Universal coordinates for Schwarzschild black holes
aa r X i v : . [ g r- q c ] J a n Universal coordinates for Schwarzschild black holes
W. G. Unruh
CIAR Cosmology and Gravity ProgramDept. of PhysicsUniversity of B. C.Vancouver, Canada V6T 1Z1email: [email protected]
A variety of historical coordinates in which the Schwarzschild metric is regular over the whole ofthe extended spacetime are compared and the hypersurfaces of constant coordinate are graphicallypresented. While the Kruscal form (one of the later forms) is probably the simplest, each of theothers has some interesting features.
For years after Schwarzschild[1] found a solution for spherically symmetric metrics to Einstein equations, ds = (1 − Mr ) dt − dr − Mr − r ( dθ + sin( θ ) dφ ) (1)the status of the singularity at r = 2 M (in units where c = 1 G = 1) confused many, including Einstein[2]. It wasonly in 1933, when Lemaˆıtre[7] found his coordinate transformation that he explicitly stated that that singularity inthe metric was an artifice introduced because of the coordinates that Schwarzschild had used. It had already beenrecognized by Lanczos in 1922 that the status of singularities in a metric was unclear because singularities could beintroduced by making a singular choice of coordinates. However, the application of this to the r = 2 M singularity notappreciated. In 1921, both Gullstrand and Painleve[6] had found new, spherically symmetric solutions to Einstein’sequation, ds = (1 − Mr ) dτ − r Mr dtdr − r ( dθ + sin( θ ) dφ ) (2)In the following I will refer to this as the PG form of the metric. They, however, did not recognize that this solutionis simply a coordinate transformations of Schwarzschild’s solution, nor did they recognize the implication for theSchwarzschild singularity, believing that coordinates themselves held physical significance.In the Kruskal[9] paper, the claim is made that Kasner[3] in 1921 showed that the r = 2 M singularity was a just acoordinate singularity. This is not true. Kasner embedded the Schwarzschild solution into a 6 dimensions (signature4+2) flat spacetime but that embedding is singular at r = 2 M – it covers only the region r > M .In 1922, Eddington[4] found an explicit coordinate transformation which gave the metric ds = (1 − Mr )( d ˜ t + dr ) − d ˜ tdr − dr − r ( dθ + sin( θ ) dφ ) (3)which is regular at r = 2 M , but did not recognize (or at least did not comment on ) the implication that this had forthe Schwarzschild singularity. (This coordinate transformation and metric were rediscovered in 1954 by Finkelstein[5]who certainly did recognize that this implied that the Schwarzschild singularity was purely a coordinate artifact.What is now called the Eddington-Finkelstein (EF) form of the metric is obtained from their form by replacing t by v = ˜ t + r to give ds = (1 − Mr )( dv ) − dvdr − r ( dθ + sin( θ ) dφ ) (4)but this null form was never actually written down by either of them.)In the following I will chose spatial units so that 2 M = 1. Thus the Schwarzschild metric becomes ds = (1 − r ) dt − dr − r − r ( dθ + sin( θ ) dφ ) (5)Both the PG metric and the EF metric are coordinate transformations of each other, with a transformation isregular for all values of r >
0. In particular, if we take τ = v − r + √ r (6)we turn the PG into the EF form of the metric.In 1933, Lemaˆtre, concerned about cosmological solutions to Einstein’s equations, introduced his form of theSchwartzschild metric. He was interested in the solution in which one embeds a Schwarzschild solution in a De-Sitteruniverse, but also took the limit as the cosmological constant was zero. ds = dτ − Mr ( σ − t ) dσ − r ( σ − t ) ( dθ − sin( θ ) dφ ) (7)where r ( σ − τ ) = M (cid:0) ( σ − τ ) (cid:1) and τ is the same time coordinate as in the PG form of the metric. Lemaˆıtre wasthe one that showed, in passing that this was simply a