Black Holes. Tom Charnock

Transcription

1 Black Holes Tom Chanock

2 Contents 1 Why Study Black Holes? 2 2 Relativity The Metic Chistoffel Connections Cuves Geodesics Kinematics Einstein s Equations Hypesufaces Schwazschild Geomety Exteio Schwazschild Spacetime Eddington-Finkelstein Coodinates Kuskal Coodinates Killing Vectos 13 5 Singulaities 15 6 Stuctue of a Sta Pefect Fluid Static Sta Constant Density Sta Bikhoff s Theoem Redshift Sta Collapse Non-Radial Null Rays Spacetime Diagams Minkowski

3 Chapte 1 Why Study Black Holes? Black holes exist. Pobably. Although they ae not diectly obsevable they ae believed to be at the cente of galaxies. The object at the cente of the Milky Way has a mass of appoximately 10 6 M, which is contained in a elatively small adius. This is infeed fom the obits of stas about the galactic cente. Some galaxies have black holes up the mass of M. Thee ae also individual (collapsed stas) stella mass dak objects seen as pat of a double sta. Acceted Matte X-Rays Sta Dak Patne ys+ Thee ae 10 of these systems documented whee the dak patnes ae believed to be black holes. One of these systems is Cygnus X-1. In tems of cosmology then one of the dak matte candidates is the black hole, although this is not the most popula model cuently. Thee may also be mini-black holes which existed in the ealy univese and influenced the expansion. The poblem with this is that thee is no known mechanism fo ceating black holes at this peiod and thee has been no obsevations of them. Black holes began as a theoetical pediction of Geneal Relativity in 1916 by Schwazschild. In 1965 then the tem black hole aose and a pope undestanding of them was initiated. A black hole is an exteme example of cuved spacetime which has inteesting mathematical popeties. One example of this is the singulaity. Inside a Schwazschild black hole then the contaction of the Riemann cuvatue tenso is R abcd R abcd. The tidal foces gow to infinite stengths and so the desciption of spacetime beaks down. Paticle physics and othe well known models beak down at the singulaity and so it is known that Einsteins equations ae no longe good nea the singulaity. This can be ectified by eplacing Einsteins classical theoy of gavity by a quantum theoy of gavity. Thee is no cuent quantum theoy of gavity. If one was designed then it would need to ecove geneal elativity in the coect limit and would also find Hawking Tempeatue. Hawking Pediction 1974 Quantum matte is placed on a collapsing sta spacetime. At late times then the black hole adiates themally. Fo the Schwazschild black hole, with mass M, then: T = c3 8πMk B 2

4 This can be seen to be a quantum effect since is pesent. This is not a quantum gavity but shows that quantum matte feels the effect of the cuvatue. Fo M = M then T 10 7 K which is totally unobsevable fo a sola mass black hole. Say if M = kg then T 10 9 K. This would be visible, but this mass would be equivalent to a kilomete cubed vessel full of sea wate, o a small mountain. These ae vey small and so would be had to obseve and have not been so fa. The hint that Hawkings pediction is coect because black holes, which now have a tempeatue, can be bought into discussions of themodynamics. Without Hawking adiation then the second law of themodynamics could be violated since entopy would educe due to black holes. Hawking adiation aises due to pai ceation on the hoizon of a black hole so that one of the paticle falls into the black hole and the othe escapes to infinity. The paticle which falls in can be thought of as binging with it an enegy debt fom the Heisenbeg uncetainty pinciple and the black holes pays the debt by dissipating some of its mass. This dissipation is slow, with the lifetime of the black hole being: Black Hole Lifetime = ( yeas ) ( M kg So fo most easonable sized black holes then this is odes of magnitudes longe than the length of the univese. The yeas aises fom the type of quantum matte being adiated. ) 3 3

