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The geometry of photon surfaces

The photon sphere concept in Schwarzschild space-time is generalized to a definition of a photon surface in an arbitrary space-time. A photon sphere is then defined as an SO(3)xR-invariant photon surface in a static spherically symmetric space-time. It is proved, subject to an energy condition, that a black hole in any such space-time must be surrounded by a photon sphere. Conversely, subject to an energy condition, any photon sphere must surround a black hole, a naked singularity or more than a certain amount of matter. A second order evolution equation is obtained for the area of an SO(3)-invariant photon surface in a general non-static spherically symmetric space-time. Many examples are provided.


Introduction
The exterior region of the maximally extended Schwarzschild space-time is described by the metric g = − 1 − 2m r dt 2 + 1 − 2m r −1 dr 2 + r 2 dθ 2 + sin 2 θdφ 2 , r > 2m . (1) For any null geodesic in this exterior region the null geodesic equations give where λ is an affine parameter along the geodesic. The right side here is evidently positive for r > 3m and negative for r : 2m < r < 3m. It follows that any future endless null geodesic in the maximally extended Schwarzschild space-time starting at some point with r > 3m and initially directed outwards, in the sense that dr/dλ is initially positive, will continue outwards and escape to infinity. Any future endless null geodesic in the maximally extended Schwarzschild space-time starting at some point with r : 2m < r < 3m and initially directed inwards, in the sense that dr/dλ is initially negative, will continue inwards and fall into the black hole. The hypersurface {r = 3m}, known as the Schwarzschild photon sphere, thus distinguishes the borderline between these two types of behaviour; any null geodesic starting at some point of the photon sphere and initially tangent to the photon sphere will remain in the photon sphere. (See Darwin [1,2] for a detailed analysis of the behaviour of null and timelike geodesics in Schwarzschild space-time.) The Schwarzschild photon sphere also has physical significance for massive bodies. For any timelike geodesic in the exterior region the geodesic equations give where s is arc length along the geodesic. At any point with r > 3m one may arrange for the two terms on the right of (3) to cancel and so obtain a timelike geodesic at constant r. For r : 2m < r < 3m the right hand side of (3) is evidently negative. Thus any future endless timelike geodesic in the maximally extended Schwarzschild space-time starting at some point between the event horizon at r = 2m and the photon sphere at r = 3m and initially directed inwards, in the sense that dr/ds is initially negative, will continue inwards and fall into the black hole. Any observer who traverses a Schwarzschild photon sphere must therefore engage some form of propulsion or else be drawn in to the black hole to meet an inevitable fate.
A photon sphere has been defined by Virbhadra & Ellis [3] as a timelike hypersurface of the form {r = r 0 } where r 0 is the closest distance of approach for which the Einstein bending angle of a light ray is unboundedly large. These authors subsequently [4] considered the Einstein deflection angle for a general static spherically symmetric metric and obtained an equation for a photon sphere. The existence of a photon sphere in a space-time has important implications for gravitational lensing. In any space-time containing a photon sphere, gravitational lensing will give rise to relativistic images [3].
The Schwarzschild photon sphere may be usefully be compared with the concept of a closed trapped surface. Any null geodesic originating from any point on a closed trapped surface in Schwarzschild space-time is drawn into the singularity at r = 0. By contrast, any null geodesic originating from any point on the photon sphere will be drawn into the singularity if and only if it is initially directed inwards.
The main objectives of the present paper are to give a geometric definition of a photon surface in a general space-time and of a photon sphere in a general static spherically symmetric space-time.
An evolution equation is obtained for the cross-sectional area of a photon surface in a dynamic spherically symmetric space-time. It is shown, subject to suitable energy conditions, that in any static spherically symmetric space-time a black hole must be surrounded by a photon sphere, and a photon sphere must surround either a black hole, a naked singularity or more than a certain amount of matter. Many examples are given of photon spheres in static spherically symmetric spacetimes. Photon surface evolution is considered for the dynamic space-time example of Vaidya null dust collapse to a naked singularity.

Photon surfaces
The hypersurface S := {r = 3m} in Schwarzschild space-time has two main properties, first that any null geodesic initially tangent S will remain tangent to S, and second that S does not evolve with time. The following general definition of a photon surface is based on only the first of these properties. A more restrictive class of photon surfaces may be defined when the space-time admits a group of symmetries (see Definition 2.3).
Definition 2.1. A photon surface of (M, g) is an immersed, nowhere-spacelike hypersurface S of (M, g) such that, for every point p ∈ S and every null vector k ∈ T p S, there exists a null geodesic γ : (−ǫ, ǫ) → M of (M, g) such thatγ(0) =k, |γ| ⊂ S.
