Notes for the standard central, single mass metric in Kruskal coordinates

Transcription

1 Notes fo the stana cental, single mass metic in Kuskal cooinates I. Relation to Schwazschil cooinates One oiginally elates the Kuskal cooinates to the Schwazschil cooinates in the following way: u = /2m 1 e /4m cosh(t/4m), v = /2m 1 e /4m sinh(t/4m),, 2m, u = 1 /2m e /4m sinh(t/4m), v = 1 /2m e /4m cosh(t/4m),, 2m. (1.1) As it tuns out, when one ceates the manifol escibe above whee we label the quaant whee 2m as quaant I below an label the quaant whee 2m as quaant II, the manifol that esults is incomplete. To complete it we must appen an aitional copy of it, whee u an v ae given as the negatives of the expessions above. We efe to the copy of the two oiginal quaants as quaant III, when 2m but u < 0, an also quaant IV, when 2m an v < 0. Ou eath of couse lives in quaant I, whee we began. The iagam shows the Kuskal cooinates u as spatial an v as timelike, ignoing θ an φ. Cuves of constant ae hypebolae, as is seen fom equations below: 1

2 ( ) { 2 4m tanh 1 u 2 v 2 = (/2m 1) e /2m (v/u), I & III, = (1 2m/), t = 4m f tanh 1 (u/v), II & IV. (1.2) On the gaph those cuves of constant > 2m ae the plum-coloe hypebolae in eithe quaant I o quaant III. Fo 0 < < 2m these ae the blue hypebolae, in quaant II, o the geen hypebolae in quaant IV. Howeve the value = 0 ceates the black hypebolae in both quaants II an IV, which ae the singulaities at = 0, beyon which no cuves may pass. One shoul then notice the pai of staight lines which pass the (u, v) oigin. These coespon to the Schwazschil-cooinate values of = 2m an the value t = ±. The Schwazschil metic then has the following equivalent foms, in the two cooinate systems: g = 2 1 2m/ + 2 Ω 2 (1 2m/) t 2 = 32m3 e /2m (u 2 v 2 ) + 2 Ω 2, (1.3) whee it is convenient to give the coefficient of u 2 a name. We efine f() so that the u, v-pat of the metic is now just f 2 (u 2 v 2 ). It is then staightfowa to see that u = 1 4m v = 1 4m { { 4m e /4m e = 4m /2m /2m /2m, (1.4) /2m e /4m cosh(t/4m) + } /2m 1 e /4m sinh(t/4m) t /2m 1 /2m e /4m sinh(t/4m) + } /2m 1 e /4m cosh(t/4m) t /2m 1 (1.5a) fom which we may quickly veify that the two foms of the metic ae equivalent. On the othe han, we may invet the equations, to etemine an t: = f ( u ω 4m û v ˆv ω ), t = f ( ) /2m ( v ω 4m /2m 1 û + u ˆv ω ). (1.5b) A possibly useful/inteesting comment is that u,t = v an v,t = u, so that,, t = v u + u v = f(vẽû + uẽˆv ). (1.5c) 2

3 Since t is a Killing vecto, this suggests that this is the fom of the Killing vecto in these cooinates, so that the following shoul be a constant of the motion: A = u t = g( t, ũ) = f(v uû + u uˆv ) = f(v uû u uˆv ). (1.5) Since it is ifficult to esolve Eqs. (1.2) fo explicitly such explication woul involve Lambet s function, as escibe, fo instance, by Hille we use the given fom of the equation to etemine,u an,v, which will be neee. Diffeentiating that equation by u, o by v, an then iviing appopiately we may etemine,u = (2m) 2 2u e /2m = u 4m f 2 (),,v = (2m) 2 2v e /2m = v 4m f 2 (). (1.6) It is also useful to go ahea an explicitly etemine eivatives of ou function f: f f = 1 + /2m (/2m) 3/2 e /4m = 1 + /2m /2m f 4m = 1 + /2m 2 f. (1.7) 2. Cuvatue an geoesics in these cooinates We now efine an othonomal basis in ou Kuskal cooinates: ω û f u, ω ˆv f v, ω ˆθ θ, ω ˆφ sin θ φ, ẽû = 1 f u, ẽˆv = 1 f v, ẽˆθ = 1 θ, ẽ ˆφ = 1 sin θ φ. (2.1) One may then ask GRTensoII to calculate the connections an cuvatues: Γûˆθ ˆθ = u f 4m = Γ û ˆφ ˆφ, Γˆv ˆθ ˆθ = + v f 4m = Γˆv ˆφ ˆφ, Γûˆvû = + v f (1 + /2m) 2 4m, Γ ûˆvˆv = u f (1 + /2m) 2 4m, Γˆθ ˆφ ˆφ = cot θ. (2.2) As well we etemine the components of the Riemann tenso, whee one nees to eplace u 2 v 2 by its equal (/2m 1) e /2m seveal times: Rûˆθûˆθ = m 3 = R û ˆφû ˆφ = Rˆv ˆθˆv ˆθ = Rˆv ˆφˆv ˆφ, Rˆθ ˆφˆθ ˆφ = 2 m 3 = R ûˆvûˆv. (2.3) 3