coordinate transformation of the PG metric, and that the PGmetric itself was just a coordinate transformation of Schwarzschild’s form.What is interesting about all three forms of the metric (PG,EF, and Lemaˆıtre) is that while they do demonstratethat the Schwarzschild singularity is a coordinate artifact, and in all three, the metric is regular (has a well definedinverse everywhere including at r = 2 M ) they come in two forms. We can define two possibilities for the coordinatetransformation. For the EF metric t = u ± ± ( r + 2 M ln (cid:18) r − M M (cid:19) (8) ds = (1 − Mr ) du ± ± du ± dr − r ( dθ + sin( θ ) dφ ) (9)For the PG metric t = τ ± ( √ M r + 2 M ln √ M r − M √ M r + 2 M ! (10) ds = (1 − Mr ) dτ ± ± r Mr dτ ± dr − dr − r ( dθ + sin( θ ) dφ ) (11)and for the Lemˆıtre metric, the PG transformation plus the extra transformation r √ r = τ ± ± σ ± (12) ds = dτ ± − r dσ ± − r ( dθ + sin( θ ) dφ ) (13)In all three cases the two solutions, labelled by ± are not the same solution. While they are just coordinatetransformations of each other for r > M , the spacetime covered is different for r <
0. This can be most easily seenby looking at the radial null geodesics.In the EF case, the null geodesics are u ± = u ± (14) u ± = u + 0 ± ± r + 2 M ln( r − M M ) (15)The first equation has a regular solution for u ± for all values of r while the second equation has u ± go to − ± ∞ as r → M . But for the u + , r the first represent null rays which are travelling outward, While the second is nullrays which travel inward. Thus for the u + the ingoing null rays have no representation for r < M . For u − it isthe opposite. The first represents null rays which travel inward, while the second singular solution is null rays whichtravel outward. Thus for the u + , r coordinates, the region r < u − , r it is where ingoing null rays go to. Thus the regions r < τ ± = τ ± − ( ± r Mr + 1) (16) τ ± = τ ± − ( ± r Mr −
1) (17)while the second, irregular solution is if one changes the sign of v or τ one obtains a different solution of the Einsteinequations. While outside r > M this new metric is simply a coordinate transformation of the Schwarzschild, inside r < M it is not, the two forms cover different spacetimes.The ingoing null geodesics in the EF metric are give by v constant, which is clearly regular for all values of r > drdv = 12 (1 − Mr ) (18) v − v = 2 (cid:18) r + 2 M ln (cid:18) r − M M (cid:19)(cid:19) (19)with v going to −∞ as r approaches 2 M . In the u, r coordinates obtained from this form by setting u = − v (ormaking the coordinate transformation from Schwartzschild of t = u + r + 2 M ln( r M − u constant, everywhere down to r = 0 while the ingoing null geodesics u − u = 2( r + 2 M ln( r M − r → M .Similarly in the PG form of the metric, the outgoing null geodesics are given by (cid:18) drdτ (cid:19) + 2 r Mr drdτ − (1 − Mr ) = 0 (20)or drdτ = − r Mr ± r = 2 M for the minus sign, but divergent solutions there for the plus sign. Againnull geodesics going into the horizon are well behaved through the horizon, while those coming out are badly behaved.This is reversed with the other PG solution obtained when τ → − τ .Finally, the Lemaˆıtre form is more mysterious. Not only is the metric diagonal but the metric looks completelyregular at r = 2 M (or rather σ − τ = 3 M ) The null geodesics are given by dσdτ = ± r ( σ − τ )2 M However, writing this interms of the variable r rather than σ we obtain exactly the PG null geodesics which we know are singular at r = 2 M .Is there a set of coordinates for which the only singularities occur at r=0, and in which the null geodesics are allregular at r = 2 M ? The answer is of course yes, and the best known answer is the Kruskal-Szekeres form. However,such a coordinate system was first given by Synge[8] in 1950.In the following I will choose units for my coordinates so that 2 M = 1 so factors of 2 M do not have to be draggedalong through all of the equations.Write the Schwarzschild metric in terms of the proper distance to the horizon R = Z r dr q − r = √ r − √ r + asinh(r − √ r − (cid:18) √ r + asinh( √ r − √ r − (cid:19) (22)We have 1 − r = 2 R r (cid:16) √ r + asinh( √ r − √ r − (cid:17) (23) ds = F ( r ( R )) R dt − dR − r ( R ) ( dθ + sin( θ ) dφ ) (24)where F ( r ) = 4 r (cid:16) √ r + asinh( √ r − √ r − (cid:17) (25)The function F ( r ( R )) looks singular at r = 1 but is not. √ r is analytic for r >
