THE GEOMETRY OF PHOTON SURFACES

Transcription

1 THE GEOMETRY OF PHOTON SURFACES CLARISSA-MARIE CLAUDEL 1 Mathematical Institute, Univesity of Umeå, S Umeå, Sweden K. S. VIRBHADRA 2 andg.f.r.ellis 3 Depatment of Applied Mathematics, Univesity of Cape Town, Rondebosch 7701, South Afica. Abstact. The photon sphee concept in Schwazschild space-time is genealized to a definition of a photon suface in an abitay space-time. A photon sphee is then defined as an SO3) R-invaiant photon suface in a static spheically symmetic space-time. It is poved, subject to an enegy condition, that a black hole in any such space-time must be suounded by a photon sphee. Convesely, subject to an enegy condition, any photon sphee must suound a black hole, a naked singulaity o moe than a cetain amount of matte. A second ode evolution equation is obtained fo the aea of an SO3)-invaiant photon suface in a geneal non-static spheically symmetic space-time. Many examples ae povided. 1. Intoduction The exteio egion of the maximally extended Schwazschild space-time is descibed by the metic g = 1 2m ) dt m ) 1 d dθ 2 +sin 2 θdφ 2), > 2m. 1) Fo any null geodesic in this exteio egion the null geodesic equations give { d 2 ) 2 ) } 2 dθ dφ = 3m) +sin 2 θ 2) dλ2 dλ dλ whee λ is an affine paamete along the geodesic. The ight side hee is evidently positive fo > 3m and negative fo : 2m < < 3m. It follows that any futue endless null geodesic in the maximally extended Schwazschild space-time stating at some point with > 3m and initially diected outwads, in the sense that d/dλ is initially positive, will continue outwads and escape to infinity. Any futue endless null geodesic in the maximally extended Schwazschild space-time stating at some point with :2m<<3m and initially diected inwads, in the sense that d/dλ is initially negative, will continue inwads and fall into the black hole. The hypesuface { = 3m}, known as the Schwazschild photon sphee, thus distinguishes the bodeline between these two types of behaviou; any null geodesic stating at some point of the photon sphee and initially tangent to the photon sphee will emain in the photon sphee. See Dawin [1, 2] fo a detailed analysis of the behaviou of null and timelike geodesics in Schwazschild space-time.) 1 claissa@abel.math.umu.se 2 shwetket@maths.uct.ac.za 3 ellis@maths.uct.ac.za 1

2 PHOTON SURFACES 2 The Schwazschild photon sphee also has physical significance fo massive bodies. Fo any timelike geodesic in the exteio egion the geodesic equations give { dθ d 2 ds 2 = m ) 2 ) } 2 dφ + 3m) +sin 2 θ 3) 2 ds ds whee s is ac length along the geodesic. At any point with >3mone may aange fo the two tems on the ight of 3) to cancel and so obtain a timelike geodesic at constant. Fo :2m<<3m the ight hand side of 3) is evidently negative. Thus any futue endless timelike geodesic in the maximally extended Schwazschild space-time stating at some point between the event hoizon at =2m and the photon sphee at =3m and initially diected inwads, in the sense that d/ds is initially negative, will continue inwads and fall into the black hole. Any obseve who taveses a Schwazschild photon sphee must theefoe engage some fom of populsion o else be dawn in to the black hole to meet an inevitable fate. A photon sphee has been defined by Vibhada & Ellis [3] as a timelike hypesuface of the fom { = 0 } whee 0 is the closest distance of appoach fo which the Einstein bending angle of a light ay is unboundedly lage. These authos subsequently [4] consideed the Einstein deflection angle fo a geneal static spheically symmetic metic and obtained an equation fo a photon sphee. The existence of a photon sphee in a space-time has impotant implications fo gavitational lensing. In any space-time containing a photon sphee, gavitational lensing will give ise to elativistic images [3]. The Schwazschild photon sphee may be usefully be compaed with the concept of a closed tapped suface. Any null geodesic oiginating fom any point on a closed tapped suface in Schwazschild space-time is dawn into the singulaity at = 0. By contast, any null geodesic oiginating fom any point on the photon sphee will be dawn into the singulaity if and only if it is initially diected inwads. The main objectives of the pesent pape ae to give a geometic definition of a photon suface in a geneal space-time and of a photon sphee in a geneal static spheically symmetic space-time. An evolution equation is obtained fo the coss-sectional aea of a photon suface in a dynamic spheically symmetic space-time. It is shown, subject to suitable enegy conditions, that in any static spheically symmetic space-time a black hole must be suounded by a photon sphee, and a photon sphee must suound eithe a black hole, a naked singulaity o moe than a cetain amount of matte. Many examples ae given of photon sphees in static spheically symmetic spacetimes. Photon suface evolution is consideed fo the dynamic space-time example of Vaidya null dust collapse to a naked singulaity. 2. Photon sufaces The hypesuface S := { = 3m} in Schwazschild space-time has two main popeties, fist that any null geodesic initially tangent S will emain tangent to S, and second that S does not evolve with time. The following geneal definition of a photon suface is based on only the fist of these popeties. A moe estictive class of photon sufaces may be defined when the space-time admits a goup of symmeties see Definition 2.3).

3 PHOTON SURFACES 3 Definition 2.1. A photon suface of M,g) is an immesed hypesuface S of M,g) such that, fo evey point p S and evey null vecto k T p S, thee exists a null geodesic γ : ɛ, ɛ) M of M,g) such that γ0) =k, γ S. A photon suface is nowhee spacelike since no spacelike hypesuface can contain any null geodesic that extends beyond a single point. Any null hypesuface is tivially a photon suface. Photon sufaces ae confomally invaiant stuctues. If S is a photon suface of M,g) thens is a photon suface of M,Ω 2 g) fo any smooth function Ω : M 0, ). Note that Definition 2.1 is entiely local. In paticula, a photon suface S need contain no endless null geodesics of M,g). Moeove, a photon suface need only be immesed, athe than embedded in M, and so may have self-intesections. If M,g) is of dimension n +1 n 2) then, though each point p of a photon suface S in M,g), thee is an n 2)-paamete family of null geodesics of M,g) that lie entiely in S. The pape will be pincipally concened with photon sufaces in space-times of dimensions. The exceptions ae Examples 1 and 3 which give photon sufaces in space-times of dimension and espectively. Example 1 Minkowski 3-space). In Minkowski 3-space M 3, conside the singlesheeted hypeboloid S given by t 2 + x 2 + y 2 = a 2 4) fo some constant a>0. This suface is doubly uled, the ulings being given by γ ± θ t) :=a 0, cos θ, sin θ)+at 1, sin θ, ± cos θ) 5) <t<, 0 θ<2π), whee θ identifies the intesection points with {t =0} and t is the paamete along the uling lines. The tangents γ ± θ t) to the uling lines ae null with espect to the M 3 metic. Clealy they ae geodesics in M 3.Ateach point of S thee can be just two null diections tangent to S. These must theefoe be the diections of the two uling lines though that point. Hence S is a photon suface in the sense of Definition 2.1 see Fig. 1). Note that fo any cicle of the fom C = {t 0,x 0 + cos θ, y 0 + sin θ}, > 0, 6) and any futue-diected timelike vecto field X along C that espects the symmety of C, in the sense of X = X t,x cos θ, X sin θ ), 7) fo constant X t > 0, X such that X t ) 2 > X ) 2, thee is a unique single-sheeted hypeboloid S though C such that X is tangent to S along C. In the case a = 0, equation 4) gives the null cone though the oigin. The complement of p in this null cone is a null photon suface of M 3. Example 2 Minkowski 4-space). One may genealize Example 1 to Minkowski 4- space M 4 as follows. Let S be a timelike hypesuface in M 4 of the fom t 2 + x 2 + y 2 + z 2 = a 2 8) fo some constant a>0. The two-paamete 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) 9)

