Carter-Penrose diagrams and black holes

Transcription

1 Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example of computation fo cuved spacetime has been povided, with a confomal diagam fo Robetson-Walke univese. A way of futhe coodinate tansfomations needed to extend given manifold has been povided and a Penose diagam fo Schwazschild black hole has been constucted. 1. Penose diagams 1.1. Obtaining Penose diagams fom Minkowski space Cuved spacetime manifolds can be often appoximated by manifolds with high degees of symmety. It would be useful to be able to daw spacetimes diagams that captue global popeities and casual stuctue of sufficiently symmetic spacetimes. What is needed to be done is a confomal tansfomation which bings entie manifold onto a compact egion such that we can fit the spacetime ie. its infinities) on a finite 2-dimensional diagam, known as Penose-Cate diagam o Cate-Penose diagam o just confomal diagam). Let us stat with the Minkowski space, with metic in pola coodinates: ds 2 = dt 2 + d dω 2, 1) whee dω 2 = dθ 2 + sin 2 ΘdΦ 2 is a metic on a unit two-sphee and anges of timelike and spacelike coodinates ae: < t <, 0 <. 2) In ode to get coodinates with finite anges, let us switch to null coodinates: u = t, v = t + 3) with coesponding anges Fig.1.): < u <, < v <, u v. 4) 1)

2 2 Cate-Penose diagams pinted on July 6, 2010 Fig. 1. Each point epesents a 2-sphee of adius = 1 2 v u). The Minkowski metic in null coodinates is given by ds 2 = 1 2 dudv + dvdu) v u)2 dω 2. 5) We can bing infinity into a finite coodinate value by using the actangent: with anges U = actanu), V = actanv), 6) π/2 < U < π/2, π/2 < V < π/2, U V. 7) Afte easy calculations one gets that the metic given in 5) in this coodinates takes fom: ds 2 1 [ = 4 cos 2 U cos 2 2dUdV + dv du) + sin 2 V U)dΩ 2]. 8) V R: Tansfoming back to the timelike coodinate T and adial coodinate with finite anges T = U + V, R = V U 9) 0 R < π, T + R < π, 10)

3 Cate-Penose diagams pinted on July 6, the metic is given by: whee ds 2 = ω 2 T, R) dt 2 + dr 2 + sin 2 RdΩ 2). 11) ωt, R) = 2 cos U cos V = cos T + cos R. 12) Finally, the oiginal Minkowski metic, which we denoted ds 2, is elated by a confomal tansfomation to the new metic: ds 2 = ω 2 T, R)ds 2 = dt 2 + dr 2 + sin 2 RdΩ 2. 13) This descibes the manifold R S 3, whee 3-sphee is maximally symmetic and static. Fig. 2. The Einstein static univese, R S 3, potayed as a cylinde. The shaded egion is confomally elated to Minkowski space see: Fig.3.). The stuctue of the confomal diagam allows us to subdivide confomal infinity into a few diffeent egions Fig.2.): i + = futue timelike infinity T = π, R = 0) i 0 = spatial infinity T = 0, R = π) i = past timelike infinity T = π, R = 0) I + = futue null infinity T = π R, 0 < R < π) I = past null infinity T = π + R, 0 < R < π)

4 4 Cate-Penose diagams pinted on July 6, 2010 Fig. 3. The confomal diagam of Minkowski space. Light cones ae ±45 thoughout the diagam. Note that i +, i 0, and i ae actually points, since R = 0 and R = π ae the noth and south poles of S 3. Meanwhile, I + and I ae null sufaces, with the topology of R S 2. The confomal diagam fo Minkowski spacetime contains a numbe of impotant featues: adial null geodesics ae at the ±45 angle in the diagam. All timelike geodesics begin at i and end at i +. All null geodesics begin at I and end at I + ; all spacelike geodesics both begin and end at i 0. On the othe hand, thee can be non-geodesic timelike cuves that end at null infinity if they become asymptotically null ) Examples When we put pola coodinates on space, the metic becomes: ds 2 = dt 2 + t 2q d dω 2) 14) with 0 < q < 1. The cucial diffeence between this metic and the one of Minkowski space is a singulaity at t = 0, what esticts the ange of coodinates: 0 < t <, 0 <. 15)

5 Cate-Penose diagams pinted on July 6, We choose new time coodinate η called confomal time, which satisfies: dt 2 = t 2q dη 2 16) with ange same as of t: 0 < η <. 17) This allows us to bing out the scale facto as an oveall confomal facto times Minkowski: ds 2 = [1 q)η] 2q/1 q) dη 2 + d dω 2) 18) Afte the same sequence of coodinate tansfomations as peviously, one gets η, ) to T, R) with anges: The metic 18) becomes: 0 R, 0 < T, T + R < π. 19) ds 2 = ω 2 T, R) dt 2 + dr 2 + sin 2 RdΩ 2) 20) Once again we expessed ou metic as a confomal facto times that of the Einstein static univese. The diffeence between this case and that of flat spacetime is that timelike coodinate ends at singulaity T = 0 Fig. 4. Confomal diagam fo a Robetson-Walke univese.