0. The function asinh( √ r − √ r − is also ananalytic function of r everywhere for r >
0. It is an even funtion in the argument √ r − r for r > F ( r ) is also monotonic in r and thus R is an analytic monotonic function of r for r > r ( R ).Also F ( r = 1) = 1 and we can thus write the metric as ds = ( F ( r ( R )) − R dt + R dt − dR − r ( R ) ( dθ + sin( θ ) dφ ) (26)Now defining T = R sinh( t/
2) (27) ξ = R cosh( t/
2) (28)and thus R = ξ − T , we have the regular metric ds = ( F ( r ( R )) − ξdT − T dξ ) + dT − dξ − r ( p T − ξ ) ( dθ + sin( θ ) dφ ) (29)This metric is singular for T − ξ = π (which corresponds to r = 0) but is regular everywhere else. This is the Syngeform of the Schwarzschild metric, the first of the metric forms whose coordinates cover all of the analytically extendedspacetime (all geodesics either end in a genuine singularity, corresponding to one of the r = 0 singularities, or extendto infinity.) Note also that the lines of ξ, θ, φ constant are not necessarily timelike lines. for ξ sufficiently large and r sufficiently small, F (( r ) − ξ + 1 can be negative of r < ds = G ( r ( ρ ))( ρ α dt − dρ ) − r ( ρ ) ( dθ + sin( θ ) dφ ) (30)where α is a constant. This leads to dρdr = α ρ − r (31)or ρ = ( r − α e α r (32)Choosing α = we have ds = e − r ( ρ )2 r ( ρ ( dt M ) − dρ ) − r ( ρ ) ( dθ + sin( θ ) dφ ) (33)Defining τ = ρ sinh( t χ = ρ cosh( t ds = e − r r ( dτ − dχ ) − r ( τ − χ ) ( dθ + sin( θ ) dφ ) (36)There is another way of arriving at the same result. Writing the EF metric ds = (1 − r ) du ± ± du ± dr − − r ( dθ + sin( θ ) dφ ) (37)with u ± = t ± ( r + ln( r −
1) (38)to give r − e u + − u −− r r − r ( dθ + sin( θ ) dφ ) (39)to give ds = e − r r ( e u +2 du + )( e − u − du − ) − r ( dθ + sin( θ ) dφ ) (40)Defining U ± = ± M e ± u ± and τ = ( U + + U − ) / χ = ( U + − U − ) / τ ± = t ± (2 √ r + ln (cid:18) √ r − √ r + 1 (cid:19) (43)we have dτ ± ± q r − r dτ ± dr − r ( dθ + sin( θ ) dφ ) (44)In terms of these ”times” we have ds = − r − r ( dτ + dτ − ) + ( r ) − r dτ + dτ − − r ( dθ + sin( θ ) dφ ) (45)Defining Ξ ± = e τ ± / and y = √ r , we have ds = (2 M ) (cid:20) − e − y ( y + 1) y (Ξ d Ξ − + Ξ − d Ξ ) + − e − y ( y + 1)( y + 1) y d Ξ + d Ξ − (cid:21) (46)where r (Ξ + Ξ − ) is defined by Ξ + Ξ − = y − y + 1 e y (47)This is again a regular metric everywhere where r > + Ξ − > − + =const or Ξ − =const are flat spacelike surfaces– ie it foliates the extended Schwarzschild spacetimewith a series of intersecting flat spatial slices.Another interesting metric is obtained by taking the Lemaˆıtre metric, obtained from the Schwarzschild by thecoordinate transformation τ = t + Z sqrt /r − r dr = √ r + ln (cid:18) √ r − √ r + 1 (cid:19) (48) σ = τ + 23 r √ r (49)which gives the metric ds = dτ − dσ r − r ( dθ + sin( θ ) dφ ) (50)where r = (cid:0) ( σ − τ ) (cid:1) .Again, taking τ → − τ gives another solution which covers a different sector of the spacetime than does the abovemetric. Taking τ ± as two coordinates leads to the same metric as the above extended PG metric. However we canalso take Σ ± = exp ( (cid:18) (cid:18) r + √ r + ln (cid:18) √ r − √ r + 1 (cid:19) ± t (cid:19)(cid:19) (51)from which we find Σ + Σ − = √ r − √ r + 1 e r + √ r (52)Σ + Σ − = e t (53)and the metric becomes ds = e − r + √ r ( √ r + 1) (Σ d Σ − + Σ − d Σ − d Σ − d Σ + ) − r ( dθ + sin( θ ) dφ ) (54)In this case the surfaces of either Σ + or Σ − constant are timelike surfaces and the lines in those surfaces of θ and φ constant are time-like geodesics in the Schwarzschild metric.As a final example, we can look at a coordinate system related to the global embedding of the Schwarzschild metricfound by Fronsdale.Define the funtion ˆ R by ˆ R = 4(1 − r ) (55)ˆ R runs from −∞ ( r = 0) to 0 ( r = ∞ ). Then we can write ds = ˆ R (cid:18) dt (cid:19) − d ˆ R − r + r + r r dr − r ( dθ + sin( θ ) dφ ) (56)= ˆ R (cid:18) dt (cid:19) − d ˆ R − − ( ˆ R ! ˆ R d ˆ R − − ˆ R ( dθ + sin( θ ) dφ ) (57)As before, define Θ = ˆ R sinh( t Y = ˆ R cosh( t R = Y − Θ (60)This gives ds = d Θ − dY − ( Y dY − Θ d Θ) − Y − Θ ! − − Y − Θ ( dθ + sin( θ ) dφ ) (61)These are related to the global embedding of the Schwarzschild metric in a 6-dimensional flat spacetime, firstsuggested by Fronsdale[10]. Defining the Z coordinate by Z = Z r r ′ + r ′ + 1 r ′ dr ′ (62)the metric becomes ds = d Θ − dY − dZ − dr − r ( dθ + sin( θ ) dφ ) (63)with the above definition of Θ , Y, Z as functions of t, r giving the embedding functions of the 4 dimensional surfacein the 6 dimensional flat spacetime. I. RELATIONS BETWEEN COORDINATES
Since the SK coordinates are the most standard, let us compare the other two coordinate systems to the PKcoordinates graphically.Let us first look at the generalised PG coordinates to the SK coordinates. The extended PG coordinate surfaces ofconstant Xi ± to those of the SK coordinates. Using the SK coordinates U = τ − ρ and V = τ + ρ we haveΞ + Ξ − = e t M = VU (64)Ξ + Ξ − = p r M − p r M + 1 e √ r M (65) U V = ( r M − e r/ M (66) FIG. 1: The Ξ constant coordinate surfaces in the Kruskal coordinates. Each of those surfaces is a flat spatial slice. All beginat the r=0 singularity and go out to infinity. Note that both the Ξ + and the Ξ − constant surfaces are spatial surfaces. Ie, Ξ + Ξ − is a function of U V given parametrically by the last two equations.The diagram indicates the graph of constant Ξ + and Ξ − spacelike hyperspace’s for a few values of each.Note that as r → ∞ , both Ξ + and Ξ − (for suitable values) asymptote to the same line. in the U V plane. Ie, theΞ + , Ξ − coordinates become degenerate as r → ∞ .1Then the Synge coordinates are plotted vs the SK coordinates. The surfaces of constant Synge time T are given interms of the SK coordinates parametrically by V + U T ) = T e r √ r + asinh( √ r − √ r − (67) V − U T ) = r ( V + U + ( r − e r (68)where r must be large enough that V − U is real.The ξ coordinate constant surfaces are given by V − U ξ ) = ξe r √ r + asinh( √ r − √ r − (69) V + U ξ ) = ± r ( ( V − U )2 ) − ( r − e r (70)where the parameter r is chosen small enough so that V + U ( ξ ) is real. -4 -2 0 2 4-4-2024 FIG. 2: The
T, ξ constant coordinate surfaces plotted in Kruskal coordinates. Note that while the T constant hypersurfacesare spacelike hypersurfaces, the ξ constant one as not everywhere timelike. In particular near and within the horizon thesesurface become timeline for large enough values of ξ . In figure 2 we have the plot of the T and ξ constant surfaces in the SK coordinates.The Lemaˆıtre coordinates are interesting because they look, at first, as though they are regular coordinates alreadywhich cover the whole spacetime. The metric ds = dτ − r ( σ − t ) dσ − r ( σ − t ) ( dθ − sin( θ ) dφ ) (71)looks regular everywhere.except at r = 0 of t = σ . But if we look at the null geodesics dσdτ = ± p ( r ( τ − σ )) = ± ( 32 ( σ − τ )) (72)we find for the + sign, taking z = σ − τ that dzdτ = ± ( 32 z ) − z = and τ goes to ∞ if we take the + sign in the equation for z . Ie, the null geodesicscoming out of the black hole come from τ → −∞ . Had one taken the other solution ( with τ → − τ ) for theLemaˆıtre metric, it would be the ingoing null geodesics which would