Black Holes. Volker Perlick General Relativity text-books with detailed sections on black holes:

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1 Black Holes Volke Pelick Summe Tem 2015, Univesity of Bemen Lectues: Tue 12 14, NW1, S1260 Fi 12 13, NW1, N1250 Tutoials: Fi 13 14, NW1, N1250 Video ecodings of the lectues ae available at Geneal Relativity text-books with detailed sections on black holes: L. Ryde: Intoduction to Geneal Relatvity Cambidge Univesity Pess 2009 N. Staumann: Geneal Relativity and Relativistic Astophysics Spinge 1984 [moe ecent editions ae available] C. Misne, K. Thone and J. Wheele: Gavitation Feeman 1973 R. Wald: Geneal Relativity Chicago Univesity Pess 1984 Monogaphs: V. Folov and I. Novikov: Black Hole Physics Kluwe 1998 S. Hawking and G. Ellis: The Lage-Scale Stuctue of Space-time Cambidge Univesity Pess 1973 B. O Neill: The Geomety of Ke Black Holes A. K. Petes 1995 S. Chandasekha: The Mathematical Theoy of Black holes Oxfod Univesity Pess 1984 Contents 1. Histoic intoduction 2 2. Bief eview of geneal elativity 3 3 Schwazschild black holes 6 4. Spheically symmetic gavitational collapse Othe spheically symmetic and static black holes Ke black holes Black holes in astophysics Black hole theoems 102 1

2 1. Intoduction: Histoic Notes 1783 J. Michell speculates in a lette to H. Cavendish if thee might exist dak bodies that ae so dense that light cannot escape fom thei suface see Woksheet P. S. Laplace calculates, independently of J. Michell, unde what condition the escape velocity fom the suface of a body is bigge than the velocity of light. Laplace s calculation can be found in an Appendix of the book by Hawking and Ellis A. Einstein publishes the field equation of geneal elativity K. Schwazschild finds the spheically symmetic and static solution to Einstein s vacuum field equation. The same solution is also found, independently and only a little late, by H. Doste H. Reissne and two yeas late G. Nodstöm find the spheically symmetic and static solution to the Einstein-Maxwell equations, known as the Reissne-Nodstöm solution R. Oppenheime and H. Snyde calculate the gavitational collapse of a spheically symmetic dust cloud D. Finkelstein explains the chaacte of the sphee = S in the Schwazschild solution as a one-way membane i.e., as an event hoizon. Related aticles appea in the yeas 1958 to 1960 by G. Szekees, M. Kuskal and C. Fønsdal R. Ke finds a solution to Einstein s vacuum field equation that descibes, in moden teminology, a otating black hole E. Newman and collaboatos find the solution to the Einstein-Maxwell equation that descibes otating chaged black holes, known as the Ke-Newman solution R. Penose and S. Hawking pove a seies of theoems to the effect that the fomation of a singulaity is geneic fo solutions to Einstein s field equation whee the enegy-momentum tenso satisfies cetain enegy conditions J. Wheele uses the wod black hole fo the fist time in a talk in New Yok. The fist page of the witten vesion of this talk, with Wheele s annotations, can be found in the book by Folov and Novikov Fo black holes with mass, angula momentum and electic chage, uniqueness theoems no-hai theoems ae poven R. Penose fomulates the cosmic censoship hypothesis, saying that unde cetain physically easonable assumptions gavitational collapse will neve esult in a naked singulaity, i.e., a singulaity without a hoizon J. Bekenstein and S. Hawking fomulate the themodynamical laws of black holes. S. Hawking pedicts that black holes decay by emitting black-body adiation if quantum effects ae taken into account Hawking adiation. 2

3 Obsevational evidence fo the existence of black holes: Supemasive black holes M : Thee is vey good evidence fo thei existence, as we will discuss late. The best candidate is the object at the cente of ou Galaxy, associated with the adio souce Sg A, with M. It is widely believed that most, if not all, galaxies host black holes at thei centes. Intemediate black holes M : They ae much moe hypothetical than supemassive black holes. They may exist at the centes of some globula clustes. Stella black holes 3 30M : Fo this class of black holes the obsevational evidence fo thei existence is almost as good as fo the supemassive ones. The best candidate fo a stella black hole is associated with the X-ay souce Cyg X-1, believed to be a black hole of 15M in a binay system. We will discuss the obsevational featues of stella black holes late. Mini-black-holes M oless: They aehypothetical. Theymayhavebeenpoduced at a vey ealy stage of the univese, in which case they ae called pimodial black holes. Even moe speculative is the idea that they may be poduced fom cosmic ays in ou atmosphee o in acceleatos. If they exist, they ae expected to decay quickly by emitting Hawking adiation. 2. Bief eview of geneal elativity A geneal-elativistic spacetime is a pai M, g whee: M is a fou-dimensional manifold; local coodinates will be denoted x 0,x 1,x 2,x 3 and Einstein s summation convention will be used fo geek indices µ,ν,σ,... = 0,1,2,3 and fo latin indices i,j,k,... = 1,2,3. g is a Loentzian metic on M, i.e. a covaiant second-ank tenso field, g = g µν dx µ dx ν, that is a symmetic, g µν = g νµ, and b non-degeneate with Loentzian signatue, i.e., fo any p M thee ae coodinates defined nea p such that g p = dx 0 2 +dx 1 2 +dx 2 2 +dx 3 2. As the metic is non-degeneate, we may intoduce contavaiant metic components by g µν g νσ = δ µ σ. We use g µν and g στ fo aising and loweing indices, e.g. g ρτ A τ = A ρ, B µν g ντ = B µ τ. The metic contains all infomation about the spacetime geomety and thus about the gavitational field. In paticula, the metic detemines the following. 3

4 The causal stuctue of spacetime: A cuve s xs = x 0 s,x 1 s,x 2 s,x 3 s is called spacelike lightlike timelike g > 0 µν xs ẋµ sẋ ν s = 0 < 0 timelike lightlike spacelike Timelike cuves descibe motion at subluminal speed and lightlike cuves descibe motion at the speed of light. Spacelike cuves descibe motion at supeluminal speed which is fobidden fo signals. Fo timelike cuves we can choose the paametisation such that g µν xτ ẋµ τẋ µ τ = c 2. The paamete τ is then called pope time. The motion of a mateial continuum, e.g. of a fluid, can be descibed by a vecto field U = U µ µ with g µν U µ U ν = c 2. The integal cuves of U ae to be intepeted as the woldlines of the fluid elements. The geodesics: By definition, the geodesics ae the solutions to the Eule-Lagange equations of the Lagangian d Lx, ẋ ds ẋ µ Lx,ẋ x µ = 0 L x,ẋ = 1 2 g µνxẋ µ ẋ ν. These Eule-Lagange equations take the fom ẍ µ + Γ µ νσxẋ ν ẋ σ = 0 whee Γ µ νσ = 1 2 gµτ ν g τσ + σ g τν τ g νσ ae the socalled Chistoffel symbols. The Lagangian Lx,ẋ is constant along a geodesic see Woksheet 1, so we can speak of timelike, lightlike and spacelike geodesics. Timelike geodesics L < 0 ae to be intepeted as the woldlines of feely falling paticles, and lightlike geodesics L = 0 ae to be intepeted as light ays. The Chistoffel symbols define a covaiant deivative that takes tenso fields into tenso fields, e.g. ν U µ = ν U µ + Γ µ ντu τ, ν A µ = ν A µ Γ ρ νµa ρ. 4

5 The cuvatue. The Riemannian cuvatue tenso is defined, in coodinate notation, by R τ µνσ = µ Γ τ νσ ν Γ τ µσ + Γ ρ νσγ τ µρ Γ ρ µσγ τ νρ. The cuvatue tenso detemines the elative motion of neighbouing geodesics: If X = X µ µ is a vecto field whose integal cuves ae geodesics, and if J = J ν ν connects neighbouing integal cuves of X i.e., iftheliebacket between X andj vanishes, then the equation of geodesic deviation o Jacobi equation holds: X µ µ X ν ν J σ = R σ µνρx µ X ν J ρ. X J If the integal cuves of X ae timelike, they can be intepeted as woldlines of feely falling paticles. In this case the cuvatue tem in the Jacobi equation gives the tidal foce poduced by the gavitational field. The spacetime metic is detemined, in tems of its souces, by Einstein s field equation The cuvatue quantity R µν R 2 g µν + Λg µν = κt µν. G µν = R µν R 2 g µν is called the Einstein tenso field, Λ is called the cosmological constant, and κ is called Einstein s gavitational constant. Based on cosmological obsevations we believe that we live in a univese with a positive cosmologicalconstantthatisoftheodeofλ m 2 sothatitcanbeneglectedonnon-cosmological scales. Einstein s gavitational constant is elated to Newton s gavitational constant G accoding to κ = 8πG/c 2 as follows fom the Newtonian limit of Einstein s theoy. The enegy-momentum tenso T µν on the ight-hand side of the field equation depends on the matte model that is used fo the souce of the gavitational field. The most impotant cases ae the following. Vacuum: T µν = 0 Then the field equation simplifies to R µν = Λg µν, as can be veified by calculating the tace of the field equation and then e-inseting the esult into the field equation. R µν = 0 is a system of ten scala second-ode non-linea patial diffeential equations fo the ten independent metic coefficients g µν. Among the solutions to Einstein s vacuum field equation to be discussed in this couse ae the Schwazschild solution and the Ke solution. 5

6 Pefect fluid: T µν = µ + p U c 2 µ U ν + pg µν Fo solving Einstein s field equation with a pefect-fluid souce, one has to specify an equation of state linking the pessue p to the mass density µ. The simplest equation of state is that fo a dust, p = 0. Then Einstein s equation togethe with the Eule equation gives a system of patial diffeential equations fo the g µν, the fou-velocity U ρ and the density µ. Pefect fluid solutions without a cosmological constant ae of inteest as models fo the inteio of stas. The inteio Schwazschild solution is an example; it descibes a spheically symmetic static sta with constant mass density µ. The socalled Fiedmann solutions, which ae the simplest cosmological models of ou univese, ae pefect fluid solutions with a cosmological constant. The athe pathological Goedel univese Kut Goedel s bithday pesent to Einstein on occasion of his 70th bithday in 1949 is a dust solution with a non-vanishing cosmological constant. In this couse we will conside, among othe things, a spheically symmetic collapsing ball of dust. Electovacuum: T µν = F µα F ν α 1 4 g µνf αβ F αβ In this case Einstein s field equation togethe with Maxwell s equations gives a system of patial diffeential equations fo the g µν and the electomagnetic field stength F µν. The best-known electovacuum solutions without a cosmological constant ae the Reissne- Nodstöm solution field outside of a chaged spheically symmetic static object and the Ke-Newman solutionfield of a chaged and otating black hole which will be teated late in this couse. 3. Schwazschild black holes 3.1 The Schwazschild metic in Schwazschild coodinates The Schwazschild metic, found by Kal Schwazschild in 1916, is the unique spheically symmetic solution to Einstein s vacuum field equation without a cosmological constant, R µν = 0. The deivation is a subject fo a fist couse on Geneal Relativity and will not be epeated hee. The metic eads g = 1 S c 2 dt 2 + d dϑ 2 + sin 2 ϑdϕ 2. S It depends on a paamete S, called the Schwazschild adius, with the dimension of a length. Fo a celestial body of adius t the metic is valid in the exteio egion whee we have vacuum, i.e., the ange of the coodinates is t R, ] t, [, ϑ,ϕ S 2. The meaning of the paamete S follows fom compaison with the Newtonian limit: S = 2GM c 2 6

7 whee G is Newton s gavitational constant, c is the vacuum speed of light and M is the mass of the cental body. Fo stas, planets etc. we have S < t so that in these cases thee is no poblem with the zeo in the denominato of g at = S. The idea of a body with adius smalle than S leads to the notion of a Schwazschild black hole as will be discussed below. We list some popeties of the Schwazschild metic. The Schwazschild metic is asymptotically flat, i.e., it appoaches the Minkowski metic fo. Indeed, if S / is negligibly small in compaison to 1, the Schwazschild metic is g c 2 dt 2 + d dϑ 2 + sin 2 ϑdϕ 2. which is the Minkowski metic in spheical pola coodinates. The Schwazschild metic is static in the domain > S, i.e., it admits a timelike Killing vecto field that is hypesuface-othogonal. Indeed, the vecto field t which is timelike in the domain > S is i a Killing vecto field, t g µν = 0, and ii othogonal to the hypesufaces t = constant, g t = g tϑ = g tϕ = 0. The fact that a spheically symmetic solution to Einstein s vacuum field equation is necessaily static was poven by G. Bikhoff in 1923 and is known as Bikhoff s theoem. Schwazschild had assumed staticity in his deivation. The coodinate has the following geometic meaning. Fo a cicle C in the equatoial plane t = t 0, = 0, ϑ = π/2, 0 < ϕ < 2π the cicumfeence can be ead fom the metic, dx µ dx ν l 0 = g µν ds ds ds = C C dϕ 2ds g ϕϕ ds dϕ 2ds 2π = 0 2 = 0 dϕ = 2π 0. C ds 0 Hence, the length of a ope laid out along this cicle is given by the fomula fo the cicumfeence of a cicle that is familia fom Euclidean geomety. Similaly, one finds that the aea of a sphee t = t 0, = 0 equals 4π0 2 which is again the usual Euclidean fomula. Fo this eason, is sometimes called the aea coodinate. By contast, fo a adial line segment S the length is t = t 0, 1 < < 2, ϑ = ϑ 0, ϕ = ϕ 0 l 1 2 = = S 2 g µν dx µ ds 1 g d = dx ν ds ds = 2 1 S d 2ds g ds d s This demonstates that cannot be intepeted as a distance fom a cente. 7

8 Fo a hypothetical sta with a adius smalle than S, the metic in the exteio has a singulaity at = S. The coect intepetation of this singulaity was an unsolved poblem until the late 1950s. It is the metic coefficient g = g, that diveges to infinity at = S. This does not necessaily indicate a pathology of the metic; it could vey well be that the metic is pefectly egula at = S, and that it is the coodinate basis vecto field that causes the divegence. Then we would only have a coodinate singulaity at = S that could be emoved by a coodinate tansfomation. In this context it is helpful to calculate cuvatue invaiants i.e., scalas that ae fomed out of the cuvatue tenso. If a cuvatue invaiant becomes infinite, this indicates a tue singulaity, a socalled cuvatue singulaity, that cannot be emoved by any coodinate tansfomation. If, howeve, all cuvatue invaiants emain finite at a point whee some metic coefficients divege, then it might be just a coodinate singulaity. Fo the Schwazschild metic, calculation e.g. of the socalled Ketschmann scala yields R µνρσ R µνρσ = 42 S 6. This demonstates that thee is a cuvatue singulaity at = 0 if we extend the vacuum Schwazschild solution that fa, but it gives us some hope that at = S we might have only a coodinate singulaity. This is indeed tue. We will discuss below fou coodinate systems Eddington-Finkelstein, Kuskal, Painlevé-Gullstand and Lemaîte which ae egula at = S. We will then see that a spheically symmetic sta with adius t < S necessaily foms a black hole. 3.2 The Schwazschild metic in othe coodinates a Isotopic coodinates Stating fom Schwazschild coodinates, we define a coodinate tansfomation that changes only the adial coodinate, t,,ϑ,ϕ t,,ϑ,ϕ, whee Then = S + S 2 d = + S 4, = S 4 d 2 + S 4 and substituting fo and d in the Schwazschild metic yields + S g = 1 S 2 c 2 S S 2d S 2 dt dϑ 2 + sin 2 ϑdϕ 2 + S S 4 = S 4 + S c 2 dt S 4 d dϑ 2 + sin 2 ϑdϕ

9 The inteesting popety of isotopic coodinates is that now the spatial pat of the metic is confomaltothe3-dimensionaleuclideanmeticd dϑ 2 +sin 2 ϑdϕ 2. Asaconfomalfacto does not change angles, the angles between spatial cuves epesented in isotopic coodinates coespond to the physical angles i.e., to the ones as measued with the physically coect metic. The pathological value = S coesponds to = S /4. At this point, thee is no zeo in the denominato but the metic coefficient g tt vanishes, so the matix g µν becomes degeneate. Thus, in isotopic coodinates the pathological value of the adius coodinate is just shifted to anothe coodinate position but nothing is gained in view of extending the metic. b Totoise coodinates Again we stat fom Schwazschild coodinates and tansfom only the adial coodinate into a new adial coodinate, t,,ϑ,ϕ t,ˆ,ϑ,ϕ, whee ˆ = + S ln 1. S This gives a tanscendental equation fo as a function of ˆ, i.e., the invese tansfomation cannot be witten in tems of elementay functions, but it is well defined: Thee is a bijective elation between unning fom S to and ˆ unning fom to. ˆ was called the totoise coodinate by John Wheele, alluding to Zeno s paadox of Achilles ace against the totoise. Just as accoding to Zeno s agument Achilles neve eaches the totoise, the coodinate ˆ neve eaches the point whee = S because it is shifted to. The totoise coodinate is impotant, among othe things, fo solving wave equations on the Schwazschild backgound. Just as the isotopic adius coodinate, it does not help extending the metic beyond = S. c Eddington-Finkelstein coodinates As befoe, we stat out fom Schwazschild coodinates, but this time we tansfom the time coodinate, t,,ϑ,ϕ t,,ϑ,ϕ whee t depends on t and. We will do this in such a way that ingoing adial lightlike geodesics ae mapped onto staight lines in the new coodinates. We will see that in these new coodinates the metic coefficients ae egula in the whole domain 0 < <. A adial lightlike cuve has to satisfy the equations g µν dx µ ds dx ν ds = 0, dϑ ds = dϕ ds = 0. Owing to the symmety, any such cuve must be a geodesic if the paamete s is chosen appopiately, i.e., the woldline of a classical photon. If we inset the g µν of the Schwazschild metic, we get 0 = c 2 1 S dt ds 1 S d 2, ds 9

10 d ds 2 = c S, ds dt d dt = ±c 1 S, Hee the uppe sign holds fo outgoing photons and the lowe sign holds fo ingoing photons. We integate the last expession. ±c dt = d 1 = S S + S d S = d + S ±ct = + S ln S + C, S < <. d S, It is convenient to wite the integation constant in the fom C = S ln S + ct 0. Then the equations fo adial lightlike geodesics ead ±ct = + S ln 1 + ct 0 S whee we ecognise the totoise coodinate on the ight-hand side. The equations fo adial lightlike geodesics hold on the domain S < < and on the domain 0 < < S if we assume that thee is vacuum. If we appoach S fom above, we have t along outgoing and t along ingoing lightlike geodesics. The diagam shows ingoing and outgoing adial lightlike geodesics in the exteio egion S < < and in the inteio egion 0 < < S. In eithe egion the Schwazschild metic is egula. Howeve, the two egions ae sepaated by the suface = S which shows a singula behaviou in the Schwazschild coodinates. None of ou lightlike geodesics eaches this suface at a finite coodinate time. As the angula coodinates ae not shown, any point in this diagam epesents a sphee. In the inteio egion and t have intechanged thei causal chaacte: is a time coodinate, g < 0, and t is a space coodinate, g tt > 0. While in the exteio egion t cannot stand still along an obseve s woldline, in the inteio egion cannot stand still along an obseve s woldline. As the Killing vecto field t is not timelike in the inteio, in this egion the Schwazschild metic is not static. 0 < < S S < < timelike spacelike t spacelike timelike 10

11 We now tansfom fom Schwazschild coodinates t,,ϑ,ϕ to ingoing Eddington-Finkelstein coodinates t,,ϑ,ϕ, ct = ct + S ln 1, S cdt = cdt + Sd S. This tansfomation maps ingoing adial lightlike geodesics onto staight lines, ct = + ct 0. By contast, the outgoing adial lightlike geodesics ae now given by the equation ct = + 2 S ln 1 + ct 0. S We will demonstate now that in the ingoing Eddington-Finkelstein coodinates the metic coefficients ae egula fo all values 0 < <. We have thus found an analytical extension of the Schwazschild spacetime acoss the suface = S. The spacetime diagam above shows the adial lightlike geodesics in this extended spacetime. We calculate the metic in the new coodinates t,,ϑ,ϕ, whee we wite, fo the sake of bevity, dαdβ = 1 2 dα dβ +dβ dα and dω 2 := sin 2 ϑdϕ 2 + dϑ 2. g = 1 S c 2 dt 2 + d 2 1 S + 2 dω 2 = S cdt Sd S 2 + d 2 1 S + 2 dω 2 = S c 2 dt 2 + 2c S S dt d S 2 S S d 2 S 2 S + d2 + 2 dω 2 = 1 S c 2 dt c S dt d S 1 S d dω 2. S 11

12 In the new coodinates the Schwazschild metic is, indeed, egula on the whole domain 0 < <. Also the invese metic exists on this whole domain, as 1 S S 0 0 det S 1 + S 0 0 g µν = det = { 1 S 1 + S sin 2 ϑ } 2 S 4 sin 2 ϑ = 4 sin 2 ϑ 2 is non-zeo fo all > 0, apat fom the familia coodinate singulaity on the axis, whee sinϑ = 0. Eddington-Finkelstein coodinates wee intoduced by Athu Eddington aleady in Howeve, he did not use them fo investigating the behaviou of the Schwazschild metic at = S but athe fo compaing Einstein s geneal elativity to an altenative gavity theoy of Whitehead. The same coodinates wee independently ediscoveed by David Finkelstein in 1958 who claified, with thei help, the natue of the suface = S. We discuss now the popeties of the extended Schwazschild spacetime that is coveed by the ingoing Eddington-Finkelstein coodinates. Themeticisegulaonthewholedomain0 < <. Itiscleathatthespacetimecannot be extended into the domain of negative -values, as = 0 is a cuvatue singulaity. We havealeadynoticedthatthecuvatueinvaiantr µνστ R µνστ goestoinfinity fo 0. As the cuvatue tenso detemines the elative acceleation of neighbouing geodesics ecall the geodesic deviation equation, this means that nea = 0 any mateial body will be ton apat by infinitely stong tidal foces. It is widely believed that a tue undestanding of what is going on nea = 0 equies a not yet existing quantum theoy of gavity. At = S the spacetime is pefectly egula. The tidal foces ae finite thee. By local expeiments nea = S, an obseve would not notice anything unusual. Howeve, the hypesuface = S plays a paticula ole in view of the global stuctue of the spacetime: Fom the ct diagam one can ead that it is an event hoizon fo all obseves in the domain > S, i.e., that no signal fom the domain < S can each an obseve at > S. Inpaticula, photonscannot tavel fomthedomain < S tothe domain > S. Fo this eason, the spacetime coveed by the ingoing Eddington-Finkelstein coodinates is called a Schwazschild black hole. As the angula coodinates ϑ and ϕ ae suppessed, each point in ou spacetime diagam epesents a sphee. Coespondingly, in the diagam each light signal epesents an ingoing o outgoing spheical wave font. In the domain > S the adius coodinate is inceasing fo outgoing sphees and deceasing fo ingoing sphees, as it should be in accodance with ou geometic intuition. In the domain 0 < < S, howeve, we ead fom the diagam that is deceasing fo ingoing and fo outgoing sphees. As 4π 2 gives the 12