Any null hypersurface is trivially a photon surface. Photon surfaces are conformally invariant structures. If S is a photon surface of (M, g) then S is a photon surface of (M, Ω 2 g) for any smooth function Ω : M → (0, ∞).
Note that Definition 2.1 is entirely local. In particular, a photon surface S need contain no endless null geodesics of (M, g). Moreover, a photon surface need only be immersed, rather than embedded in M , and so may have self-intersections. If (M, g) is of dimension n + 1 (n ≥ 2) then, through each point p of a photon surface S in (M, g), there is an (n − 2)-parameter family of null geodesics of (M, g) that lie entirely in S.
The paper will be principally concerned with photon surfaces in space-times of 3 + 1 dimensions. The exceptions are Examples 1 and 3 which give photon surfaces in space-times of dimension 2 + 1 and 4 + 1 respectively.
Example 1 (Minkowski 3-space). In Minkowski 3-space M 3 , consider the singlesheeted hyperboloid S given by for some constant a > 0. This surface is doubly ruled, the rulings being given by where θ identifies the intersection points with {t = 0} and t is the parameter along the ruling lines. The tangentsγ ± θ (t) to the ruling lines are null with respect to the M 3 metric. Clearly they are geodesics in M 3 . At each point of S there can be just two null directions tangent to S. These must therefore be the directions of the two ruling lines through that point. Hence S is a photon surface in the sense of Definition 2.1 (see Fig. 1).
Note that for any circle of the form C = {t 0 , x 0 + r cos θ, y 0 + r sin θ} , r > 0, (6) and any future-directed timelike vector field X along C that respects the symmetry of C, in the sense of X = X t , X r cos θ, X r sin θ , for constant X t > 0, X r such that (X t ) 2 > (X r ) 2 , there is a unique single-sheeted hyperboloid S through C such that X is tangent to S along C.
In the case a = 0, equation (4) gives the null cone through the origin. The complement of p in this null cone is a null photon surface of M 3 .
Example 2 (Minkowski 4-space). One may generalize Example 1 to Minkowski 4space M 4 as follows. Let S be a timelike hypersurface in M 4 of the form for some constant a > 0. The two-parameter family of lines γ ± θ,φ (t) = a (0, cos θ, sin θ sin φ, sin θ cos φ) + at (1, ∓ sin θ, ± cos θ sin φ, ± cos θ cos φ) foliate S and are null geodesics with respect to the M 4 metric. For each p ∈ S, the tangents at p to those γ ± θ,φ (t) that pass through p can be shown to generate the null cone of T p S. Hence S is a photon surface in the sense of Definition 2.1. In terms of the double null coordinates for r := (x 2 + y 2 + z 2 ) 1/2 , equation (8) assumes the simple form In the future direction S tends asymptotically to the null hypersurface {v = 0}, whilst in the past direction S tends asymptotically to the null hypersurface {u = 0}.
Example 3 (De Sitter space). De Sitter space-time may be regarded [5] as a singlesheeted hyperboloid in Minkowski 5-space M 5 . By analogy with Examples 1 and 2, de Sitter space-time is thus realized as a photon surface in M 5 .
Example 4 (The Robertson-Walker models). Since all Robertson-Walker models are conformally flat and therefore locally conformally transformable to Minkowski space, the photon surfaces of any such model may thus be obtained, at least locally, by conformal transformations of Minkowski space.
Theorem 2.2. Let S be a timelike hypersurface of (M, g). Let n be a unit normal field to S and let h ab be the induced metric on S. Let χ ab be the second fundamental form on S and let σ ab be the trace-free part of χ ab . Then the following are equivalent: i) S is a photon surface; ii) χ ab k a k b = 0 ∀ null k ∈ T p S ∀p ∈ S; iii) σ ab = 0; iv) every affine null geodesic of (S, h) is an affine null geodesic of (M, g).
(iii) ⇒ (iv). For any curve in S with null tangent k one has where || denotes covariant differentiation in S with respect to h. The second term on the right of (15) vanishes by hypothesis. If k is tangent to an affine null geodesic of (S, h) then the term on the left of (15) also vanishes and so k is tangent to an affine null geodesic of (M, g).
Condition (iii) of Theorem 2.2 is equivalent to a requirement that χ ab is pure trace in the sense of where Θ := h cd χ cd is the expansion of the unit normal to S. For Example 1 one has Θ = 2/a; for Example 2 one has Θ = 3/a. (Note that, by a standard abuse of notation, h ab denotes both the induced metric on S and the symmetric tensor field of rank (0, 2) along S in M which satisfies h ab n b = 0 and pulls back to the induced metric on S.) It is clear from condition (iii) of Theorem 2.2 that a space-time must be specialized in some respect in order to admit any timelike photon surfaces in the sense of Definition 2.1. For this reason it is helpful to restrict attention to space-times which admit groups of symmetries. Definition 2.3. Suppose (M, g) admits a group G of isometries. A photon surface S of (M, g) that is invariant under G, in the sense that each g ∈ G maps S onto itself, will be called a G-invariant photon surface.