4 PHOTON SURFACES 4 Figue 1. A photon suface in Minkowski 3-space. The two families of uling lines ae null geodesics with espect to both the Minkowski 3-metic and the induced 2-metic. The lines may be egaded as the space-time paths of pulsed lase beams. foliate S and ae null geodesics with espect to the M 4 metic. Fo each p S, the tangents at p to those γ ± θ,φ t) that pass though p can be shown to geneate the null cone of T p S. Hence S is a photon suface in the sense of Definition 2.1. In tems of the double null coodinates u := t + 10) v := t 11) fo := x 2 + y 2 + z 2 ) 1/2, equation 8) assumes the simple fom uv = a 2. 12) In the futue diection S tends asymptotically to the null hypesuface {v = 0}, whilst in the past diection S tends asymptotically to the null hypesuface {u =0}. Example 3 De Sitte space). De Sitte space-time may be egaded [5] as a singlesheeted hypeboloid in Minkowski 5-space M 5. By analogy with Examples 1 and 2, de Sitte space-time is thus ealized as a photon suface in M 5. Example 4 The Robetson-Walke models). Since all Robetson-Walke models ae confomally flat and theefoe locally confomally tansfomable to Minkowski space, the photon sufaces of any such model may thus be obtained, at least locally, by confomal tansfomations of Minkowski space.

5 PHOTON SURFACES 5 Theoem 2.2. Let S be a timelike hypesuface of M,g). Letn be a unit nomal field to S and let h ab be the induced metic on S. Letχ ab be the second fundamental fom on S and let σ ab be the tace-fee pat of χ ab. Then the following ae equivalent: i) S is a photon suface; ii) χ ab k a k b =0 null k T p S p S; iii) σ ab =0; iv) evey affine null geodesic of S, h) is an affine null geodesic of M,g). Poof. i) ii). Suppose S is a photon suface. Let p S and let k T p S be null. Thee exists an affine null geodesic γ : ɛ, ɛ) M of M,g) such that γ 0) = k, γ S. One has χ ab γ a γ b = n a;b γ a γ b =n a γ a ) ;b γ b = 0 13) along γ. Atp this gives χ ab k a k b =0. ii) iii). Let p S. By ii) one has σ ab k a k b = χ ab k a k b =0 null k T p S. Let {e 0), e 1), e 2) } be an othonomal basis fo T p S with e 0) timelike and e 1), e 2) spacelike. Any null k T p S, nomalized such that gk, e 0) )= 1, has components k a = 1, cos ψ, sin ψ) with espect to {e 0), e 1), e 2) } fo some ψ [0, 2π). A calculation gives σ ab k a k b =σ σ σ 22)+2σ 01 cos ψ +2σ 02 sin ψ σ 11 σ 22 )cos2ψ + σ 12 sin 2ψ. 14) This must vanish fo all ψ [0, 2π). One thus has σ 01 = σ 02 = σ 12 =0and σ 00 = σ 11 = σ 22. Since σ ab is tace-fee one must also have σ 00 = σ 11 + σ 22. Thee follows σ ab =0. iii) iv). Fo any cuve in S with null tangent k one has k a bk b = h a ck c ;bk b = k a ;bk b +σ bc k b k c )n a 15) whee denotes covaiant diffeentiation in S with espect to h. The second tem on the ight of 15) vanishes by hypothesis. If k is tangent to an affine null geodesic of S, h) then the tem on the left of 15) also vanishes and so k is tangent to an affine null geodesic of M,g). iv) i). Let p S and let k T p S be null. Let γ : ɛ, ɛ) S be an affine null geodesic of S, h) such that γ0) = k. Then, by iv), γ is an affine null geodesic of M,g) such that γ0) = k, γ S. Condition iii) of Theoem 2.2 is equivalent to a equiement that χ ab is pue tace in the sense of χ ab = 1 3 Θh ab, 16) whee Θ := h cd χ cd is the expansion of the unit nomal to S. Fo Example 1 one has Θ = 2/a; fo Example 2 one has Θ = 3/a. Note that, by a standad abuse of notation, h ab denotes both the induced metic on S and the symmetic tenso field of ank 0, 2) along S in M which satisfies h ab n b = 0 and pulls back to the induced metic on S.) It is clea fom condition iii) of Theoem 2.2 that a space-time must be specialized in some espect in ode to admit any timelike photon sufaces in the sense of Definition 2.1. Fo this eason it is helpful to estict attention to space-times which admit goups of symmeties.

6 PHOTON SURFACES 6 Definition 2.3. Suppose M,g) admits a goup G of isometies. A photon suface S of M,g) that is invaiant unde G, in the sense that each g G maps S onto itself, will be called a G-invaiant photon suface. Clealy any G-invaiant null hypesuface is a G-invaiant photon suface. In paticula, if G = R o G = S, then any Killing hoizon [6, 7] is a G-invaiant photon suface. 3. Dynamic Spheical symmety: Geneal theoy By definition, a geneal spheically symmetic space-time admits an SO3) isomety goup fo which the goup obits ae spacelike 2-sphees. The following esult descibes the evolution of the coss-sectional aea of an SO3)-invaiant photon suface in a spheically symmetic space-time. Theoem 3.1. Let M,g) be a spheically symmetic space-time. Let S be an SO3)-invaiant timelike hypesuface of M,g) and let X be the SO3)-invaiant unit futue-diected timelike tangent vecto field along S othogonal to the SO3)- invaiant 2-sphees in S. Let beonesuchso3)-invaiant 2-sphee in S and let s be the SO3)-invaiant 2-sphee in S at ac-length s fom along the integal cuves of X. Then S is a photon suface of M,g) iff the aea 2) A s of s satisfies d 2 2) ds 2 A s = 1 ) 2 d 2) A 4 2) s + 2) A 1 s A s ds 3 Θ2 G ab n a n b) 4π 17) whee n a istheunitnomaltos, Θ is the expansion of n a and G ab := R ab 1 2 Rg ab is the Einstein tenso of M,g). Poof. Let h ab be the induced Loentzian 3-metic on S and, fo each s, let 2) h ab be the induced Riemannian 2-metic on s. The expansion of X in S, h) isgivenby 2) Θ= 2) h ab X a;b whee the covaiant deivative is that of M,g). Since X is both shea-fee and voticity-fee in S, h), the Raychaudhui equation fo X in S, h) assumes the fom d 2) Θ= 1 ds 2 2) Θ) 2 3) R ab X a X b 18) whee 3) R ab is the Ricci tenso of S, h). Fom fist pinciples one has 2) R = 3) R +2 3) R ab X a X b 2) χ a a) 2 + 2) χ a b 2) χ b a 19) 3) R = R 2R ab n a n b +χ a a) 2 χ a bχ b a 20) whee 2) χ ab is the second fundamental fom of each s in S, h). Since X a is shea-fee and voticity fee in S, h) one has 2) χ ab = 1 2 Θ 2) h ab. The second fundamental fom of S admits the canonical decomposition χ ab = 1 3 Θh ab + σ ab. Equations 19) and 20) theefoe give 2) R = 3) R +2 3) R ab X a X b 1 2 Θ 2 21) 3) R = R 2R ab n a n b Θ2 σ a bσ b a 22) which combine to yield 2 3) R ab X a X b = 2) R +2G ab n a n b 2 3 Θ ) Θ) 2 + σ a bσ b a. 23)