13 aea of a sphee, as measued with the metic, this means that both the ingoing and the outgoing spheical wave fonts have deceasing aea. In a teminology intoduced by Roge Penose, they ae called closed tapped sufaces. The existence of closed tapped sufaces is an impotant indicato fo a black hole and plays a majo ole in the Hawking-Penose singulaity theoems. Quite geneally, the bounday of the egion whee closed tapped sufaces exist is called the appaent hoizon. In the Schwazschild spacetime, the appaent hoizon coincides with the event hoizon. In moe geneal spacetimes this is not the case. Along any futue-oiented timelike cuve in the domain < S, the -coodinate deceases monotonically, as can be ead fom the ct diagam. If an obseve was foolhady enough to ente into the egion 0 < < S, he will end up in the singulaity at = 0. In the next section we will calculate the pope time that elapses between cossing the hoizon and aiving at the singulaity. We will see that this time is maximal fo a feely falling obseve that is dopped fom est at the hoizon; in any case, it is finite. We have emphasised seveal times that the Schwazschild metic applies only to the exteio egion of a spheically symmetic celestial body, > t, because only thee is the vacuum field equation satisfied. We may conside a sta whoseadius tisbiggethan S at the beginning and then shinks beyond S. As soon as the adius is smalle than S, the sta is doomed. It will collapse to a point inafinite time. The diagam shows this phenomenon, which is known as gavitational collapse, in ingoing Eddington-Finkelstein coodinates. Wewilldiscusslateindetailthegavitationalcollapseofastathatismodelledasaballofdust. In paticula, we will discuss how a distant obseve sees the suface of the sta appoaching the hoizon, becoming moe and moe edshifted. Instead of ingoing Eddington-Finkelstein coodinates, we could intoduce the outgoing Eddington- Finkelstein coodinates t,,ϑ,ϕ, whee ct = ct S ln 1 S, cdt = cdt S d. S In these coodinates the outgoing adial lightlike geodesics ae mapped onto staight lines. In complete analogy to the ingoing Eddington-Finkelstein coodinates, also in these coodinates the metic becomes egula on the whole domain 0 < < S. In this way we get anothe ana- 13

14 lytic extension of the Schwazschild metic fom the domain S < < to the domain 0 < <. By constuction, it is obvious that it is just the image unde time-eflection of the extension we got fom the ingoing Eddington- Finkelstein coodinates. Now the hypesuface = S is an event hoizon fo obseves in the egion 0 < < S : Signals can coss this hypesuface only fom the inside to the outside, but not fom the outside to the inside. Fo this eason, one speaks of a Schwazschild white hole. Up to now, thee is no indication fo the existence of white holes in Natue. d Kuskal coodinates The maximal analytic extension of the Schwazschild metic was found independently by Matin Kuskal and by Gyögy Szekees in the late 1950s and also, with diffeent mathematical techniques, by Chistian Fønsdal. This maximal analytic extension, which is pobably only of mathematical inteest, can be found if one tansfoms on the domain S < < fom Schwazschild coodinates t,, ϑ, ϕ to Kuskal-Szekees coodinates u, v, ϑ, ϕ via u = S 1 e /2 S cosh ct 2 S, v = S 1 e /2 S sinh ct 2 S. These equations cannot be solved fo in tems of elementay functions, but they implicitly detemine and t as functions of u and v. This puts the Schwazschild metic into the following fom: g = 43 S e / S du 2 dv dω 2, whee is to be viewed as a function of u and v, implicitly given by the equations above. The metic is egula on the domain v 2 u 2 < 1. This maximal domain coves two copies I und I of the exteio egion S < <, a black hole inteio egion II and a white hole inteio egion II, see the diagam on the next page. The bounday of the Kuskal extension is given by the equation v 2 u 2 = 1 which coesponds to = 0. The two exteio egions I and I meet at the point at the cente of the diagam which is actually a sphee. Recall fom Woksheet 2 that the hypesufaces t = constant in the domain S < < ae the Flamm paaboloids. At the cente of the diagam the Flamm paaboloids of egion I ae glued togethe with the Flamm paaboloids of the egion I to fom the socalled Einstein-Rosen bidge. This is a womhole, but it is non-tavesible in the sense that an obseve cannot tavel at subluminal velocity fom egion I into egion I o vice vesa, as can be ead fom the diagam. In the letteing of the diagam it is m = GM/c 2, hence S = 2m. 14

15 In the u v diagam Kuskal-Szekees diagam light signals go unde 45 degees, du = ±dv. If a light signal entes into the black-hole inteio egion II by cossing one of the hoizons, it will end up in the singulaity at = 0. In the white-hole inteio egion II all light signals stat at the singulaity. They leave this egion ove one of the hoizons. e Painlevé-Gullstand coodinates Recall that in isotopic coodinates the spatial pat of the metic becomes confomally flat. We will now discuss a coodinate tansfomation fom standad Schwazschild coodinates to new coodinates, t,,ϑ,ϕ t,,ϑ,ϕ with t depending on t and, such that the hypesufaces t = constant become not only confomally flat but even flat. We ty the ansatz Then the Schwazschild metic eads = 1 S t = t+f, dt = dt+f d. g = 1 S = 1 S c 2 dt 2 +2f 1 S c 2 dt 2 + d2 1 S + 2 dω 2 c 2 dt f d 2 d dω 2 S { c 2 1 dtd + 1 S 1 S c 2 f 2 }d dω 2. 15

16 We have achieved ou goal if we choose f such that the culy backet equals unity, 1 1 S 1 S We choose the uppe sign. Then and, upon integation, c 2 f 2 = 1, f = dt = dt + In the new coodinates the metic eads g = ±1 c 1 S S 1 1 S S. d c 1 S t = t + 2 S S c c ln + S. S 1 S c 2 dt 2 +2c 2c 2 f 2 = 1 S, S dtd+d2 + 2 dω 2. This solution to the vacuum field equation was found independently by Paul Painlevé 1921 and Allva Gullstand Neithe of them ealised that it was just the Schwazschild metic in new coodinates. In Painlevé-Gullstand coodinates the Schwazschild metic is egula on the whole domain whee > 0. Indeed, it is obvious that the g µν ae finite at = S, and the following calculation shows that also the invese metic exists eveywhee on the domain whee > 0: 1 S 0 0 det g µν = det S S sin 2 ϑ = 4 sin 2 ϑ 1+ S S = 4 sin 2 ϑ. The Painlevé-Gullstand coodinates cove the same black-hole spacetime as the ingoing Eddington-Finkelstein coodinates. By constuction, the 3-dimensional hypesufaces t = constant ae flat. We will now discuss the family of obseves fo whom these hypesufaces ae the est spaces. We will demonstate that they ae ingoing feely falling paticles. Had we chosen the lowe sign fo f above, the dtd tem in the metic would have a minus sign. Then we would have a white-hole metic, as with the outgoing Eddington-Finkelstein coodinates, and the associated obseves would be outgoing. To that end we ecall that the woldlines of feely falling paticles ae timelike geodesics xτ paametised by pope time τ. Such cuves satisfy g µν xẋ µ ẋ ν = c

17 whee the ovedot means deivative with espect to τ, and whee 0 = d Lx, ẋ dτ ẋ µ Lx,ẋ x µ 2 Lx,ẋ = 1 2 g µνxẋ µ ẋ ν. We conside adial motion i.e., ϑ = ϕ = 0 in the Schwazschild spacetime in Painlevé- Gullstand coodinates. Then we get the two equations 1 S c 2ṫ2 S +2c ṫṙ +ṙ 2 = c 2 1 and 0 = d Lx, ẋ dτ ṫ Lx,ẋ t { = d dτ 1 S } c 2ṫ + c S ṙ whee we have witten only the t-component of the Eule-Lagange equation 2. We divide 1 by c 2, and we denote by E the constant of motion which is given by 2 : 1 S ṫ2 S 2 ṫ ṙ c ṙ2 c = 1, 2 1 E = 1 S c 2ṫ c S ṙ. 2 We want to conside paticles that ae dopped fom est at infinity, i.e., with Then fom 1 we find that ṙ = = 0. ṫ = = ±1 whee we choose the uppe sign to have τ and t both unning in the futue diection, and with that we find fom 2 that E = c 2ṫ = = c 2. With E detemined this way, 1 and 2 can be ewitten as Equation 1 is equivalent to 1 = ṫ2 1 = ṫ S S S ṫ + ṙ 2, 1 c ṫ + ṙ. 2 c 2 S ṫ + ṙ 2 = c ṫ 1 ṫ+1. 17

18 With 2 inseted on the ight-hand side, we get S ṫ + ṙ 2 S S = c and, afte eainging tems, S { ṫ + ṙ c ṫ + ṙ ṫ+1 c S ṫ + ṙ c S ṫ = 0. As ou paticles come in fom infinity, ṙ is negative, so the culy backet cannot be zeo. Hence, the ound backet must be zeo, i.e. S ṫ = ṙ c. If we einset this esult into 2 above we find that ou feely falling paticles must satisfy ṙ ṫ = 1, c = S. The fist equation says that along the woldlines of ou feely falling paticles the coodinate t coincides with pope time. The equation fo ṙ can be integated: d dt = c S, cdt = d, ct = constant. S 3 S These cuves, which ae called the woldlines of the Painlevé-Gullstand obseves, ae plotted in the figue below. The 4-velocity of the Painlevé-Gullstand obseves is hence g S U, = g t c, U = ṫ t +ṙ = t c = g t c S S, } S g S = c c S = 0. This demonstates that the hypesufaces ct t = constant ae indeed othogonal to the woldlines of the Painlevé-Gullstand obseves, i.e., that they ae what these obseves conside as simultaneous. The figue on the ight shows the hypesufaces t = constant and the woldlines of the Painlevé- Gullstand obseves which become vetical fo and hoizontal fo 0. Nothing paticula happens with these woldlines at = S. S 18

19 f Lemaîte coodinates Stating fom the Painlevé-Gullstand coodinates we pefom a tansfomation t,, ϑ, ϕ t,,ϑ,ϕ with being a function of t and. We want to choose this function such that is constant along the woldline of each Painlevé-Gullstand obseve. With the equation fo these woldlines as given in Section e, we see that this is achieved by = ct S Then the Schwazschild metic eads g = c 2 dt 2 + S d2 + 2 dω 2 whee is to be viewed as a function of t and, = 3 2/3 S ct. 2 This fom of the Schwazschild metic was found by Geoge Lemaîte in He clealy saw that, as in these coodinates the metic is egula at = S, the singulaity in the Schwazschild coodinates at = S is a mee coodinate singulaity. Howeve, he did not undestand the chaacte of = S as a hoizon. In Lemaîte coodinates the singulaity at = 0 is epesented as a diagonal line, ct =. The hoizon at = S is epesented as a paallel diagonal line, ct = 2 S /3. The disadvantage of the Lemaîte coodinates is in the fact that now the metic is no longe manifestly static. The hypesufaces t = constant now cay the 3-metic which is time-dependent because depends not only on but also on t. g 3 = S d2 + 2 dω 2 ct The figue on the ight shows the singulaity at = 0 thick line and the hoizon at = S dashed line in Lemaîte coodinates. The vetical lines ae the woldlines of the Painlevé-Gullstand obseves. The hoizontal lines ae the hypesufaces t = constant with the angle coodinates not shown. 19

20 3.3 Timelike geodesics in the Schwazschild spacetime Recall that timelike geodesics, if paametised by pope time, satisfy the equations whee 0 = d Lx,ẋ dτ ẋ µ Lx,ẋ x µ Lx,ẋ = 1 2 g µνxẋ µ ẋ ν and 1 2 g µνxẋ µ ẋ ν = c 2. Fo the Schwazschild metic, which is spheically symmetic, we can specify without loss of geneality to the case that the motion is in the equatoial plane, ϑ = π/2, ϑ = 0. Then Lx,ẋ = 1 1 S c 2 ṫ 2 + ṙ ϕ 2 s and the t and ϕ components of the Eule-Lagange equation give us two constants of motion, d 1 s c 2 ṫ 0 = 0, E = 1 s c 2 ṫ = constant, A dτ d ϕ dτ 2 and the definition of pope time equies 1 S 0 = 0, L = ϕ = constant, B 2 c 2 ṫ 2 + ṙ 2 1 S + 2 ϕ 2 = c 2. C The thee equations A, B and C detemine the obits, because one can check that the component of the Eule-Lagange equation is a consequence of these thee equations. We solve the thee equations fo the velocities, ṫ = E c 2 1, A S ϕ = L 2, B ṙ 2 = 1 1 S c S 2 E 2 c L 2 c 2 S 4 = E2 c 2 1 S L c2. C 20

21 a Radial obits Fo adial motion we must have ϕ = 0, hence L = 0. Then we only have to deal with the equations A and C, E ṫ = c 2 1, A S ṙ 2 = E2 c 2 1 S c 2. C E is detemined if we fix an initial condition. We want to assume that the paticle is dopped fom est at a adius 0, i.e. ṙ = 0, =0 Then C equies 0 = E2 c 2 1 S 0 c 2, hence E 2 = c 4 1 S 0. Clealy, this equation can hold only if 0 > S. This is in ageement with ou ealie obsevation that beyond the hoizon the t coodinate is spacelike, so motion at subluminal speed cannot have ṙ = 0 thee. With E detemined this way, C eads ṙ 2 = 1 S 0 c 2 1 S c 2 = c 2 S S 0 We want to integate this equation fo the limiting case that the paticle is dopped fom the hoizon, 0 S, S ṙ = c 1 whee we have chosen the minus sign because the motion is ingoing. Then. cdτ = d S 1 and, upon integation ove the obit fom cossing the hoizon at S until aiving at the singulaity at = 0, 0 d S d c τ = = = S S 1 S 0 [ S ] S + S actan = S actan 0 = Sπ, S

22 hence τ = π 2 S c. Note that S /c is the time that it takes to tavel the distance S at the speed of light. This τ is actually the maximal time an obseve can spend in the domain < S befoe ending up in the singulaity. To pove this, we conside an abitay timelike cuve in this domain, paametised by pope time, c 2 dx µ dx ν = g µν dτ dτ, c 2 = c 2 1 S dt dτ 1 S Multiplication with 1 yields fo 0 < < S : c 2 = c 2 1 S dt 2 + }{{ dτ } 0 c 2 1 S 1 1 S 1 d dτ d sin 2 ϑ dτ d 2 2 dτ 2, d dτ dϕ dτ 2 dϑ 2 + dτ dϕ 2 dϑ 2 sin 2 ϑ +. dτ dτ }{{} 0 2 c 2 S 1 As, fo < S, the ight-hand side is stictly bigge than zeo, d/dτ cannot change sign, so must be a monotonic function of τ. Hence cdτ d S This is the same integal as above. Integating ove the maximal domain 0 < < S gives c τ S π 2 Fom ou calculation above we know that equality holds fo the motion in fee fall fom est at the hoizon. Intuitively one would assume that it is a good idea to acceleate away fom the cente, with the help of a ocket engine, to avoid the singulaity at = 0 as long as possible. Actually, any such action makes the lifetime shote. So the best stategy is to appoach the cente in fee fall. Fo a black hole with 4 million Sola masses, as the one in the cente of ou Galaxy, we have S km and thus τ π S 2 c 63seconds. So the maximal lifetime of an obseve afte cossing the hoizon is only about one minute. Fo a stella black hole of about 20 Sola masses, the maximal τ is less than a Millisecond. We will late see that fo a otating black hole an obseve who has cossed the hoizon need not end up in the singulaity

23 b Non-adial obits Fo studying the shapes of non-adial obits, we may use the angula coodinate ϕ as the cuve paamete. Fom B and C above we find the obit equation fo feely falling paticles d 2 ṙ 2 = dϕ ϕ = 4 E 2 L2 2 L 2 c 2 + SL 2 c 2 + c2 S 2 3 = E2 c 4 c 2 L c2 S L S =: 2V E,L. Wehaveintoduced theeffective potential V E,L insuch awaythatakindof enegyconsevation law holds, 1 d 2 + VE,L = 0. 2 dϕ An obit with constants of motion E and L must be confined to the egion whee V E,L 0; the bounday points, whee V E,L = 0, ae tuning points of the obit whee d/dϕ = 0. So, with the help of the effective potential we can detemine fo which values of E,L bound obits exist. Similaly, we can detemine fo which values of E, L thee ae stable o unstable cicula obits. V E,L bound obit V E,L stable cicula obit V E,L unstable cicula obit p a It is ou goal to chaacteise the totality of bound obits in the Schwazschild spacetime. As an impotant step fo achieving this goal, we fist discull cicula obits which ae also of some inteest by themselves. Fo a cicula obit at adius we must have V E,L = 0 and V E,L = 0, i. e. 0 = V E,L = c4 E 2 2c 2 L 2 0 = V E,L = 2c4 E 2 c 2 L 2 4 c2 S 2L S 2, 3 3c2 S 2L S 2. Multiplying the fist equation with 4 and the second with and subtacting the latte fom the fome esults in 0 = c2 S 2L S 2. 23

24 Solving fo L 2 yields L 2 = c2 S S. Upon inseting this esult into the equation V E,L = 0 we find 0 = 2c4 E S c 4 S 2 3 c2 S S 2 c 2 S 2, 0 = 2 S 2 3 S 2E2 2 3 S c 4 S 2+4 S, 2E S = c S +4 2 S, E 2 = 2c4 S S. As L 2 and E 2 cannot be negative, cicula obits exist only fo those -values that satisfy the inequality > 3 2 S. We will see in the next subsection that at the limiting adius = 3 s /2 thee is a cicula lightlike geodesic. Fo < 3 S /2 the cicula obital velocity is bigge than the velocity of light which means that a cicula obit cannot be ealised at such a adius, neithe by a feely falling massive paticle no by a photon. We now check stability of the cicula obits at > 3 S /2. Fo stability we must have V E,L > 0. Fom diffeentiating the effective potential we find c 2 L 2 V E,L = 6c4 E 2 2 3c 4 S + c 2 L 2 = 3c 4 2 S + c4 S 2 12 S S = c S +3S 2 +S S +S S = c4 32 S 2 3 S + S 12 S S = c4 S 3S. 2 3 S The stability condition V E,L > 0 is, thus, satisfied fo > 3 S. In the adius inteval 3 S /2 < < 3 S ciculaobitsdoexist; howeve, theyaeunstablewhichmeansthatpactically they cannot be ealised, as any small deviation fom the initial condition would lead eithe to an escape obit towads infinity o to a plunge obit towads the singulaity. The limiting case = 3 S is known as the Innemost Stable Cicula Obit ISCO. Matte obiting a Schwazschild black hole cannot be at a adius value smalle than 3 S. Keep in mind that we ae talking about geodesic motion. With a ocket engine one can obit a Schwazschild black hole at any adius bigge than S. 24

25 We summaise ou esults on cicula timelike geodesics in the following table: S < < 3 S /2 cicula obits do not exist 3 S /2 < < 3 S cicula obits do exist, but they ae unstable 3 S < < cicula obits do exist and ae stable Recall that in the Newtonian theoy, i.e., fo a paticle moving in the Keple potential, cicula stable obits exist at all adius values 0 < <. We now tun to non-cicula bound obits. It is clea that bound obits can exist only in the domain S < <, because in the domain 0 < < S of a Schwazschild black-hole spacetime the adius coodinate is monotonically deceasing along any paticle woldline. Fo a bound obit it is necessay that the potential V E,L has in V E,L the inteval S < < two successive zeos with a local minimum inbetween, see thefigueon p.23. As V E,L is a fouth ode polynomial with the popeties that V E,L 0 = 0 and V E,L 0 = s /2 < 0, this is possible only if 0 a p V E,L has exactly thee positive zeos, 0 < 0 < p < a, see the figue on the ight. The bound motion takes place between the peicenteat p andtheapocente at a. Fo the same values of E and L thee is also a plunge obit of a paticle that eaches its maximum adius at 0 and falls into the black hole. Fo the paticle on the bound obit, 0 has no geometic meaning. Fo any values of E and L whee the potential V E,L has exactly thee positive zeos, 0 < 0 < p < a, thee is a bound obit. The limiting cases ae eached if p coincides with a o with 0. We will discuss these limiting cases in a minute. We will fist demonstate that, instead of E and L, we can use p and a fo labeling the potential. This is useful because p and a have a diect geometic meaning associated with the bound obit we want to chaacteise. It is even moe convenient to use, instead of the peicental adius p and the apocental adius a, the socalled semi-latus ectum p and the eccenticity e which ae defined by p = p 1+e, a = p 1 e. 25

26 These quantities ae familia fom the case of an elliptical obit but note that they ae welldefined fo any bound obit. We will now demonstate that E, L and 0 can be expessed in tems of p and e. To that end we compae the equation by which the effective potential was intoduced, V E,L = c4 E 2 4 c2 S 2c 2 L 2 2L S 2, with the epesentation in tems of its zeos, V E,L = c4 E 2 2c 2 L 2 p p 0. 1 e 1+e Compaing coefficients of 3, 2 and yields thee equations, c 2 S 2 L = c4 E e 2 +2p 2 2 c 2 L, 2 1 e = c4 E 2 p 2 +2p 0, 2 c 2 L 2 1 e 2 S 2 = c4 E 2 2c 2 L 2 0 p 2 1 e 2, which can be solved fo E, L and 0. Fom the last two equations we find 0 = Sp p 2 s which, inseted into the othe two equations, yields E 2 = c 2 4 S 21 e2 +2pp 2 S p 2p 3 S S e 2, L2 = c 2 S p 2 2p 3 S S e 2. Wewillnowdiscussfowhichvaluesofpandeboundobitsdoexist. Fothesakeofcompaison, note that in the Newtonian case i.e., fo Keple ellipses all values 0 < p < and 0 e < 1 ae possible. Fo finding the allowed values in the case of Schwazschild geodesics, we ecall that fo a bound obit we need thee positive zeos of the potential V E,L, see the plot on p.25. Clealy, the limiting values of p and e coespond to the limiting cases whee two of these thee zeos coincide, i.e., if eithe a = p o p = 0. The fist case is easily undestood. If p = a we have a stable cicula obit at this adius value, see figue on the ight. Fo the same values of E and L we also have a plunge obit between 0 and the singulaity at = 0, but this is ielevant fo the discussion of bound obits. Fom the calculation above we know that a stable cicula obit, and thus the situation depicted in the plot, is possible fo all values p = a > 3 S. V E,L 26 0 a = p