Clearly any G-invariant null hypersurface is a G-invariant photon surface. In particular, if G = R or G = S, then any Killing horizon [6,7] is a G-invariant photon surface.

Dynamic Spherical symmetry: General theory
By definition, a general spherically symmetric space-time admits an SO(3) isometry group for which the group orbits are spacelike 2-spheres. The following result describes the evolution of the cross-sectional area of an SO(3)-invariant photon surface in a spherically symmetric space-time.
where n a is the unit normal to S, Θ is the expansion of n a and G ab := R ab − 1 2 Rg ab is the Einstein tensor of (M, g).
Proof. Let h ab be the induced Lorentzian 3-metric on S and, for each s, let (2) h ab be the induced Riemannian 2-metric on T s . The expansion of X in (S, h) is given by (2) where the covariant derivative is that of (M, g). Since X is both shear-free and vorticity-free in (S, h), the Raychaudhuri equation for X in (S, h) assumes the form where (3) R ab is the Ricci tensor of (S, h). From first principles one has where (2) χ ab is the second fundamental form of each T s in (S, h). Since X a is shear-free and vorticity free in (S, h) one has (2) fundamental form of S admits the canonical decomposition χ ab = 1 3 Θh ab + σ ab . Equations (19) and (20) therefore give which combine to yield One may now substitute for the second term on the right of (18) to obtain From first principles one has (2) Θ = d ds ln (2) A, and the Gauss-Bonnet theorem gives (2) R (2) A = 8π. Substituting for (2) Θ and (2) R in (24) one obtains This agrees with (17) iff σ a b σ b a = 0. Construct, for the tangent bundle T S of S, an orthonormal basis field of the form {X, e (1) , e (2) }, with e (1) and e (2) unit spacelike. With respect to this basis one has By spherical symmetry the vector field σ a b X b must be proportional to X a . Hence one has σ 1 0 = σ 2 0 = 0. The vanishing of σ a b σ b a is thus equivalent to the vanishing of σ ab . One has σ ab = 0 iff S is a photon surface.
A spherically symmetric metric is locally expressible in the form with respect to coordinates x 0 , x 1 , θ, φ adapted to the spherical symmetry, where g 00 , g 01 = g 10 , g 11 and g θθ > 0 depend only on x 0 and x 1 . It is often convenient to introduce a radial coordinate r, depending only on x 0 and x 1 , such that g θθ is a function of r only. One is free to specify g θθ as a function of r alone since to do so is, in effect, a definition of the coordinate r. This will be assumed to be done throughout this paper.
The following result is useful in the locating of SO(3)-invariant photon surfaces in dynamic spherically symmetric space-times.
Lemma 3.2. Let (M, g) be a spherically symmetric space-time. Let S be an SO(3)invariant timelike hypersurface of (M, g) and let X be the SO(3)-invariant unit future-directed timelike tangent vector field along S that is orthogonal to the SO(3)invariant 2-spheres in S. Then S is a photon surface of (M, g) iff holds along S, where n a is the unit normal field to S in (M, g).
Proof. By spherical symmetry, and since X is unit timelike, the vector field ∇ X X must be proportional to n. Hence it suffices to show that S is a photon surface iff along S one has or equivalently Construct for T S a local orthonormal basis field of the form {X, e (θ) , e (φ) }. With respect to this basis field the components of χ a b form a diagonal matrix with Equation (30) is thus equivalent to χ 0 0 = χ θ θ = χ φ φ which is in turn equivalent to σ 0 0 = σ θ θ = σ φ φ . In view of the trace-free property of σ a b , equation (30) is thus equivalent to σ a b = 0. From Theorem 2.2 one has that σ a b = 0 holds along S iff S is a photon surface.
Let us continue to work with respect to the coordinate system {x 0 , x 1 , θ, φ} employed in (27). Let x a (s) be an integral curve of the vector field X in Lemma and equation (28) becomes The x 0 and x 1 components of equation (33) thus combine to give where the components of n are given by is the determinant of the time-space part of g ab in (27). Equation (35) is the coordinate equivalent of (28) and provides for the easy determination of SO(3)invariant photon surfaces (see Example 10).

Static Spherical Symmetry: General Theory
By definition, a spherically symmetric space-time is static if it admits an SO(3)× R group of isometries such that the R orbits are generated by a Killing field K which is both hypersurface orthogonal and orthogonal to the SO(3) orbits. The present section will be concerned with SO(3) × R-invariant photon surfaces in static spherically symmetric space-times. Such surfaces may be termed photon spheres because, as will be seen, they are a natural generalization to general static spherically symmetric space-times of the Schwarzschild photon sphere concept. The term "photon sphere" will be regarded as applicable only in static spherically symmetric spacetimes. For clarity, the term "SO(3) × R-invariant photon surface" will usually be employed in preference to "photon sphere".