7 PHOTON SURFACES 7 One may now substitute fo the second tem on the ight of 18) to obtain d 2) Θ= 3 ds 4 2) Θ) Θ2 1 2) 2 R G ab n a n b 1 2 σa bσ b a. 24) Fom fist pinciples one has 2) Θ= d ds ln 2) A, and the Gauss-Bonnet theoem gives 2) R 2) A =8π. Substituting fo 2) Θand 2) R in 24) one obtains d 2 2) ds 2 A s = 1 ) 2 d 2) A 4 2) s + 2) A 1 s A s ds 3 Θ2 G ab n a n b 1 2 σa bσ b ) a 4π. 25) This agees with 17) iff σ a bσ b a =0. Constuct, fo the tangent bundle TS of S, an othonomal basis field of the fom {X, e 1), e 2) },withe 1) and e 2) unit spacelike. With espect to this basis one has σ a bσ b a =σ 0 0) 2 +σ 1 1) 2 +σ 2 2) 2 +2σ 1 2) 2 2σ 1 0) 2 2σ 2 0) 2. 26) By spheical symmety the vecto field σ a bx b must be popotional to X a. Hence one has σ 1 0 = σ 2 0 = 0. The vanishing of σ a bσ b a is thus equivalent to the vanishing of σ ab. One has σ ab =0iffS is a photon suface. A spheically symmetic metic is locally expessible in the fom g 00 g g ab = g 10 g g θθ 0 27) g θθ sin 2 θ with espect to coodinates x 0,x 1,θ,φ ) adapted to the spheical symmety, whee 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 intoduce a adial coodinate, depending only on x 0 and x 1, such that g θθ is a function of only. One is fee to specify g θθ as a function of alone since to do so is, in effect, a definition of the coodinate. This will be assumed to be done thoughout this pape. The following esult is useful in the locating of SO3)-invaiant photon sufaces in dynamic spheically symmetic space-times. Lemma 3.2. Let M,g) be a spheically symmetic space-time. Let S be an SO3)- invaiant timelike hypesuface of M,g) and let X be the SO3)-invaiant unit futue-diected timelike tangent vecto field along S that is othogonal to the SO3)- invaiant 2-sphees in S. Then S is a photon suface of M,g) iff X a ;bx b = 1 2 gθθ n b b g θθ )n a 28) holds along S, wheen a is the unit nomal field to S in M,g). Poof. By spheical symmety, and since X is unit timelike, the vecto field X X must be popotional to n. Hence it suffices to show that S is a photon suface iff along S one has n X X = 1 2 gθθ n b b g θθ 29) o equivalently χ ab X a X b = 1 2 gθθ n a a g θθ. 30)

8 PHOTON SURFACES 8 Constuct fo TS a local othonomal basis field of the fom {X, e θ), e φ) }.With espect to this basis field the components of χ a b fom a diagonal matix with χ θ θ = χ φ φ = n θ ;θ = 1 2 gθθ n a a g θθ. 31) Equation 30) is thus equivalent to χ 0 0 = χ θ θ = χ φ φ which is in tun equivalent to σ 0 0 = σ θ θ = σ φ φ. In view of the tace-fee popety of σ a b, equation 30) is thus equivalent to σ a b = 0. Fom Theoem 2.2 one has that σ a b = 0 holds along S iff S is a photon suface. Let us continue to wok with espect to the coodinate system {x 0,x 1,θ,φ} employed in 27). Let x a s) be an integal cuve of the vecto field X in Lemma 3.2. One has dx a ds = Xa 32) and equation 28) becomes Since X = dx0 ds 1, dx1 dx 0, 0, 0 d 2 x a dx b ds 2 +Γa bc ds ) is unit timelike one has dx c ds = 1 2 gθθ n b b g θθ )n a. 33) ) 2 ds dx a dx b dx 0 = g ab dx 0 dx 0. 34) The x 0 and x 1 components of equation 33) thus combine to give d 2 x 1 ) dx 0 ) 2 = 1 2 gθθ n a a g θθ n 1 dx1 dx b dx c ) dx 1 dx dx 0 n0 g bc dx 0 dx 0 + dx 0 Γ0 ab Γ 1 a dx b ab dx 0 dx 0 35) whee the components of n ae given by n 0 dx a = ψg 1a dx 0 ; dx a n1 = ψ g 0a dx 0 36) fo ψ := ) 1 dx 2 g a dx b ) 1 2 ab dx 0 dx 0 37) whee :=g 00 g 11 g 01 ) 2 38) is the deteminant of the time-space pat of g ab in 27). Equation 35) is the coodinate equivalent of 28) and povides fo the easy detemination of SO3)- invaiant photon sufaces see Example 10). 4. Static Spheical Symmety: Geneal Theoy By definition, a spheically symmetic space-time is static if it admits an SO3) R goup of isometies such that the R obits ae geneated by a Killing field K which is both hypesuface othogonal and othogonal to the SO3) obits. The pesent section will be concened with SO3) R-invaiant photon sufaces in static spheically symmetic space-times. Such sufaces may be temed photon sphees because, as will be seen, they ae a natual genealization to geneal static spheically symmetic space-times of the Schwazschild photon sphee concept. The tem photon