27 The second limiting case of bound obits is moe V E,L intiguing. If 0 = p, thee is an unstable cicula obit at this adius coodinate, see figue on the ight. Fo the same values of E and L, thee ae two additional obits that appoach 0 = p asymptotically. One is an obit that spials away fom 0 = p in the diection of deceasing and ends up in the singulaity. This has nothing to do with bound o 0 = p a bits. The othe one is an obit that spials away fom 0 = p in the diection of inceasing, eaches a maximum adius at a and then spials back towads 0 = p. Such an obit is called homoclinic. Quite geneally, in the theoy of dynamical systems an obit that asymptotically stats and ends at the same equilibium point is called homoclinic, while it is called heteoclinic if it connects two diffeent equilibium points. In the Schwazschild spacetime thee ae no heteoclinic obits. Heteoclinic obits occu, e.g., in the Kottle spacetime, i.e., in the Schwazschild spacetime with a cosmological constant, as we will biefly discuss late. A homoclinic obit makes infinitely many tuns aound the cente while asymptotically appoaching the limiting cicle at 0 = p. The homoclinic obits give us a bounday line in the p,e-plane fo bound obits. The equation fo this bounday line is given by 0 = p, S p = p p 2 S 1+e, S p 1+e = p p 2 S, p 3 S S e = 0. This gives us the egion of bound obits in the p,e-plane which is shown in the figue on the next page. It is bounded fom above by the line e = 1 and bounded fom below by the stable cicula obits, e = 0 and 3 S < p <. It is bounded on the left by the homoclinic obits, p 3 S S e = 0. This bounday line is often called the sepaatix. The sepaatix is a staight line connecting the points p,e = 3 s,0 and p,e = 4 S,1. The fome is the ISCO, the latte is an obit which asymptotically spials fom p = 2 S to a =. Along the sepaatix, the adius coodinate 0 = p of the unstable limit cuve vaies monotonically fom 3 S to 2 S. We see that only the unstable cicula obits in this adius inteval can seve as limit cuves fo homoclinic obits; the ones between = 3 S /2 and = 2 S ae limit cuves fo obits that spial out to infinity. 27

28 e 1 sepaatix bound obits 3 S 4 S p Bound obits nea the sepaatix ae known as zoom-whil obits. Such an obit peiodically zooms out to its apocente, which can be abitaily fa away because e can be abitaily close to 1. In between, it makes a lage numbe of whils nea its peicente p which lies close to a limit cuve of a homoclinic obit, i.e., between 2 S and 3 S. The pictue on the ight shows the potential V E,L fo a choice of E and L whee such a zoomwhil obit occus between p and a. Fo the same values of E and L thee is also a plunge obit with many whils nea 0. If a body obits a black hole, it follows appoximately a geodesic as long as its mass is much smalle than the mass of the black hole. If such a body would be on a zoom-whil obit, it would poduce a chaacteistic gavitational-wave signal. V E,L 0 p a 28

29 3.4 Lightlike geodesics in the Schwazschild spacetime Fo a discussion of the lightlike geodesics in the Schwazschild spacetime we poceed in a simila fashion as fo the timelike ones. Again, we estict to the equatoial plane and conside the Eule-Lagange equation 0 = d Lx,ẋ Lx,ẋ ds ẋ µ x µ with the Lagangian Lx,ẋ = S c 2 ṫ 2 + ṙ2 1 s + 2 ϕ 2 but now the dot denotes deivative with espect to an affine paamete s and g µν xẋ µ ẋ ν = 0. The t and ϕ components of the Eule-Lagange equation ae the same as befoe, d 1 s c 2 ṫ 0 = 0, E = 1 s c 2 ṫ = constant, dτ d 2 ϕ 0 = 0, dτ L = 2 ϕ = constant, B while the thid equation now eads 1 S c 2 ṫ 2 ṙ ϕ 2 = 0. S C The thee equations A, B and C completely detemine the lightlike geodesics, because one can check that the component of the Eule-Lagange equation is a consequence of these thee equations. We solve the thee equations fo the velocities, ṫ = E c 2 1, A S ϕ = L 2, B ṙ 2 = 1 1 S c S 2 E 2 c L 2 S 4 = E2 1 S L 2 c 2. 2 C As we have discussed the adial light ays aleady above, when intoducing Eddington-Finkelstein coodinates, we conside now only non-adial ones. Then we can use ϕ as the cuve paamete. Fom B and C above we find the obit equation fo light ays d 2 ṙ 2 = dϕ ϕ = 4 2 L 2 E 2 L2 c 2 + SL = E2 c 2 L S., A OL

30 a Cicula light ays We will fist inquie if thee ae cicula lightlike geodesics. To that end we wite the obit equation d 2 E 2 4 = dϕ c 2 L S and the ϕ-deivative of this equation, 2 d d 2 4E 2 dϕ dϕ = d c 2 L 2 S dϕ. Fo a cicula lightlike geodesic we must have d dϕ = 0 and d2 = 0, which gives us the following dϕ2 two equations: 0 = E2 4 c 2 L S, 0 = 4E c 2 L 2 S. To eliminate E 2 /L 2, we multiply the fist equation with 4/ and subtact the second equation. This esults in 0 = S = 3 2 S = 3GM. c 2 We have thus shown that thee is a cicula lightlike geodesic o photon cicle at the adius value 3GM/c 2. As we can choose any plane though the oigin as ou equatoial plane ϑ = π/2, thee is actually a photon sphee at this adius value in the sense that evey geat cicle on this sphee is a lightlike geodesic. The photon sphee at = 3 S /2 does, of couse, not exist fo stas with > 3 S /2. It is elevant only fo black holes and fo hypothetical ultacompact stas whee S < < 3 S /2. We will show in the 5th woksheet that the photon cicles at = 3 S /2 ae unstable in the following sense: A lightlike geodesic with an initial condition that deviates slightly fom that of a photon cicle at = 3 S /2 will spial away fom = 3 S /2 and eithe go to infinity o to the hoizon. Fo late convenience, we also calculate the value of the constant of motion L 2 /E 2 that coesponds to a photon cicle: If we inset the value = 3 S /2 into the equation we find E 2 c 2 L 2 = 0 = E2 3 c 2 L 2 + S 3 2 S S S = S. b Exact deflection angle Fom the obit equation of light ays we can deive an exact fomula fo the deflection angle. We want to conside a light ay that comes in fom infinity, goes though a minimum adius value at = m and then escapes back to infinity. We want to expess the deflection angle δ in tems of m and the mass of the cental body. 30

31 We stat out fom the obit equation OL. E 2 /L 2 is detemined by the condition that 0 = d dϕ 2 =m = E2 c 2 L 2 4 m 2 m + S m δ m = E 2 c 2 L 2 = 1 2 m S 3 m. We can, thus, ewite the obit equation OL as dϕ = 1 2 m ±d S m 3 S. Integation ove the light ay esults in ϕ0 +π+δ ϕ 0 dϕ = m + m 1 2 m d S m 3 S whee the signs of the two integals on the ight-hand side had to be chosen in ageement with the fact that ϕ is always inceasing. We have thus found an exact fomula, π + δ = 2 m m d 1 S 4 m m 2 S m, fo the deflection angle δ in tems of an elliptic integal. Fom the deivation it is clea that the integand has a singulaity at the lowe bound = m, so the evaluation of the integal needs some cae. A moe detailed analysis shows that the integal is finite fo all values of m that ae bigge than 3 S /2. If we conside a sequence of light ays with m appoaching 3 S /2 fom above, the deflection angle δ becomes bigge and bigge which means that the light ays make moe and moe tuns aound the cente. In the limit m 3 S /2 the integal goes to infinity and the limiting light ay spials asymptotically towads a cicle at = 3 S /2. This is a geneal featue of lightlike geodesics in spheically symmetic and static spacetimes: If an unstable photon cicle is appoached, the deflection angle goes to infinity. The exact fomula fo the bending angle is plotted on the next page. 31

32 δ 6π 4π 2π 3 S /2 m Plot of the bending angle δ against the minimum adius m. c Shadow of a Schwazschild black hole We fix an obseve at adius O and conside all light ays that go fom the position of this obseve into the past. To put this anothe way, we conside all light ays that aive at the position of the obseve. They fall into two categoies: Categoy I consists of light ays that go out to infinity, categoy II consists of light ays that go to the hoizon at = S. The bodeline case that sepaates the two categoies is given by light ays that asymptotically spial towads the light sphee at = 3 S /2. θ 0 O Now assume that thee ae light souces distibuted eveywhee in the spacetime but not between the obseve and the black hole. Then the initial diections of light ays of categoy I coespond to points at the obseve s sky that ae bight, and the initial diections of light ays of categoy II coespond to points at the obseve s sky that ae dak, known as the shadow of the black hole. The bounday of the shadow coesponds to light ays that spial towads = 3 S /2. It is ou goal to calculate the angula adius θ 0 of the shadow, in dependence of S and O. 32

33 Fo any light ay, the initial diection makes an angleθwithespecttotheaxis that is given, accoding to the pictue, by tanθ = lim x 0 y x. Fom the Schwazschild metic in the equatoial plane, g = 1 S c 2 dt 2 + d2 1 S we can ead the length x and y in the desied limit, dϕ tanθ = 1 S 1/2d y + 2 dϕ 2, = O. θ x d/dϕ can be expessed with the help of the obit equation OL fom p.29, hence tan 2 O 2 1 S θ = O O S E 2 O 4 = c 2 L 2 2 O + E 2 3. O S O c 2 L 2 O + S By elementay tigonomety, sin 2 θ = sin 2 θ sin 2 θ+cos 2 θ 1 = cot 2 θ + 1 = 1 E 2 3 O S 1 O c 2 +L 2 O + S + 1 O S c 2 L 2 O S = E 2 O 3 =. E c 2 L 2 O + S + O 2 3 O S The angula adius θ 0 of the shadow is given by the angle θ fo a light ay that spials towads = 3 S /2. This light ay must have the same constants of motion E and L as a cicula light ay at = 3 S /2 because the tangent vectos of these two light ays come abitaily close to each othe, O as we have calculated on p.30. c 2 L 2 E 2 = S 33

34 This gives us θ 0 in dependence of S = 2GM/c 2 and O, sin 2 θ 0 = 272 S O S 4 3 O. This fomula was found by J. Synge [Mon. Not. Roy. Aston. Soc. 131, ]. He did not use the wod shadow, howeve, because he consideed the time-evesed situation and calculated what he called the escape cones. If the obseve is fa away fom the black hole, O S O, Synge s fomula can be appoximated by tanθ 0 sinθ S 2 O. Up to a facto of 3, θ 0 is then the angula adius unde which a sphee of adius 3 S /2 is seen fom a distance O accoding to Euclidean geomety. This means that a naive Euclidean estimate gives quite a easonable idea of the diamete of the shadow if the obseve is fa away. Note that O : θ 0 0 i.e., the shadow vanishes. O = 3 S /2: θ 0 = π/2 i.e., the shadow coves half of the sky. O S : θ 0 π i.e., the shadow coves the whole sky. The shadow may be visualised in tems of the socalled escape cones. Fo each obseve position, the ed cone indicates the pat of the sky that is bight: O = 1.05 S O = 1.3 S O = 3 S /2 O = 2.5 S O = 6 S Fo the shadow of the black hole at the cente of ou galaxy M M, O 8.5kpc Synge s fomula gives an angula diamete of 2θ 0 54µas. One expects that this shadow will be seen with submillimete adio telescopes in the nea futue, using Vey Long Baseline Intefeomety VLBI. Thee ae two dedicated pojects, called the Event Hoizon Telescope and the BlackHoleCam, which ae now in the pepaing stage. The name Event Hoizon Telescope is is a bit misleading: Fist, it is not one telescope but athe a system of seveal existing and planned VLBI stations aound the Eath. Second, it will not make the event hoizon diectly visible; what is meant is to each an angula esolution compaable to the size of the event hoizon. Note that the shadow would exist not only fo a black hole, but in exactly the same way also fo an ultacompact sta S < < 3 S /2, povided the sta is dak. It is the light sphee at = 3 S /2 and not the hoizon at = S that is elevant fo the fomation of the shadow. 34

35 Ou calculation was based on the Schwazschild metic, so it does not apply to a otating black hole. The latte is to be descibed by the Ke metic; then the shadow tuns out to be noncicula. In any case, ou calculation with the Schwazschild metic gives the coect ode of magnitude fo the size of the shadow. We will calculate the shadow of a otating black hole late. d Multiple imaging of a Schwazschild black hole In this section we will biefly discuss the qualitative popeties of multiple imaging in the Schwazschild spacetime. We fix a static obseve at adius O and a static light souce at adius L. We exclude the case that obseve and light souce ae exactly aligned i.e., that they ae on a staight line though the oigin of the coodinate system which would give ise to Einstein ings instead of point images. We want to detemine how many images the obseve sees of the light souce. If one thinks of each lightlike geodesic as being suounded by a thin bundle that is focussed into the eye of the obseve with a lens, evey lightlike geodesic fom the light souce to the obseve gives ise to an image. L L O O The qualitative imaging featues follow fom the fact that the bending angle gows monotonically to infinity fo light ays that appoach the photon sphee at = 3 S /2. As a consequence, fo any intege n = 0,1,2,3,... thee is a light ay fom the light souce to the obseve that makes n full tuns in the clockwise sense, and anothe light ay fom the light souce to the obseve that makes n full tuns in the counte-clockwise sense. Hence, thee ae two infinite sequences of light ays fom the light souce to the obseve, one in the clockwise sense left pictue and one in the counte-clockwise sense ight pictue. Eithe sequence has as its limit cuve a light ay that spials asymptotically towads = 3 S /2. The pictues ae not just qualitatively coect; they show numeically integated lightlike geodesics in the Schwazschild spacetime. One sees that fo each sequence the light ays with n = 1,2,3,... lie pactically on top of each othe. Coespondingly, the obseve sees infinitely many images on eithe side of the cente. Each sequence apidly appoaches the shadow. In the pictue, which is again the esult of a calculation, the shadow is shown as a big black disc. On eithe side only the outemost image n = 0 can be isolated, all the othe ones clump togethe and they ae vey close to the bounday of the shadow. If thee ae many light souces, thei highe-ode images will be visible as a bight ing aound the shadow. 35

36 It can be shown that the outemost images ae bighte than all the othe ones combined. Of the two outemost images, the bighte one is called the pimay image and the othe one is called the seconday image. All the othe ones, which coespond to light ays that make at least one full tun, ae known as highe-ode images. Highe-ode images have not been obseved so fa. Just as with the obsevation of the shadow, thee is some hope that they might be seen nea the black hole at the cente of ou galaxy in the foeseeable futue. The Vey Lage Telescope in Chile will soon be equipped with a new instument, called GRAVITY, fo infaed intefeomety. With this instument it should be possible to obseve stas that ae much close to the black hole than the ones that have been obseved so fa. 4. Spheically symmetic gavitational collapse Stas ae stable as long as the pessue balances the gavitational attaction. If the nuclea fuel at the coe of a sta is used up, the sta becomes unstable, possibly blows away some of its mass in a nova o supenova explosion and then collapses. Accoding to pesent knowledge, thee ae thee possible end states of a sta. A sta could end up as a white dwaf, whee the electons fom a degeneate Femi gas. The electon degeneacy pessue can balance the gavitational attaction. White dwafs have a adius of about 5000 km, i.e., they ae simila in size to the Eath. It was shown by S. Chandasekha in the ealy 1930s that a white dwaf must have a mass M 1.4M. This wok won him the physics Nobel pize in Anothe possible end state is a neuton sta. Neuton stas ae much moe difficult to undestand than white dwafs. Roughly speaking, they consist of extemely densely packed neutons. It is the neuton degeneacy pessue that balances gavity so that a stable object esults. Typically, a neuton sta has a adius between 10 and 20 km. Similaly to the Chandasekha limit fo white dwafs, thee is an uppe limit fo the mass of a neuton sta, but it is not yet pecisely known. The most massive neuton sta that has been found so fa has a little bit moe than 2 Sola masses. Neuton stas with 3 o maybe even 5 Sola masses ae not completely uled out. A sta that is so massive that it cannot end up as a white dwaf o a neuton sta is believed to undego gavitational collapse and to fom a black hole at the end of its life. A spheically symmetic sta that has fallen though its Schwazschild adius is doomed. The whole sta must end up in a singulaity at = 0 in a finite time. This follows fom the consideation in the peceding section, because a mass element on the suface of the sta must move on a timelike cuve in the ambient vacuum Schwazschild spacetime. In this chapte we want to discuss the dynamical pocess of the collapse. In analytical tems, this is possible only fo the simplest case of a spheically symmetic ball of dust. In a sense, this is a tivial situation, because it is clea that in the case of pefect spheical symmety the entie sta must collapse into a singulaity unde the influence of its own gavity if thee is no pessue. Howeve, it is emakable that the metic inside the sta can be detemined fully analytically as an exact solution to Einstein s field equation with a dust souce, and that an exact analytical fomula can be given fo the adius of the sta as a function of time. 36

37 The following calculation follows J. R. Oppenheime and H. Snyde [Phys. Rev. 56, ]. It is based on the exact solution of Einstein s field equation fo a spheically symmetic dust which had been found by R. Tolman [Poc. Nat. Acad. Sci. 20, ] aleady a few yeas ealie. It is shown in a fist couse on geneal elativity that any spheically symmetic metic can be witten, in spheical pola coodinates t, ρ, ϑ, ϕ, as g = c 2 e νt,ρ dt 2 + e λt,ρ d 2 + Yt,ρ 2 dϑ 2 +sin 2 ϑdϕ 2. We want to solve Einstein s field equation with the enegy-momentum tenso of a dust, R βγ R 2 g βγ = κt βγ T βγ = µu β U γ whee µ is the mass density, depending on t and ρ, and U = U β x β is the fou-velocity field of the dust. We will choose the coodinates such that the dust paticles move on t-lines, U β = αδ β t. The facto α is detemined by the nomalisation condition hence c 2 = g βγ U β U γ = c 2 α 2 e νt,ρ, α = e νt,ρ/2, U = e νt,ρ/2 t. Fom the field equation it follows that β T βγ = 0 which implies that the dust paticles move on geodesics, U U = 0. Again, we assume that this esult is known fom a fist couse in geneal elativity. Claim: U U = 0 implies that the metic function ν is independent of ρ. Poof: Fom U U = 0 we have 0 = g U U, / ρ = U g U, / ρ g U, U / ρ hence ν/ ρ = 0. = e ν/2 t e ν/2 g / t, / ρ e ν g / t, / t / ρ } {{ } =g tρ=0 = e ν g / t, / ρ / t = e ν 1 2 ν c2 = e 2 ρ eν ν c2 ν = e eν 2 ρ = c ρ g / t, / t = e ν 1 2 ν ρ, ρ g tt

38 We can thus intoduce a new time coodinate τ by dτ = e νt/2 dt, τ = In the coodinates τ,ρ,ϑ,ϕ, the metic eads t t 0 e ν t/2 d t. g = c 2 dτ 2 +e λτ,ρ dρ 2 +Yτ,ρ dϑ 2 +sin 2 ϑdϕ 2, M the fou-velocity of the dust is and the enegy-momentum tenso is U = c τ T βγ = µδ τ β δτ γ. Fom these expessions we ead that τ is pope time of the dust paticles. This fom of the metic is known as a spheically symmetic dust in comoving coodinates. We will now evaluate the field equation R βγ R 2 g βγ = κt βγ. This equies calculating the Chistoffel symbols and, theeupon, the components of the Ricci tenso of the metic M. One finds that thee ae fou non-tivial equations. The τ τ component: 2 Y 2 Y ρ Y The ρρ component: λ Y ρ ρ 1 Y 2 e λ + 1 Y 2 ρ c 2 Y Y τ λ τ + 1 Y 2+ 1 c 2 Y 2 τ Y = 2 κµc2. F1 1 Y 2e λ + 2 Y 2 ρ c 2 Y 2 Y τ + 1 Y c 2 Y 2 τ Y = 0. 2 F2 The ϑϑ component which equals the ϕϕ component: 1 2 Y Y ρ + 1 Y λ e λ + 1 Y 2 2Y ρ ρ 2c 2 Y τ λ τ λ c 2 τ + 1 λ c 2 τ c 2 Y 2 Y τ 2 = 0. F3 The τ ρ component: 1 cy Y λ ρ τ + 2 cy 2 Y τ ρ = 0. All the othe components educe to the identity 0 = 0. To solve this system of diffeential equations, we begin with F 4. F4 2 Y Y 2 λ τ = τ ρ ρ Y 2 = ρ Y 2 τ ρ Y 2 = Y 2, τ ln ρ ρ 38

39 hence Y λ+ln τ ρ τ e λ Y ρ 2 2 = 0, = 0, sothefunctionundethedeivativeontheleft-handsidedependsonρonly. Itwillbeconvenient to wite this function in the following fom: e λ = Y 2 ρ, 1 εfρ 2 ε = 0,±1. G1 We inset this esult into F2: 1 εfρ2 Y c 2 Y εfρ 2 = 2Y 2 Y τ + 1 Y c 2 Y 2 τ Y = 0, 2 2 Y c 2 τ + 1 Y 2. 2 c 2 τ Fo the following calculation we keep ρ fixed and view the last equation as an odinay diffeential equation fo Y as a function of τ. We substitute which implies u = 1 dy 2 c 2 dτ d du uy = u + Y dy dy = 1 dy 2 2Y dy + c 2 dτ c 2 dτ d 2 Y dτ dτ 2 dy = εfρ2, and, afte esubstituting fo u, uy = εfρ 2 Y +Fρ, 1 dy 2 = εfρ 2 + Fρ c 2 dτ Y. This equation can be integated, whee we have to distinguish the case ε = 0 fom the cases ε = ±1. Fo ε = 0 the last equation implies Y 3/2 = ± 3c 2 1 c dy dτ = ± FρY 1/2, Y 1/2 dy = ±c Fρdτ, Fρ τ τ 0 ρ, ε = 0. G2a 39

40 Fo ε = ±1, we substitute the time coodinate τ η, Then 1 dy c 2 dη 1 dy c 2 dη dη = ± fρ Y cdτ. dη dτ 2 fρ 2 c 2 dy dη 2 = εfρ 2 + Fρ Y, = εfρ 2 + Fρ Y 2 Y, 2 = εy 2 + FρY fρ, 2 dy = ±dη. FρY fρ εy 2 2 This is a elementay integal that can be found in an integation table, Y = Fρh ε η 2fρ 2, h ε η = { η sinη fo ε = +1, sinhη η fo ε = 1, G2b whee η and τ ae inteelated by dη = ± 2fρ3 cdτ Fρh εη, h εη = ± 2fρ3 c Fρ τ τ 0 ρ. Inseting G2a o G2b, espectively, into G1 gives λ. It can then be checked that F3 is identically satisfied, while F 4 gives the mass density µ, κµc 2 = F ρ Y 2 Y ρ. G4 This gives us the geneal speically symmetic solution to Einstein s field equation with a dust souce as dyτ,ρ 2 g = c 2 dτ 2 dρ 2 + dρ 1 εfρ2 +Yτ,ρ2 dϑ 2 +sin 2 ϑdϕ 2 whee Y is given by G2a o G2b and the coesponding mass density is given by G4. Oppenheime and Snyde took this solution fom a 1934 pape by R. Tolman. The same solution had also be found, inadiffeent fom, by G. Lemaîte in 1933and it was futhe investigated by H. Bondi in It is theefoe known as the Lemaîte-Tolman o the Lemaîte-Tolman-Bondi solution. It involves a paamete ε that can take the values 0, +1 o 1, and thee abitay functions fρ, Fρ and τ 0 ρ. As the metic peseves its fom unde a tansfomation of the ρ coodinate, thee ae actually only two abitay functions with a physical meaning involved. They can be chosen, e.g., to be the initial density and the initial adial velocity. 40