Although the space-times of Examples 1 and 2 are static and spherically symmetric, the photon surfaces in these space-times are not SO(3) × R-invariant and so are not photon spheres. Of the Robertson-Walker space-times of Example 4, only the Einstein cylinder is both spherically symmetric and static. None of the photon surfaces of the Einstein cylinder are SO(3)× R-invariant. Thus the Einstein cylinder has no photon spheres.
One may characterize an SO(3) × R-invariant photon surface, or photon sphere, in a static spherically symmetric space-time by means of the following special case of Theorem 3.1.
where A is the area of T , n a is the unit normal to S and Θ is the trace of the Proof. Note that the unit future-directed timelike tangent field X along S in Theorem 3.1 is proportional to the restriction to S of the Killing field K. Suppose first that there exists an SO(3)-invariant 2-sphere T ⊂ S such that (39) holds. The quantities A, Θ and G ab n a n b remain constant as T is mapped along the flow lines of the Killing field K. So they also remain constant as they are mapped along the flow lines of X. Hence (17) holds with the term on the left and the first term on the right both zero. Thus, by Theorem 3.1, S is a photon surface of (M, g). By hypothesis S is SO(3) × R-invariant.
For the converse, suppose that S is an SO(3) × R-invariant photon surface of (M, g). Then (17) holds for every SO(3)-invariant 2-sphere T s ⊂ S. Since K induces groups of local isometries, the area A s of T s is independent of the parameter s. Hence the term on the left and the first term on the right of (17) both vanish and one obtains (39).
with equality holding iff G ab n a n b = 0 along S.
If the Einstein equations hold with a zero cosmological constant then, in the corollary to Theorem 4.1, the hypothesis G ab Y a Y b ≥ 0 for all vectors Y is equivalent to T ab Y a Y b ≥ 0 for all vectors Y. This is a physically reasonable energy condition. In particular, for a perfect fluid with density ρ and pressure p, it is equivalent to a condition that ρ and p are both non-negative. More generally, the condition holds for an energy tensor with a single timelike eigenvector (type I in the classification of Hawking & Ellis [5]) iff each energy tensor eigenvalue is non-negative.
The characterization of timelike photon surfaces provided by Theorem 4.1 involves derivatives of the metric components up to second order. The following result (Theorem 4.2) provides an entirely different characterization of SO(3) × Rinvariant photon surfaces in terms of derivatives of the metric components up to only first order.
Let a general static spherically symmetric metric g be expressed in the form (27) with g θθ a function of r only. The Killing equation K (a;b) = 0 and the orthogonality of K to ∂ θ and ∂ φ gives K a ∂ a g θθ = 0 and hence ∇ K r = 0 where r is to be regarded as a scalar field on M . Since r is independent of θ and φ it follows that any SO(3) × R-invariant hypersurface S of (M, g) must be of the form {r = const.}. If S is also a timelike hypersurface then r ;a is a spacelike vector field along S and K is a timelike vector field in a neighbourhood of S.
If the coordinates x 0 and x 1 implicit in (27) are chosen such that K = ∂ x 0 then the Killing equation gives that all the metric components g ab in (27) are independent of x 0 . Since r is constant along the integral curves of K, the coordinate x 1 must be a function of r only. A natural choice is x 1 = r. One may redefine x 0 according to x 0 → x 0 − (g 01 /g 00 )dx 1 . This diagonalizes the time-space part of g ab and leaves the components of g ab independent of x 0 . Furthermore the curves {x 1 , θ, φ = const.} are unchanged except that they are re-parametrized. The vector field ∂ x 0 then becomes a conformal Killing field.
Define the tensor field The second fundamental form of S is given by The vector fields form an orthonormal frame field along S, with e (θ) and e (φ) unit spacelike and X unit timelike. One has Condition (iii) of Theorem 2.2 holds iff χ ab is proportional to h ab and hence iff This is equivalent to which is in turn equivalent to (42).
Equation (42) can have solutions such that {r = const.} is a spacelike hypersurface and therefore not a photon surface (see e.g. Example 7). It is therefore always necessary in the use of Theorem 4.2 to check that the hypersurface {r = const.} is in fact timelike or null.
Note that in a region of space-time where the Killing field K is spacelike, for example behind the event horizon of Schwarzschild space-time, the hypersurfaces {r = const.} are necessarily spacelike and so cannot be photon spheres. Case 1. (x 1 := r, components of g ab independent of x 0 .) As discussed previously, for a static spherically symmetric metric it is possible to choose x 1 := r, with g θθ depending only upon r and with all the components of g ab independent of x 0 . In this case (42) reduces to g 00 ∂ r g θθ = g θθ ∂ r g 00 .