9 PHOTON SURFACES 9 sphee will be egaded as applicable only in static spheically symmetic spacetimes. Fo claity, the tem SO3) R-invaiant photon suface will usually be employed in pefeence to photon sphee. Although the space-times of Examples 1 and 2 ae static and spheically symmetic, the photon sufaces in these space-times ae not SO3) R-invaiant and so ae not photon sphees. Of the Robetson-Walke space-times of Example 4, only the Einstein cylinde is both spheically symmetic and static. None of the photon sufaces of the Einstein cylinde ae SO3) R-invaiant. Thus the Einstein cylinde has no photon sphees. One may chaacteize an SO3) R-invaiant photon suface, o photon sphee, in a static spheically symmetic space-time by means of the following special case of Theoem 3.1. Theoem 4.1. Let M,g) be a static spheically symmetic space-time with Killing field K and let S be an SO3) R-invaiant timelike hypesuface of M,g). Then S is an SO3) R-invaiant photon suface of M,g) if thee exists an SO3)- invaiant 2-sphee S satisfying A Θ 2 3G ab n a n b) =12π 39) whee A is the aea of, n a istheunitnomaltos and Θ is the tace of the second fundamental fom of S. Convesely, if S is an SO3) R-invaiant photon suface of M,g) then 39) holds fo evey SO3)-invaiant 2-sphee S. Poof. Note that the unit futue-diected timelike tangent field X along S in Theoem 3.1 is popotional to the estiction to S of the Killing field K. Suppose fist that thee exists an SO3)-invaiant 2-sphee S such that 39) holds. The quantities A, ΘandG ab n a n b emain constant as is mapped along the flow lines of the Killing field K. So they also emain constant as they ae mapped along the flow lines of X. Hence 17) holds with the tem on the left and the fist tem on the ight both zeo. Thus, by Theoem 3.1, S is a photon suface of M,g). By hypothesis S is SO3) R-invaiant. Fo the convese, suppose that S is an SO3) R-invaiant photon suface of M,g). Then 17) holds fo evey SO3)-invaiant 2-sphee s S. Since K induces goups of local isometies, the aea A s of s is independent of the paamete s. Hence the tem on the left and the fist tem on the ight of 17) both vanish and one obtains 39). Coollay. If one has G ab Y a Y b 0 vectos Y and S is an SO3) R-invaiant timelike photon suface of M,g), then fo any SO3)-invaiant 2-sphee S one has AΘ 2 12π 40) with equality holding iff G ab n a n b =0along S. Fo Schwazschild space-time whee see Example 5) the only timelike photon sphee is at =3m, one has A =4π3m) 2,Θ=1/ 3m) andg ab =0which veifies 39) and 40) fo this case. If the Einstein equations hold with a zeo cosmological constant then, in the coollay to Theoem 4.1, the hypothesis G ab Y a Y b 0 fo all vectos Y is equivalent to T ab Y a Y b 0 fo all vectos Y. This is a physically easonable enegy condition. In paticula, fo a pefect fluid with density ρ and pessue p, itis

10 PHOTON SURFACES 10 equivalent to a condition that ρ and p ae both non-negative. Moe geneally, the condition holds fo an enegy tenso with a single timelike eigenvecto type I in the classification of Hawking & Ellis [5]) iff each enegy tenso eigenvalue is non-negative. The chaacteization of timelike photon sufaces povided by Theoem 4.1 involves deivatives of the metic components up to second ode. The following esult Theoem 4.2) povides an entiely diffeent chaacteization of SO3) R- invaiant photon sufaces in tems of deivatives of the metic components up to only fist ode. Let a geneal static spheically symmetic metic g be expessed in the fom 27) with g θθ a function of only. The Killing equation K a;b) = 0 and the othogonality of K to θ and φ gives K a a g θθ = 0 and hence K = 0 whee is to be egaded as a scala field on M. Since is independent of θ and φ it follows that any SO3) R-invaiant hypesuface S of M,g) mustbeofthefom{ = const.}. If S is also a timelike hypesuface then ;a is a spacelike vecto field along S and K is a timelike vecto field in a neighbouhood of S. If the coodinates x 0 and x 1 implicit in 27) ae chosen such that K = x 0 then the Killing equation gives that all the metic components g ab in 27) ae independent of x 0. Since is constant along the integal cuves of K, the coodinate x 1 must be a function of only. A natual choice is x 1 =. One may edefine x 0 accoding to x 0 x 0 g 01 /g 00 )dx 1. This diagonalizes the time-space pat of g ab and leaves the components of g ab independent of x 0. Futhemoe the cuves {x 1,θ,φ = const.} ae unchanged except that they ae e-paametized. The vecto field x 0 then becomes a confomal Killing field. Define the tenso field ɛ ab := ) 1/ ) on M, whee the components ae given with espect to the coodinate basis employed in 27) and is the deteminant of the time-space pat of g in 27), as in 38). Theoem 4.2. Let M,g) be a static spheically symmetic space-time with g of the fom 27), with g θθ a function of the coodinate only. Let S be an SO3) R- invaiant timelike hypesuface of M,g) and suppose that is nowhee-zeo along S. Then S is an SO3) R-invaiant photon suface of M,g) iff 2g θθ ɛ ab ɛ cd ;ac ;b ;d + ;a ;a ;c c g θθ = 0 42) holds along S. Poof. Since M,g) is both spheically symmetic and static, the suface S is of the fom { = const.}. The unit spacelike nomal to S is theefoe given by n a = η ;a 43) fo η := ;a ;a ) 1/2. 44) The second fundamental fom of S is given by χ ab := ηh c a h d b ;cd. 45)

11 PHOTON SURFACES 11 The vecto fields X a := g bc ;b ;c ) 1/2 ;1, ;0, 0, 0) 46) e θ) := g θθ ) 1/2 θ 47) e φ) := g θθ sin 2 θ) 1/2 φ 48) fom an othonomal fame field along S, withe θ) and e φ) unit spacelike and X unit timelike. One has χ ab X a e b θ) = χ abx a e b φ) = χ abe a θ) eb φ) = 0 49) χ ab e a θ) eb θ) = χ abe a φ) eb φ) = η;a a g θθ 2g θθ 50) χ ab X a X b = η 3 ɛ ab ɛ cd ;ac ;b ;d. 51) Condition iii) of Theoem 2.2 holds iff χ ab is popotional to h ab and hence iff χ ab X a X b = χ ab e a θ) eb θ) = χ abe a φ) eb φ). 52) This is equivalent to η 2 ɛ ab ɛ cd ;ac ;b ;d = ;a a g θθ 53) 2g θθ which is in tun equivalent to 42). Equation 42) can have solutions such that { = const.} is a spacelike hypesuface and theefoe not a photon suface see e.g. Example 7). It is theefoe always necessay in the use of Theoem 4.2 to check that the hypesuface { = const.} is in fact timelike o null. Note that in a egion of space-time whee the Killing field K is spacelike, fo example behind the event hoizon of Schwazschild space-time, the hypesufaces { = const.} ae necessaily spacelike and so cannot be photon sphees. Case 1. x 1 :=, components of g ab independent of x 0.) As discussed peviously, fo a static spheically symmetic metic it is possible to choose x 1 :=, with g θθ depending only upon and with all the components of g ab independent of x 0. In this case 42) educes to g 00 g θθ = g θθ g ) This agees with an equation obtained by Vibhada & Ellis [4] on the basis of a diffeent definition [3] of a photon sphee. Note that even though the components g,g 0 = g 0 do not appea in 54), they ae not assumed to vanish. A paticula sub-case of inteest is that of time-space coodinates, with t := x 0 timelike in the sense of g tt < 0. Anothe sub-case of inteest is that of single null adiation) coodinates, with u := x 0 null in the sense of g uu =0. Case 2. Double null coodinates u, v.) Let x 0 := u, x 1 := v be double null coodinates in the sense of g uu = g vv = 0. The adial coodinate is to be egaded as a function of u and v. Then 42) assumes the fom g θθ { ;uu ;v ) 2 2 ;uv ;u ;v + ;vv ;u ) 2 } +2 ;u ;v ) 2 g θθ =0. 55) The metic components g uv = g vu ente hee though the covaiant deivatives of.