5 Chapte 2 Relativity 2.1 The Metic Spacetime is possibly cuved. Fo Minkowski space then the line element is the distance measue and is given by: ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 c = 1 can be set by choice of natual units. In cuved spacetime this is genealised to: ds 2 = g ab (x)dx a dx b g ab ae the components of the metic tenso and is symmetic so g ab = g ba. This means that it is always diagonisable at a point in spacetime. It is also non-degeneate so that it has an invese g ab such that g ab g bc = δ a c. The eigenvalues of g ab ae (-+++). dx a dx b is the symmetic poduct of diffeential foms: g ab dx a dx b = 1 2 g ab(dx a dx b + dx b dx a ) = g ab dx a dx b The dimension of the line element is [ds 2 ] = L 2. Thee is some choice in whethe the dimension is in the coodinate, say d 2 o c 2 dt 2 o in the metic, say the 2 component in font of dϑ 2 in a line element witten in spheical polas. Vectos ae given by v a and one-foms can be obtained using the metic tenso: v a = g ab v b and v a = g ab v b The vectos and one-foms can be classified by the value of thei squae. If v a v a > 0 then the vecto v a o one-fom v a is called spacelike. v a v a = 0 indicates a null vecto o one-fom and v a v a < 0 has vectos o one-foms called timelike Chistoffel Connections Γ a bc = 1 2 gad (g bd,c + g cd,b g bc,d ) These connections ae tosion-fee and so ae symmetic on the lowe indices. This is used to define a covaiant deivative: Scala a f = a f Vecto a v b = a v b + Γ b acv c One-Fom a v b = a v b Γ c ab v c 4

6 With the Chistoffel connection then c g ab = 0 and c g ab = 0. Anothe popety is that: c (g ab v b ) = ( c g ab )v b + g ab c v b = g ab ( c v b + Γ b cd vd ) = c v a + g ab g bd Γ b cd v b = c v a Γ b acv b 2.2 Cuves = c v a x 2 X a (λ) λ = 3 λ = 1 λ = 2 x 1 λ is a paamete which incease monotonically along the cuve. A tangent vecto is given by: u a = dxa dλ This tangent vecto can be classified as timelike, spacelike o null eveywhee and defines timelike, spacelike o null cuves. The pope time is defined fo timelike cuves by: τ = ua u a dλ This is the natual nomalisation of the paamete along the cuve and descibes the woldline of a massive paticle. Fo spacelike cuves this paamete is: l = ua u a dλ This is the pope length and descibes the woldline of imaginay mass paticles. Null cuves have: ua ua u a dλ = 0 = u a dλ Thee is no notion of pope time o length. This descibes the tajectoy of massless paticles. 2.3 Geodesics The action integal fo the distance can be witten down as: S = 1 dλg ab (x) dxa 2 dλ dx b dλ 5

7 The geodesic is the stationay cuve of S. Thee is no squae oot which has some advantages, such as the ability to not have to know whethe the integal is valid because of the classification of the cuve since it is always valid. The Eule-Lagange equations can be solved to get the equations of motion. These can always be eaanged into the fom: d 2 x c dλ 2 This is a covaiant equation since it can be witten as: + dx a dx b Γc ab dλ dλ = 0 u d d u c = 0 This explains that the tangent vecto to the cuve is paallel tanspoted along the cuve. λ is an affine paamete. This is tue because if λ aλ + b whee a and b ae constant then the geodesic equation emains unchanged. On a geodesic then: d dλ (u au a ) = dxb d dλ dx b (u au a ) = u b b (u a u a ) = 2u a u b ( b u a ) = 2u a (0) = 0 This means that u a u a is constant so that a timelike geodesic stays timelike and similaly fo the spacelike and null geodesics. Because this is tue then fo timelike geodesics then λ = τ can be chosen abitaily, and fo spacelike geodesics λ = l. Fo null geodesics then affine paamete needs to be used. Geodesics ae the tajectoies of test of fee paticles. 2.4 Kinematics Suppose X a (λ) in a timelike cuve then λ = τ can be chosen and u a u a = 1 whee u a is the fou-velocity. The fou-acceleation is obtained by: A a = u b b u a To see what this means then conside: d dτ (ua u a ) = d dτ ( 1) = 0 And also: d dτ (ua u a ) = 2u b b u a = 2u a A a The contaction of a timelike vecto with the acceleation is zeo so they ae othogonal. This means that the acceleation must be spacelike o null. The scala pope acceleation is: A = A a A a 6