have terminated at r = 1. Ie, again the Lemaˆıtrecoordinates cover only a part of the complete spacetime. The extended Lemaˆıtre coordinates (Σ ± ) do cover the wholeof the spacetime.From the two graphs, of the extended PG coordinates, and the extended Lemaˆıtre coordinates, we can see theproblem with the original Lemaˆaitre coordinates. The latter are essentially using the Ξ − and the Σ − coordinates. FIG. 3: The Lemaitre extended coordinates plotted on the SK extended coordinates. the problem with these is they become degenerate along the past horizon, where both are equal to zero. Ie, these (and the original Lemaˆıtre coordinates which are the logarithm of these coordinates) coordinates do not cover the pasthorizon. However, if we choose for example the Σ + and the Ξ − coordinates, these do cover the whole of the extendedspacetime, with no degeneracies. We have Σ + Ξ − = √ r − √ r + 1 e √ r ( r +1) (74)Σ + Ξ − = e t e r √ r (75)or √ r ( r + 2)2( r − dr = d Σ + Σ + + d Ξ − Ξ − (76) dt + 12 √ rdr = d Σ + Σ + − d Ξ − Ξ − (77)to give ds = √ r + 1( r + 1) h ( √ r + 1) e − ( r/ √ r (Ξ − d Σ − Σ d Ξ − ) + 4 d Ξ − d Σ + i + r ( dθ + sin( θ ) dφ (78)This shares with the original Lemaˆıtre coordinates that each of the Σ constant hypersurfaces are flat three dimensionalspatial metrics, while each of the Ξ , θφ constant lines are timelike geodesics which have zero velocity at infinity. Unlikethe original Lemaˆıtre coordinates however, they cover the whole of the analytic extension of Schwarzschild spacetime.0 FIG. 4: The Θ and Y constant hypersurfaces for the Fronsdale embedding of Schwarzschild into a flat 6 dimensional spacetime,While the Y constant coordinates seem to hit the r = 0 singularity are various points, those surfaces actually skirt (as spacelikesurfaces) extremely close to the singularity before finally all hitting it at the same point. They are thus just as simply married to the flat Robertson Walker dust universe model as were the original Lemaitrecoordinates.Finally, using the Fronsdale coordinates Θ , Y we plot the Θ constant and Y constant hypersurfaces. Note that theseΘ constant hypersurfaces surfaces do not run into the r = 0 singularity. On the other hand, all of the Y constantlines originate at T = ± , ξ = 0 points on the singularity, with the Y constant lines only being timelike for certainvalues of Y < Y constant coordinate in these “Fronsdale” coordinates isvery badly behaved near the r = 0 singularity while the Θ const. coordinate surfaces are nicely behaved. [1] Schwarzschild, K. (1916). ” ¨Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”. Sitzungs-berichte der K¨oniglich Preussischen Akademie der Wissenschaften 7: 189-196. For the modern form of this metric seeDroste, J. (1917). ”On the field of a single centre in Einstein’s theory of gravitation, and the motion of a particle in thatfield”. Proceedings Royal Academy Amsterdam 19 (1): 197?215.[2] See for example Eisenstaedt, J “The Schwarzschild Solution” in Einstein and the History of General Relativity edD. Howard, J. Staechel (1989) Birkh¨auser (Boston)[3] E. Kasner “Finite Representations of the Solar Gravitational Field in flat space of six dimension” Am. J. Math. Astron. Fys. 16(8), 1-15 (1922)[7] G. Lemaˆıtre (1933). Annales de la Soci´et´e Scientifique de Bruxelles A53: 51-85[8] J. L. Synge, Proc. Roy. Irish Acad., 50, 83 (1950).[9] G. Szekeres,Publicationes Mathematicae Debrecen 7, 285 (1960) [submitted May 26, 1959] reprinted in Gen. Rel. Grav., 34,2001 (2002) Kruskal, M. (1960). ”Maximal Extension of Schwarzschild Metric”. Physical Review 119 (5): 1743. [submittedDec 21, 1959 although reported at conference June 1959] Both reference the Synge paper above.[10] C. Fronsdal, Phys. Rev.,
778 (1959). See also S.A. Paston, A.A. Sheykin ”Embeddings for Schwarzschild metricclassification and new results” Class. Quant. Grav.29