12 PHOTON SURFACES 12 Equations 42), 54) and 55) may be efeed to as photon sphee equations since they give the location of timelike photon sphees in static spheically symmetic space-times. In ode to facilitate futhe pogess, a geneal static spheically symmetic metic will be witten in such a fom as to cast the Einstein tenso in a paticulaly simple and convenient fom. One has fom pevious emaks that a geneal static spheically symmetic metic is locally expessible in the fom g = g tt dt 2 + g d dθ 2 +sin 2 θdφ 2 ) 56) whee g tt and g ae functions of only. Let m) := ) g 57) µ) := ln g tt g ). 58) Then the metic assumes the fom g = 1 2m) ) e µ) dt m) ) 1 d dθ 2 +sin 2 θdφ 2 ) 59) and the Einstein tenso is given by G a b =8π ρ) p 1 ) p 2 ) p 2 ) 60) fo 8πρ) := 2m ) 2 61) 8πp 1 ) := 1 2 { 2m))µ ) 2m )} 62) 8πp 2 ) := 1 { 2 + m) 3m 4 2 ))µ )+ 2m))µ )) 2 4m )+2 2m))µ )}. 63) whee a pime denotes diffeentiation with espect to. Fo the ight sides of 61), 62) and 63) to be defined at some adius ˆ >0 one evidently needs m) andµ) to be twice diffeentiable at =ˆ. Equations 61) and 62) combine to give µ ) = 8πρ)+p 1)) ) 64) 1 2m) wheeby one may ewite 63) in the moe convenient fom ) m) 2 p 2) p 1 )) = p 1 )+ +4πp 2 1 ) ρ)+p 1 )) ). 65) 1 2m)

13 PHOTON SURFACES 13 In the pefect fluid case p 1 ) =p 2 ) =:p) equation 65) educes to the Tolman- Oppenheime-Volkoff equation ) m) p +4πp) ρ)+p)) ) = 2 ). 66) 1 2m) By means of equations 61) and 64), the photon sphee equation 54) becomes 1 3m) 4π 2 p 1 ) =0. 67) Fo such that 2m) <, and hence such that the hypesuface { = const.} is timelike, equation 67) gives the location of the SO3) R-invaiant timelike photon sufaces fo the metic 59). Equation 67) is the basis fo the following esult which shows that, subject to a suitable enegy condition, any black hole in a static spheically symmetic space-time must be suounded by an SO3) R-invaiant photon suface. Fo the pupose of this and subsequent esults, a function f : R I R on an inteval I will be said to be piecewise C if I is the disjoint union of a locally finite collection of intevals I i such that f I i is C. Each inteval I i may be open, closed o half-open. Theoem 4.3. Suppose the metic g has the fom 59) fo 0 <<, fo some 0 > 0, withm) and µ) both C 0, piecewise C 2 functions of 0, ). Suppose the following hold: 1) ρ) and p 1 ) ae bounded functions of 0, ); 2) 2m) < 0, ); 3) ρ) 0, p 1 ) 0 0, ); 4) lim 4π 2 p 1 ) = lim 4π 2 ρ) =0; 5) fo each value of t the 2-sufaces t, := {t =const.} { =const.}, 0 <<, ae such that t := lim 0 t, exists as an embedded spacelike 2-sphee in M,g) and is maginally oute tapped. Then M,g) admits an SO3) R-invaiant timelike photon suface of the fom { = 1 } fo some 1 0, ). Poof. Fix t and let k be the outwad futue-diected null nomal field along each ) 1/2 t,, 0 <<, nomalized such that gk, n) = 1, whee n = 1 2m) is the outwad adial unit tangent to {t = const.}. Since k is paallelly popagated along each of the geodesic integal cuves of n, one has that lim 0 k is a welldefined, nowhee-zeo null vecto field along t := lim 0 T t,. Fo 0, ) the vecto field k has the fom k a =k t,a)k t, 0, 0) fo a) := g tt /g ) 1/2 = 1 2m) ) e µ)/2. 68) The expansion of k is given by Θ out = 2) h b ak a ;b = 2a) k t. 69) The condition that t is maginally oute tapped theefoe implies 0 = lim a) = lim 1 2m) ) e µ)/2. 70) 0 0

14 PHOTON SURFACES 14 The non-negativity of ρ) andp 1 ) gives, by means of 61) and 64), that m) and µ) ae non-deceasing functions of 0, ). Thus 70) holds iff at least one of lim 1 2m) ) = 0 ; lim µ) = 71) 0 0 holds. Suppose the fist of 71) fails. Then the second must hold and one has lim 1 2m) ) 1 <. 72) 0 Fom the boundedness of ρ) andp 1 ) on 0, ) one has, by means of 64) and 72), that lim sup 0 µ ) is finite. This is incompatible with the second of 71). Hence the fist of 71) must hold. Let f : 0, ) R be the left side of 67). By the non-negativity of p 1 ) and the fist of 71) one has lim 0 f) 1 2. By condition 4), equation 61) and l Hôpital s ule one has lim m)/ = lim 2 p 1 ) = 0 and hence lim f) = 1. Hence thee exists some 1 0, ) such that f 1 ) = 0. The hypesuface { = 1 } is an SO3) R-invaiant photon suface of M,g). Condition 3) of Theoem 4.3 may be expessed moe succinctly as G ab Y a Y b 0 vectos Y. With egad to condition 5) of Theoem 4.3, to have equied T t to be contained in the hypesuface {t = const.} would have been too stong since, fo Schwazschild space-time, no spacelike hypesuface of the fom {t = const.} in the exteio egion contains a maginally oute tapped 2-suface. Theoem 4.3 may be intepeted to the effect that, subject to the enegy conditions expessed in condition 3), any static spheically symmetic black hole must be suounded by an SO3) R-invaiant timelike photon suface. The following esult may then be egaded as a patial convese in that it shows, subject to a suitable enegy condition, that if thee exists an SO3) R-invaiant timelike photon suface then thee must be a naked singulaity o a black hole, o moe than a cetain amount of matte. Poposition 4.4. Suppose the metic g has the fom 59) fo 0 <<, with m) and µ) both C 0, piecewise C 2 functions of 0, ). If the following all hold: 1) ρ) is a non-inceasing, bounded function of 0, ); 2) lim 0 m) =0; 3) 4m) < 0, ); 4) p 1 ) ρ)/3 0, ), then M,g) can contain no SO3) R-invaiant timelike photon sufaces. Poof. By conditions 1) and 2) with equation 61) one has m) 4π/3) 3 ρ) >0. By condition 4) one theefoe has 4π 2 p 1 ) 4π/3) 2 ρ) m)/ >0. The left side of 67) is thus bounded fom below by 1 4m)/ >0. This is positive by condition 3). The left side of 67) is theefoe non-vanishing fo all >0. Note that this esult is valid even fo negative p 1 ) andρ). Condition 2) of Poposition 4.4 pohibits any cuvatue singulaity at = 0. Condition 3) may