8 2.5 Einstein s Equations The Riemann tenso is R a bcd. It is defined using paallel tanspot. Contacting this tenso gives the Ricci tenso, R a bad = R bd which is symmetic and fom which the Ricci scala can be obtained, g ab R ab = R a a = R. The Einstein tenso is then given by: G ab = R ab 1 2 Rg ab This satisfies a G ab = 0. Einstein s equations ae then given by: G ab + Λg ab = 8πGT ab Hee Λ is a cosmological constant. 2.6 Hypesufaces Hypesufaces, Σ, ae ceated by a constant function f whose exteio deivative is a f 0 which means that thee ae no intesections o cusps. a f = n a is the nomal one-fom. The nomal vecto is obtained fom the metic. Fo a Minkowski spacetime the dimensional plane is given by ds 2 = dt 2 + dx 2 and is the hypesufaces ae ceated by f = αt + x whee α is a constant then the nomal one-fom is n a = a f = ( α, 1) and so the nomal vecto is n a = (α, 1): t v a n a n a v a α = 1 x α > 1 n a = v a 0 < α < 1 7

9 When 0 < α < 1 then n a n a = α > 0 so n a is spacelike and so Σ is timelike. As expected the tangent vecto and the nomal vecto ae othogonal, v a n a = 0, eventhough the angle does not look like a ight angle on the gaph. This is because the geomety is Loentzian and not Euclidean. Including the othe two coodinates it can be seen that the signatue of the induced metic is ( + +). When α > 1 then n a n a = α 2 < 0 and so n a is timelike and hence σ is spacelike. The induced metic has signatue (+ + +). When α = 1 then n a n a = α = 0 and so n a is null and the hypesuface is also called null. The nomal vecto is also the tangent vecto and so the tangent vecto is (0 + +) and as such the metic on Σ is degeneate. 8

10 Chapte 3 Schwazschild Geomety The Schwazschild line element is: ( ds 2 = 1 2M ) dt 2 d M/ + ( 2 dϑ 2 + sin 2 ϑdφ 2) The Schwazschild geomety is the only static matte-λ fee solution of the Einstein equation, G ab = 0, which is non-flat, R abcd 0. M is the mass paamete and has dimensions of length. It can be equated to the physical mass by: If M 0 then Minkowski space is ecoveed. M = G c 2 M phys 3.1 Exteio Schwazschild Spacetime The Schwazschild spacetime has M > 0 and > 2M and when then 2M/ 0 so spheical Minkowski spacetime metic is ecoveed. This means that it is called asymptotically flat o asymptotically Minkowski. Schwazschild spacetime is also spheically symmetic and since g tt < 0 and t g ab = 0 then the metic is time-independent. Thee is a pseudo-singulaity when 2M since the fomula, as witten hee, does not define a metic since the tem diveges. The light cones of this metic ae: t Outgoing Radial Null Geodesics 2M Ingoing Radial Null Geodesics 9

