4. Kruskal Coordinates and Penrose Diagrams.

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1 4. Kuskal Coodinates and Penose Diagams Removing a coodinate ingulaity at the chwazschild Radius. The chwazschild metic has a singulaity at = whee g 0 and g. Howeve, 00 we have aleady seen that a fee falling obseve acknowledges a smooth motion without any peculiaity when he passes the hoizon. This suggests that the behaviou at the chwazschild adius is only a coodinate singulaity which can be emoved by using anothe moe appopiate coodinate system. This is in GR always possible povided the tansfomation is smooth and diffeentiable, a consequence of the diffeomophism of the spacetime manifold. Instead of the 4-dimensional chwazschild metic we study a -dimensional t,-vesion. The spheical symmety of the chwazschild BH guaanties that we do not loose geneality = 1 dt 1 d (4.1) To descibe outgoing and ingoing null geodesics we divide though d λ and set = 0. 1 o ewitten 1 t& 1 & = 0 (4.) dt d = 1 (4.3) Note that the angle of the light cone in t,-coodinate.deceases when appoaches Afte integation the outgoing and ingoing null geodesics of chwazschild satisfy t = ± * + const.. (4.4) * is called totoise coodinate and defined by * = + ln 1 (4.5) so that d * = 1 d 1. (4.6) As anges fom to, * goes fom - to +. We intoduce the null coodinates which have the diection of null geodesics by υ = t + * and u = t * (4.7) u, υ Fom (4.7) we obtain and fom (4.6) 1 dt = υ + ( d du) (4.8) 8

2 1 d = 1 d* = 1 ( d υ du) (4.9) Inseting (4.8) and (4.9) in (4.1) we find = 1 dudυ (4.10) Fig This plot of t = υ vesus is called a Finkelstein diagam. When the suface of the sta appoaches the light cones distot. Instead of outgoing null geodesics they coincide with the hoizon. Theefoe we can say: the hoizon is geneated by the null geodesics.. You will often find the Finkelstein diagam used to illustate a collapsing sta (David Finkelstein 1958). One may also add the φ-coodinate to constuct a 3-dimensional diagam of the same kind.. 9

3 Fig. 4.. Anothe epesentation of a collapsing sta. Each cicle in a section paallel to the,φplane at t = constant is in eality a sphee. 4.. Kuskal-zegees Coodinates. Consideing equ. (4.5) we find that is now a function of u and υ 1 * = + ln 1 = ( u) υ (4.11) We now ewite the expession in paenthesis in (4.10) with 1 = ln 1 ( u) = υ and use (4.11) to eplace 1 by exponentials. We obtain exp = exp ( υ u) / dudυ (4.1) which is now of the fom = g1 (, u, υ) dudυ. Note that g is non singula at = 1 while u andυ. We may absob the second exponential of (4.1) in the coodinates and define new coodinates U = exp u and V = exp υ (4.13) The metic becomes now in tems of U and V 4 3 = exp du dv (4.14) 30

4 The final tansfomation bings the ccodinates in the fom ( U V ) 1 T = + and X ( V U ) = 1 (4.15) and the -dimensional metic becomes 3 = exp ( dt dx ) (4.16) This metic was fist intoduced by Matin Kuskal nad Geoge zekees in The elation between old (t,) and new coodinates is as follows 1 exp = X T (4.17) and t ( T ) 1 = tgh (4.18) X The Kuskal metic is initially defined fo T < 0 and X > 0 but it can be extended by analytic continuation to T > 0 and X < 0. The fome coodinate singulaity = coespon in Kuskal coodinates to UV = 0, that is eithe T = 0 o X = 0. The singulaity at = 0 now coespon to TX = 1 and is plotted as hypebola with banches in the nd and 4 th egion Fig The analytic extension of the chwazschild spacetime by Kuskal coodinates. Each point = constant is a -sphee. It is epesented in the diagam as hypebolae with the X-axis as symmety axis. taight lines coespond to a constant time t. Howeve, at the two 45 diagonal lines = which epesents a limiting case whee a timelike line goes ove in a spacelike line. 31