41 If we choose Fρ to be a constant, the mass density vanishes by G4, so in this case the Lemaîte-Tolman solution gives a spheically symmetic vacuum solution. By the Bikhoff theoem, this must be equal to the Schwazschild solution, fo any choice of fρ and τ 0 ρ. This gives anothe family of coodinate epesentations of the Schwazschild vacuum solution. Following Oppenheime and Snyde, we model the inteio of a collapsing sta by selecting a paticula solution of the Lemaîte-Tolman class that gives a spatially constant mass density. This can be achieved by choosing fρ = ρ, Fρ = Aρ 3, τ 0 ρ = 0, whee A is a constant. Then we find fo ε = 0 that 9 1/3 Yτ,ρ = K 0 τρ, K 0 τ = 4 c2 Aτ 2 and fo ε = ±1 that A Yτ,ρ = K ε τρ, K ε τ = A 1 cosη 2 h ε η = 2 A coshη 1 2 whee the elation between η and τ is given by fo ε = +1, fo ε = 1, dη = 2cdτ Ah εη, 2cτ A = h εη = { η sinη fo ε = +1, sinhη η fo ε = 1. Hee we have chosen the integation constant such that τ = 0 coesponds to η = 0. The metic simplifies to g = c 2 dτ 2 +K ε τ 2 { dρ 2 1 ερ 2 +ρ2 dϑ 2 +sin 2 ϑdϕ 2}. Note that ρ is a dimensionless adius coodinate. The expession in the culy backet is the 3-metic of constant cuvatue: Fo ε = 0 it is the 3-dimensional Euclidean flat metic in spheical pola coodinates, fo ε = +1 it is the metic of a 3-sphee, and fo ε = 1 it is the metic of 3-dimensional hypebolic space. If ρ anges ove its maximal domain, this gives a cosmological dust solution of the field equation known as Fiedmann solution. Hee, howeve, we ae inteested in the case that this metic is valid only fo 0 < ρ < ρ 0, whee ρ 0 is the adius coodinate of the suface of a ball of dust, and that the metic is joined at ρ = ρ 0 to an exteio Schwazschild vacuum. The suface of the sta must be given by a timelike geodesic of the ambient Schwazschild metic. We know that then the Schwazschild aea coodinate has to satisfy the equation 1 d 2 S = const.+ c dτ 2 at the suface of the sta, ecall p. 21. Inside the sta, the aea adius coodinate is given = Yτ,ρ, as can be ead fom the metic, and we know that Y satisfies the equation 1 Y 2 = εfρ 2 + Fρ c 2 τ Y. 41

42 At the suface of the sta, whee ρ = ρ 0 and = Yτ,ρ 0, these two expessions must coincide, i.e. Fρ 0 = S. This elates the constant A to the Schwazschild adius S, and gives us the density as Aρ 3 0 = S = 2GM c 2 With κ = 8πG/c 4 this can be ewitten as µτ = κµτc 2 = 3A K ε τ 3 = 6GM c 2 ρ 3 0K ε τ. 6 GM c 4 c 4 ρ 3 0K ε τ 3 8π G = 3M 4πYτ,ρ 0 = M π3 0 So µτ is given by the usual Euclidean fomula fo the density of a mass M distibuted homogeneously ove a sphee of adius 0 = Yτ,ρ 0. As M was defined asymptotically by compaison with the Newtonian theoy, and as the spatial geomety is non-flat unless ε = 0, this esult was not to be expected and can be consideed only as a concidence. Ou line of easoning shows that the condition Fρ 0 = S is necessay fo matching ou dust solution to an exteio Schwazschild vacuum. It does not actually pove that the matching is possible. Such a poof equies to veify that the junction conditions that follow fom the field equation ae satisfied at the suface of the sta. This is indeed tue, but we will not wok out the details hee. We plot the suface of the sta in an cτ diagam, whee = Yτ,ρ is the aea adius coodinate. Fo ε = 0 we have 9 1/3ρ0 9 1/3. = Yτ,ρ 0 = 4 c2 Aτ 2 = 4 c2 S τ 2 Fo ε = 1 we use a paametic epesentation η,τη given by = Yτ,ρ 0 = K 1 τρ 0 = A 2 h 1 ηρ 0 = S 1 cosη, 2ρ 2 0 Similaly, fo ε = 1 we have τ = A 2c h 1η = S η sinη. 2cρ 3 0 = Yτ,ρ 0 = K 1 τρ 0 = A 2 h 1ηρ 0 = S coshη 1, 2ρ 2 0 τ = A 2c h 1η = S sinhη η. 2cρ

43 The plots ae shown in the pictue below. cτ S = Yτ,ρ cτ m ε = +1 ε = 1 ε = 0 Fo ε = 0 the suface of the sta coincides with woldlines of Painlevé-Gullstand obseves, i.e., it descibes fee fall fom est at infinity. Fo ε = 1 the sta falls fom infinity with an inwads diected asymptotic initial velocity. The physically most elevant case is the case ε = +1. This descibes fee fall fom est with a finite initial adius if we choose the initial hypesuface at τ = τ m which coesponds to the maximum value of the function K 1 η, that is η = π/2. We concentate on this case in the following and detemine the adial lightlike geodesics inside the sta, to see how the hoizon is fomed. Fom the metic we ead that the adial lightlike geodesics inside the sta ae given by 0 = c 2 dτ 2 +K 1 τ 2 dρ 2 1 ρ 2, ±c dτ dη dη = K dρ 1τ, 1 ρ 2 ±c Ah 1η 2c dη = Ah 1η 2 ±dη = dρ 1 ρ 2 dρ, 1 ρ 2 The aea adius coodinate is ±dη = d acsinρ, ±η η 0 = acsinρ, ρ = sin ± η η 0. = Yτ,ρ = K 1 τρ = A 2 h 1ηρ = S 1 cosηsin ±η η 2ρ

44 Togethe with the equation τ = A 2c h 1η = S η sinη 2cρ 3 0 this gives the adial lightlike geodesics in paametised fom η,τη. The diagam below shows thee outgoing adial lightlike geodesics i.e., with the + sign as dashed cuves. As the angle coodinates ae not shown, each of these cuves epesents a sphee s woth of lightlike geodesics. A adial lightlike geodesic stating at the cente of the sta can do one of two things afte cossing the suface of the sta: Eithe it goes to infinity lowe dashed cuve o it goes to the singulaity at = 0 uppe dashed cuve. The bodeline case is fomed by geodesics that continue on the suface = S afte leaving the sta middle dashed cuve. It can be ead fom the pictue that this middle dashed cuve gives the event hoizon fo the total spacetime, consisting of the collapsing sta and the vacuum exteio: It is impossible to send a signal at subluminal speed fom inside the hoizon to an obseve outside. The hoizon comes into existence at the cente of the sta, expands until it eaches the collapsing suface at = S and then continues in the vacuum egion at this fixed value. cτ S = Yτ,ρ cτ m Recall that the appaent hoizon is defined as the bounday of the egion whee closed tapped sufaces exist, i.e., whee both the ingoing and the outgoing adial lightlike geodesics go into the diection of deceasing aea coodinate. As the uppe dashed cuve goes fo the most pat into the diection of inceasing coodinate, although lying beyond the event hoizon, the 44

45 diagam demonstates that in the case of ou collapsing dust ball the appaent hoizon does not coincide with the event hoizon. The visual appeaance of a collapsing dust ball to an outside obseve follows fom Poblem 3 of Woksheet 3: It fades out of view because the suface of the sta becomes infinitely edshifted. The calculation of this poblem applies to the case ε = 0 fee fall fom est at infinity, but the qualitative featues ae the same fo ε = ±1. 5. Othe spheically symmetic and static black holes In this chapte we will biefly discuss a few spheically symmetic and static black-hole metics othe than Schwazschild, i.e., metics that do not solve the vacuum field equation without a cosmological constant. We will summaise only a few main featues without going too much into detail because these metics ae usually thought to be of less astophysical elevance. 5.1 Kottle black holes The Kottle metic is the unique spheically symmetic solution to Einstein s vacuum field equation with a cosmological constant, R µν = Λg µν. The deivation is completely analogous to the Schwazschild metic and will not be given hee. In paticula, the Bikhoff theoem is valid also in the case with a non-vanishing cosmological constant. The metic eads g = c 2 1 S Λ 3 2 dt 2 + d 2 1 S Λ + 2 dϑ 2 +sin 2 ϑdϕ This solution was found by F. Kottle in 1918 and almost simultaneously by H. Weyl. Fo Λ = 0, it educes of couse to the Schwazschild metic. Fo S = 0, it educes to the de Sitte solution if Λ > 0 and to the anti-de Sitte solution if Λ < 0. Fo this eason, the Kottle metic is also known as the Schwazschild-de Sitte metic in the case Λ > 0 and as the Schwazschildanti-de Sitte metic in the case Λ < 0. The name Schwazschild metic with a cosmological constant is also used occasionally. Just as the Schwazschild metic, the Kottle metic contains an integation constant S. The most impotant diffeence in compaison to the Schwazschild solution is in the fact that the metic is not asymptotically flat, i.e., it does not appoach the Minkowski metic fo. Theefoe, S cannot be detemined by consideing the egion S and compaing with the Newtonian appoximation. Howeve, if Λ is small, thee is a egion whee S and still 2 Λ 1. In this domain, the compaison with the Newtonian theoy is possible and the identification S = 2GM/c 2 is justified. 45

46 The physical elevance of the Kottle metic is dubious. It is tue that we believe to live in a univese with a small but non-zeo positive cosmological constant, Λ km 2. Howeve, fo this value of Λ, the tem Λ 2 /3 is negligibly small in compaison to S / unless 3 S km 2. Foplanets, stasandstella osupemassive blackholesthismeansthattheλ tem becomes elevant only at vey lage distances whee the appoximation of the gavitational field as being spheically symmetic is no longe valid because of the pesence of neighbouing masses. That is to say, as long as it is justified to model the gavitational field aound an astonomical object as spheical, the Λ tem plays no ole. Nonetheless, the Kottle metic displays a few inteesting mathematical featues which we will discuss in the following. As the contibution fom the cosmological constant vanishes fo 0, the Kottle metic has a cuvatue singulaity at = 0, just as the Schwazschild metic. This implies that the metic cannot be extended fom the domain > 0 into the domain < 0. One may conside the domain < 0 as a spacetime in its own ight. The Kottle metic with a constant S > 0 on the domain < 0 is the same as the Kottle metic with a constant S < 0 on the domain > 0, as is obvious fom a coodinate tansfomation. We will not conside this possibility hee and athe estict to the domain > 0 assuming that S > 0, as suggested by ou identification S = 2GM/c 2. Keeping a positive value fo S fixed, we want to investigate the stuctue of the spacetime in dependence of Λ which is allowed to take any value between and. Fom the metic we ead that hoizons ae whee 1 S Λ 3 2 = 0, i.e., at zeos of the thid-ode polynomial The deivative of this function is f = Λ S. f = Λ 2 1. Fo Λ < 0 we have f < 0. As f0 = S > 0 and f fo, the function f has pecisely one eal positive zeo, f h = 0, see the plot below. Hence, thee is pecisely one hoizon. h is situated between 0 and S. We have h 0 fo Λ and, of couse, h S fo Λ 0 f S Λ < 0 h Fo Λ > 0, the function f stats at = 0 with a positive value, f0 = S, and a negative deivative, f 0 = 1. As f fo, the function f must have a minimum at 46

47 some adius value between 0 and. As f is a thid-ode polynomial, it can have only one minimum. At this minimum value, f may be positive, zeo o negative. In the fist case f has no positive eal zeos, in the second case it has a double zeo, and in the thid case it has two diffeent positive eal zeos. The citical value Λ c whee a double zeo occus is found by setting f and f equal to zeo, Solving fo Λ c yields 0 = Λ c S, 0 = Λ c 2 1. Λ c = S and einseting this esult into f = 0 shows that the double zeo occus at = 3 S /2. Fo 0 < Λ < Λ c thee ae two hoizons, fo Λ = Λ c thee is a degeneate hoizon at = 3 S /2, and fo Λ c < Λ thee is no hoizon, see the plots below. f S Λ c < Λ Λ = Λ c 0 < Λ < Λ c h1 3 2 S h2 To demonstate that the zeos of f can be emoved by a coodinate tansfomation and have, indeed, the chaacte of a hoizon, we poceed as fo the Schwazschild metic. Radial lightlike geodesics in the Kottle spacetime ae given by 0 = 1 S Λ 3 2 c 2 dt 2 + ±cdt = d 1 S Λ. 3 2 d 2 1 S Λ 3 2, 47

48 We tansfom to genealised ingoing Eddington-Finkelstein coodinates, t whee c t = ct + d 1 S Λ. 3 2 Hee the integal can actually be witten in tems of the zeos of the thid-ode polynomial f but we will not wite this out because the solution fomulas fo cubic equations ae athe awkwad. In any case, the integal can be easily evaluated numeically fo any value of Λ. In ingoing Eddington-Finkelstein coodinates the ingoing adial light ays ae given as c t = + constant and the outgoing ones ae given as c t = + 2 d 1 S Λ. 3 2 Fo Λ < 0, we get a black-hole spacetime with an event hoizon at a adius h ] 0, S [. The qualitative featues ae quite simila to the Schwazschild case Λ = 0. The plots below show the adial light ays in Schwazschild coodinates, t and in ingoing Eddington-Finkelstein coodinates, t, whee the little ellipses indicate the futue light-cones. If we use outgoing athe than ingoing Eddington-Finkelstein coodinates, we get of couse a white-hole spacetime which is just the time evesed vesion of the black-hole spacetime. ct c t h h 48

49 Fo 0 < Λ < Λ c we have two hoizons. The inne one at = h1 is known as the blackhole hoizon while the oute one at = h2 is known as the cosmological hoizon. Again, the plots below show the adial light ays in Schwazschild and in ingoing Eddington-Finkelstein coodinates with the little ellipses indicating the futue light-cones. / t is timelike between the two hoizons, i.e., in this egion the spacetime is static and an obseve is fee to move in the diection of inceasing o deceasing. In the othe two egions / t is spacelike. An obseve is foced to move in the diection of deceasing. Fom the static egion, an obseve can send signals to the inteio non-static egion 0 < < h1 but he cannot eceive signals fom thee. By contast, he can eceive signals fom the exteio non-static egion h2 < < but he cannot send signals to this egion. Again, an analogous constuction with outgoing Eddington-Finkelstein coodinates gives a white hole. ct c t h1 h2 h1 h2 If Λ appoaches Λ c fom below, the two hoizons mege at = 3 S /2. Fo Λ c < Λ < thee is no hoizon. / t is eveywhee spacelike, i.e., the spacetime is non-static. The metic is egula on the entie domain 0 < < in Schwazschild coodinates, so thee is no need to intoduce othe coodinates. The timelike vecto field / can be intepeted asfutue-pointing o as past-pointing. In the fist case, the cuvatue singulaity at = 0 sucks in all signals and mateial bodies, but it is not a black hole because thee is no hoizon. As the singulaity is not visible to any obseve until he eally aives thee, one wouldn t call this a naked singulaity. In the second case all signals and mateial bodies move away fom the singulaity, so the singulaity is visible to an obseve at an abitaily lage distance fom the cente and the singulaity is indeed naked. The plot on the next page shows the adial light ays in Schwazschild coodinates. If the little ellipses indicate the futue light cones, the singulaity is epellent and not naked. 49

50 ct We will now discuss the lightlike and timelike geodesics in the equatoial plane of the Kottle spacetime. They ae to be deived fom the Lagangian Lx,ẋ = 1 1 S 2 Λ 3 2 c 2 ṫ 2 ṙ s Λ + 2 ϕ The t and ϕ components of the Eule-Lagange equation give us two constants of motion, E = 1 S Λ 3 2 c 2 ṫ, We conside the lightlike geodesics fist. Fo them we have and thus ṙ 2 1 S Λ 3 2 c 2 ṫ ϕ 2 1 S Λ = 3 2 ṙ 2 L = 2 ϕ. 1 S Λ + 2 ϕ 2 = S Λ 3 2 c 2 ṫ2 ϕ + ṙ2 1 2 ϕ 2 1 S Λ + 2 = 0, S Λ c 2 E c 4 1 S Λ 2L d 2 E 2 4 = dϕ c 2 L S Λ 3 2, 50 2,

51 Diffeentiation with espect to ϕ yields 1 d 2 E 2 = 4 dϕ c 2 L + Λ S 2. 3 KOL 2 d 4 dϕ d 2 dϕ 4 d 3 2 d = 2 5 dϕ 3 dϕ 3 S d 4 dϕ, d 2 dϕ 2 d 2 3 = 2 dϕ 2 S. This second-ode diffeential equation detemines the lightlike geodesics, giving a unique solution fo any initial condition 0, d/dϕ0. We see that the cosmological constant has completely dopped out. Hence, the lightlike geodesics in the Kottle spacetime ae given by pecisely the same cuves in the coodinate pictue as without Λ tem. Until quite ecently it was thus thought that the cosmological constant had no effect on the lensing popeties of spheical objects. This conclusion, howeve, is wong, as pointed out by M. Ishak and W. Rindle [Phys. Rev. D 76, ]: As the angle between two light ays, o between a light ay and some efeence diection, has to be measued with the metic, and as the metic does depend on Λ, the lensing featues ae affected by Λ in a measuable way. We will demonstate this below by showing that the angula adius of the shadow of a Kottle black hole does depend on Λ. Asapepaationfothat, welookfociculalightlikegeodesics. Theyhavetosatisfyd/dϕ = 0 and d 2 /dϕ 2 = 0 which educes the second-ode diffeential equation to 0 = 3 S 2. So thee is a cicula light ay at exactly the same coodinate adius as in the Schwazschild spacetime. This is a conseuqnece of the fact that the diffeential equation fo light obits is independent of Λ. Howeve, ou calculation demonstates only that = 3 S /2 is necessay fo a cicula light ay. To demonstate that this light ay actually exists, we have to make sue that the coesponding constant of motion E 2 /L 2 is indeed non-negative. Fom the Kottle obit equation KOL we find with d/dϕ = 0 and = 3 S /2 that 0 = E2 c 2 L 2 + Λ S + 8, 27S 2 E 2 c 2 L 2 = S Λ 3 KCL which demonstates that the cicula light obit at = 3 S /2 exists as long as Λ Λ c = 4. 9S 2 This should not come as a supise. We aleady knew that fo Λ > Λ c no timelike o lightlike cuve could stay at a fixed adius value. With this knowledge of the cicula obit we can now calculate the angula diamete of the shadow just as in the Schwazschild case. We fix an obseve at adius O and conside all light ays that go fom the position of this obseve into the past. 51

52 They leave the obseve at an angle θ with espect to the adial line that satisfies y tanθ = lim x 0 x. Fom the Kottle metic we ead that x and y satisfy, in the desied limit, tanθ = dϕ 1 S Λ 1/2 3 2 d = O d/dϕ can be expessed with the help of the obit equation KOL, hence tan 2 θ = E 2 2 O c 2 L + Λ 2 3 By elementay tigonomety, 1 S Λ O 3 2 O = 4O 2O + S O. y O S Λ 3 3 O E 2. 3O O + S c 2 L 2 + Λ 3 θ x O sin 2 θ = sin 2 θ cos 2 θ+sin 2 θ = 1 cot 2 θ + 1 = 1 E 2 c 2 L + Λ 2 O 3 3 O + S O S Λ 3 3 O + 1 = O S Λ 3 3 O E 2 O 3 + Λ c 2 L O O + S + O S Λ 3 3 O = 1 S O Λ 3 2 O E 2 c 2 L 22 O The angula diamete of the shadow is found be equating E 2 /L 2 to the constant of motion that coesponds to the cicula light ay at = 3 S /2. Substituting fom KCL yields. sin 2 θ 0 = 1 S Λ O 3 2 O 4 Λ 27S 2 O 2 3 = 1 S Λ O 3 2 O Λc 3 Λ. O 2 3 As we assume that the obseve is at a fixed -coodinate, he must be in the static egion, i.e., the numeato must be positive. As a static egion exists only fo Λ < Λ c, the denominato must also be positive, so it is assued that indeed sin 2 θ 0 0. In addition, we have to make sue that sin 2 θ 0 1, i.e. we have to check if 1 S Λ? O 3 2 O Λ c 3 2 O Λ 3 2 O, f O = Λ c 3 3 O O + S 52? 0.

