100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

114 P. T. ChruścielRobinson–Trautman C 5 extensions k of (M ,g) through H + , which are obtainedby gluing to (M ,g) any other Robinson–Trautman spacetime withthe same mass parameter m, as shown in Figure 6. Each such extensionleads to a black-hole space-time, in which H + becomes a black hole eventhorizon. (There also exist an infinite number of C 117 vacuum RT extensionsof (M ,g) through H + — one such extension can be obtained by gluingacopyof(M ,g) to itself. Somewhat surprisingly, no extensions of C 123differentiability class exist in general.)u = u 0❅❅❅❅❅u = ∞r =0❅❅❅ r = ∞❅❅(M ,g) r = ∞u = u 0r =0Fig. 6. Vacuum RT extensions beyond H +5. Initial Data Sets Containing Trapped, or MarginallyTrapped, SurfacesLet T be a compact, (n−1)–dimensional, spacelike submanifold in a (n+1)–dimensional space-time (M ,g). We assume that there is a continuous choicel of a field of future directed null normals to T , which will be referred toas the outer one. Lete i , i =1, ··· ,n− 1bealocalONframeonT ,onesetsn−1∑θ + = g(∇ ei l, e i ) .i=1Then T will be called future outer trapped if θ + > 0, and marginally futureouter trapped if θ + = 0. A marginally trapped surface lying within aspacelike hypersurface is often referred to as an apparent horizon.k By this we mean that the metric can be C 5 extended beyond H + ; the extension canactually be chosen to be of C 5,α differentiability class, for any α