11 If thee is a geodesic which begins adially, ϑ = φ = 0 then it stays adial. Radial null geodesics have: ( 0 = 1 2M ) dt 2 d M/ These can be solved by integating: t = ± d 1 2M/ The lightcones get naowe the close to the hoizon they get. At = 2M then the metic becomes degeneate since the dt tem disappeas Eddington-Finkelstein Coodinates This can be ectified by changing to the coodinate: w = t + 2M ( 2M 1 ) Finding dt in this coodinates allows the line element to be witten as: ( ds 2 = 1 2M ) dw 2 + 4M ( dwd M ) d dω 2 Radial null geodesics have ds 2 = dω 2 = 0 and so thee ae two families of geodesics. The ingoing adial null ays ae descibed by d + dw = 0 and the outgoing ones ae: ( 1 2M ) ( d 1 2M ) dw = 0 The lightcone diagam is given by: w 2M Outgoing Radial Null Geodesics v Ingoing Radial Null Geodesics The ingoing lightcones actually cay on staight though = 2M. This is because the metic is egula since the components ae all alight fo > 0 and the signatue emains ( + ++) via continuity. The exteio Schwazschild spacetime has been extended into the egion 0 < < 2M by the Eddington- Finkelstein extension. Thee is also a adial null ay which stays at = 2M. The Schwazschild coodinates have time invesion symmety such that t t is an isomety. The Eddington-Finkelstein coodinates do not have time invesion symmety and instead have two diffeent 10

12 metics to descibe tavelling into the futue and tavelling into the past. The ingoing coodinates given by w can be witten in tems of v = w + so that: ( ) v = t + + 2M ln 2M 1 Unde t t then the outgoing Eddington-Finkelstein coodinates ae obtained and given in tems of: ( ) u = t 2M ln 2M 1 The lightcone diagam fo these coodinates is:? Outgoing Radial Null Geodesics Ingoing Radial Null Geodesics 2M Kuskal Coodinates Anothe extension is the Kuskal-Szekees-Fousdal extension, moe often efeed to as the Kuskal extension. Beginning with the ingoing and outgoing Eddington-Finkelstein then algebaically escaling them gives: U = e u/4m and V = e v/4m When > 2M then U < 0 and V > 0. The whole metic is then given by: ds 2 = 32M 2 e /2M dudv + 2 dω 2 This = (U, V ) is a function of U and V. This can be found using: ( ) UV = 2M 1 e /2M This is not an elementay function but is almost elementay and good enough to do calculations with. Thee is still a 1 tem in the metic and so thee is a singulaity when = 0. UV is valid as long as it is geate than 1 o indeed UV < 1. Radial null ays in Kuskal coodinates ae given by du dv = 0 which gives eithe U =constant o V =constant. 11

13 U t = constant V UV = 1 = 2M t = constant = 2M UV = 1 = constant I: Oiginal Exteio, > 2M The Schwazschild time coodinate is given by t = 2M ln( V/U) and sufaces of constant t ae spacelike and to t is a timelike coodinate. The sufaces of constant ae timelike and so the Schwazschild coodinate is spacelike. II: Black Hole Inteio, 0 < < 2M This is the egion descibed by the ingoing Eddington- Finkelstein coodinates. An exteio Schwazschild-like time component t can be defined t = 2M ln(v/u) and is used to descibe a metic: ( ds 2 = 1 2M ) d t 2 d M/ + 2 dω 2 Since 0 < < 2M then it can be seen just fom the metic that has become the timelike component and so sufaces of constant ae spacelike and t has become the spacelike component and so sufaces of constant t ae timelike. III: Second Black Hole Exteio, > 2M then thee is spatial invesion symmety. Since the metic has (U, V ) (V, U) as an isomety IV: White Hole Inteio, 0 < < 2M Finkelstein coodinates. This is the egion descibed by the outgoing Eddington- The t( t) coodinates ae highly singula at the oigin which is why the singulaity at = 2M occus in the Schwazschild coodinates. 12