(54) [3] of a photon sphere. Note that even though the components g rr , g 0r = g r0 do not appear in (54), they are not assumed to vanish. A particular sub-case of interest is that of time-space coordinates, with t := x 0 timelike in the sense of g tt < 0. Another sub-case of interest is that of single null (radiation) coordinates, with u := x 0 null in the sense of g uu = 0.

This agrees with an equation obtained by Virbhadra & Ellis [4] on the basis of a different definition
Case 2. (Double null coordinates u, v.) Let x 0 := u, x 1 := v be double null coordinates in the sense of g uu = g vv = 0. The radial coordinate r is to be regarded as a function of u and v. Then (42) assumes the form g θθ {r ;uu (r ;v ) 2 − 2r ;uv r ;u r ;v + r ;vv (r ;u ) 2 } + 2(r ;u r ;v ) 2 ∂ r g θθ = 0 .
The metric components g uv = g vu enter here through the covariant derivatives of r.
Equations (42), (54) and (55) may be referred to as photon sphere equations since they give the location of timelike photon spheres in static spherically symmetric space-times.
In order to facilitate further progress, a general static spherically symmetric metric will be written in such a form as to cast the Einstein tensor in a particularly simple and convenient form.
where a prime denotes differentiation with respect to r. For the right sides of (61), (62) and (63) to be defined at some radiusr > 0 one evidently needs m(r) and µ(r) to be twice differentiable at r =r. Equations (61) and (62) combine to give µ ′ (r) = 8πr(ρ(r) + p 1 (r)) whereby one may rewrite (63) in the more convenient form 2 r (p 2 (r) − p 1 (r)) = p ′ 1 (r) + m(r) In the perfect fluid case p 1 (r) = p 2 (r) =: p(r) equation (65) reduces to the Tolman-Oppenheimer-Volkoff equation By means of equations (61) and (64), the photon sphere equation (54) becomes For r such that 2m(r) < r, and hence such that the hypersurface {r = const.} is timelike, equation (67) gives the location of the SO(3) × R-invariant timelike photon surfaces for the metric (59). Equation (67) is the basis for the following result which shows that, subject to a suitable energy condition, any black hole in a static spherically symmetric space-time must be surrounded by an SO(3) × R-invariant photon surface. For the purpose of this and subsequent results, a function f : R ⊃ I → R on an interval I will be said to be piecewise C r if I is the disjoint union of a locally finite collection of intervals I i such that f |I i is C r . Each interval I i may be open, closed or half-open. Theorem 4.3. Suppose the metric g has the form (59) for r 0 < r < ∞, for some r 0 > 0, with m(r) and µ(r) both C 0 , piecewise C 2 functions of r ∈ (r 0 , ∞). Suppose the following hold: 1) ρ(r) and p 1 (r) are bounded functions of r ∈ (r 0 , ∞); 2) 2m(r) < r ∀r ∈ (r 0 , ∞); 3) ρ(r) ≥ 0, p 1 (r) ≥ 0 ∀r ∈ (r 0 , ∞); 4) lim r→∞ 4πr 2 p 1 (r) = lim r→∞ 4πr 2 ρ(r) = 0; 5) for each value of t the 2-surfaces T t,r := {t = const.}∩{r = const.}, r 0 < r < ∞, are such that T t := lim rցr0 T t,r exists as an embedded spacelike 2-sphere in (M, g) and is marginally outer trapped. Then (M, g) admits an SO(3) × R-invariant timelike photon surface of the form {r = r 1 } for some r 1 ∈ (r 0 , ∞).