53 This is indeed tue, fo all O > 0, because Λ = Λ c was the case whee the function f has a double zeo and does not become negative anywhee in the domain 0 < <. To summaise, thee is a shadow fo any obseve in the static egion of a Kottle black hole with < Λ Λ c. We see that, fo an obseve at a given adius coodinate = O in a spacetime with a given S, the angula diamete of the shadow depends on Λ. This is an effect of Λ that can be measued if the mass and thus S is known. Recall that the adius coodinate can be detemined in pinciple by measuement, as a cicle at = O has cicumfeence 2π O and the sphee at = O has aea 4π 2 O. Now we tun to timelike geodesics. Fo them we use the pope time paametisation, i.e. 1 S Λ 3 2 c 2 ṫ 2 + ṙ 2 1 S Λ + 2 ϕ 2 = c We expess ṫ and ϕ in tems of the constants of motion E and L, 1 S Λ c 2 E 2 ṙ 2 2 L c 4 1 S Λ S Λ + = c 2, ṙ 2 = E2 L 2 c 2 2 +c2 1 S Λ 3 2 =: 2U E,L. As befoe, we have intoduced an effective potential U E,L such that we get a kind of enegy consevation law 2ṙ2 1 +U E,L = 0. Note that hee we find it convenient to wok with the deivative ṙ = d/dτ athe than with d/dϕ, as we did in the Schwazschild case. Fo this eason we use a diffeent symbol fo the potential, U E,L athe than V E,L. We fist investigate adial motion, L = 0. Then ṙ 2 = E2 c 2 c 2 1 S Λ 3 2, OTR 2ṙ = c 2 S 2 2Λ 3 ṙ, = c2 S Λ 3 c2 =: a. a gives the adial acceleation. We see that fo Λ < 0 we have a < 0, i.e., the acceleation is always pointing inwads as usual. Fo Λ > 0, howeve, the acceleation is pointing outwads fo big. This eflects the fact that a positive cosmological constant has a epellent effect. Thee is a adius value eq whee this epulsion balances the gavitational attaction, 0 = a eq = c2 S 2 2 eq + Λ 3 c2 eq, 3 eq = 3 S 2Λ. A paticle that is placed at this adius coodinate will stay put. Actually, as this equilibium is unstable, the paticle will move eithe inwads o outwads as soon as it is petubed a little bit. 53

54 As we know that fo Λ > Λ c it is impossible fo a paticle to stay at a fixed adius coodinate, the equilibium can exist only fo 0 < Λ Λ c. This can indeed be veified fom ou equation fo eq because OTR implies E 2 = 1 S 1+ Λ3 eq c 4 eq 3 S and, as E 2 /c 4 must be non-negative, 1 0 = E2 c 2 1 S Λ c 2 eq 3 2 eq, 3 3 S 27 = 1 2 eq 8 = 1 S 1+ 1 eq 2 S 3 eq 3 = 92 S Λ 4 = = Λ Λ c. Next we conside cicula timelike geodesics. They ae chaacteised by ṙ = 0 and = 0, i.e., by U E,L = 0 and U E,L = 0. These two equations ead 0 = E2 c 2 L 2 2 +c2 1 S Λ 3 2, 0 = 2L2 1 S 3 Λ L 2 S c2 2Λ 2 3. As E 2 /c 2 0, the fist equation equies that f 0 and thus Λ Λ c. This was clea aleady befoe because we know that timelike cuves can stay at a constant value only in the static egion. The second equation can be solved fo L 2, L 2 = To assue L 2 0, we must have eithe o c 2 S 1 2Λ3 3 S 2 3 S 1 2Λ3 3 S 0 and < 3 2 S 1 2Λ3 3 S 0 and > 3 2 S. The fist pai of conditions is in contadiction with Λ Λ c, so the second pai of conditions must be satisfied. Fo Λ 0, it educes to > 3 S /2. Fo 0 < Λ Λ c, it can be ewitten as 3 2 S < eq with eq fom above. These two bounday values fo cicula timelike geodesics ae easily undestood: Fo 3 S /2 the velocity appoaches the velocity of light, while fo = eq the velocity becomes zeo. 54. S eq

55 In Woksheet 7 we will demonstate the following: In the case of a positive cosmological constant, stable cicula timelike geodesics exist only fo Λ < Λ b = 16/5625S 2. They ae situated between an innemost stable cicula obit ISCO and an outemost stable cicula obit OSCO. Fo Λ 0 we have ISCO 3 S and OSCO, while fo Λ Λ b both ISCO and OSCO tend to 15 S /4. We will also show in Woksheet 7 that fo 0 < Λ < Λ b thee is a heteoclinic obit, i.e., an obit that appoaches an unstable cicula obit at some adius 1 fo t and anothe unstable cicula obit at some adius 2 fo t, with 1 2. In the neighbouhood of this heteoclinic obit thee ae obits that whil altenately nea 1 and nea Reissne-Nodstöm black holes The Reissne-Nodstöm metic is the unique spheically symmetic and asymptotically flat solution to the Einstein equation without a cosmological constant with the enegy-momentum tenso of a Maxwell field. It is a static metic that descibes the spacetime aound a spheically symmetic chaged object. Just as in the Schwazschild case, the field is static even fo a pulsating souce. We do not give a deivation of the Reissne-Nodtsöm metic which equies solving the coupled system of the Einstein and the Maxwell equations unde the assumption of spheical symmety. The metic eads g = c 2 1 S + 2 Q 2 dt 2 + d 2 1 S + 2 Q dϑ 2 +sin 2 ϑdϕ 2. It was found independently by H. Reissne 1916, H. Weyl 1917 and G. Nodstöm It contains two integation constants S and Q with the dimension of a length. Compaison with the Newtonian theoy fo lage shows that, as in the Schwazschild metic, S = 2GM c 2 whee M is the mass of the cental object. The intepetation of Q can be eadfom the electic field that is associated with the metic: 2 Q = GQ2 4πε 0 c 4 whee Q is the electic chage. Hee we assume that the cental object caies an electic chage. If one believes in the existence of magnetic monopoles, one could also conside a magnetic chage. We can match the Reissne-Nodstöm solution at some appopiate adius value to an inteio solution, e.g. to a chaged pefect fluid, to get a model fo a static chaged sta. Hee we ae inteested in Reissne-Nodstöm black holes, so we will assume that the electo-vacuum Reissne-Nodstöm metic is valid all the way down to = 0. Just as the Schwazschild metic, the Reissne-Nodtsöm metic has a cuvatue singulaity at = 0. We will now investigate if this singulaity is hidden behind a hoizon. Fom the metic we ead that a coodinate singulaity of the same kind as the one at = S in the Schwazschild case occus whee 1 S + 2 Q 2 = 0. 55

56 This gives a quadatic equation fo, 2 S + 2 Q = 0. Any eal and positive solution of this equation gives a hoizon. Solving the quadatic equation yields ± = 1 S ± S Q. H We have a two hoizons if 4 2 Q < 2 S black hole, b one degeneate hoizon if 4 2 Q = 2 S c no hoizon if 4 2 Q > 2 S naked singulaity. extemal black hole, and We conside case a in some detail and teat the othe two cases only biefly late. So let us assume that 4 2 Q < 2 S which implies that H gives us two eal adius values, 0 < < +. To intoduce a genealised totoise coodinate and, theeupon, genealised ingoing Eddington- Finkelstein coodinates, we conside the equation fo adial light ays, 0 = c 2 1 S + 2 Q 2 dt 2 + d ±cdt = 1. S + 2 Q 2 The genealised totoise coodinate is thus to be defined as 2 d ˆ = 2 S +Q 2 d 2 1 S + 2 Q 2, The adial light ays ae then given by = ln ln. ±ct t 0 = ˆ. If we define ingoing Eddington-Finkelstein coodinates t,,ϑ,ϕ by ct = ct + ˆ, the ingoing adial light ays ead and the outgoing ones ead c t t 0 = c t t 0 = 2ˆ. 56

57 The two plots on the ight show the adial light ays in the Reissne- Nodstöm black-hole spacetime, fist in the Schwazschild-like coodinates and then in ingoing Eddington-Finkelstein coodinates. The egion I is simila to the exteio pat of the Schwazschild spacetime. Hee t is timelike and an obseve can move in the diection of deceasing o inceasing. No signal fom the othe side of the hoizon at = + can each an obseve in egion I, so it is indeed justified to speak of a black hole. The egion II is non-static. Just as in the inteio pat of the Schwazschild spacetime, an obseve must move in the diection of deceasing. Howeve, in contast to the Schwazschild spacetime this egion does not extend up to the singulaity at = 0. Thee is anothe static egion III between = 0 and =. In this egion an obseve is again fee to move in the diection of deceasing o inceasing. As indicated by the futue light cones, thee ae timelike cuves that end in the singulaity, but thee ae also timelike cuves that escape fom the singulaity. We will show that, in paticula, timelike geodesics neve each the singulaity. ct ct III III II + II + I I Note the diffeences between the Reissne-Nodstöm spacetime and the Kottle spacetime: The cosmological constant modifies the Schwazschild metic at big adii, while the chage modifies the Schwazschild spacetime at small adii. In cases whee thee ae two hoizons, in the Kottle spacetime we have a static egion between two non-static egions, while in the Reissne-Nodtsöm spacetime we have a non-static egion between two static egions. We will now demonstate that in the Reissne-Nodtsöm spacetime a timelike geodesic cannot each the singulaity. As always in spheically symmetic spacetimes, we need only conside timelike geodesics in the equatoial plane. We use the constants of motion E = Lx,ẋ ṫ = c 2 1 S + 2 Q 2 ṫ 57

58 and L = Lx,ẋ ϕ = 2 ϕ, and we paametise the timelike geodesic by pope time, c 2 = c 2 1 S + 2 Q 2 ṫ2 + ṙ 2 1 S + 2 Q 2 We substitute the constants of motion into the last equation: c 2 = E 2 c 2 1 S + 2 Q 2 + ṙ 2 1 S + 2 Q ϕ 2. + L2 2, ṙ 2 = E2 c 2 L 2 2 +c2 1 S + 2 Q 2, ṙ 2 + L2 2 1 S + 2 Q 2 + c 2 = E2 c 2 +c2 S c2 2 Q 2. The tem in the backet on the left-hand side is positive between = 0 and =, so the left-hand side is positive, hence E 2 c 2 +c2 S > c2 2 Q 2, E 2 c 2 2 +c 2 S > c 2 2 Q. Fo 0, the left-hand side appoaches 0. Howeve, the ight-hand side is bigge than 0 if Q 0, so must be bounded away fom 0. So an obseve in fee fall will neve aive at the singulaity. One can show that in egion III any feely falling obseve eaches t = in a finite pope time. So thee is the possibility to extend the spacetime fom egion III to the futue. Such a possibility did not exist fo the Schwazschild spacetime because in the latte case thee was no egion III. We will now constuct the maximal extension of the Reissne-Nodstöm black-hole spacetime. Fo the sake of compaison, we ecall the maximal extension of the Schwazschild black-hole spacetime which was given by the Kuskal diagam of p.15. In this diagam, light ays go unde 45 o with espect to the hoizontal. We can peseve this popety, and at the same time map the entie spacetime onto a compact set, if we use coodinates w 1,w 2 defined by w 1 = actanu+v, w 2 = actanu v. This gives us the following diagam fo the Kuskal extension of the Schwazschild metic: 58

59 w 2 = 0 = = = S II = S I I = S II = S = = = 0 w 1 Such a spacetime diagam whee a evey point epesents a sphee, i.e., the angle coodinates ae suppessed, b light ays go unde 45 o, and c the entie spacetime is mapped to a compact set, is called a Penose diagam. The constuction of a Penose diagam is possible fo any spheically symmetic and static spacetime. Now we want to constuct the Penose diagam fo the extension of the Reissne-Nodstöm black-hole spacetime. We begin with the domain coveed by ingoing Eddington-Finkelstein coodinates, i.e., with egions I, II and III as shown in the spacetime diagams on the pevious page. The hoizons at and + ae geneated by adial light ays, so they must make an angle of 45 o with the hoizontal. The singulaity at = 0 is vetical which eflects the fact that it can be avoided by timelike cuves, see pictue on the ight. = 0 III = II = + I = 59

60 We can glue a egion II, which is isometic to egion II, to the futue of egion III, so that obseves can escape fom egion III into this egion II. We can futhe extend the spacetime by gluing a egion I, which is isometic to egion I, to the futue of egion II. Fo an obseve in egion I, the singulaity at the cente of egion III is hidden behind a hoizon, so fo him it is a black hole. Fo an obseve in egion I, howeve, it is a white hole. = I = + = II = 0 III = II = + I = In the middle of the figue we can mach anothe copy of egion III, and we can infinitely extend the diagam to the futue and to the past, see the pictue on the next page. Evey singulaity is at the cente of a black hole fo obseves in some asymptotically flat egions and at the cente of a white hole fo obseves in some othe asymptotically flat egion. In this sense, evey Reissne-Nodstöm black hole is at the same time a white hole. If we think of a Reissne-Nodstöm black hole as being the esult of gavitational collapse, the past extension of the oiginal spacetime I, II and III has no physical elevance, similaly to the lowe left-hand potion of the Kuskal diagam; one would have to eplace this pat by an inteio solution. The futue extension, howeve, would not be cut away by the inteio of the collapsing sta, so the obsevation that one can escape fom egion III would still be tue. 60

61 The pesence of the egion III and the possibility of extending the spacetime towads the futue of this egion is a completely new featue which has no analogue in the Schwazschild o Kottle case. While in the Schwazschild spacetime any obseve that has cossed the hoizon will inevitably end up in the singulaity, a chaged black hole seems to be moe benign: Only those obseves who delibeately acceleate towads = 0 will hit the singulaity; all othe obseves that have enteed into egion II, in paticulaly the feely falling ones, will escape though egion III into anothe asymptotically flat spacetime egion I. At least that s what the mathematical model of the Reissne-Nodstöm spacetime descibes. Howeve, we have to take into account that, accoding to ou pesent knowledge, all celestial bodies have a net chage that is vey small. Theefoe, we expect fo all black holes that is vey close to 0. As the cuvatue becomes infinite fo 0, this means that the tidal foces ae so stong at that a macoscopic obseve will be ipped apat befoe enteing the egion III. The situation is diffeent fo elementay paticles. Howeve, even in this case thee is a minimum adius beyond which the classical spacetime desciption is no longe applicable and a yet-to-be-found quantum theoy of gavity would have to take ove. It is believed that this beak-down of the classical spacetime theoy cetainly takes place at a distance fom = 0 of the ode of the Planck length, l P. Following this line of thought, one may conclude that fo elementay paticles the escape though egion III is possible if l P. We will biefly estimate the chage fo a black hole that coesponds to = l P. Fom l P = 1 2 S S 2 42 Q we find S 2l P 2 = 2 S 42 Q, l P S l 2 P = 2 Q. 61

62 To give a numeical example, conside a stella black hole of 10 Sola masses. Then S = 30km and with l P = m gives The coesponding chage Q satisfies 2 Q = m 2. Q 2 = 4πε 0c 4 G 2 Q. With c = m/s, G = m 3 /kg 3 s 2 and ε 0 = C 2 s 2 /m 3 kg, this gives Q 82C. As fa as we know, stas have a net chage of a few hunded Coulombs at most. So if such a sta collapses to an object that is popely descibed by a Reissne-Nodstöm black hole, and if thee ae no additional pocesses that would chage the black hole, then is not much bigge than the Planck length. Note that by the chage, Q, that coesponds to = l P is popotional to M as long as S l P. So fo a supemassive black hole of 10 9 Sola masses, like the one at the cente of M87, we have Q C. It is not easy to think of a pocess that would give a supemassive black hole a net chage that is consideably bigge than this value. To sum up, we believe that fo chaged non-otating black holes in Natue the inne hoizon is vey close to the singulaity, pobably so close that the classical theoy is not applicable thee. We now biefly conside the case of an extemal Reisne-Nodstöm black hole, 2 S = 42 Q. Then the adial light ays satisfy 0 = c 2 1 S + 2 S 4 2 dt 2 + d 2 1 S + 2 S 4 2 d ±cdt = 1 2. S 2 Again, we define ingoing Eddington-Finkelstein coodinates t,,ϑ,ϕ by, whee now ct = ct + ˆ, ˆ = 4 2 d 2 S 2 As befoe, the ingoing adial light ays ead = S + S ln 2 S. and the outgoing ones ead c t t 0 = c t t 0 = 2ˆ. 62

63 The two plots below show the adial light ays in the extemal Reissne-Nodstöm black-hole spacetime, fist in the Schwazschild-like coodinates and then in ingoing Eddington-Finkelstein coodinates. It is simila to the non-extemal black hole, just with the egion II missing: The static egion I is sepaated fom the static egion III by a degeneate hoizon. ct III I S /2 ct III I S /2 63

64 With the egion II gone, the Penose diagam fo the extemal Reissne-Nodstöm black hole has the following shape. As in the non-extemal black-hole case, the figue extends up and down to infinity. Finally, we look at the case of a naked singulaity, 4Q 2 > 2 S. Then the adial light ays ae given by ±c d t t 0 = 1 S + 2 Q 2 = + S 2 ln 2 S +Q 2 22 Q 2 S 4Q 2 2 S actan 2 S. 4Q 2 2 S 64

65 Thee ae no hoizons, so thee is no need fo intoducing othe coodinates. A plot of the light ays is shown below. ct The spacetime is static eveywhee. Obseves can feely move in the diection of inceasing o deceasing. The singulaity at = 0 is exposed to the eyes of obseves anywhee in the spacetime. 6. Ke black holes All known celestial bodies otate, so it is vey likely that also black holes ae otating. A black hole that otates at a constant ate is descibed by a spacetime that is stationay and axisymmetic, i.e., in appopiately chosen coodinates t,, ϑ, ϕ the metic coefficients ae independent of t and ϕ. If we ignoe the cosmological constant, and if we assume that the black hole is unchaged, we have thus to look fo a solution to Einstein s vacuum field equation, R µν = 0, that is stationay and axisymmetic. Fo spheically symmetic vacuum solutions the situation is quite simple: Thee is only a onepaamete family of such solutions, given by the Schwazschild metic. By contast, the class of stationay and axisymmetic vacuum solutions is vast. The exteio egions aound two otating stas with diffeent inteio stuctue will be completely diffeent fom each othe and fom the egion aound a otating black hole. It is fa fom obvious by which popety the egion aound a otating black hole is distinguished among all stationay and axisymmetic vacuum solutions. The vacuum solution that actually descibes otating black holes was found by R. Ke in 1963 almost half a centuy afte Schwazschild had found the spheically symmetic solution. It 65

66 tuned out that it is distinguished among all stationay and axisymmetic vacuum solutions by the popety that the eigenvalues of the cuvatue tenso ae in a cetain sense degeneate. The technical tem is: The metic is of Petov type D. A systematic deivation of the Ke metic would have to be based on this popety, see e.g. S. Chandasekha: The Mathematical Theoy of Black Holes. This deivation cannot be given hee. Instead, we will biefly sketch how the Ke metic can be found by a pocess of tial and eo, making a cetain ansatz and modifying it only whee necessay and in the most simple way. We will be guided by thee assumptions: 1. The metic depends on two paametes: the mass M and the angula momentum J of the cental object. 2. In the weak field the metic appoaches the Minkowski metic. 3. In the case of vanishing angula momentum, J = 0, the metic educes to the Schwazschild solution. A maximally symmetic object which otates foms an oblate spheoid. Appopiate coodinates ae, theefoe, oblate spheoidal coodinates, ϑ, ϕ which ae connected to Catesian coodinates x,y,z by x = 2 +a 2 cosϕsinϑ, y = 2 +a 2 sinϕsinϑ, z = cos θ. Hee a has the dimension of a length, which suggests that it is elated to the angula momentum J by a = J/Mc. The pictue shows the the sufaces = constant ellipsoids and the sufaces ϑ = constant hypeboloids. z x 2 +y 2 66

67 To detemine the Minkowski metic in the oblate spheoidal coodinates we need the diffeentials dx, dy, dz, dx = dy = 2 +a 2 cosϕsinϑd 2 +a 2 sinϕsinϑdϕ+ 2 +a 2 cosϕcosϑdϑ, 2 +a 2 sinϕsinϑd + 2 +a 2 cosϕsinϑdϕ+ 2 +a 2 sinϕcosϑdϑ, dz = cosϑd sinϑdϑ. If we inset this into the Minkowski metic we get η = c 2 dt 2 +dx 2 +dy 2 +dz 2 = c 2 dt 2 + ρ a 2 sin 2 ϑdϕ 2 +ρ 2 dϑ 2 2 +a 2d2 whee we have defined ρ 2 := 2 +a 2 cos 2 ϑ. We will use now this fom of the metic as a basis to get an idea of the fom of the Ke metic. As the Ke metic should educe to the Schwazschild metic it seems easonable to extend g in M to the fom g = 2 +a 2 cos 2 ϑ 2 +a 2 2m = ρ2 whee we defined := 2 +a 2 2m with m := GM = S c 2 2. Fo the the g tt component it seems easonable to use the Schwazschild expession but to eplace 2 by ρ 2, which is also suggested by g ϑϑ = ρ 2 in M. This gives g tt = c 2 1 2m ρ 2 = c 2 2 +a 2 cos 2 ϑ 2m ρ 2 M = c 2 a2 sin 2 ϑ ρ 2. As we want to descibe the metic of a otating souce, we expect that we need a non-vanishing g tϕ component: A otating souce would twist the integal cuves of t so that they ae no longe othogonal to the hypesufaces t = constant. So the simplest ansatz would be to take the g, g tt, g ϑ,ϑ and g ϕϕ fom above and supplement it with an unspecified g tϕ. If one plugs this ansatz into the vauum field equation, one finds that it cannot be solved unless am = 0. The next ty would be to add an unspecified tem to g ϕϕ. With this ansatz, the vacuum field equation can indeed be satisfied, fo any m and any a. The solution is the Ke metic in Boye-Lindquist coodinates, g = c 2 1 2m ρ 2 dt 2 + ρ2 d2 +ρ 2 dϑ 2 +sin 2 ϑ 2 +a 2 + 2ma2 sin 2 ϑ dϕ 2 4masin2 ϑ cdtdϕ. ρ 2 ρ 2 67

68 The Ke metic was found in 1963 by R. Ke [ Gavitational field of a spinning mass as an example of algebaically special metics, Phys. Rev. Lett. 11, ]. The epesentation of the Ke metic in tems of the coodinates t,,ϑ,ϕ which we have given on the peceding page was intoduced fou yeas late by R. Boye and R. Lindquist [ Maximal analytic extension of the Ke metic, J. Math. Phys. 8, ]. The Boye-Lindquist coodinates ae best adapted to the symmeties of the Ke metic and allow the most convenient epesentation of the geodesics, although fo some othe puposes othe coodinates might be moe appopiate. In the following section we discuss some geneal featues of the Ke metic in Boye Lindquist coodinates; then we study in some detail the lightlike and timelike geodesics which ae cucial fo linking the Ke metic to obsevations. 6.1 Popeties of the Ke metic a Asymptotic flatness Fo, the Ke metic appoaches the Minkowski metic in odinay spheical pola coodinates, i.e., the metic is asymptotically flat. This obsevation is cucial fo the intepetation of the two paametes m and a on which the Ke metic depends. We have aleady anticipated that m = GM c 2, a = J Mc whee M is the mass and J is the angula momentum of the souce. Keep in mind that both m and a have the dimension of a length. Indeed, if we take the deviation fom the Minkowski metic up to fist ode in 1/ into account, we get the Schwazschild metic with 2m = S. This demonstates that the identification of m with GM/c 2 can be justified by compaison with the Newtonian theoy in the same way as fo the Schwazschild metic. The identfication of a with J/Mc can be justified in a simila way. Ke made this identification by compaison with a cetain appoximation method Einstein-Infeld-Hoffmann appoximation fo spinning paticles. This was late cooboated by a caeful analysis of angula-momentum multipoles fo stationay and axisymmetic spacetimes. As a should vanish fo a souce with J = 0, the identification a = J/Mc can be guessed, up to a possible numeical facto, just by a dimensional analysis. Note the following special cases of the Ke metic in Boye-Lindquist coodinates: a = 0 and m 0: Schwazschild metic, a 0 and m = 0: Minkowski metic in spheoidal coodinates, a = 0 and m = 0: Minkowski metic in spheical pola coodinates. ThefistcasecoespondstosendingJ 0withM 0keptfixed. Thesecondcasecoesponds to sending J 0 and M 0 in such a way that the quotient J/M emains fixed. The thid case coesponds to sending J 0 and M 0 in such a way that the quotient J/M appoaches zeo. b The ange of the coodinates The similaity with spheical pola coodinates suggests to have t unning ove all of R, ϑ and ϕ coodinatising the 2-sphee in the usual fashion, and unning fom 0 to. Howeve, as to the ange of the coodinate thee is actually no eason fo this estiction unless a = 0. Fom the metic we ead that t = t 0, = 0 is a 2-sphee, paametised by ϑ and ϕ, that has 68