5 4.3. The supising stuctue of the extended spacetime. The stuctue of the extended chwazschild spacetime is divided in fou egions: 1) Region I is the oiginal spacetime which is obsevable by physical instuments. It is ou wold. Radial infalling matte cosses the hypebolae and finally hits the line T = X whee it cosses the hoizon. ) Infalling matte entes egion II (at T = X ) and will fall into the singulaity at = 0. Any light signal fom egion II will emain thee and also fall in the singulaity. Region II descibes the BH. 3) Region III is the time evesal of egion II. An obseve pesent in III must have been oiginated in the singulaity and must leave egion III again to egion I. Theefoe III is called a white hole. In the sixties some astonomes speculated that Quasas might be fuelled by white holes. Howeve, obsevations at high esolution have unambiguously shown that the intense emission is due to matte which moves to the BH and finally vanishes thee. Besides these obsevational evidences the existence of white holes would cause sevee themodynamic poblems. 4) Region IV has popeties identical with those of egion I, i.e. epesents an asymtotically flat egion which lies inside (!!) of the adius =. 5) The singulaity at = 0 cannot be emoved. The components of the Riemann 3 cuvatue tenso divege thee as and tidal foces become infinitely lage. In the oiginal chwazschild epesentation coect fo > the egion IV spacetime is left out (see e.g. Fig. 4.5). We ae going to illustate this by embedding the elevant space into a 3-dimensional flat space. The metic in cylinde coodinates looks as follows dz d σ d dz dφ d = + + = 1+ + dφ (4.19) d When we compae this with chwazschild we find dσ 1 = d 1 + dφ dz d = 1! 1 (4.0) Afte integation the non-euclidian hypesuface (a d-hypeboloid) is embedded in the 3d euclidian space by z = ± 1 fo > (4.1) All allowed points lie on the suface of the hypeboloid. The space points inside the hoizon < ) ae left out. ( 3

6 Fig.4.4. The spheical geomety of the hypesuface at t = 0 shown as it is embedded in flat space. The figue contains all space points but all points < ae lacking Do we have to take this discussion seiously, when only egion I is obsevationally accessible? We have to admit that all egions discussed so fa ae valid solutions of Einstein s equations fo the chwazschild poblem. But which of those solutions ae ealized in natue. In ode to decide this question we cetainly need obsevations but also the consideation of othe physical laws which should not contadict those solutions. We will discuss elated questions when we pesent the themodynamics of BHs in lectue Penose diagams. When you ead papes on subjects concened with GR o on a special metic you will often find that the causal stuctue is discussed in a Penose diagam, which allows to conside the espective geomety in a compactified fom. As an example we conside the Minkowski metic ( c = 1) ( dt + d) ( dt d) Ω = dt d dω = d Fig A plot of the function tanh(x) which appoaches +1 fo x and -1 fo x The tansfomation to be found should 1) peseve the light cone and ) map the entie infinite space to a finite potion of the d-plan. 33

7 The expessions dt ± d = 0 descibe the popagation on the light cone. The tansfomation should have the fom + Y = F( t + ) and Y = F( t ) (4.) The function tanh(x ) has the equested popetiy. Theefoe we set + Y = tanh( t + ) and Y = tanh( t ) (4.3) The entie spacetime is mapped on the tiangle bounded by and + Y = 1 with ( t + ) in the diagam called I + Y = 1 with ( t ) in the digam called I - We can tansfom the Kuskal diagam in the same way by applying the tansfomation of (4.3) to the coodinates T and X of equ. (4.16) and Fig Light ay (null geodesics) ae + going paallel to Y and Y. Fig The Cate-Penose diagam of the Minkowski spacetime. 34

8 Fig The Cate-Penose diagam of a static BH obtained fom Kuskal coodinates as epesentated in Fig The axes assigned with t and in the diagam coespond to T and X used in ection 4.. The infinities of the Kuskal diagam ( T ± ) appea in Fig.4.7. as finite points foming the uppe and lowe peaks. The hypebolae of the singulaity at = 0 ae compessed to a hoizontal lines. The hoizon is a global popety and foms a lightlike suface which sepaates the spacetime in an inne and oute egion. All events in the oute egion (egion I) can send signals (light ays) to I + and timelike tajectoies to T =. But any light ay which is emitted in the inne egion (egion II) will neve each the futue asymptotic infinity ( T = ) no can matte each the oute egion I Poblems Conside the following metic t Is the singulaity at t = 0 a coodinate singulaity? Hint: Use the tansfomation ~ 1 t = t to investigate the metic (4.4) and discuss the extension of coodinates. The spacetime geomety is geodesically complete, when all the geodesics appoaching t = 0 extend to abitay lage values of the affine paamete (e.g. τ) tat with the adial collapse of fig.3.4. Make a hand dawing of the tajectoy of a collapsing mass in the oiginal chwazschild coodinates (,t). Now make copies of the diagams in fig.4.3 and fig Daw qualitatively the espective woldlines of infalling matte in Kuskal and in Penose coodinates Find aguments why the existence if white holes appeas implausible. 1 = dt d (4.4) 4 35

From Gravitational Collapse to Black Holes

From Gravitational Collapse to Black Holes Fom Gavitational Collapse to Black Holes T. Nguyen PHY 391 Independent Study Tem Pape Pof. S.G. Rajeev Univesity of Rocheste Decembe 0, 018 1 Intoduction The pupose of this independent study is to familiaize