Proof. Fix t and let k be the outward future-directed null normal field along each T t,r , r 0 < r < ∞, normalized such that g(k, n) = 1, where n = 1 − 2m(r) r 1/2 ∂ r is the outward radial unit tangent to {t = const.}. Since k is parallelly propagated along each of the geodesic integral curves of n, one has that lim rցr0 k is a welldefined, nowhere-zero null vector field along T t := lim rցr0 T t,r . For r ∈ (r 0 , ∞) the vector field k has the form k a = (k t , a(r)k t , 0, 0) for a(r) : The expansion of k is given by The condition that T t is marginally outer trapped therefore implies The non-negativity of ρ(r) and p 1 (r) gives, by means of (61) and (64), that m(r) and µ(r) are non-decreasing functions of r ∈ (r 0 , ∞). Thus (70) holds iff at least one of holds. Suppose the first of (71) fails. Then the second must hold and one has From the boundedness of ρ(r) and p 1 (r) on r ∈ (0, ∞) one has, by means of (64) and (72), that lim sup rցr0 µ ′ (r) is finite. This is incompatible with the second of (71). Hence the first of (71) must hold. Let f : (r 0 , ∞) → R be the left side of (67). By the non-negativity of p 1 (r) and the first of (71) one has lim rցr0 f (r) ≤ − 1 2 . By condition (4), equation (61) and l'Hôpital's rule one has lim r→∞ m(r)/r = lim r→∞ r 2 p 1 (r) = 0 and hence lim r→∞ f (r) = 1. Hence there exists some r 1 ∈ (r 0 , ∞) such that f (r 1 ) = 0. The hypersurface {r = r 1 } is an SO(3) × R-invariant photon surface of (M, g).  (3), any static spherically symmetric black hole must be surrounded by an SO(3) × R-invariant timelike photon surface. The following result may then be regarded as a partial converse in that it shows, subject to a suitable energy condition, that if there exists an SO(3) × R-invariant timelike photon surface then there must be a naked singularity or a black hole, or more than a certain amount of matter. Proposition 4.4. Suppose the metric g has the form (59) for 0 < r < ∞, with m(r) and µ(r) both C 0 , piecewise C 2 functions of r ∈ (0, ∞). If the following all hold: 1) ρ(r) is a non-increasing, bounded function of r ∈ (0, ∞); 2) lim r→0 m(r) = 0; 3) 4m(r) < r ∀r ∈ (0, ∞); 4) p 1 (r) ≤ ρ(r)/3 ∀r ∈ (0, ∞), then (M, g) can contain no SO(3) × R-invariant timelike photon surfaces.
Note that this result is valid even for negative p 1 (r) and ρ(r). Condition (2) of Proposition 4.4 prohibits any curvature singularity at r = 0. Condition (3) may be interpreted as a requirement that there is no black hole and less than a certain amount of matter. The result shows that one of these two conditions must fail if there is an SO(3) × R-invariant timelike photon surface and conditions (1) and (4) both hold.
When the matter is a perfect fluid it is possible to improve condition (3) of Proposition 4.4 to condition (3) of the following result.
Note that, as for Proposition 4.4, Theorem 4.5 is valid even for negative pressure and density.
One would like to remove the insufficient matter parts of condition (3) of Proposition 4.4 and condition (3) of Theorem 4.5, in other words to weaken these to a no-black-hole condition 2m(r) < r ∀r > 0. But no result to this effect is forthcoming. On the other hand no counterexample is known.
To conclude this section it will be shown that the physical significance of the photon sphere in Schwarzschild space-time, as discussed in the Introduction, carries over to the general static spherically symmetric case. Suppose the metric has the form (59) for r 0 < r < ∞ and is asymptotically flat in the limit r → ∞. Assume p 1 (r) ≥ 0, m(r) ≥ 0 ∀r > r 0 . The matter need not be a perfect fluid. Denote the left side of (67) by f (r). The condition of asymptotic flatness gives lim r→∞ f (r) = 1 so, if there are any SO(3) × R-invariant timelike photon surfaces, there will be an outermost such surface S. For simplicity assume f ′ (r) = 0 at S. Let R ext be the connected component of {q ∈ M : f (q) > 0} that has S as its inner boundary and extends to r = ∞. Let R int be the connected component of {q ∈ M : f (q) < 0} that has S as its outer boundary.
Consider first the case of a future endless affine null geodesic γ(λ). The null geodesic equations for the metric (59) give At any point p ∈ |γ| ∩ R ext such that dr/dλ = 0 one has d 2 r/dλ 2 > 0. At any point p ∈ |γ| ∩ R int such that dr/dλ = 0 one has d 2 r/dλ 2 < 0. Thus if γ starts outside S (i.e. in R ext ) and is initially directed outwards, in the sense that dr/dλ is initially positive, then γ will continue outwards. If γ starts in R int and is initially directed inwards, in the sense that dr/dλ is initially negative, then γ will continue inwards until it falls either into a singularity or through an SO(3) × R-invariant photon surface other than S. Consider now a unit speed timelike geodesic ξ(s). The timelike geodesic equations give For any point p ∈ R ext one can arrange for the first three terms on the right of (77) to cancel and so obtain a unit speed timelike geodesic ξ(s) through p ∈ R ext at constant r. For p ∈ R int the first three terms on the right of (77) are evidently negative. For p ∈ |ξ| ∩ R int such that dr/ds = 0 one has d 2 r/ds 2 < 0. Thus if ξ starts in R int and is initially directed inwards, in the sense that dr/ds is initially negative, then ξ will continue inwards until it falls either into a singularity or through an SO(3) × R-invariant photon surface other than S.

Spherical symmetry: Examples
The One finds that equation (54) reduces to r = 3m, independent of the value of Λ. This is surprising since Schwarzschild and Schwarzschild-de Sitter space-times are not conformally related.