69 a finite aea, see Woksheet 8. It is not a point as in the case of spheical pola coodinates on flat space. Of couse, one has to make sue that all the metic coefficients ae non-singula on this sphee. This is indeed tue, unless in the equatoial plane ϑ = π/2, see item c below. If we emove the ing at ϑ = π/2 fom the 2-sphee t = t 0, = 0, the two emaining hemisphees ae pefectly egula with a finite aea. This allows extending the spacetime to the domain of negative values. A convenient way of plotting this situation is by using e /m fo the adial coodinate, see the pictue below. Then = 0 is epesented as a sphee with the domain of positive values on the exteio and the domain of negative values on the inteio. The oigin coesponds to =. In summay, the ange of the Boye-Lindquist coodinates of the Ke metic with a 2 > 0 is t R, R, ϑ,ϕ S 2. Fo a = 0, thee is a pointlike singulaity at = 0 and the coodinate is esticted by 0 < <. The domain < < 0 could be consideed as anothe spacetime which is disconnected fom the domain 0 < <. The Schwazschild metic with positive m and anging ove the negative half-axis is isometic with the Schwazschild metic with negative m and anging ove the positive half-axis. As in this spacetime the cuvatue singulaity at = 0 is naked, it is usually believed to be unphysical. c The ing singulaity The metic coefficient g tt of the Ke metic becomes singula at ρ 2 = 0. In the Schwazschild limit a 0 the equation ρ 2 = 0 educes to 2 = 0 which is known to be a cuvatue singulaity. One would theefoe expect that ρ 2 = 0 is a cuvatue singulaity also in the Ke metic with a 0. This is indeed tue as can be veified by calculating e.g. the Ketschmann scala R µνστ R µνστ which e /m diveges fo ρ 2 0. As ρ 2 = 2 +a 2 cos 2 ϑ, the condition ρ 2 = 0 is equivalent to 2 = 0 and ϑ = π 2 if a 0. This is a ing, see pictue on the ight. The cicumfeence of this ing can be easily calculated fom the metic as we will demonstate in Woksheet 8. = 0 d The hoizons The metic coefficient g becomes singula if = 0. In the Schwazschild case whee we estict to > 0 this coesponds to 1 2m/ = 0, i.e., to the coodinate singulaity at = S = 2m. We know that this coodinate singulaity in the Schwazschild metic can be emoved by a tansfomation e.g. to Eddington-Finkelstein coodinates and that in the coespondingly extended spacetime the suface = 2m plays the ole of a hoizon. Theefoe, it is natual to assume that also in the case a 0 the equation = 0 gives a hoizon. We will demonstate late, when we have the equation fo lightlike geodesics at ou disposal, that this is indeed tue. 69

70 The equation is a quadatic equation fo with the solution We distinguish thee cases: 0 = = 2 2m +a 2 ± = m± m 2 a 2. 0 < a 2 < m 2 : Thee ae two eal solutions, 0 < < +, i.e., two hoizons. The oute hoizon at + is an event hoizon that hides its inteio with the ing sinulaity fo an outside obseve. So in this case we have a Ke black hole. 0 < a 2 = m 2 : Thee is a double solution = + = m which is eal and positive if m > 0, i.e., thee is one degeneate hoizon. Again, this is an event hoizon fo an obseve in the exteio domain. In this case we speak of an exteme Ke black hole. m 2 < a 2 : Thee ae no eal solutions and thus no hoizons. The ing singulaity is exposed to the eyes of any obseve, even fo an obseve at an abitaily lage distance. In this case we speak of a Ke naked singulaity. In the black-hole case, the egion > + is called the domain of oute communication because this is the egion fom which an obseve who is fa away fom the black hole can eceive signals. Fo planets, stas o galaxies, a = J/Mc is usually bigge than m = GM/c 2. This seems to suggest that such an object would end up as a naked singulaity if it undegoes gavitational collapse. Howeve, this does not take into account that gavitational collapse is believed to be peceded by a kind of explosion. Duing such an explosion the body can lose not only mass but also angula momentum so that eventually it may end up as a black hole. In 1969 Roge Penose fomulated the socalled Cosmic Censoship Hypothesis accoding to which any gavitational collapse that stats out fom physically ealistic initial conditions leads to a black hole and not to a naked singulaity. The name efes to the idea that Natue acts like a censo pohibiting naked singulaities. Although seveal patial esults have been achieved, the Cosmic Censoship Hypothesis has not been made into a mathematical theoem so fa. Of couse, the cucial point is to give a mathematically pecise fomulation of which initial conditions should be consideed as physically ealistic. d The Killing vecto fields t and ϕ The metic coefficients ae independent of t and of ϕ, so the vecto fields t and ϕ ae Killing vecto fields. On a domain whee t is timelike g tt < 0, we may intepet the tansfomation t t+constant as a time tanslation; invaiance unde this tansfomation then means that the metic is stationay. Similaly, on a domain whee ϕ is spacelike g ϕϕ > 0, we may intepet the tansfomation ϕ ϕ+ constant as a spatial otation about the z axis; invaiance unde this tansfomation then means that the metic is axisymmetic. Fom the fact that the metic appoaches the Minkowski metic fo it follows that the conditions g tt < 0 and g ϕϕ > 0 ae indeed satisfied fo lage. In the inne pat of the spacetime, howeve, these conditions ae violated. The egion whee g tt > 0 is bounded by two sufaces which ae known as the stationaity limit sufaces. They ae detemined by 0 = g tt = 1 2m ρ 2, 0 = 2 +a 2 cos 2 ϑ 2m, a 2 sin 2 ϑ =. 70

71 Clealy, these sufaces lie in the egion whee 0 and equality, = 0, holds exactly at the poles sinϑ = 0 if a 0. In the black-hole case 0 < a 2 m 2 one stationaity limit suface lies in the egion > + and touches the oute hoizon at the poles, while the othe stationaity limit suface lies in the egion < and touches the inne hoizon at the poles. The egion between the oute stationaity limit suface and the oute hoizon is known as the egoegion. This name efes to the fact that enegy can be extacted fom a Ke black hole by dopping a paticle into the egoegion, splitting it into two paticles thee and getting one of them back with a highe enegy than the compound had had befoe. This socalled Penose pocess will be discussed late when we have the equations fo timelike geodesics in the Ke spacetime at ou disposal. In the egoegion a timelike vecto must have a nonvanishing ϕ component, as can be ead fom the metic, i.e., all obseves must otate. In the Schwazschild case a 0 the oute stationaity limit suface meges with the hoizon at = 2m while the inne one meges with the cuvatue singulaity at = 0, so thee is no egoegion in this case. The pictue below shows fo a Ke black hole with a = 0.9m the egoegion dak shaded and the est of the domain whee g tt > 0 light shaded. We have indicated the sphee = 0 by a ed dashed line, the ing singulaity by a ed blob, the hoizons by solid lines, and the stationaity limit sufaces by dotted lines. The pictue is otationally symmetic about the vetical axis. = + e /m = = 0 71

72 The egion whee g ϕϕ < 0, i.e. 2 +a 2 ρ 2 +2ma 2 sin 2 ϑ < 0, lies completely in the domain < 0 and touches the ing singulaity. It is known as the causality violating egion. In this egion the ϕ lines, which ae cicles in the spacetime, ae timelike. An obseve can move along such a line and theeby etun to his own past. Closed timelike cuves lead to the familia kind of paadoxa, e.g., to the possibility of killing one s paents befoe being bon. In the case of a Ke black hole the egion with closed timelike cuves is accessible only fo those obseves who actually jump into the black hole; fo an obseve in the domain of oute comunication it is without elevance. The plot on the ight shows the causality violating egion shaded fo a Ke black hole with a = 0.9m. As befoe, we have maked the sphee = 0 by a ed dashed line andthe ingsingulaitybyaedblob. Onthebounday of the causality violating egion the ϕ lines ae lightlike. = 0 e /m Although we have seen that, in the black-hole case 0 < a 2 m 2, the vecto field t fails to be timelike inside the egoegion, it is nonetheless justified to say that the spacetime is stationay on the entie domain of oute communication. The eason is that, nea any adius value 0 with + < 0 <, it is possible to find a constant Ω such that the vecto field t +Ω ϕ is timelike nea 0. So thee is always a family of obseves, defined on a spheical shell 0 ε < < 0 +ε, who see a time-independent metic. The vecto field t is distinguished among all vecto fields of the fom t +Ω ϕ by the fact that it is hypesuface-othogonal twist-fee in the limit. e Stationay obseves and ZAMOs Obseves whose woldlines ae t lines ae called stationay. They exist only on the domain whee g tt < 0, i.e. between the oute stationaity limit suface and = and between the inne stationaity limit suface and =. In paticula, stationay obseves do not exist in the egoegion. By contast, the hypesufaces t = constant ae spacelike on the domain whee g > 0, i.e., whee > 0. In the black hole case, this coves the entie domain of oute communication including the egoegion. If the hypesufaces t = constant ae spacelike, the woldlines othogonal to these hypesufaces ae timelike, i.e., they ae possible woldlines of obseves. These obseves ae known as Zeo Angula Momentum Obseves, often abbeviated as ZAMOs. In the limit ± the stationay obseves and the ZAMOs coincide. 72

73 6.2 Lightlike and timelike geodesics of the Ke metic The geodesics of the Ke metic have a vey ich stuctue. Even afte having woked with them fo many yeas one still discoves featues one hadn t known befoe. Theefoe we can give only a few selected esults hee. In view of physical intepetation, we ae inteested in lightlike geodesics light signals o classical photons and in timelike geodesics feely falling massive paticles. Recall that fo any metic the geodesics ae the solutions to the Eule-Lagange eqations with the Lagangian Lx,ẋ = 1 2 g µνxẋ µ ẋ ν. The Lagangian itself is always a constant of motion g µν xẋ µ ẋ ν = ε whee ε = 0 fo lightlike geodesics and ε = c 2 fo timelike geodesics paametised by pope time. In the following we will not discuss spacelike geodesics but they ae also coveed by ou equations if we allow ε to be a negative constant. Fo the Ke metic in Boye-Lindquist coodinates x = t,,ϑ,ϕ, the metic coefficients ae independent of t and of ϕ. The coesponding components of the Eule-Lagange equation give us two constants of motion, E = Lx,ẋ ṫ = g tt ṫ g tϕ ϕ and L = Lx,ẋ ϕ = g tϕ ṫ+g ϕϕ ϕ. Up to dimensional factos, E is to be intepeted as the enegy of the classical photon o the massive paticle, while L is to be intepeted as the z component of its angula momentum. It would be moe appopiate to wite L z instead of L, but fo the sake of bevity we stick with L. The equations fo E and L can be solved fo ṫ and ϕ, We calculate g ϕϕ E +g tϕ L = g tt g ϕϕ g 2 tϕ ṫ, g tϕ E +g tt L = g tt g ϕϕ g 2 tϕ ϕ. D := g tt g ϕϕ gtϕ 2 = c2 1 2m sin 2 ϑ 2 +a 2 + 2ma2 sin 2 ϑ 4m2 2 a 2 c 2 sin 4 ϑ ρ 2 ρ 2 ρ 4 = c2 sin 2 ϑ ρ 2 2m 2 +a 2 ρ 2 +2ma 2 sin 2 ϑ 4m 2 2 a 2 sin 2 ϑ ρ 4 = c2 sin 2 ϑ ρ a 2 ρ 42 ρ 2 2ma 2 sin 2 ϑ+2m 2 +a 2 ρ 2 ρ2 = c2 sin 2 ϑ ρ 2 2 +a 2 +2m 2 +a 2 cos 2 ϑ = c 2 sin 2 ϑ. 73

74 Upon inseting this expession into the equations fo ṫ and ϕ we get ρ 2 ṫ = sin 2 ϑ 2 +a 2 ρ 2 +2ma 2 sin 2 ϑ E 2ma sin 2 ϑcl c 2 sin 2, G1 ϑ ρ 2 ϕ = c2masin 2 ϑe +c 2 ρ 2 2m L c 2 sin 2 ϑ. G2 Fo any choice of the constants of motion, these two equations detemine t and ϕ as functions of the cuve paamete if and ϑ ae known as functions of the cuve paamete. Howeve, the thid constant of motion Lx,ẋ = ε/2 gives us only one equation fo the two unknown functions and ϑ. In the case of spheical symmety we could estict to the equatoial plane, ϑ = π/2, without loss of geneality. Then we had thee constants of motion E, L and L fo a dynamical system with thee degees of feedom t, ϕ and, and the Eule-Lagange equations educed to thee fist-ode equations that could be solved fo ṫ, ϕ and ṙ. The geodesics could then be witten in tems of integals ove the metic coefficients and the constants of motion. A dynamical system with n degees of feedom allows such a eduction to fist-ode fom wheneve thee ae n independent constants of motion with paiwise vanishing Poisson backets. If this is the case the dynamical system is called completely integable o integable in the sense of Liouville. Fo the Ke metic, which is only axisymmetic but not spheically symmetic, we cannot estict to the equatoial plane; the geodesics ae not in geneal contained in a plane. So ou dynamical system has fou degees of feedom t, ϕ, and ϑ while fom the symmeties of the spacetime we get only thee constants of motion E, L and L. This seems to indicate that the geodesic equation in the Ke metic fails to be completely integable. Fotunately, this is not the case. It was discoveed by Bandon Cate that thee is a fouth constant of motion which is not elated to any symmety of the spacetime. Togethe with the constants of motion E, L and L the Cate constant K secues complete integability of the Ke geodesic equation. It is tue that evey constant of motion is elated with a symmety tansfomation on the phase space, i.e., on the cotangent bundle ove spacetime, but not in geneal to a symmety of the spacetime. A constant of motion is elated to a symmety of the spacetime, i.e., to a Killing vecto field, if and only if it is linea in the velocities ẋ µ o, equivalently, in the momenta ẋ µ. The Cate constant is quadatic in the ẋ µ, as we will see. Cate found the constant of motion that is named afte him by woking in the Hamiltonian, athe than the Lagangian, fomalism. The Cate constant made its appeaance as a sepaation constant fo the Hamilton-Jacobi equation. We will follow this path now in detail. Fo the geodesic equation of any spacetime, we can pass fom the Lagangian to the Hamiltonian fomalism by intoducing the canonical momenta p µ = Lx,ẋ ẋ µ = g µν xẋ ν. This equation can be solved fo the velocities by multiplying with g σµ, ẋ σ = g σµ xp µ. 74

75 The Hamiltonian is Hx,p = ẋ µ p µ Lx,ẋ = ẋ µ p µ 1 2 g τσxẋ τ ẋ σ = g µν xp ν p µ 1 2 g τσxg τµ p µ g σν p ν = g µν xp ν p µ 1 2 δµ σp µ g σν xp ν = 1 2 gµν xp ν p µ. Note that Hx,p = Lx,ẋ = 1 2 ε, i.e. the Hamiltonian is given by the same quantity as the Lagangian, just expessed in tems of the momenta instead of the velocities. This is a consequence of the fact that the Lagangian is a quadatic fom in the velocities. In paticula, the Hamiltonian is a constant of motion which is always tue if the Hamiltonian does not explicitly depend on the cuve paamete. It is then possible to solve the equations of motion with the help of the time-independent Hamilton-Jacobi equation whee time efes to the cuve paamete, H x, S/ x = 1 2 ε. Hee the notation means that in the agument of the Hamiltonian we have to eplace p µ with the patial deivative S/ x µ of a function Sx = St,ϕ,,ϑ that is to be detemined. The goal is to find a complete integal of this patial diffeential equation, i.e., a solution Sx that involves as many independent constants as the system has degees of feedom. In ou case, we need thee constants of motion in addition to ε. The usual fist ty to find such a solution is with a sepaation ansatz, St,ϕ,,ϑ = S t t+s ϕ ϕ+s +S ϑ ϑ. Fom classical mechanics we know that it is always possible to sepaate off a cyclic coodinate, i.e., a coodinate that does not occu in the Hamiltonian. Then the coesponding momentum is a constant of motion and the function S is linea in the cyclic coodinate, with the constant of motion as the pe-facto. As we know that t and ϕ ae cyclic coodinates, with coesponding constants of motion p t = E and p ϕ = L, we specify the sepaation ansatz accoding to St,ϕ,,ϑ = Et+Lϕ+S +S ϑ ϑ. We have to plug this ansatz into the Hamilton-Jacobi equation. Fo that pupose we need the contavaiant metic coefficients g µν. The matix of covaiant components g µν of the Ke metic is of the fom g tt g tϕ 0 0 gµν = g tϕ g ϕϕ g g ϑϑ 75.

76 The invese matix eads g µν = D 1 g ϕϕ D 1 g tϕ 0 0 D 1 g tϕ D 1 g tt g g 1 ϑϑ g tt g tϕ 0 0 = g tϕ g ϕϕ g g ϑϑ whee D denotes the deteminant we have calculated above, gtt g D = det tϕ = g g tϕ g tt g ϕϕ gtϕ 2 = c2 sin 2 ϑ. ϕϕ So the Hamilton-Jacobi equation eads g tt E 2 2g tϕ EL+g ϕϕ L 2 +g S 2 +g ϑϑ S ϑ ϑ2 = ε, g ϕϕ E 2 D + 2g tϕel D + g ttl 2 D +g 1 S 2 +g 1 ϑϑ S ϑ ϑ2 = ε, sin 2 ϑ 2 +a 2 + 2ma2 sin 2 ϑ E 2 ρ 2 c 2 sin 2 ϑ + 4ma sin 2 ϑ cel + ρ 2 c 2 sin 2 ϑ c 2 1 2m ρ 2 L 2 c 2 sin 2 ϑ + ρ 2S ρ 2S ϑ ϑ2 = ε, E 2 +a 2 2 +a 2 a 2 sin 2 ϑ 2ma 2 sin 2 2 ϑ c 2 + 4maEL c + 2 +a 2 a 2 sin 2 ϑ 2m L 2 sin 2 ϑ + S 2 +S ϑϑ 2 = ερ 2, 76

77 2 +a 2 2 E2 c 2 + a2 sin 2 ϑe 2 + 4maEL c 2 c + L2 sin 2 ϑ a2 sin 2 ϑl 2 sin 2 ϑ + S 2 +S ϑϑ 2 = ε 2 +a 2 cos 2 ϑ, 1 2 +a 2 E 2 c al 2 2 +a 2 ael L + c sinϑ a 2 c sinϑe 2asinϑEL 4maEL + + csinϑ c + S 2 +S ϑ ϑ2 = ε 2 εa 2 cos 2 ϑ. L S ϑϑ 2 + sinϑ a 2 c sinϑe +εa 2 cos 2 ϑ = S a 2 E 2 c al ε 2 =: K. The fist expession is independent of wheeas the second expession is independent of ϑ. This implies that K depends neithe on no on ϑ, so it is a constant. K is the Cate constant. With S ϑ ϑ = p ϑ and S = p we have found that p 2 ϑ = K L sinϑ a c sinϑe 2 εa 2 cos 2 ϑ, p 2 = K a 2 E c al 2 ε 2. Now we have all fou components of the equation of motion in fist-ode fom: The equations G1 and G2 above give us ṫ and ϕ, ρ 2 ṫ = 2 +a 2 ρ 2 +2ma 2 sin 2 ϑ E 2macL c 2, G1 ρ 2 ϕ = 2masin2 ϑe +c ρ 2 2m L c sin 2 ϑ. G2 The last two equations, with p ϑ = g ϑϑ ϑ = ρ 2 ϑ and p = g ṙ = ρ 2 ṙ/, give us the emaining two components: L ρ 4 ϑ2 = K sinϑ a 2 c sinϑe εa 2 cos 2 ϑ, ρ 4 ṙ 2 = K + G1, G2, G3 and G4 detemine the geodesics. G3 2 +a 2 E c al 2 ε 2. G4 77

78 Note that the ight-hand side of G3 depends only on ϑ while the ight-hand side of G4 depends only on. So the only coupling of the motion to the ϑ motion is though the ρ 4 tem on the left-hand side. One can completely decouple the two equations by intoducing the Mino paamete, λ, which is elated to the affine paamete s we have used fo the paametisation by ds dλ = ρ2. If we denote the deivative with espect to λ by a pime, to distinguish it fom the deivative with espect to s which is denoted by a dot, we have ρ 2 =. If we wite the left-hand sides of G1, G2, G3 and G4 with the pime deivatives, the motion is completely decoupled fom the ϑ motion. Fo any choice of the constants of motion which detemine the initial velocities and any choice of initial conditions ϑ0 and 0 we can solve G3 and G4 to get ϑλ and λ. This can be done by sepaation of vaiables, esulting in elliptic integals. Upon inseting these esults into G1 and G2 we can detemine tλ and ϕλ by integating these equations with initial conditions t0 and ϕ0. Finally, we can expess the esult tλ,ϕλ,ϑλ,λ in tems of the affine paamete s i.e., pope time in the case of timelike geodesics instead of the Mino paamete λ if we like to do so. In this way we get all the geodesics in the Ke spacetime. We will see in the following that, actually, fo seveal special classes of geodesics the solution can be found without using the Mino paamete. We will always use ε,e,l,k fo the constants of motion. Of couse, one is fee to choose any fou independent combinations of these quantities instead. Fo some applications it is advantageous to use ε,e,l,q whee Q = K L ae/c 2. The name Cate constant is fequently used fo Q as well. K has a moe diect intepetation than Q because in the case of a non-spinning black hole a = 0 it educes to L 2 whee up to a dimensional facto L is the angula momentum vecto. Keep in mind that, by abuse of notation, we use the lette L fo the z component of L, i.e., L 2 = L 2 only fo motion in the equatoial plane. We will now study some aspects of lightlike and timelike geodesics sepaately. a Lightlike geodesics We begin ou discussion of lightlike geodesics ε = 0 with the ones that have vanishing Cate constant, K = 0. They ae known as pincipal null geodesics. Recall that null is often used as a synonym fo lightlike. We will see in a minute that they educe to adial lightlike geodesics in the non-spinning case, a = 0. In the spinning case, a 0, thee ae no adial geodesics because a geodesic that stats tangentially to the adial diection will be dagged along by the otating souce so that it peels off fom the adial diection. 78

