Schwarzschild spacetime and the Penrose–Kruskal diagram

In relativity, light rays, the quasi-classical trajectories of photons, are null geodesics. In special relativity, this is quite obvious, since in Minkowski space the geodesics are straight lines and ‘null’ just means v = c. A more rigorous argument involves the solution of the Maxwell equations for the vacuum and the

Figure 1.1. Penrose diagram of Minkowski spacetime.

subsequent determination of the normals to the wave surface (rays) which turn out to be null geodesics. This remains valid in general relativity. Null geodesics can be easily obtained by integrating the equation 0 = ds. We find for the Schwarzschild metric, specializing to radial light rays with dφ = 0 = dθ, that

 (1.1)

If we denote by r₀ the solution of the equation r + 2m ln | r/2m− 1| = 0, we have for the t-coordinate of the light ray t (r₀) =: v. Hence, if r = r₀, we can use v to label light rays. In view of this, we introduce v and u¹

 (1.2)

 (1.3)

Then ingoing null geodesics are described by v = constant, outgoing ones by u = constant, see figure 1.2. We define ingoing Eddington–Finkelstein coordinates by replacing the ‘Schwarzschild time’ t by v. In these coordinates

Figure 1.2. In- and outgoing Eddington–Finkelstein coordinates (where we introduce t ' with v = t ' + r , u = t ' − r ).

(v, r, θ, φ), the metric becomes

 (1.4)

For radial null geodesics ds² = dθ = dφ = 0, we find two solutions of (1.4), namely v = constant and v = 4m ln |r/2m − 1| + 2r + constant. The first one describes infalling photons, i.e. t increases if r approaches 0. At r = 2m, there is no longer any singular behavior for incoming photons. However, for outgoing photons, ingoing Eddington–Finkelstein coordinates are not well suited. Ingoing Eddington–Finkelstein coordinates are particularly useful for describing the gravitational collapse. Analogously, for outgoing null geodesics take (u, r, θ, φ) as the new coordinates. In these outgoing Eddington–Finkelstein coordinates the metric reads:

 (1.5)

Outgoing light rays are now described by u = constant, ingoing light rays by u = −(4m ln |r/2m −1|+ 2r )+constant. In these coordinates, the hypersurface r = 2m (the ‘horizon’) can be recognized as a null hypersurface (its normal is null or lightlike) and as a semi-permeable membrane.

Next we try to combine the advantages of in- and outgoing Eddington– Finkelstein coordinates in the hope of obtaining a fully regular coordinate system for the Schwarzschild spacetime. Therefore we assume coordinates (u, v, θ, φ). Some (computer) algebra yields the corresponding representation of the metric:

 (1.6)

Unfortunately, we still have a coordinate singularity at r = 2m. We can get rid of it by reparametrizing the surfaces u = constant and v = constant via

 (1.7)

In these coordinates, the metric reads (r = r ( ˜ u, ˜ v) is implicitly given by (1.7) and (1.3), (1.2)):

 (1.8)

Again, we go back from ữ and ṽ to time- and spacelike coordinates:

 (1.9)

In terms of the original Schwarzschild coordinates we have

 (1.10)

 (1.11)

The Schwarzschild metric

 (1.12)

in these Kruskal–Szekeres coordinates (˜t, ˜r, θ, φ), behaves regularly at the gravitational radius r = 2m. If we substitute (1.12) into the Einstein equation (via computer algebra), then we see that it is a solution of it for all r > 0. Equations (1.10), (1.11) yield

 (1.13)

Thus, the transformation is only valid for regions with |˜r | > ˜t . However, we can find a set of transformations which cover the entire (˜t, ˜r )-space. They are valid in different domains, indicated here by I, II, III, and IV, to be explained later:

(1.14)

 (1.15)

 (1.16)

 (1.17)

The inverse transformation is given by

 (1.18)

 (1.19)

The Kruskal–Szekeres coordinates (˜t , ˜r, θ, φ) cover the entire spacetime (see figure 1.3). By means of the transformation equations we recognize that we need two Schwarzschild coordinate systems in order to cover the same domain. Regions (I) and (III) both correspond each to an asymptotically flat universe with r > 2m. Regions (II) and (IV) represent two regions with r < 2m. Since ˜t is a time coordinate, we see that the regions are time reversed with respect to each other. Within these regions, real physical singularities (corresponding to r = 0) move along the lines ˜t ² − ˜r ² = 1. From the form of the metric we can infer that the lightlike geodesics (and therewith the light cones ds = 0) are lines with slope 1/2. This makes the discussion of the causal structure particularly simple.

Finally, we would like to represent the Schwarzschild spacetime in a manner analogous to the Penrose diagram of the Minkowski spacetime. To this end, we proceed along the same line as in the Minkowskian case. First, we again switch to null coordinates v' = ˜t+˜r and u' = ˜t−˜r and perform a conformal transformation which maps infinity into the finite (again, by means of the tangent function). Finally we return to a timelike coordinate ˆt and a spacelike coordinate ˆr. We perform these transformations all in one go:

(1.20)

 (1.21)

Figure 1.3. Kruskal-Szekeres diagram of the Schwarzschild spacetime.

The Schwarzschild metric then reads:

 (1.22)

where the function r (ˆt , ˆr ) is implicitly given by

 (1.23)

The corresponding Penrose–Kruskal diagram is displayed in figure 1.4.

Figure 1.4. Penrose–Kruskal diagram of the Schwarzschild spacetime.