Example 6 (Schwarzschild interior solution). The Schwarzschild interior solution describes a spherically symmetric distribution of perfect fluid of radius R, bounded pressure p and constant density ρ 0 > 0. The solution is to have a metric of the form (59) and is to be matched at r = R to a Schwarzschild vacuum solution in such a way that the pressure is a continuous function of r. One thus has p 1 (r) = p 2 (r) =: p(r) for all r : 0 ≤ r < ∞, ρ(r) = ρ 0 for all r : 0 ≤ r ≤ R, ρ(r) = 0 for all r : R < r < ∞ and p(r) = 0 for all r : R ≤ r < ∞. The pressure p(r) for r : 0 ≤ r ≤ R is to be obtained by an integration of the Tolman-Oppenheimer-Volkoff equation (66) subject to the boundary condition p(R) = 0. This yields for The spherically symmetric system described by the Schwarzschild interior solution can exist in a state of stable equilibrium iff m(R)/R < 4/9 (see in Stephani [8]). This condition is equivalent to 8πρ 0 R 2 < 8/3, which implies p(r) ≥ 0 ∀r ≥ 0, and implies that the absence of a black hole is a general feature of the Schwarzschild interior solution.
The left side of equation (67) now assumes the form For m(R)/R < 1/3 one has that (84) is positive for all r > 0, so there are no timelike photon spheres. For m(R)/R = 1/3 there is a single timelike photon sphere which lies at the boundary r = R of the matter. For 1/3 < m(R)/R < 4/9 there is one timelike photon sphere outside the matter at r = 3m(R) > R and one timelike photon sphere inside the matter at For fixed R the radius of the outer photon sphere is a strictly increasing function of m(R)/R whilst the radius of the inner photon sphere is a strictly decreasing function of m(R)/R. Thus a Schwarzschild interior solution matched to a Schwarzschild vacuum exterior solution contains no black hole, and contains one timelike photon sphere iff 1/3 = m(R)/R and two timelike photon spheres iff m(R)/R lies in the range 1/3 < m(R)/R < 4/9. However for such values of m(R)/R the space-time is unphysical in that the pressure at the center is p(0) ≥ ρ 0 / √ 3 > ρ 0 /3. Therefore under the reasonable energy condition p(r) ≤ ρ 0 /3 ∀r : 0 ≤ r < ∞ there are no photon spheres in this example.
The energy condition 0 ≤ p(r) ≤ ρ(r)/3 ∀r ∈ [0, ∞) is in fact satisfied iff m(R)/R lies in the range 0 ≤ m(R)/R ≤ 5/18. The corresponding space-times (in view of 5/18 < 7/24) satisfy all of conditions (1) to (4) of Theorem 4.5, except the smoothness of ρ(r) at r = R. Thus considering a smoothed family of solutions approximating the solution above and having it as a strict limit, but each with smooth ρ(r) at r = R, we can apply Theorem 4.5 to show no photon spheres exist in all the cases discussed in this section where this energy condition is satisfied. On the other hand Proposition 4.4 is directly applicable without smoothing, but only applies to the subset of these cases with 0 ≤ m(R)/R < 1/4.
where m is the ADM mass and e is the electric charge. We assume m > 0. There is an event horizon at r = r + := m + √ m 2 − e 2 and a Cauchy horizon at r = r − := m − √ m 2 − e 2 . The event horizon exists for 0 ≤ (e/m) 2 ≤ 1, and the Cauchy horizon exists for 0 < (e/m) 2 ≤ 1. For (e/m) 2 = 1 they both lie at r = m. There can be no timelike photon spheres between the event horizon and the Cauchy horizon because the Killing field K is spacelike there. Outside the event horizon the Killing field K is timelike, so every hypersurface of the form {r = const.} which lies outside the event horizon and satisfies the photon sphere equation (54) is necessarily a timelike photon sphere. Equation (54) assumes the form which has solutions r ± ps given by The hypersurface S + := {r = r + ps } exists for 0 ≤ (e/m) 2 ≤ 9/8 and lies outside the event horizon and is therefore a timelike photon sphere. The hypersurface S − := {r = r − ps } exists for 0 < (e/m) 2 ≤ 9/8 but lies outside the event horizon only for 1 < (e/m) 2 ≤ 9/8, and so is a timelike photon sphere only then. The hypersurfaces S + and S − coincide for (e/m) 2 = 9/8. The Cauchy horizon and event horizon are always null photon spheres. Figure 2. The radii of the event horizon, Cauchy horizon and photon spheres for Reissner-Nordström space-time.