15 PHOTON SURFACES 15 be intepeted as a equiement that thee is no black hole and less than a cetain amount of matte. The esult shows that one of these two conditions must fail if thee is an SO3) R-invaiant timelike photon suface and conditions 1) and 4) both hold. When the matte is a pefect fluid it is possible to impove condition 3) of Poposition 4.4 to condition 3) of the following esult. Theoem 4.5. Suppose the metic g has the fom 59) fo 0 <<, withm) and µ) both C 1, piecewise C 2 functions of 0, ). If the following all hold: 1) the matte is a pefect fluid with pessue p) and density ρ); 2) lim 4π 2 p) = lim 4π 2 ρ) =0; 3) 24/7)m) < 0, ); 4) p) ρ)/3 0, ), then M,g) can contain no SO3) R-invaiant timelike photon sufaces. Poof. Let f :0, ) R be the left side of equation 67). Since m) andµ) ae C 1, piecewise C 2 functions of 0, ), one has by equation 62) that f) is a C 0, piecewise C 1 function of 0, ). The function f ) is then a piecewise C 0 function of 0, ) which, by means of of the Tolman-Oppenheime-Volkoff equation 66), is given by f ) =3 m) Fo = 1 0, ) such that 12π 2 ρ) 8π 2 p)+ m) ) +4π 2 p) 1 2 m) ) 4π 2 ρ)+p)). 0=f 1 )=1 3 m 1) 4π1 2 p 1) 74) 1 equation 73) educes to 1 f 1 )=1 8π1ρ 2 1 )+p 1 )). 75) Fom condition 3) and equation 74) one has 4π 2 1p 1 ) > 1/8, whence by condition 4) one has 4π 2 1 ρ 1) > 3/8. Thus 75) gives f 1 ) < 0. By condition 2), equation 61) and l Hôpital s ule one has lim m)/ = lim 4π 2 p) = 0 and hence lim f) = 1. Since it has been established that f ) is negative fo all 0, ) such that f) = 0, one must theefoe have f) > 0 fo all 0, ). Hence the space-time can contain no SO3) R- invaiant timelike photon sufaces. Note that, as fo Poposition 4.4, Theoem 4.5 is valid even fo negative pessue and density. One would like to emove the insufficient matte pats of condition 3) of Poposition 4.4 and condition 3) of Theoem 4.5, in othe wods to weaken these to a no-black-hole condition 2m) < >0. But no esult to this effect is fothcoming. On the othe hand no counteexample is known. To conclude this section it will be shown that the physical significance of the photon sphee in Schwazschild space-time, as discussed in the Intoduction, caies ove to the geneal static spheically symmetic case. Suppose the metic has the fom 59) fo 0 << and is asymptotically flat in the limit. Assume 73)

16 PHOTON SURFACES 16 p 1 ) 0, m) 0 > 0. The matte need not be a pefect fluid. Denote the left side of 67) by f). The condition of asymptotic flatness gives lim f) =1 so, if thee ae any SO3) R-invaiant timelike photon sufaces, thee will be an outemost such suface S. Fo simplicity assume f ) 0atS. Let R ext be the connected component of {q M : fq) > 0} that has S as its inne bounday and extends to =. Let int be the connected component of {q M : fq) < 0} that has S as its oute bounday. Conside fist the case of a futue endless affine null geodesic γλ). The null geodesic equations fo the metic 59) give d 2 dλ 2 = f) { dθ dλ ) 2 +sin 2 θ ) } 2 dφ µ ) dλ 2 ) 2 d. 76) dλ At any point p γ ext such that d/dλ = 0 one has d 2 /dλ 2 > 0. At any point p γ int such that d/dλ = 0 one has d 2 /dλ 2 < 0. Thus if γ stats outside S i.e. in ext ) and is initially diected outwads, in the sense that d/dλ is initially positive, then γ will continue outwads. If γ stats in int and is initially diected inwads, in the sense that d/dλ is initially negative, then γ will continue inwads until it falls eithe into a singulaity o though an SO3) R-invaiant photon suface othe than S. Conside now a unit speed timelike geodesic ξs). The timelike geodesic equations give d 2 ds 2 = m) 2 4πp 1 )+f) { dθ ) 2 +sin 2 θ ds ) } 2 dφ µ ) ds 2 ) 2 d. ds 77) Fo any point p ext one can aange fo the fist thee tems on the ight of 77) to cancel and so obtain a unit speed timelike geodesic ξs) though p R ext at constant. Fo p R int the fist thee tems on the ight of 77) ae evidently negative. Fo p ξ int such that d/ds = 0 one has d 2 /ds 2 < 0. Thus if ξ stats in R int and is initially diected inwads, in the sense that d/ds is initially negative, then ξ will continue inwads until it falls eithe into a singulaity o though an SO3) R-invaiant photon suface othe than S. 5. Spheical symmety: Examples The following ae some examples of SO3) R-invaiant and SO3)-invaiant photon sufaces in familia space-times. Example 5 Schwazschild space-time). The metic of Schwazschild space-time in single null adiation) coodinates has the fom g = 1 2m ) du 2 +2dud + 2 dθ 2 +sin 2 θdφ 2). 78) In this case equation 54) educes to =3m. The timelike hypesuface { =3m} is thus an SO3) R-invaiant photon suface, o photon sphee, as expected. Thee ae no othe SO3) R-invaiant timelike photon sufaces. Fo a non-zeo cosmological constant Λ, the Schwazschild metic 78) genealizes to the Schwazschild-de Sitte metic

17 PHOTON SURFACES 17 g = 1 ) 2m Λ2 + du 2 +2dud + 2 dθ 2 +sin 2 θdφ 2). 79) 3 One finds that equation 54) educes to =3m, independent of the value of Λ. This is supising since Schwazschild and Schwazschild-de Sitte space-times ae not confomally elated. Example 6 Schwazschild inteio solution). The Schwazschild inteio solution descibes a spheically symmetic distibution of pefect fluid of adius R, bounded pessue p and constant density ρ 0 > 0. The solution is to have a metic of the fom 59) and is to be matched at = R to a Schwazschild vacuum solution in such a way that the pessue is a continuous function of. One thus has p 1 ) =p 2 ) =:p) fo all :0 <, ρ) =ρ 0 fo all :0 R, ρ) = 0 fo all : R<< and p) = 0 fo all : R <. The pessue p) fo :0 R is to be obtained by an integation of the Tolman-Oppenheime-Volkoff equation 66) subject to the bounday condition pr) = 0. This yields m) = 4π 3 ρ 0 3, 0 R, 80) e µ) = 3 u))2 4u 2 ), 0 R, 81) p) = u) 1 3 u) ρ 0, 0 R, 82) fo 3 8πρ0 2 ) 1 2 u) :=, 0 R. 83) 3 8πρ 0 R 2 The spheically symmetic system descibed by the Schwazschild inteio solution can exist in a state of stable equilibium iff mr)/r < 4/9 see in Stephani [8]). This condition is equivalent to 8πρ 0 R 2 < 8/3, which implies p) 0 0, and implies that the absence of a black hole is a geneal featue of the Schwazschild inteio solution. The left side of equation 67) now assumes the fom 1 3m) 1 8πρ 0 2 :0< R, 4π 2 p) = 3 u) 1 3mR) 84) : R. Fo mr)/r < 1/3 one has that 84) is positive fo all >0, so thee ae no timelike photon sphees. Fo mr)/r = 1/3 thee is a single timelike photon sphee which lies at the bounday = R of the matte. Fo 1/3 <mr)/r < 4/9 thee is one timelike photon sphee outside the matte at = 3mR) > Rand one timelike photon sphee inside the matte at 1 3πρ0 R 2 ) 1/2 = πρ 0 3 8πρ 0 R 2 = 2R ) 3 1 9mR) 1 2mR) R 4R ) mr) R 1/2 <R. 85)