4 We may then wite own the fom of a timelike tangent vecto, ũ, nomalize to have length 1 = ũ 2 : ũ = = u u + v v + θ θ + φ φ = uû ẽû + uˆv ẽˆv + uˆθ ẽˆθ + u ˆφ ẽ ˆφ, = uû = f u, uˆv = f v, uˆθ = θ, u ˆφ = sin θ φ. (2.4) The basic geoesic equations then say that uû = Γ ûˆθ ˆθ (uˆθ) 2 Γû ˆφ ˆφ (u ˆφ ) 2 Γûˆvû uˆv uû Γûˆvˆv (uˆv ) 2 uˆv = u f [(uˆθ) 2 + (u ˆφ ) 2 ] vuû uuˆv 4m 2 (1 + /2m) f 4m uˆv, = + (uˆθ) Γˆv ˆθ ˆθ 2 + Γˆv ˆφ ˆφ (u ˆφ ) 2 + Γˆvûû (uû) 2 + Γˆvûˆv uûuˆv = + v f [(uˆθ) 2 + (u ˆφ ) 2 ] vuû uuˆv (1 + /2m) f 4m 2 4m uû, uˆθ = cot θ (u ˆφ ) 2 uuû vuˆv f uˆθ, 4m u ˆφ = cot θ uˆθu ˆφ uuû vuˆv f 4m u ˆφ. (2.5) We may check that the nomalization of the 4-velocity is inee peseve, i.e., multiplying each equation fo u α / by u α an summing, we may check that the sum as up to just zeo, as esie fo ( 1)/. Looking at the last two of these equations, we easily see that thee o exist obits that emain in the equatoial plane if they begin thee; i.e., we consie θ 0 = π/2 an uˆθ = 0, which tells us that uˆθ/ = 0. All single wollines may be consiee of this type, as we may oient the equatoial plane so that it contains that wolline. On the othe han, thee ae, in paticula, aial obits in that plane: if we set u ˆφ = 0, then we have u ˆφ / = 0. Raial Obits: The nomalization conition, aleay consiee, now says that (uˆv ) 2 (uû) 2 = +1 = f 2 [(v ) 2 (u ) 2 ]. (2.6a) The consevation law, because of the timelike Killing vecto, fom Eq. (1.5), says that f(u uˆv v uû) = A = f 2 [u v v u ]. (2.6b) 4

5 To give a moe iect poof that this enegy-like quantity is tuly conseve, we e-wite it in the fom A = [ fu uˆv fv uû] uˆv = fu = (v uû u uˆv )(1 + /2m) f 2 uû fv + f uˆv u f uûv + (u uˆv f v uû) 8m (u uû v uˆv ) (u uˆv v uû)(1 + /2m) f 2 8m (u uû v uˆv ) = 0, (2.7) whee we have use Eqs. (1.7) an (1.5b) to calculate f/, an the two cental tems, among the 6 tems in the fist line, have cancelle since fu = uû an fv = uˆv. It is pehaps also inteesting to e-wite the equations in a moe cooinate-base point of view. We may intouce a name fo the exta pat of the equations not involving f, but involving iectly, an fist e-wite the equations aleay given: uû = Af 3 (vu uv )v, uˆv = Af 3 (vu uv )u, f = Af 3 (uu vv ), A 1 + /2m 2m. (2.8) But now using the fact that uû fu an uˆv fv, we may obtain the following: u = Af 2 {u[(u ) 2 + (v ) 2 ] 2vu v }, v = Af 2 {v[(u ) 2 + (v ) 2 ] 2uu v }. (2.9) Next we may note that the facto may be e-witten in many ways: Af 2 = 1 + /2m 4e /2m (1 /2m /2m = 1ρ ) 4 2 u 2, ρ /2m. (2.10) v2 Given the two conseve quantities isplaye in Eqs. (2.6), an noticing, fom the efinition of f = f(), that as 0, 1/f 2 vanishes like, it follows that (v ) 2 appoaches (u ) 2, which is the same as saying that the spee of light is being appoache. If we also notice that v > 0 while u < 0, then we may say that v appoaches u. This then also allows us to see that u + v appoaches zeo like. As an asie I note that the aial equations in the {, t} cooinates ae given as follows: ( ) 2 = A m, t = A 1 2m/, (2.11) 5

6 an have (paametic) solution, fo the case whee A 1: = R cos 2 (η/2) = 1 2R(1 + cos η), τ whee R is the maximum value of. = 1 2 R R 2m (η + sin η), On a blowup of the figue in Kuskal cooinates on the fist page we can now plot an astonaut falling into the hole with no ifficulties in the equations at all as she passes = 2m an also a aio ay that passes he location exactly at the point of he being at = 2m. At that point the values of u an v ae of couse equal, because they ae on the 45 line, with u = 0.60 = v. He wolline is the obvious timelike cuve shown, shown in e pio to = 2m an in geen aftewa. The oange line is the aio message mentione above. One can see that, in tems of the timelike cooinate v, the light ay aives at the singulaity pio to the astonaut s aival at that singulaity. Since light ays must tavel on 45 lines, one can also notice that any aio signal that she sens out afte passing = 2m has no chance of enteing back into quaant I. The bown cuve tying this is shown, heae outwa. 6