6 6 Cate-Penose diagams pinted on July 6, Black holes The confomal diagam gives us an idea of the casual stuctue of the spacetime, e.g. whethe the past o futue light cones of two specified points intesect. In Minkowski space this is always tue fo any two point, but the situation becomes much moe inteesting in cuved spacetimes. A good example is Schwazschild solution, which descibes spheically symmetic vacuum spacetimes. Schwazschild metic is given by: ds 2 = 1 2GM ) dt GM ) 1 d dω 2 21) This is tue fo any spheically symmetic vacuum solution to Einstein s equations; M functions as a paamete. Note that as M 0 we ecove Minkowski space, which is to be expected. It is also woth noting, that the metic becomes pogessively Minkowskian as we go to ; this popety is known as asymptotic flatness. One way of undestanding a geomety is to exploe its causal stuctue, as defined by the light cones. We theefoe conside adial null cuves, those fo which θ and φ ae constant and ds 2 = 0: ds 2 = 0 = 1 2GM ) dt GM ) 1 d 2, 22) fom which we see that dt d = ± 1 2GM ) 1 23) This measues the slope of the light cones on a spacetime diagam of the t- plane. Fo lage the slope is ±1, as it would be in flat space, while as we appoach = 2GM we get dt/d ±, and the light cones close up The poblem with ou cuent coodinates is that pogess in the diection becomes slowe and slowe with espect to the coodinate time t. We suspect that ou coodinates may not have been good fo the entie manifold. Thus, lets have a close look at EddingtonFinkelstein coodinates, a pai of coodinate systems which ae adapted to adial null geodesics fo a Schwazschild geomety. By changing coodinate t to the new one ũ, which has the nice popety that if we decease along a adial cuve null cuve ũ = constant, we go ight though the event hoizon = 2GM without any poblems. The egion 2GM is now included in ou spacetime, since physical paticles can easily each thee and pass though - the appaent singulaity at the Schwazschild adius is not a physical singulaity but only a coodinate one. Still, thee ae othe diections in which we can extend ou manifold.

7 Cate-Penose diagams pinted on July 6, Fig. 5. Light ay which appoaches = 2GM neve seems to get thee, at least in this coodinate system; instead it seems to asymptote to this adius. 3. Penose diagams fo Schwazschild black holes Kuskal coodinates ds 2 = 32G3 M 3 e /2GM dv 2 + du 2 ) + 2 dω 2 24) whee is defined implicitly fom ) u 2 v 2 ) = 2GM 1 e /2GM 25) We stat with the null vesion of the Kuskal coodinates, in which the metic takes the fom ds 2 = 16G3 M 3 e /2GM du dv + dv du ) + 2 dω 2, 26) whee is defined implicitly via ) u v = 2GM 1 e /2GM. 27) Then essentially the same tansfomation as was used in flat spacetime suffices to bing infinity into finite coodinate values: u u ) v = actan, v ) = actan, 28) 2GM 2GM

8 8 Cate-Penose diagams pinted on July 6, 2010 with anges π/2 < u < +π/2, π/2 < v < +π/2, π < u + v < π. 29) The u, v ) pat of the metic that is, at constant angula coodinates) is now confomally elated to Minkowski space. In the new coodinates the singulaities at = 0 ae staight lines that stetch fom timelike infinity in one asymptotic egion to timelike infinity in the othe. Fig. 6. Penose diagams fo Schwazschild black holes 4. Conclusions Penose diagams captue the causal elations between diffeent points in spacetime, with the confomal facto chosen in a way that entie infinite spacetime is tansfomed into a diagam of finite size. This gives a vey intuitive pictue of the whole spacetime and its signulaities, thus while compaing diffeent spacetimes it is much easie to compae thei confomal diagams than metics themselves. REFERENCES [1] Sean Caoll, Spacetime and Geomety: an Intoduction to Geneal Relativity, 2004.

9 Cate-Penose diagams pinted on July 6, [2] Leonad Susskind, James Lindesay, An Intoduction to Black Holes, Infomation and The Sting Theoy Revolution, [3] Sean M. Caoll, Lectue Notes on Geneal Relativity, [4] Jezy Kowalski-Glikman, Wpowadzenie do Teoii Gawitacji, 2004.