14 Chapte 4 Killing Vectos Minkowski spacetime has ds 2 = dt 2 +dx 2 + and is invaiant unde tanslations of the fom x = x +a whee a is a constant. This means that the metic has space tanslation symmety o isomety. It also has time tanslation isomety t = t + b whee b is a constant. These isometies ae obvious fom just plugging in these new values into the metic. Killing vectos ae a way of seeing these isometies in a coodinate fee desciption. Self-isomety can be descibed as: Φ g = g A one paamete goup of isometies is given by Φ t = e tξ whee ξ is a vecto field. This family of isometies have Φ g = g t. This can also be denoted by Killings equation using the Lie deivative: O altenatively as: L ξ g = ξ c c g ab + g ac b ξ c + g cb a ξ c = 0 a ξ b + b ξ a = 0 Fo Minkowski space then the Killing vecto which tanslates acoss the x diection is ξ = x and fo time tanslation then ξ = t. Boosts ae witten as ξ = x t + t x. Suppose thee ae coodinates (x 1, x 2, x 3, x 4 ) such that x 1g ab = 0 then ξ = k x 1 is Killing, whee k is a constant. This could be witten as ξ a = (1, 0, 0, 0). Now suppose X a (λ) is a geodesic and ξ is Killing and u a = Ẋa : d dλ (ξ au a ) = u c c (ξ a u a ) = u c c (ξ a u a ) = u c c (ξ a u a ) = u a u c ( c ξ a ) + ξ a u c ( c u a ) = 1 2 ua u c ( c ξ a ) uc u a ( a ξ c ) + (0 fom geodesic equation) = ua u c 2 = 0 This means that ξ a u a =constant. ( c ξ a + a ξ c ) If thee is a two dimensional manifold with a vecto field then sufaces which ae eveywhee othogonal 13

15 to the vecto field can be constucted. In moe than two dimensions then this is not geneally tue. A one-fom v a is hypesuface othogonal if it is othogonal to sufaces of f =constant fo some f: v dv = 0 In less mathematical (diffeential geomety) language then this can be witten as: v [a b v c] = ( v a b v c + v b c v a + v c a v b v a c v b v c b v a v b a v c ) = 0 Fo the Chistoffel connections then this is equivalent to: v [a b v c] = 0 Stationay spacetime has a timelike Killing vecto. Thee exists coodinates (t, x 1, x 2, x 3 ) such that t g ab = 0 and g tt < 0, but g ti may be non-zeo. Static spacetime has a hypesuface-othogonal Killing vecto and has no coss tems in the time pat of the metic, g ti = 0. This means that thee is time evesal invaiance. 14

16 Chapte 5 Singulaities The two dimensional Euclidean plane ds 2 = dx 2 + dy 2 is not singula because infinite distances can be tavelled in any diection and no bounday will be eached. The two-sphee S 2 in R 3 is also not singula because any diection can be tavesed to an infinite distance (even though steps may be etaced) and an edge will neve be found. A bumped sphee again has no way to fall off the suface and so is not singula. In Loentzian signatues such as Minkowski space then positive and negative infinite pope distances can be eached as can pope times. The poblem aises with null geodesics. This has no concept of pope distance o pope time but still a positive o negative infinite affine paamete can be eached. Since pope time and pope distance ae affine then it can be seen that Minkowski spacetime is non-singula fo all affine paamete. A complete geodesic is one whee < λ <. A geodesically complete spacetime is one whee integating along a geodesic can be done anywhee in the spacetime. Convesely, a geodesically incomplete spacetime cannot be integate along a geodesic continuously in the spacetime. If the integal falls of the spacetime then it is called geodesically incomplete. Geodesically incomplete spacetime can be atificial say fo R 2 {0, 0} and so this is not consideed singula. Singula spacetime must be geodesically incomplete and the spacetime cannot be extended into a geodesically complete one. Schwazschild spacetime is singula. This can be seen be cause the Cuvatuve Scala R abcd R abcd at = 0. Fo a cone then R abcd = 0 but this space is still singula so the cuvatue scala does not have to become infinite. Suppose the metic is ds 2 = dt 2 + t 2 dφ 2 with t > 0 and φ = φ + 2π. This is the same as a FRW univese and so is an expanding cosmology. The singulaity occus at t = 0 which is identified to the big bang. The Riemann cuvatue tenso is R abcd = 0. This is also an example of a singula spacetime. 15

