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28.1 Robinson–Trautman vacuum solutions 423where m is an arbitrary function of integration (independent of r). R 34 =0is then satisfied identically. R 13 =0=R 23 tells us that m is a function ofu alone. The last equation R 33 = 0 leads to∆∆ ln P +12m(ln P ) ,u − 4m ,u =0. (28.7)Theorem 28.1 The general vacuum solution admitting a geodesic,shearfree, twistfree but diverging null congruence is the Robinson–Trautman metric (Robinson and Trautman 1962)ds 2 2r 2[=P 2 (u, ζ, ζ) dζdζ − 2du dr − ∆lnP − 2r(ln P ) ,u − 2m(u) ]du 2 ,r∆∆(ln P )+12m(ln P ) ,u − 4m ,u =0, ∆ ≡ 2P 2 ∂ ζ ∂ ζ. (28.8)This line element is invariant with respect to the transformations (27.38),i.e. tou ′ = F (u), r = r ′ F ,u , ζ ′ = f(ζ), P ′ = P |f ,ζ | F,u −1 , m ′ = mF,u −3 .(28.9)They can be used e.g. to give a non-zero m the values ±1.In the Robinson–Trautman line element (28.8), r is the affine parameteralong the rays of the repeated null eigenvector (r ,i is not necessarily spacelike!),and u is a retarded time. The surfaces r, u = const may be thoughtof as distorted spheres (if they are closed); the solutions (28.8) are thereforeoften referred to as describing spherical gravitational radiation. Ofcourse, no exact spherical gravitational waves exist, since spherical symmetrywould imply ∆ ln P = K(u), and, in the gauge m = 1, (28.8) showsthat the metric is then static (for m = 0, see below under type N). Insome special cases, e.g. in the static Schwarzschild metric contained here(see below under type D), the parameter m has the physical meaningof the system’s mass. Using a Lyapunov-functional argument, it can beshown (see Lukács et al. (1984), and e.g. Chruściel (1991) for results andfurther references) that for rather general initial values on u = const andζ– ζ- surfaces diffeomorphic to a sphere, the Robinson–Trautman solutionsradiate and then settle down to the static Schwarzschild solution. For axisymmetricRobinson–Trautman solutions, Hoenselaers and Perjés (1993)found that, for almost all initial conditions, the final state of the solutionis the C-metric.For the Robinson–Trautman metric (28.8), the surviving componentsof the Weyl (curvature) tensor areΨ 2 = −mr −3 , 2Ψ 3 = −r −2 P (∆ ln P ) ,ζ,[ { }(28.10)Ψ 4 = r −2 P 2 12 ∆lnP − r(ln P ) ,u ,ζ],ζ .