For 0 ≤ (e/m) 2 ≤ 1 the curvature singularity at r = 0 is locally naked but hidden behind an event horizon which lies strictly inside the only timelike photon sphere. For 1 < (e/m) 2 the singularity at r = 0 is globally naked and is surrounded by two timelike photon spheres in the case 1 < (e/m) 2 < 9/8, one timelike photon sphere in the case (e/m) 2 = 9/8 and by no photon spheres, either timelike or null, in the case (e/m) 2 > 9/8 (see Fig. 2).
Example 8 (Janis-Newman-Winicour space-time). The most general static spherically symmetric solution to the Einstein massless scalar field equations for a scalar field Φ satisfying Φ = 0 was obtained by Janis, Newman and Winicour [9]. The Ricci tensor has the form R ab = 8πΦ ;a Φ ;b . The solution is known [10] to be expressible in the form for r : b < r < ∞ where the constants b, ν are related to the ADM mass m and scalar charge q by We assume b > 0. There is a curvature singularity at r = b. In order to obtain m ≥ 0 one must assume 0 ≤ ν ≤ 1. For q = 0 the solution reduces to the Schwarzschild solution.
Since all hypersurfaces of the form {r = const.} are timelike, one has from the photon sphere equation (54) that the only timelike photon sphere is at which exists only for ν : 1 2 < ν ≤ 1, i.e. for 0 ≤ q 2 < 3m 2 . For 1 2 < ν ≤ 1 it is known [11] that a photon coming from infinity is deflected through an unboundedly large angle, i.e. the photon passes increasingly many times around the singularity as the closest distance of approach tends to the right side of (93).
Example 9 (Charged dilaton space-time). A static spherically symmetric spacetime with a charged dilaton field was obtained by Horne & Horowitz [12]. It comprises a metric g, a dilaton field Φ and an electromagnetic field F ab given by where r + and r − are related to the ADM mass m and electric charge e by r + + ωr − = 2m (97) and β is a free parameter which controls the coupling strength between the dilaton and Maxwell fields, with ω defined in terms of β 2 by We assume m > 0. For β = 0 the solution reduces to the Reissner-Nordström solution considered in Example 7. For β = 0 and e = 0 the solution reduces to the Schwarzschild solution. The solution also reduces to the Schwarzschild solution for e = r − = 0 and arbitrary β. Here we shall consider the case β 2 = 1. In this case one has (r + , r − ) = (2m, e 2 /m). There is an event horizon at r = r + = 2m and a curvature singularity at r = r − = e 2 /m. For 0 ≤ (e/m) 2 < 2 the singularity at r = r − lies inside a black hole whilst for (e/m) 2 > 2 it is globally naked.
For 0 ≤ (e/m) 2 < 2 one has r − ps ≤ r − < r + < r + ps so there is a single timelike photon sphere. For (e/m) 2 = 2 one has r − ps = r − = r + = r + ps so there are no timelike photon spheres. For 2 < (e/m) 2 < 18 both r + ps and r − ps are complex so there are no timelike photon spheres. For (e/m) 2 ≥ 18 one has r − ps ≤ r + ps < r − so there are again no well-known there is a curvature singularity at r = 0 and that for λ : 0 < λ ≤ 1 16 the part of this singularity at u = 0 is locally naked. Fix λ : 0 < λ ≤ 1 16 . For u > u 1 equation (103) gives, as expected, that there is a photon surface at r = 3m 1 . We seek to evolve this photon surface backwards in time, though the in-falling null dust, to obtain a maximally extended photon surface S. The boundary conditions are r = 3m 1 dr du = 0 at u = u 1 .  The results are shown in Fig. 4 for λ = 1/16 and λ = 1/32 for selected values of u 1 . One sees that in all cases S tends in the past direction to a null hypersurface of the form {u = const.}. The conformal diagram must therefore be of the form sketched in Fig. 5. It is evident from Fig. 5 that the naked central singularity is enclosed within the photon surface S in the sense that any partial Cauchy surface H extending to spatial infinity must intersect S in a 2-sphere. The physical significance of this may warrant further investigation.

Concluding remarks
The definition of a photon surface given in Section 2 is valid in an arbitrary spacetime. However the result that a photon surface must have a second fundamental form which is pure trace indicates that a space-time must be specialized in some respect if it is to contain any photon surfaces. For spherically symmetric space-times there are always photon surfaces that respect the spherical symmetry. For spacetimes that are not spherically symmetric, the definitions of a photon surface and Ginvariant photon surface may seem too restrictive. The problem is that, in general, one may not have orbiting null geodesics at a fixed radius. In Kerr space-time for example, although there are orbiting null geodesics in the equatorial plane, those null geodesics which move in the direction of rotation do so at a different radius than those which move in the opposite direction. But it seems implausible that a concept which is physically important in the case of exact spherical symmetry should become invalid when even a small amount of angular momentum is introduced. A nontrivial generalization of the concepts of photon surface and G-invariant photon surface, at least to axially symmetric space-times, is thus required.