79 With ε = 0 and K = 0 the equations G1, G2, G3 and G4 educe to ρ 2 ṫ = 2 +a 2 ρ 2 +2ma 2 sin 2 ϑ E 2macL c 2, G1 ρ 2 ϕ = 2masin2 ϑe +c ρ 2 2m L c sin 2. ϑ G2 L ρ 4 ϑ2 = sinϑ a 2, c sinϑe G3 ρ 4 ṙ 2 = 2 +a 2 E c al 2. G4 As the left-hand side of G3 is non-negative and the ight-hand side is non-positive, eithe side must be zeo, ϑ = 0, L = ae c sin2 ϑ. Inseting this expession fo L into G4 gives ρ 4 ṙ 2 = E2 c 2 2 +a 2 a 2 sin 2 ϑ 2, ṙ = ± E c. Similaly, with the expession fo L we find fom G1 that 2 ρ 2 +a 2 ρ 2 + 2ma 2 sin 2 ϑ E 2ma 2 Esin 2 ϑ ṫ =, ṫ = 2 +a 2 E c 2 c 2 and fom G2 that ρ 2 ϕ = 2masin 2 ϑe + ρ 2 2m aesin 2 ϑ, ϕ = aesinϑ c sinϑ c. Clealy, a pincipal null geodesic must have E 0 because fo E = 0 the above equations educe to ϑ = 0, ṙ = 0, ṫ = 0, ϕ = 0 which gives a point in spacetime athe than a cuve. Theefoe, we can divide by ṙ to get dϑ d = ϑ ṙ = 0, dt d = ṫ ṙ = ± 2 +a 2 c, dϕ d = ϕ ṙ = ± asinϑ. We see that in the case a = 0 the pincipal null geodesics satisfy dϑ/d = 0 and dϕ/d = 0, so they educe indeed to the adial lightlike geodesics as aleady anticipated. In the case a = 0 we have demonstated that the metic is egula at the hoizon by tanfoming to ingoing o outgoing Eddington-Finkelstein coodinates which whee defined by the popety that they map the ingoing o outgoing adial lightlike geodesics onto staight lines. So it is a vey natual idea to do a simila constuction in the Ke case with the adial lightlike geodesics eplaced by the pincipal null geodesics. The new featue is that we do not only have to tansfom the t coodinate, to emove the singulaity of dt/d at the hoizons, but also the ϕ coodinate to emove the singulaity of dϕ/d. 79,

80 To constuct a black hole as opposed to a white hole, we have to map the ingoing pincipal null geodesics onto staight lines, i.e., the ones chaacteised by the lowe sign. This can be achieved by intoducing ingoing Ke coodinates t,ϕ,ϑ, defined by cdt = cdt + 2 +a 2 d d, dϕ = dϕ + ad. This maps indeed the ingoing pincipal null diections onto staight lines, cdt = d, dϕ = 0, wheeas the outgoing pincipal null geodesics ae diveging even stonge at the hoizons, cdt = 22 +a 2 d d, dϕ = 2ad. Note that the pincipal null geodesics have ϑ = constant, and that the expessions fo dt/d and dϕ/d ae independent of ϑ. It is thus not necessay to show the ϑ coodinate in a plot of the pincipal null geodesics. We just have to keep in mind that in the case of ϑ = π/2 the pincipal null geodesics ae blocked at the ing singulaity, wheeas fo all othe values of ϑ they pass though one of the thoats at = 0. In the following pictue the motion of ingoing and outgoing pincipal null geodesics is plotted in an, ct diagam, not showing the ϕ motion, fo ϑ π/2. ct III II I 0 + The diagam efes to the case 0 < a 2 < m 2. In this case thee ae thee egions I, II and III which, in the Boye-Lindquist coodinates, ae sepaated by coodinate singulaities. In the exteme case 0 < a 2 = m 2 thee is no egion II. 80

81 The following plot shows the same situation in an,ct diagam, whee t,ϕ,ϑ, ae ingoing Ke coodinates. We had claimed befoe that thee ae hoizons at = and = +. Fom the diagam we can now clealy ead that ou claim was coect. Again, we emphasise that the ϕ motion is not shown in the diagam. Keep in mind that only the ingoing pincipal null geodesics have ϕ = constant while the outgoing ones appoach the hoizons in an infinite whil. ct III II I 0 + In ingoing Ke coodinates the metic eads g = c 2 dt 2 +d 2 +ρ 2 dϑ 2 +sin 2 ϑ 2 +a 2 dϕ 2 2asin 2 ϑddϕ + 2m cdt +d asin 2 ϑdϕ 2 ρ 2 which is egula eveywhee except at the ing singulaity. Thee ae of couse analogously defined outgoing Ke coodinates which join the egions I, II and III to poduce a Ke white-hole spacetime. In the non-spinning case, a = 0, Ke coodinates educe to Eddington-Finkelstein coodinates. As the Ke spacetime is not spheically symmetic, we cannot daw a = Penose diagam in the stict sense. Howeve, if we estict to motion along the axis ϑ = 0, we can daw III a Penose diagam fo this dimensional spacetime. Note that ingoing and outgoing pincipal null = geodesics that stat tangentially to II the axis emain tangential eveywhee. The figue on the ight shows the Penose diagam fo the axis of a = + Keblackholewith0<a 2 < m 2. In I the exteme case, a 2 = m 2, thee is no egion II. The -coodinate anges fom to. As we ae on the = axis, the ing singulaity is not met. 81

82 The maximal analytic extension of the Ke black-hole spacetime esticted to the axis gives a Penose diagam with infinitely many copies of the egions I, II and III. = + = = = = II = = III = + = + I = = = = II = = III = + = + I = = = = II = = = + Having claified the hoizon stuctue of the Ke black-hole spacetime with the help of pincipal null geodesics, we now tun to the spheical lightlike geodesics which ae cucial fo constucting the shadow of a Ke black hole. Recall that in the Schwazschild case the bounday cuve of the shadow was detemined by lightlike geodesics that asymptotically spial towads the photon sphee at = 3 S /2 = 3m. As the spheical symmety is boken in the Ke spacetime, we cannot expect that thee still is a photon sphee. We will see that the photon sphee is eplaced by a photon egion which is filled with lightlike geodesics each of which stays on a sphee = constant. We call such geodesics spheical. Spheical lightlike geodesics have to satisfy ṙ = 0 and = 0. Fom G3 with ε = 0 we see that this gives us the following two equations: 0 = K 2 +a 2 2m + 0 = K 2 2m a 2 E c al 2, 2 +a 2 E 2E c al. c 82

83 We multiply the fist equation with m and the second with = 2 +a 2 2m. Then the diffeence of the two equations yields 0 = 2 +a 2 E { c al 2 +a 2 E c al 2E m 2 +a 2 2m }, c al m = E 2 +a 2 m 2, c ac L E = 2 +a 2 2 m. S1 To detemine K, we inset this esult into the second equation fo spheical lightlike geodesics, 0 = K m + 2 +a 2 ac L 2E 2. E c 2 c 2 K m = 2 + a 2 2 a , E 2 m c 2 K E 2 = 42 m 2. S1 and S2 detemine the constants of motion fo spheical lightlike geodesics at adius coodinate. To find out at which values spheical lightlike geodesics actually exist, we need to evaluate G3. Astheleft-handsideofthisequationisthesquae ofaealquantity, itcannotbenegative, hence L K sinϑ a 2 c sinϑe 0, acl 2 c 2 K E E a2 sin 2 ϑ 2 a 2 sin 2 0, ϑ a m m a2 sin 2 ϑ 2 a 2 sin 2 0, ϑ 4 2 a 2 sin 2 ϑ m 2 ρ , m ρ 2 m a 2 sin 2 ϑ. P This inequality detemines the photon egion. Fo a = 0, the ight-hand side vanishes, so the left-hand side must be equal to zeo, 0 = 2 m 2 2 2m = 3 2 m 3. In this case the photon egion educes to the photon sphee, = 3m = 3 S /2. In the case a 0, howeve, it is not a 2-dimensional suface in space, but athe a 3-dimensional egion with a bounday. Though each point of the photon egion thee is a spheical lightlike geodesic. Along each of these geodesics, the ϑ coodinate oscillates between a maximal and a minimal value; the tuning points of the ϑ motion occu on the bounday of the photon egion. The ϕ motion of a spheical lightlike geodesic may be quite complicated; in the egoegion it may even be non-monotonous. S2 83

84 Fo 0 < a2 < m2, the photon egion consists of thee connected components: An exteio photon egion in the domain > + and two inteio photon egions in the domain < that ae sepaated fom each othe by the ing singulaity. In the exteio photon egion all spheical lightlike geodesics ae unstable with espect to adial petubations, so they can seve as limit cuves fo lightlike geodesics that appoach them asymptotically. In the inteio photon egions thee ae both stable and unstable spheical lightlike geodesics. Fo the fomation of the shadow, only the exteio photon egion is of elevance. The exteio photon egion has a cescent-shaped coss section, which becomes bigge and bigge with inceasing a. Fo a 0 it shinks to the photon sphee at = 3m as aleady mentioned. The pictue on the left shows the photon egions fo a Ke black hole of a = 0.75 m in a diagam whee e/m is the adius coodinate, the pictue on the ight gives an enlaged view of the inteio egion. Unstable spheical lightlike geodesics exist in the blue egion, stable ones in the geen egion. The domain between the two hoizons is shown in black. The ing singulaity is indicated by ed blobs and the thoats at = 0, cos ϑ 6= 0 by ed dashed half-cicles. The causality violating egion, whee gϕϕ < 0, is maked in oange, cf. p. 72. In each of the two pictues a black line shows the pojection of one paticula spheical lightlike geodesics. In Woksheet 8 we will discuss the special case of cicula lightlike geodesics. In a Ke spacetime with a 6= 0, thee ae only five of them, thee in the equatoial plane and two off the equatoial plane. To constuct the shadow of a Ke black hole, we fix an obseve in the domain of oute communication, i.e., at Boye-Lindquist coodinates O, ϑo with O > +. Clealy, the shape of the shadow at the obseve s celestial sphee will depend on the obseve s state of motion. We do the calculation fo one paticula obseve; fo any othe obseve that passes though the same event with a diffeent 4-velocity the shadow is then detemined by the abeation fomula. In the Schwazschild case, we have assumed that the obseve is static, i.e., that his woldline is a t line. In the Ke spacetime we will not choose an obseve moving on a t line because this would exclude the egoegion. 84

85 We find it moe convenenient to choose instead an obseve associated with the tetad e 0 = 2 +a 2 t +ac ϕ cρ, e 1 = ϑ, O,ϑ O ρ O,ϑ O e 2 = c ϕ asin 2 ϑ t cρsinϑ O,ϑ O, e 3 = ρ O,ϑ O It is eadily veified that this tetad is pseudo-othonomal, ge µ,e ν = η µν. We intepet e 0 as the 4-velocity of ou obseve, then e 1, e 2 and e 3 fom an othonomal basis in the est space of the obseve. Note that the obseve s 4-velocity is chosen such that e 0 is in the plane spanned by the ingoing and outgoing pincipal null diections. This follows immediately fom the epesentation of pincipal null geodesics given on p. 79. Fo the constuction of the shadow we have to conside past-oiented lightlike geodesics issuing fom the position of the obseve. If such a geodesic is given as a cuve ts,ϕs,ϑs,s paametised by an affine paamete s, its tangent vecto k at the obsevation event is k = ṫ t + ϕ ϕ + ϑ ϑ +ṙ. On the othe hand, as k is lightlike and past-pointing, it can be witten as k = N e 0 +cosψsinθe 1 +sinψsinθe 2 +cosθe 3 with some positive facto N. This equation detemines the celestial coodinatesψ, θ associated with the diection of the light ay. Accoding to ou choice of the tetad, θ = 0 coesponds to the diection towads the black hole. Equating the two expessions fo k yields the following fou equations. N = gk,e 0 = g ṫ t + ϕ ϕ, 2 +a 2 t +ac ϕ cρ = 2 +a 2 ṫg tt + ϕg tϕ cρ + a ṫg tϕ + ϕg ϕϕ ρ = 2 +a 2 E cρ + al ρ,. T1 N cosψsinθ = gk,e 1 = g ϑ ϑ, ϑ ρ = ϑg ϑϑ ρ = ϑρ, T2 N sinψsinθ = gk,e 2 = g = ṫg tϕ + ϕg ϕϕ ρsinϑ asinϑ ṫg tt + ϕg tϕ cρ ṫ t + ϕ ϕ, c ϕ asin 2 ϑ t cρsinϑ = L ρsinϑ + asinϑe, T3 cρ N cosθ = gk,e 3 = g ṙ, ρ = ṙ g ρ = ṙρ. T4 85

86 Hee all metic functions have to be evaluated at the position of the obseve. T2 and T3 imply { ρ 4 ϑ2 + tanψ = L ρsinϑ + asinϑe cρ ϑρ ρ 4 ϑ2 sin2 ψ L cos 2 ψ = sinϑ asinϑe 2, c L ρ 4 ϑ2 sin 2 ψ = sinϑ asinϑe 2 1 sin 2 ψ, c L sinϑ asinϑe 2 } sin 2 ψ = c, L sinϑ asinϑe 2, c and with G3 Ksin 2 E 2 cal 2, ψ = c 2 a 2 sin 2 ϑ E a2 sin 2 ϑ cal 2 sin 2 ψ = E a2 sin 2 ϑ c 2 C1 K E 2 a2 sin 2 ϑ ϑ=ϑo whee in the last line we have made explicit that the metic functions have to be evaluated at the position of the obseve. Similaly, T1 and T4 imply cosθ = ṙρ 2 +a 2 E cρ ρ 4 ṙ 2 2 +a 2 E + c With G4 this esults in + al ρ 1 sin 2 θ 2 +a 2 E c = ṙρ 2 2 +a 2 E c al 2 = ρ4ṙ 2, 2 2 +a 2 E al = c K = E2 2 +a 2 cal c 2 E 2sin 2 θ,, + al al 2sin 2 θ. c2 K sin 2 θ = E 2 2 +a 2 cal 2 C2 =O E whee, again, in thelast line we have made explicit that the metic funtions have to be evaluated at the position of the obseve. C1 and C2 detemine fo each lightlike geodesic with known constants of motion L/E and K/E 2 the celestial coodinates ψ,θ. 86

87 Now ecall fom ou discussion of the Schwazschild metic how the shadow of a black hole is defined: We assume that thee ae light souces distibuted eveywhee but not between the black hole and the obseve. We divide into two classes the light ays issuing fom the obseve into thepast: The geodesics inoneclass go outto infinity afte being deflected by theblack hole, so they can each a light souce. The geodesics of the othe class go towads the hoizon, so they do not each one of the light souces. In the Schwazschild case, the bodeline between the two classes coesponded to light ays that spialed towads the photon sphee at = 3 S /2 = 3m. In the Ke case, the cicula obits at the photon sphee ae eplaced by the spheical obits in the photon egion. In othe wods, fo detemining the bounday cuve of the shadow we have to detemine those lightlike geodesics that asymptotically appoach a spheical lightlike geodesic. If two geodesics become asymptotically tangent to each othe, they must have the same constants ofmotion. This followsfomthefactthattheconstants ofmotionaedetemined bythetangent of the geodesic at any one point. So the lightlike geodesics that coespond to the bounday cuve of the shadow must have the constants of motion given by S1 and S2, whee anges ove the adius values allowed by the inequality P fo the photon egion. Having detemined the constants of motion as a function of the paamete in this way, we plug the coesponding expessions into theight-handsidesofc1 andc2. Thisgives ustheboundayoftheshadow as a cuve paametised by on the celestial sphee of the obseve, ψ,θ. This is a fully analytic epesentation of the bounday cuve. Fo each choice of a with 0 < a 2 m 2 and of the obseve position O,ϑ O the esult can be easily plotted, whee we have to epesent the celestial sphee of the obseve in an appopiate way. It is convenient to use steeogaphic pojection with the diection towads the black hole, θ = 0, as the oigin. The two images show the shadow fo an obseve at O = 6m,ϑ O = π/2 in steeogaphic pojection. The dashed line maks the celestial equato θ = π/2. On the left, the spin is chosen to be a = 0.7m; in this case, the deviation of the shadow fom a cicula shape is not easily visible to the naked eye. Note that an obseve wouldn t see in the sky the coss-hais that indicate the diection towads the black hole! On the ight, the black hole is exteme, a = m. In this case the deviation fom the cicula shape is conspicuous. The asymmety is easily undestood fom the fact that a otating black hole dags the light ays. Fom ou analytical fomula fo the bounday cuve of the shadow we can deduce that the shadow is always symmetical with espect to a hoizontal axis, even fo an obseve off the equatoial plane, ϑ O π/2. This is a emakable esult which was not to be expected fom the symmety of the spacetime. 87

88 b Timelike geodesics We concentate on two aspects of timelike geodesics. Fistly we investigate cicula timelike obits in the equatoial plane. Secondly, we discuss some geneal featues of the ϑ motion. We conside a Ke black hole with 0 < a 2 < m 2, viewing a = 0 and a 2 = m 2 as limiting cases. We want to detemine the cicula timelike geodesics in the domain of oute communication, > +. We stat out fom equations G3 and G4 on p.77. Fo timelike geodesics ε = c 2 in the equatoial plane ϑ = π/2, ϑ = 0 these equations simplify to 0 = K L ae c ρ 4 ṙ 2 = K + 2 +a 2 E 2 c al c 2 2. Inseting the fist equation into the second one yields ρ 4 ṙ 2 = L ae c 2, a 2 E c al 2 c 2 2 =: 2V E,L. Cicula obits have to satisfy ṙ = 0 and = 0, hence V E,L = 0 and V E,L = 0. This system of two equations can be solved fo the constants of motion E and L. The calculation is elementay, but athe tedious, so that we pefe letting MATHEMATICA do it fo us. The esult is cl E = ± m3 2 +a 2 2m a 3 m am3 4m, 2m 2 a 2 m E 2 c = 3 3m 2m 2 a 2 m 2 3 5m±2am m3 2 +a 2 2m 4 2m2 a 2 m 2, whee the uppe sign holds fo co-otating and the lowe sign fo counte-otating geodesics. As we conside the domain of oute communication whee is positive, m 3 is eal. Theefoe, the only estiction on the values comes fom the condition that E 2 /c 4 0. This condition is plotted on the ight. Counte-otating cicula timelike geodesics exist at adius values above the solid line, co-otating ones above the dashed line. The limiting values ae the adii of lightlike cicula obits in the equatoial plane, see Woksheet 9. In the Schwazschild limit, cicula timelike geodesic exist fo > 3m, as we aleady know. Fo the exteme Ke black hole, a 2 = m 2, counte-otating cicula timelike geodesics exist fo > 4m while co-otating ones exist fo > m. As in the exteme case + = m, this means that in this limiting case the co-oating obits cove the entie ange of values up to the hoizon. 4m 3m m m a 88

89 We ae now going to investigate fo which adius values the timelike cicula geodesics ae stable. The condition fo stability is V E,L > 0 whee we have to inset fo E and L the values fo cicula geodesics. This condition is plotted below. Counte-otating obits ae stable at values above the solid line, while co-otating ones ae stabe above the dashed line. In the Schwazschild limit, the innemost stable cicula obit ISCO is at = 6m, as we know. The counte-otating ISCO is at a bigge adius, with a maximum at = 9m fo the exteme case. The co-otating ISCO is at a smalle value than in the Schwazschild case and appoaches = m, i.e., the adius coodinate of the hoizon, in the exteme case. This gives ise to the fact that objects on a co-otating stable obit aound a Ke black hole can suffe much stoge gavitational time dilation effects than on a stable obit aound a Schwazschild black hole, see Woksheet 10. 9m 6m m a 1 Actually, the limit a 2 m 2 is athe subtle. The Boye-Lindquist coodinates show a paticulaly pathological behaviou nea the degeneate hoizon in this case. The fact that the co-otating ISCO appoaches the same value of the coodinate as the hoizon does not mean that the two things coincide in the limit. In contast to what the Boye-Lindquist coodinates suggest, thee is still a finite distance between them. In the exteme Ke spacetime, the distance as measued with the metic along an line of the hoizon fom any point in the domain of oute communication is actually infinite, as we will see in Woksheet 10. We now tun to a discussion of some geneal featues of the ϑ motion of timelike geodesics. We use the Mino paamete λ and we denote deivative with espect to λ by a pime, = ρ 2. Then, accoding to equation G3 on p.77, the ϑ motion of timelike ε = c 2 geodesics satisfies L ϑ 2 = K sinϑ ae 2 c sinϑ c 2 a 2 cos 2 ϑ. 89

90 Fo the following it will be convenient to use the constant Q = K L ae 2 c instead of K. It was aleady mentioned that the name Cate constant is used both fo K and fo Q. In tems of Q, the ϑ equation eads ϑ 2 = Q+ L ae 2 L c sinϑ ae 2 c sinϑ c 2 a 2 cos 2 ϑ = Q+L 2 2L ae + a2 E 2 c c 2 = Q L2 1 sin 2 ϑ sin 2 ϑ sin 2 ϑ + 2L ae a2 E 2 sin 2 ϑ c 2 a 2 cos 2 ϑ c c 2 L2 Fo analysing the possible ϑ motions, we wite + a2 E 2 c 2 1 sin 2 ϑ c 2 a 2 cos 2 ϑ = Q L2 cos 2 sin 2 ϑ + a 2 E 2 c 2 c 2 a 2 cos 2 ϑ. u := cos 2 ϑ, u = 2cosϑsinϑϑ. Then u 2 = 4cos 2 ϑsin 2 ϑϑ 2 = 4cos 2 ϑsin 2 ϑ {Q L2 cos 2 a 2 sin 2 ϑ + E 2 } c 2 a 2 cos 2 ϑ c 2 = 4u 1 u { Q L2 a 2 1 u + E 2 } c 2 a 2 u. c 2 { = 4u Q 1 u a L 2 2 E 2 u+ c 2 a 2 u 1 u } c 2 With the abbeviation Tu := Q 1 u a L 2 2 E 2 u+ c 2 a 2 u 1 u c 2 this allows us to wite ϑ 2 = u 2 4cos 2 ϑsin 2 ϑ = 4uTu 4u1 u = Tu 1 u. This last equation tells us that all values of u = cos 2 ϑ whee Tu < 0 ae fobidden wheeas values whee Tu 0 ae allowed. Tuning points ϑ = 0 occu pecisely at those values of u = cos 2 ϑ whee Tu = 0. On the basis of these obsevations, we can now analyse the possible ϑ motions. We distinguish the cases Q > 0, Q = 0 and Q < 0. 90

91 Case A: Q > 0: The fist pictue shows the function Tu fo the case that L = 0. If a 2 E 2 /c 2 c 2 a 2 < Q dotted, the entie inteval 0 u 1 is allowed. Thee ae no tuning points, the motion goes though the axis. If a 2 E 2 /c 2 c 2 a 2 > Q dashed, only an inteval 0 u u 0 = cos 2 ϑ 0 with 0 < ϑ 0 < π/2 is allowed. The motion oscillates about the equatoial plane, with tuning points at π ϑ 0 and ϑ 0. In the bodeline case a 2 E 2 /c 2 c 2 a 2 = Q solid the motion asymptotically appoaches the axis in the futue and in the past. Tu Q L = 0 1 u Now we conside the case that L 0. In the pictue below the function Tu is plotted fo a 2 E 2 /c 2 c 2 a 2 < Q L 2 dotted and fo a 2 E 2 /c 2 c 2 a 2 > Q L 2 dashed. In any case, only an inteval 0 u u 0 = cos 2 ϑ 0 with 0 < ϑ 0 < π/2 is allowed. The motion oscillates about the equatoial plane, with tuning points at π ϑ 0 and ϑ 0. Tu Q L 0 1 u L 2 An oscillatoy behaviou of the ϑ motion between a value ϑ 0 and a value π ϑ 0 is what we know fom Schwazschild spacetime o fom Newtonian theoy, fo that matte. 91