17 Chapte 6 Stuctue of a Sta 6.1 Pefect Fluid Paticles tavel along woldlines and have timelike fluid fou-velocity u a which is the tangent vecto to the woldline nomalised such that u a u a = 1. Einsteins equation is: G ab = 8πGT ab T ab is the enegy momentum tenso. It is used to descibe a fluid with pessue p and density ϱ. In the case of a pefect fluid then the enegy momentum tenso is T ab = (ϱ + p)u a u b + pg ab. Using local inetial coodinates then u a can be made to be wholey in the time diection, u a = (1, 0, 0, 0) and so: ϱ T ab = p p p The time component of the tenso is the density which is what it is needed to be. The pessue is spatially isotopic and so physically thee is no intenal fiction. 6.2 Static Sta A spheically symmetic static line element can be witten as: ds 2 = e 2Φ() dt 2 + e 2Λ() d dω 2 It can seen to be spheically symmetic since the unit S 2 line element is pesent and thee ae no othe angula tems. It is static since thee is a timelike killing vecto, t, which is hypesuface-othogonal to sufaces of t =constant. Using the metic then the Einstein tenso can be found. When a fluid tavels along with nomalised velocity u a = (e Φ(), 0, 0, 0) then assuming a static fluid then then enegy momentum tenso can be found. Solutions to the Einstein equations can then be analysed. Thee ae thee independent equations, the 00 pat, the pat and the ΩΩ pat. Looking at the metic then it is convenient to ewite Λ() in tems of a now function, m(), called the mass function: The 00 pat can now be witten: The pat is also simplified to: e 2Λ() = dm() d Φ = 1 1 2m()/ = 4πG 2 ϱ m() + 4πGϱ3 ( 2m()) The ΩΩ pat is still complicated but since a G ab = 0, which implies that a T ab = 0, then it follows that: (ϱ + p)φ = dp d 16

18 If these thee equations ae satisfied then the ΩΩ equation is implied to be satisfied also. Outside the sta then thee is no matte so ϱ = 0 and p = 0 and when the diffeential equations ae solved then Schwazschild is ecoveed. When inside the sta then ϱ > 0, p 0 and a stability condition is also included: dp dϱ > 0 The Φ fom the Einstein equations can be eliminated to leave: dp d = (ϱ + p)(m() + 4πGp3 ) ( 2m() An equation of state now needs to be intoduced to descibe the stiffness of the matte. Thee ae two equations fo two unknowns, m() and p(), since p() detemines ϱ(). The solutions ae of the fom: p m() p c M = m( S ) ϱ S e 2φ() S ϱ c 1 Gas Solid S S The mass can be found by integating the dm()/d equation: M = G S In Newtonian gavity then the mass is given by: mass = ϱdv 0 ϱ(4π 2 )d = S 0 ϱ(4π 2 )d This looks simila to the equation obtained fom the diffeential equation but fo the elativistic case: S ϱ(4π 2 )d ϱdv 0 17

19 This is because when looking at the metic then the thickness is given by: (d (thickness)) 2 = This means that dv = 4π 2 d/ 1 2m()/ and so: ϱdv > d 2 1 2m()/ > d2 S 0 ϱ(4π 2 )d This means that the mass is less than expected on Newtonian gounds. Using E = mc 2 then the sta can be said to have less enegy than expected. This is because building the sta eleases enegy Constant Density Sta Take an equation of state ϱ = constant > 0. This is not vey physical but is solvable: [ M = 1 ( ) ] (ϱc /ϱ) 1 S (ϱ c /ϱ) M/ S 4/9 ϱ c /ϱ S > (2 + 1/4)M since the cuve is unde the cuve. This means that the suface of the sta is always outside the Schwazschild adius. This was fist found by Schwazschild. If the sta is squeezed then the density becomes abitaily lage, but this is not physical. This is alight in this case since the equation of state is just a toy one. 6.3 Bikhoff s Theoem Evey spheically symmetic solution to G ab is one, o pat, of: Kuskal (M > 0) Minkowski (M = 0) M < 0 Schwazschild (M < 0) The spacetime outside a spheical sta is Kuskal, even when the sta is dynamical, i.e. pulsates o collapses. Thee ae no spheically symmetic gavitational waves. The outcome of a spheically symmetic sta collapse must be pat of Kuskal. 18