18 PHOTON SURFACES 18 Fo fixed R the adius of the oute photon sphee is a stictly inceasing function of mr)/r whilst the adius of the inne photon sphee is a stictly deceasing function of mr)/r. Thus a Schwazschild inteio solution matched to a Schwazschild vacuum exteio solution contains no black hole, and contains one timelike photon sphee iff 1/3 = mr)/r and two timelike photon sphees iff mr)/r lies in the ange 1/3 <mr)/r < 4/9. Howeve fo such values of mr)/r the space-time is unphysical in that the pessue at the cente is p0) ρ 0 / 3 >ρ 0 /3. Theefoe unde the easonable enegy condition p) ρ 0 /3 :0 < thee ae no photon sphees in this example. The enegy condition 0 p) ρ)/3 [0, ) isinfactsatisfiediff mr)/r lies in the ange 0 mr)/r 5/18. The coesponding space-times in view of 5/18 < 7/24) satisfy all of conditions 1) to 4) of Theoem 4.5, except the smoothness of ρ) at = R. Thus consideing a smoothed family of solutions appoximating the solution above and having it as a stict limit, but each with smooth ρ) at = R, we can apply Theoem 4.5 to show no photon sphees exist in all the cases discussed in this section whee this enegy condition is satisfied. On the othe hand Poposition 4.4 is diectly applicable without smoothing, but only applies to the subset of these cases with 0 mr)/r < 1/4. Example 7 Reissne-Nodstöm space-time). The geneal static spheically symmetic solution to the Einstein-Maxwell equations is the Reissne-Nodstöm solution compising a metic g and an electomagnetic field F ab given by ) 2m e2 g = du 2 +2dud + 2 dθ 2 +sin 2 θdφ 2) 86) F t = F t = e, all othe components vanishing, 87) 2 whee m is the ADM mass and e is the electic chage. We assume m>0. Thee is an event hoizon at = + := m + m 2 e 2 and a Cauchy hoizon at = := m m 2 e 2. The event hoizon exists fo 0 e/m) 2 1, and the Cauchy hoizon exists fo 0 < e/m) 2 1. Fo e/m) 2 = 1 they both lie at = m. Thee can be no timelike photon sphees between the event hoizon and the Cauchy hoizon because the Killing field K is spacelike thee. Outside the event hoizon the Killing field K is timelike, so evey hypesuface of the fom { = const.} which lies outside the event hoizon and satisfies the photon sphee equation 54) is necessaily a timelike photon sphee. Equation 54) assumes the fom 2 3m +2e 2 = 0 88) which has solutions ps ± given by ps ± /m = 3± 9 8e/m) 2. 89) 2 The hypesuface S + := { = ps + } exists fo 0 e/m)2 9/8 and lies outside the event hoizon and is theefoe a timelike photon sphee. The hypesuface S := { = ps} exists fo 0 < e/m) 2 9/8 but lies outside the event hoizon only fo 1 < e/m) 2 9/8, and so is a timelike photon sphee only then. The hypesufaces S + and S coincide fo e/m) 2 =9/8. The Cauchy hoizon and event hoizon ae always null photon sphees.

19 PHOTON SURFACES 19 =m 3:0 oute photon sphee 2:0 event hoizon inne photon sphee 1:0 Cauchy hoizon 1: e=m Figue 2. The adii of the event hoizon, Cauchy hoizon and photon sphees fo Reissne-Nodstöm space-time. Fo 0 e/m) 2 1 the cuvatue singulaity at = 0 is locally naked but hidden behind an event hoizon which lies stictly inside the only timelike photon sphee. Fo 1 < e/m) 2 the singulaity at = 0 is globally naked and is suounded by two timelike photon sphees in the case 1 < e/m) 2 < 9/8, one timelike photon sphee in the case e/m) 2 =9/8 and by no photon sphees, eithe timelike o null, in the case e/m) 2 > 9/8 seefig.2). Example 8 Janis-Newman-Winicou space-time). The most geneal static spheically symmetic solution to the Einstein massless scala field equations fo a scala field Φ satisfying Φ = 0 was obtained by Janis, Newman and Winicou [9]. The Ricci tenso has the fom R ab =8πΦ ;a Φ ;b. The solution is known [10] to be expessible in the fom g = Φ= q b 4π ln 1 b ) ν dt b ) 1 b ) ν d b ) 1 ν 2 dθ 2 +sin 2 θdφ 2) 90) fo : b<< whee the constants b, ν ae elated to the ADM mass m and scala chage q by 91) ν = 2m b, b =2 m 2 + q 2. 92)

20 PHOTON SURFACES 20 We assume b>0. Thee is a cuvatue singulaity at = b. In ode to obtain m 0 one must assume 0 ν 1. Fo q = 0 the solution educes to the Schwazschild solution. Since all hypesufaces of the fom { = const.} ae timelike, one has fom the photon sphee equation 54) that the only timelike photon sphee is at b 2ν +1) =, 93) 2 which exists only fo ν : 1 2 <ν 1, i.e. fo 0 q2 < 3m 2. Fo 1 2 <ν 1itis known [11] that a photon coming fom infinity is deflected though an unboundedly lage angle, i.e. the photon passes inceasingly many times aound the singulaity as the closest distance of appoach tends to the ight side of 93). Example 9 Chaged dilaton space-time). A static spheically symmetic spacetime with a chaged dilaton field was obtained by Hone & Hoowitz [12]. It compises a metic g, a dilaton field Φ and an electomagnetic field F ab given by g = 1 ) + 1 ) ω dt ) ) ω d ) 1 ω 2 dω 2 94) e 2Φ = 1 ) 1 ω)/β 95) F t = F t = e, all othe components vanishing, 96) 2 whee + and ae elated to the ADM mass m and electic chage e by + + ω =2m 97) + = e β 2 ) 98) and β is a fee paamete which contols the coupling stength between the dilaton and Maxwell fields, with ω defined in tems of β 2 by ω := 1 β2 1+β 2. 99) We assume m>0. Fo β = 0 the solution educes to the Reissne-Nodstöm solution consideed in Example 7. Fo β =0ande = 0 the solution educes to the Schwazschild solution. The solution also educes to the Schwazschild solution fo e = =0 and abitay β. Hee we shall conside the case β 2 = 1. In this case one has +, )=2m, e 2 /m). Thee is an event hoizon at = + =2m and a cuvatue singulaity at = = e 2 /m. Fo 0 e/m) 2 < 2 the singulaity at = lies inside a black hole whilst fo e/m) 2 > 2 it is globally naked. Fo β 2 = 1 the photon sphee equation 54) educes to ps ± /m = 6 + e/m)2 )± 36 + e/m) 4 20e/m) ) 4 Fo 0 e/m) 2 < 2 one has ps < + < ps + so thee is a single timelike photon sphee. Fo e/m) 2 = 2 one has ps = = + = ps + so thee ae no timelike photon sphees. Fo 2 < e/m) 2 < 18 both ps + and ps ae complex so thee ae no timelike photon sphees. Fo e/m) 2 18 one has ps ps + < so thee ae again no timelike photon sphees. Thus in the black hole case 0 e/m) 2 < 2 thee is a