92 Case B: Q = 0 Again, we begin with the case L = 0. If a 2 E 2 /c 2 c 2 a 2 > 0 dotted, the entie inteval 0 u 1 is allowed. Thee ae no tuning points. The motion goes though the axis. As Tu 0 fo u 0, the equatoial plane is asymptotically appoached. If a 2 E 2 /c 2 c 2 a 2 < 0 dashed, the motion must be confined to the equatoial plane o to the axis. In the bodeline case a 2 E 2 /c 2 c 2 a 2 = 0 solid the motion is confined to a cone ϑ = constant. Again, this includes as special cases motion in the equatoial plane and in the axis. Tu L = 0 1 u Now we conside the case L 0. If a 2 E 2 /c 2 c 2 a 2 > L 2 dotted, an inteval 0 u u 0 = cos 2 ϑ 0 is allowed. As Tu 0 fo u 0, the equatoial plane is asymptotically appoached. If a 2 E 2 /c 2 c 2 a 2 < L 2 dashed, the motion is confined to the equatoial plane. The same is tue in the bodeline case a 2 E 2 /c 2 c 2 a 2 = L 2 solid. Tu L 0 1 u L 2 Note that motion in the equatoial plane must have Q = 0. 92

93 Case C: Q < 0 Again, we begin with the case L = 0. If a 2 E 2 /c 2 c 2 a 2 < Q dotted, an inteval u 0 = cos 2 ϑ 0 u 1 with 0 < ϑ 0 < π/2 is allowed. The motion goes though the axis. It stays eithe in the nothen hemisphee, with tuning points at ϑ 0, o in the southen hemisphee, with tuning points at π ϑ 0. If a 2 E 2 /c 2 c 2 a 2 > Q dashed o a 2 E 2 /c 2 c 2 a 2 = Q solid, the motion is confined to the axis. Tu L = 0 1 u Q Now we conside the case L 0. If a 2 E 2 /c 2 c 2 a 2 < Q L 2 dotted, an inteval u 1 = cos 2 ϑ 1 u u 2 = cos 2 ϑ 2 with 0 < ϑ 2 < ϑ 1 < π/2 is allowed. The motion stays eithe in the nothen hemisphee, oscillating between ϑ 1 and ϑ 2, o in the southen hemisphee, oscillating between π ϑ 1 and π ϑ 2. If a 2 E 2 /c 2 c 2 a 2 > Q L 2 dashed, no motion is allowed. In the bodeline case a 2 E 2 /c 2 c 2 a 2 = Q L 2 solid, the motion stays on a cone. Tu L 0 1 u Q L 2 Obits oscillating between two cones, o staying on a cone, ae called votical. They do not occu in the Schwazschild case. 93

94 6.3 Egoegion and Penose pocess Fo an obseve in the domain of oute communication i.e., outside of the oute hoizon of a Ke black hole, the most impotant new featue in compaison to a Schwazschild black hole is the pesence of the egoegion. In paticula, this gives ise to the possibility of extacting enegy fom the black hole via the socalled Penose pocess. We will now discuss this possibility in some detail. We conside a Ke black hole with 0 < a m. The estiction to positive values of a is no loss of geneality because we ae fee to tansfom the azimuthal coodinate, ϕ ϕ. In this section we conside only the domain of oute communication which we denote, as befoe, as egion I, i.e. I : > + = m+ m 2 a 2. The egoegion is that pat of egion I whee the Killing vecto field t is spacelike. We decompose the egion I into the pat outside the egoegion, I a : > + andg tt = c 2 1 2m < 0, ρ 2 and the egoegion I b : > + andg tt = c 2 1 2m > 0. ρ 2 The inteface between I a and I b, whee g tt = 0, is the stationaity limit suface. In the pictue on p.71 the egoegion I b is the dak shaded egion. We have aleady mentioned that inside the egoegion the hypesufaces ϕ = constant ae spacelike, i.e., the woldline of an obseve o a light signal cannot be tangent to such a hypesuface. This is anothe effect of the dagging by a otating black hole: Evey obseve that is close to the black hole inside the egoegion must have a velocity with a non-vanishing ϕ component, i.e., the obseve is foced to otate about the black hole whateve populsion might be used. As a pepaation fo discussing the Penose pocess, we want to quantify the allowed angula velocities fo obseves. To that end, we conside an obseve on a ϕ line i.e., on a cicle about the axis of the black hole with constant angula velocity Ω. Such an obseve is, of couse, not in geneal feely falling, i.e., he needs a ocket engine o some othe kind of populsion. The 4-velocity is c t +Ω ϕ U = g tt +2g tϕ Ω+g ϕϕ Ω 2. We want to detemine the allowed values fo the constant angula velocity Ω, depending on the and ϑ coodinate whee the obseve is cicling. These allowed values ae detemined by the condition of t +Ω ϕ being timelike, g tt +2g tϕ Ω+g ϕϕ Ω 2 < 0. This is tue fo Ω < Ω < Ω + whee Ω ± ae the solutions of the quadatic equation g tt g ϕϕ +2 g tϕ g ϕϕ Ω+Ω 2 = 0, hence Ω ± = g tϕ ± 94 g 2 tϕ g tt g ϕϕ g ϕϕ.

95 We will now detemine the signs of Ω and Ω +. Fist of all, obseve that g ϕϕ = sin 2 ϑ 2 +a 2 + 2ma2 sin 2 ϑ > 0 ρ 2 eveywhee in egion I except on the axis which is to be excluded because thee ou cicula obit degeneates into a point. Also, as we have chosen a > 0, g tϕ = 2mac ρ 2 sin 2 ϑ < 0 eveywhee in egion I except on the axis. In egion I a i.e., outside the egoegion we have g tt < 0 and thus gtϕ 2 g tt g ϕϕ > g tϕ. This implies that in this egion Ω < 0 and Ω + > 0, i.e., thee is allowed motion in the positive and in the negative ϕ diection, as we ae used to. By contast, in the egoegion I b we have g tt > 0 and thus gtϕ 2 g tt g ϕϕ < g tϕ. This implies that in this egion both Ω and Ω + ae positive, i.e., only motion in the positive ϕ diection is allowed. We will now demonstate that the values Ω and Ω + tend to the same limit Ω H if the hoizon is appoached. To that end, we ecall that we have calculated aleady on p.73 that This allows to ewite Ω ± as Ω ± = g 2 tϕ g tt g ϕϕ = c 2 sin 2 ϑ. 2cmasin 2 ϑ ±c sinϑ. ρ 2 sin 2 ϑ 2 +a 2 + 2ma2 sin 2 ϑ ρ 2 If the hoizon is appoached, +, we have 0 and thus lim Ω ± = + = 2cm + a sin 2 ϑ sin 2 ϑ 2 + +a a 2 cos 2 ϑ +2m + a 2 sin 2 ϑ 2cm + a 2m a 2 cos 2 ϑ +2m + a 2 sin 2 ϑ = ca 2 + +a 2 =: Ω H. Ω H is known as the angula velocity of the hoizon. Note that Ω H is independent of ϑ which may be intepeted as saying that the hoizon otates igidly. The vecto field t + Ω H ϕ is timelike and non-geodesic eveywhee inside the egoegion. On the hoizon = + it becomes lightlike and geodesic. In the equatoial plane this lightlike geodesic vecto field becomes tangent to the outgoing pincipal null geodesics, as can be veified by compaison with p. 79. Afte these pepaations we will now conside the motion of feely falling paticles i.e., of timelike geodesics in the egion I. Recall that, fo a timelike geodesic γτ = tτ,ϕτ,ϑτ,τ with tangent vecto γ = ṫ t + ϕ ϕ + ϑ ϑ +ṙ, 95

96 we have two constants of motion: The enegy E = L ṫ and the z component of the angula momentum = g tt ṫ g tϕ ϕ = g γ, t L = L ϕ = g ϕϕ ϕ + g ϕt ṫ = g γ, ϕ γ must be timelike and futue-pointing eveywhee. Outside of the egoegion, t is also timelike and futue-pointing, hence E = g γ, t > 0. Inside the egoegion, howeve, t is spacelike. Then the scala poduct g γ, t may be positive, negative o zeo, i.e., inside the egoegion thee exist feely falling paticles with negative enegies. Of couse, as E is a constant of motion, such a paticle cannot move into the egion I a ; it can leave the egion I b only ove the hoizon at +. As t +Ω H ϕ is timelike and futue-pointing inside the egoegion, we have As Ω H > 0, this implies 0 > g t +Ω H ϕ, γ = E +Ω H L. L < E Ω H which demonstates that paticles with negative E must also have negative L. We ae now eady fo a discussion of the Penose pocess. Assume that a feely falling paticle is eleased outside the egoegion. This paticle must have an enegy E > 0. Let this paticle ente the egoegion and decay into two paticles thee, one with an enegy E 1 > 0 and anothe one with an enegy E 2 < 0. Of couse, we assume enegy consevation, E = E 1 + E 2, hence E 1 > E. The decaying pocess may be aanged in a way that the paticle with positive enegy E 1 etuns to the egion I a while the paticle with the negative enegy E 2 is swallowed by the black hole. Then we have the following enegy balance: black hole outside paticle initial enegy E 0 E final enegy E 0 +E 2 < E 0 E 1 > E We see that it ispossible to extact enegy fomaotating black holeby this pocess. Obviously, the pocess can be epeated only until all enegy of the black hole has been used up. Actually, the pocess teminates ealie: As we have seen that paticles with negative E have negative L, the black hole loses not only enegy but also angula momentum when swallowing the infalling paticle. The Penose pocess can be epeated only until the black hole has lost all its angula momentum; then it is a Schwazschild black hole which has no egoegion. 96

97 7. Black holes in astophysics In this chapte we give a bief oveview of the black-hole candidates that ae actually obseved in the sky and we discuss why we ae quite optimistic that they can be successfully modelled as Ke o Schwazschild black holes. The pomising black-hole candidates can be divided into two classes: Stella black holes with masses of a few maybe up to 30 Sola masses, and supemassive black holes with millions o billions of Sola masses. Black holes with othe masses ae moe speculative: Mini-black-holes with masses of seveal odes ofmagnitude less than a Sola mass could have come into existence shotly afte the big bang, and intemediate black holes with a few hunded Sola masses might be haboued in globula clustes. Howeve, up to now we have no stong astophysical evidence fo thei existence. Theefoe we discuss only the othe two types. 7.1 Stella black holes a Cygn X-1 As the name indicates, Cyg X-1 is the fist X-ay souce that was detected in the constellation Cygnus Swan. It is the bightest pemanent souce of had X-ays in the sky. X-ay souces cannot be obseved with gound-based telescopes because fotunately ou atmosphee is nontanspaent fo X-ays. The fist obsevations of celestial X-ay souces wee made with the help of Aeobee sounding ockets beginning in the yea Cyg X-1 was detected in The Aeobees wee ballistic ockets that could each an altitude of about 250 km. Afte a flight of less than 10 minutes the payload fell back to the gound on a paachute. The X-ay obsevations wee made at wavelengths fom 1 to 15 Å with Geige countes whose field of view swept ove a stipe in the sky as the ocket otated about its axis. This allowed only a ough localisation of the detected souces in the sky, so it ws not possible to decide whethe thee is an optical o adio souce at the same position as Cyg X-1 In the yea 1970 the fist X-ay satellite, Uhuu, was launched. Uhuu is a Swahili wod meaning feedom. It allowed to localise Cyg X-1 to within a few acminutes which is still not accuate enough fo deciding whethe thee is an optical o adio souce at the same position. Howeve, with the Uhuu satellite one obseved time vaiations in the flux fom Cyg X-1 on a time-scale of a little bit less than a second. As diffeent pats of an emission egion can show synchonised time vaiations only if they ae causally connected, this means that the emission egion of Cyg X-1 must be so small that light can tavel though it in less than a second. This limited the diamete of the emission egion of Cyg X-1 to less than km. Fo the sake of compaison, note that the diamete of ou Sun is about km! 97

98 In the ealy 1970s the esolution of X-ay obsevations had impoved sufficiently to localise Cyg X-1 to within an acminute. Also, in the yea 1971 a adio souce was detected in the same aea that showed vaiations on a simila time scale as the X-ay souce Cyg X-1. This adio souce could be located with an accuacy of about one acsecond. This obsevation suggested that Cyg X-1 might be associated with an O sta a blue giant visible in the optical with the catalogue numbe HDE that was within the eo box of the adio obsevation. The spectal lines fom this sta show a edshift that vaies peiodically with a peiod of about 5.5 days. The obvious intepetation is that the sta foms a binay system togethe with a patne that emits the X-ays; the peiodic vaiation in the edshift comes fom the Dopple effect when the sta obits about the baycente of the binay system. The companion i.e., the X-ay souce is not visible in the optical. Moeove, it was obseved that the intensity of the blue sta vaies with the same peiod as the edshift. The commonly accepted intepetation is that the sta is not spheical but elongated in the diection towads the baycente. In the couse of one obit it shows us sometimes a smalle and sometimes a bigge coss-sectional suface. This indicates that the gavitational field of the compact companion has defomed the sta and that thee is a mass flow accetion flow fom the sta onto the companion. The X-ays that we obseve ae poduced when the acceted matte gas o plasma is stongly heated nea the compact companion. The pictue below shows an atist s impession of how the binay system may look like. Actually, an obseve would see a distoted image because of light bending, but these effects ae ignoed in the pictue. HDE is close enough fo measuing its distance with the paallax method. One found thatthesystem isabout2kpcaway fomus. Fothesake ofcompaison, notethatthecenteof ou galaxy is at a distance of about 8 kpc fom us. With the distance known, one can combine the edshift measuements and othe infomation on the obits with stella evolution models to detemine the masses of the two companions. The up-to-date values ae 18 Sola masses fo the blue sta and 15 Sola masses fo the compact companion. Recall that the companion has a diamete of less than km, so it is inded a vey compact object. It cannot be a white dwaf because thee is an uppe limit of about 1.4 Sola masses fo a white dwaf. This is the famous Chandasekha limit the discovey of which won S. Chandasekha the Nobel pize. It cannot be a neuton sta because also fo a neuton sta a mass of 15 Sola masses is out of the question. The uppe limit fo the mass of a neuton sta is not quite clea, because it depends stongly on the assumptions about the inteio stuctue, but it cannot be much moe than 3 Sola masses. The most massive neuton sta we know has a mass of just a bit moe than 2 Sola masses and most astophysicists believe that this is close to the limit. So a black hole is the most natual explanation. Othe possible explanations a boson sta, a neutino ball, a gavasta.. have been suggested, but they ae much moe speculative. 98

99 Additional suppot fo the black-hole hypothesis comes fom the obsevation of dying pulse tains fom the neighbouhood of Cyg X-1. These ae X-ay signals that die down in exactly the way as one would expect if matte spials towads an event hoizon. By contast, if matte hits a suface the emitted adiation does not die down but athe ends in a bight flash. This is stong evidence fo the existence of an event hoizon and the absence of a suface. A black hole with 15 Sola masses has a Schwazschild adius of about 45 km. At a distance of 2 kpc, the black hole associated with Cyg X-1 would cast a shadow with an angula diamete of less than 10 9 acseconds. Recall Synge s fomula fom p.34 fo the angula adius of the shadow of a Schwazschild black hole which gives the coect ode of magnitude also fo a Ke black hole. This is fa too small fo obsevations within the foeseeable futue. This is a pity because the shape of the shadow would give us unambiguous infomation on the spin of the black hole, ecall p.86/87. Up to now the spin of the black hole associated with Cyg X-1 is unknown. Recent obsevations with the X-ay satellite Chanda seem to indicate that the inne edge of the accetion disc is vey close to the hoizon which would imply that the black hole is almost exteme. Recall that in the Schwazschild limit, a 0, the ISCO of a Ke black hole tends to 6m = 3 + while fo the exteme case, a m, it appoaches m = +. Howeve, this is in disageement with ealie papes whee the authos have found indications fo a slow otation close to the Schwazschild limit. b Othe stella black holes Cyg X-1 is the oldest and best known candidate fo a stella black hole, but it not the only one. Thee ae moe than a dozen futhe candidates. All of them ae X-ay souces and they ae patnes in binay systems. Thei masses ae between 3 and 30 Sola masses. An incomplete list can be found on the Wikipedia page on stella black hole. Futhe cicumstantial evidence fo the existence of stella black holes comes fom the obsevation of gamma ay busts. The pecise mechanism behind gamma ay busts is not yet undestood, but it is widely accepted that they ae associated with the collapse of a neuton sta to a black hole. Such a collapse could be caused by matte falling onto the neuton sta fom a companion sta, theeby making the neuton sta unstable, o fom the mege of two neuton stas. 7.2 Supemassive black holes a Sg A* Aleady in the 1960s the Bitish astonomes Donald Lynden-Bell and Matin Rees conjectued that ou galaxy habous a black hole at its cente, and that the same is tue fo many othe galaxies. At that time, this idea was highly speculative. By now, thee is vey good evidence that it is tue. The immediate neighbouhood of the cente of ou galaxy, which is in the diection of the constellation Sagittaius Ache, cannot be obseved with optical telescopes because it is hidden behind dust. In the infaed and in the adio, howeve, the dust is lagely tanspaent, and the same is tue on the othe side of the optical spectum in the X-ay. In the yea 1974 a compact adio souce was detected nea the cente of ou galaxy which was called Sagittaius A*, abbeviated Sg A*. Intensive obsevations of its neighbouhood began in the 1990s. Two 99

100 goups studied the motion of a goup of stas, the so-called S stas, that obit the cente of ou galaxy with infaed cameas: A team fom the Max Planck Institute fo Extateestial Physics MPE in Gaching, headed by Reinhad Genzel, used the VLT in Chile, while a team fom the Univesity of Califonia at Los Angeles UCLA, headed by Andea Ghez, used the Keck telescope in Hawaii. The S stas have angula distances of about 0.1 to 0.5 acseconds fom the cente of ou galaxy and one of them, the sta S2, needs not moe than 15 yeas fo a evolution. The pictue below shows its obit as it was detemined by the MPE goup. TheobsevationsoftheMPEgoupandtheUCLAgoupaeinveygoodageement. AllSstas wee found to descibe pefect Keple ellipses aound a common cente seen in pespective. Fom the obits one could deduce the mass of the cental object to be Sola masses. The distance of the galactic cente fom us is about 8 kpc. This implies that the S stas obit at a distance of moe than 1000 Schwazschild adii which explains why no deviation fom a Keple ellipse, in paticula no peicente pecession, has been obseved so fa. Thee is only a vey modeate amount of X-ay adiation fom the cente of ou galaxy. If the cental mass had a suface, a lage pat of the luminosity, in paticula in the X-ay, would be emitted when acceted matte hits the suface. Howeve, this kind of adiation is not obseved. This is stong evidence fo the conjectue that thee is a black hole at the cente of ou galaxy. If the object at the cente of ou galaxy is a black hole, then it casts a shadow that is big enough fo being obseved with pesent o nea-futue instuments. A mass of Sola masses coesponds to a Schwazschild adius of S = km. If this value is inseted into Synge s fomula fo the shadow, togethe with O = 8 kpc, one finds an angula diamete of the shadow of about 53 micoacseconds. This is aleady esolvable with Vey Long Baseline Intefeomety VLBI using adio telescopes on diffeent continents. Thee ae two pojects, the Event Hoizon Telescope and the BlackHoleCam, which aim at obseving the shadow within a few yeas. Both pojects ae planning to obseve in the submillimete ange that s fa infaed because at longe wavelengths one expects that the shadow is washed out by scatteing. Thee ae seveal existing and planned instuments that can be opeated at submillimete wavelengths, among them the ALMA telescopes in Chile. We emphasise again that the shadow is not an image of the hoizon but athe of the photon sphee o, in the case of a otating black hole, of the photon egion. Theefoe, an obsevation of the shadow is not an ultimate poof that thee is a black hole, i.e., an object with an event hoizon. The most impotant infomation that can be deduced fom the shadow is the spin of the object: If the shadow is cicula, then the object is non-spinning. 100

101 The emission egion of the adio souce Sg A* has been localised to within an angula diamete of 37 micoacseconds, which is less than the expected diamete of the shadow. Also, thee is some indication that the position of Sg A* does not exactly coincide with the baycente aound which the S stas ae obiting. The intepetation is that the emission egion is pat of an accetion disc; we eceive adiation fom the pat which is moving towads us and is thus Dopple enhanced. As the offset cannot be moe than a few Schwazschild adii, this is futhe evidence fo the conjectue that thee is a black hole at the cente of ou galaxy. It is vey difficult to come up with an altenative model that concentates 4 million Sola masses within a few Schwazschild adii without a shining suface. All such models ae moe exotic, and moe speculative, than a black hole. In the yea 2012 an object was obseved nea the cente of ou galaxy in the infaed that was intepeted as a gas cloud and denoted G2. It was expected that, because of the tidal foces, G2 would disintegate when passing though its peicente and that the fagments would be swallowed by the black hole, emitting stongly in the X-ay egime when being acceleated towads the cente. Nothing like that happened. G2 has gone though its peicente in 2014 without any spectacula events. It is now widely believed that thee is a sta at the cente of G2 which keeps the matte togethe so that it will not be ipped apat by the tidal foces. The pictue below shows an atist s impession of the obit of G2 ed and, fo the sake of compaison, the obits of seveal S stas blue. In the yea 2016 a new beam combine called GRAVITY is planned to go into opeation with the VLT. With this new instument one hopes to see infaed stas, and othe objects obiting the cente of ou galaxy, with angula distances of less than 0.01 acseconds. Then genealelativistic effects, in paticula a peicente pecession, sould be obsevable. b Othe supemassive black holes If ou galaxy habous a black hole at its cente, then it is vey likely that the same is tue of most, if not all, othe galaxies. A paticulaly inteesting candidate is the object at the cente of M87 i.e., numbe 87 in the Messie catalogue. Thee is some speculation that thee might be even two black holes at the cente of M87 which ae obiting about a common baycente. M87 is the only candidate, next to Sg A*, whee the shadow might be obsevable in the nea futue. It is at a distance of about 17 Mpc which is moe than thee odes of magnitude fathe away than Sg A*. Howeve, as its mass is about Sola masses, which is almost thee odes of magnitude moe than Sg A*, the expected diamete of the shadow is about 30 micoacseconds and thus within the ange of obsevability. 101