20 6.4 Redshift γ = X a (λ) = X ( λ) + f a (λ) q γ = X a (λ) p τ u a τ v a A geneal edshift fomula can be found using a vecto field to move one geodesic to anothe close by: (Ẋa f a ) p = (Ẋa f a ) q Hee (f a ) p = u a dτ and (f a ) q = v a d τ. This means the edshift can be witten as: dτ d τ = ν obs ν souce = (Ẋa v a ) q (Ẋa u a ) p 6.5 Sta Collapse U = 2M = 0 Geodesic τ 0 V Radial Null Rays = 2M = 0 Obseve Suface of a Sta At t obs then τ = τ 0 De t obs/4m. No matte how lage t obs gets it appeas to the obseve that the suface of the sta neve cosses the Schwazschild adius. dτ = D dt obs 4M e t obs/4m This is the edshift, and so it can be seen that it is exponential. The time scale fo a scala mass is 4M 10 5 s. 19

21 6.6 Non-Radial Null Rays ds 2 = F ()dt 2 + d2 F () + 2 dω 2 Unlike adial geodesics the non-adial ones cannot have dω = 0. The angula pat can be simplified though using symmeties. The coodinates can be chosen such that ϑ = π/2 so sin 2 ϑ = 1 and dϑ = 0. This leaves the line element as: ds 2 = F ()dt 2 + d2 F () + 2 dφ 2 The Lagangian can then be witten as: L = 1 2 ] [ F ()ṫ 2 + ṙ2 F () + c2 φ2 The constants of motion ae the angula momentum, 2 φ = l and a consevation of enegy-like tem is F ()ṫ = ε. The thee diffeent types of geodesics can be consideed: dτ 2 timelike k = 1 ds 2 = 0 null k = 0 u a u a = k dτ 2 spacelike k = 1 This means that the geodesic fo the adial pat can be witten as: ṙ 2 + W () = ε 2 Whee W () = (k + l 2 / 2 )F (). Fo null geodesics in Schwazschild then F () = 1 2M/ and k = 0. W () ε 2 > l 2 /27M 2 2 U = 0 V ε 2 = l 2 /27M 2 3b 3c 3a = 2M 0 < ε 2 < l 2 /27M 2 1b 1a ε = 0 4 = 2M 4 3b 1b 2 = 0 2 1a 3a 3c 20

22 6.7 Spacetime Diagams Minkowski Timelike Geodesic U T V i + I + I + p q V = constant X i 0 i 0 Spacelike Geodesic U = constant I I i To analyse what happens at infinity in 1+1 Minkowski then fistly the line element can be changed fom ds 2 = dt 2 + dx 2 to ds 2 = dv du by coodinate tansfomation U = T X and V = T + X. The ange of all these coodinates is < T, X, U, V <. Now taking U = tan p and V = tan q then it can be seen that the ange of p and q is π/2 < p, q, π/2 and the metic can be witten as: ds 2 dpdq = cos 2 p cos 2 q A null geodesic with U =constant (left going) has a constant p and a null geodesic with V =constant (ight going) has a constant q. A timelike geodesic must coss all the (ight going fowads in time) null geodesics and so the tajectoy always begins at i and ends at i + and likewise fo a spacelike geodesic go between i 0 s. 21