21 PHOTON SURFACES 21 3:0 photon sphee 2:5 =m 2:0 1:5 event hoizon 1:0 0:5 cuvatue singulaity 0 0 0:2 0:4 0:6 0:8 1:2 1:4 e=m 1:0 Figue 3. The adii of the photon sphee, event hoizon and cuvatue singulaity ae plotted against the electic chage fo the chaged dilaton solution. single timelike photon sphee, whilst in the naked singulaity case e/m) 2 > 2thee ae no timelike photon sphees. See Fig. 3.) Example 10 Vaidya null dust collapse). The Vaidya null dust collapse model is a non-static, spheically symmetic space-time with a metic which, in tems of single null adiation) coodinates u,, θ, φ), assumes the fom g = 1 2mu) ) du 2 +2dud + 2 dθ 2 +sin 2 θdφ 2 ) 101) whee mu) is a feely specifiable function of u. Setting x 0 := u, x 1 := in 35) one obtains d 2 du 2 = 1 g uu + d ) g uu +2 d ) 3 d du du 2 du g uu 1 2 g uu g uu + u g uu ) 102) { = 1 1 3mu) ) 1 2mu) 3 d ) dmu) ) } 2 d ) du du du This is the evolution equation fo a spheically symmetic photon suface of the Vaidya collapse metic 101). Conside the special case 0 : <u<0 mu) = λu :0 u u 1 104) m 1 := λu 1 : u 1 <u< fo given constants λ>0, u 1 > 0. Fo u<0 the space-time is locally Minkowskian, fo u :0 u u 1 thee is inwad falling null dust, and fo u>u 1 the space-time is locally isometic to Schwazschild space-time with ADM mass m 1 > 0. It is well-known thee is a cuvatue singulaity at = 0 and that fo λ :0<λ 1 16 the pat of this singulaity at u = 0 is locally naked.

22 PHOTON SURFACES =m u 1 = u 1 =1 u 1 = u =m u 1 = u 1 =1 u 1 = u Figue 4. The backwads evolution of the =3m 1 Schwazschild photon suface though collapsing Vaidya null dust. The spacetime coodinates of the evolved suface ae plotted fo mu) = u/16 above) and mu) = u/32 below) fo vaiousvalues of the null time u 1 > 0 of the junction between the null dust and Schwazschild egions. In each case the evolved photon suface fails to intesect the Minkowskian egion u<0. Fix λ :0<λ Fo u>u 1 equation 103) gives, as expected, that thee is a photon suface at =3m 1. We seek to evolve this photon suface backwads in time, though the in-falling null dust, to obtain a maximally extended photon suface S. The bounday conditions ae } =3m 1 d du =0 at u = u ) The esults ae shown in Fig. 4 fo λ =1/16 and λ =1/32 fo selected values of u 1. One sees that in all cases S tends in the past diection to a null hypesuface

23 PHOTON SURFACES 23 futue singulaity I + Photon suface S naked singulaity infalling null dust Schwazschild egion =0 coodinate singulaity I patial Cauchy suface H Minkowskian egion Figue 5. The confomal diagam of the Vaidya null dust collapse model showing the photon suface S aising fom the backwads evolution of the Schwazschild photon sphee. Any patial Cauchy suface which extends to spatial infinity is cut into two components by S. of the fom {u = const.}. The confomal diagam must theefoe be of the fom sketched in Fig. 5. It is evident fom Fig. 5 that the naked cental singulaity is enclosed within the photon suface S in the sense that any patial Cauchy suface extending to spatial infinity must intesect S in a 2-sphee. The physical significance of this may waant futhe investigation. 6. Concluding emaks The definition of a photon suface given in Section 2 is valid in an abitay spacetime. Howeve the esult that a photon suface must have a second fundamental fom which is pue tace indicates that a space-time must be specialized in some espect if it is to contain any photon sufaces. Fo spheically symmetic space-times thee ae always photon sufaces that espect the spheical symmety. Fo spacetimes that ae not spheically symmetic, the definitions of a photon suface and G- invaiant photon suface may seem too estictive. The poblem is that, in geneal, one may not have obiting null geodesics at a fixed adius. In Ke space-time fo example, although thee ae obiting null geodesics in the equatoial plane, those null geodesics which move in the diection of otation do so at a diffeent adius than those which move in the opposite diection. But it seems implausible that a concept which is physically impotant in the case of exact spheical symmety should become invalid when even a small amount of angula momentum is intoduced. A nontivial genealization of the concepts of photon suface and G-invaiant photon suface, at least to axially symmetic space-times, is thus equied.

24 PHOTON SURFACES 24 Acknowledgements This eseach was suppoted by the NFR of Sweden and NRF of South Afica. Refeences [1] C.Dawin,Poc.Roy.Soc.249, ). [2] C.Dawin,Poc.Roy.Soc.263, ). [3] K. S. Vibhada and G. F. R. Ellis, Schwazschild Black Hole Lensing, asto-ph/ [4] K. S. Vibhada and G. F. R. Ellis, Gavitational Lensing by Naked Singulaities, Phys. Rev. D. To be submitted. [5] S. W. Hawking and G. F. R. Ellis, The Lage Scale Stuctue of Space-Time Cambidge Univesity Pess, Cambidge, 1973). [6] B. Cate, Phys. Rev. 141, ). [7] R.H.Boye,Poc.Roy.Soc.A311, ). [8] H. Stephani, Geneal Relativity. An Intoduction to the Theoy of the Gavitational Field Cambidge Univesity Pess, Cambidge 1982), p.222. [9] A.I.Janis,E.T.NewmanandJ.Winicou,Phys.Rev.Lett.20, ). [10] K. S. Vibhada, Int. J. Mod. Phys. A 12, ). [11] K. S. Vibhada, D. Naasimha, and S. M. Chite, Aston. & Astophys. 337, ). [12] J. H. Hone and G. T. Hoowitz, Phys. Rev. D 46, ).