold). In this supplemental material, we mainly focus on the Gauss curvature and geodesic curvature defined in two dimensional optical geometry dt 2 = ˜ g OP ij dx i dx j (the optical geometry restricted in equatorial plane θ = π/ 2), which is always a two dimensional Riemanian manifold for spherically symmetric black holes.  \nThe Gauss curvature is the intrinsic curvature of a two dimensional surface S , which measures how far this surface is from being flat intrinsically. The geodesic curvature is the curvature of a curve γ ( s ) lived in this surface, which measures how far this curve is from being a geodesic in surface S . There are many equivalent ways to define geodesic curvature and Gauss curvature. For instructive reasons, we embed the two dimensional curved surface S into higher dimensional Euclidean space. However, both Gauss curvature and geodesic curvature are intrinsic properties of two dimensional surface S , irrespective of the embedding of surface S into higher dimensional space.  \nConsider a two dimensional curved surface S embedded into three dimensional Euclidean space, which is illustrated in figure 1. For a curve γ = γ ( s ) lived in surface S with arc length parameter / spatial distance parameter s , its unit tangent vector in three dimensional space is T = d r /ds . The total curvature of γ ( s ), which is the curvature of this curve viewed in three dimensional Euclidean space, is defined to be the module of derivative d T /ds [8, 9]  \nκ = ∣ ∣ ∣ ∣ d T ds ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ d 2 r ( s ) ds 2 ∣ ∣ ∣ ∣ (6)  \nTABLE I. This table summarizes the features of geodesic curvature κ g , total curvature κ and Gauss curvature K , as well as the differences between them.  \nFor any point p along this curve, we can assign the frame fields { p ; e 1 , e 2 , e 3 } in three dimensional space such that e 1 = T is the unit tangent vector of curve γ = γ ( s ) and e 3 = n is the unit normal vector of surface S . The tangent plane of curved surface S at point p are generated by frames e 1 and e 2 . In this case, the geodesic curvature of curve γ = γ ( s ) is defined as the projection of total curvature κ = | d T /ds | on tangent space  \nκ g = d T ds · e 2 = d 2 r ( s ) ds 2 · e 2 (7)  \nFurthermore, the normal curvature κ n is defined as the projection of total curvature κ = | d T /ds | on normal vector n  \nκ n = d T ds · n = d 2 r ( s ) ds 2 · e 3 (8)  \nThe geodesic curvature κ g depends only on the intrinsic properties of surface S , while the normal curvature κ n relies on the embedding of surface S into three dimensional Euclidean space. Using geodesic curvature and normal curvature, the total curvature of a curve γ ( s ) in three dimensional Euclidean space can be expressed by the square root κ = √ κ 2 g + κ 2 n . In particular, the curves with zero geodesic curvature ( κ g = 0) are viewed as 'straight line' in surface S , and they are called geodesics in surface S . The curves with zero total curvature ( κ = √ κ 2 g + κ 2 n = 0) are exactly the straight line (geodesics) in three dimensional Euclidean space.  \nThe normal curvature κ n is the extrinsic quantity of surface S , which depends on the embedding into higher dimensional space. If we change the direction of γ ( s ) in three dimensional space, the normal curvature of this curve at point p also changes (we assume the curve γ = γ ( s ) always pass through point p during the variation). In some circumstances, the normal curvature κ n could reach its maximum and minimum. These extreme values could reflect the geometric properties of two dimensional surface S , and they are named as the principle curvatures of this surface at point p .  \nK 1 = max { κ n } K 2 = min { κ n } (9)  \nThe Gauss curvature of surface S (at point p ) is defined to be the product of principal curvatures [8, 9]  \nK ≡ K 1 K 2 (10)  \nAn elegant theorem in surface theory (the Gauss Theorema Egregium) states that: although principal curvatures K 1 and K 2 depend on the embedding of S in higher dimensional space, the Gauss curvature K = K 1 K 2 is an intrinsic geometric quantity of surface S regardless of embedding [10]. The Gauss curvature becomes zero if and only if the two dimensional surface is intrinsically flat. The features of geodesic curvature κ g , total curvature κ and Gauss curvature K are summarized in table I.  \nAs we have emphasized in table I, both geodesic curvature and Gauss curvature are intrinsic geometric quantities of two dimensional surface S , they can be calculated from the intrinsic metric of surface S naturally. If we assign the curving linear coordinates ( x 1 , x 2 ) for two dimensional surface S such that its intrinsic metric is expressed as  \nds 2 = g 11 dx 2 1 + g 22 dx 2 2 (11)  \nfor any curve γ = γ ( s ) = ( x 1 ( s ) , x 2 ( s )) lived in this surface S with arc length parameter s , its geodesic curvature can be calculated through the following equation  \nκ g ( γ ) = dα ds -1 2 √ g 22 ∂ log ( g 11 ) ∂x 2 cos α + 1 2 √ g 11 ∂ log ( g 22 ) ∂x 1 sin α (12)  \nHere, α is the angle between tangent vector of γ and coordinate axis x 1 . Since geodesic curvature measures how far a curve is from being a geodesic in two dimensional surface S , the vanishing of geodesic curvature indicates that γ = γ ( s ) is a geodesic curve in this surface.  \nThe Gauss curvature of two dimensional surface S is calculated by  \nκ g ( γ ) = 0 ⇔ ∇ T T ∣ ∣ ∣ ∣ γ = γ ( s ) = 0 ⇔ [ d 2 x i ds 2 -Γ i jk dx j ds dx k ds ] γ = γ ( s ) = 0 (13)  \nK = K 1 K 2 = R 1212 g 11 g 22 -( g 2 12 ) = -1 √ g 11 g 22 [ ∂ ∂x 2 ( 1 √ g 22 ∂ √ g 11 ∂x 2 ) + ∂ ∂x 1 ( 1 √ g 11 ∂ √ g 22 ∂x 1 )] (14)  \nwhere R 1212 is the Riemannian curvature tensor of this surface. In this supplemental material, we adopt the notation in surface theory, where covariant and contravariant coordinates are not distinguished.  \nIn the present work, we focus on the optical geometry of spherically symmetric black holes. If we restrict test particles moving in equatorial plane θ = π/ 2 and assign the coordinates ( x 1 , x 2 ) = ( r, φ ), then the two dimensional optical metric becomes  \ndt 2 = ˜ g OP ij dx i dx j = 1 [ f ( r )] 2 dr 2 + r 2 f ( r ) dφ 2 (15)  \nHere t is the arc length parameter / spatial distance parameter in optical geometry. For any circular photon orbit, the angle between this orbit and coordinate axis r is α = π/ 2, then the geodesic curvature κ g of circular photon orbit r = r ph reduces to  \nκ g ( r = r ph ) = [ dα dt -1 2 √ ˜ g OP φφ ∂ log (˜ g OP rr ) ∂φ cos α + 1 2 √ ˜ g OP rr ∂ log (˜ g OP φφ ) ∂r sin α ] r = r ph = [ 1 2 √ ˜ g OP rr ∂ log (˜ g OP φφ ) ∂r ] r = r ph (16)  \nUsing this coordinates ( x 1 , x 2 ) = ( r, φ ), the Gauss curvature of two dimensional optical geometry in equation (14) becomes  \nK ( r, φ ) = -1 √ ˜ g OP [ ∂ ∂φ ( 1 √ ˜ g OP φφ ∂ √ ˜ g OP rr ∂φ ) + ∂ ∂r ( 1 √ ˜ g OP rr ∂ √ ˜ g OP φφ ∂r )] (17)  \nwhere ˜ g OP = det (˜ g OP ij ) = ˜ g OP rr ˜ g OP φφ is the determinant of optical metric. These results coincide with equation (5) and equation (6) in the main text exactly.  \nFurthermore, in spacetime geometry (which is a four dimensional Lorentz manifold), one can pick two dimensional surface S µν with two tangent vector fields ∂ µ and ∂ ν . 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