The formation of black holes and singularities in spherically symmetric gravitational collapse. (English) Zbl 0728.53061

Christodoulou, Demetrios

Commun. Pure Appl. Math. 44, No. 3, 339-373 (1991).

The author studies spherically symmetric solutions of Einstein’s equation for a massless scalar field. He shows that (M,g) contains a curvature singularity (\(| Riem|\) diverges faster than \(r^{-3})\), provided that the following mass estimate holds: Let \(C^+_ 0\) be a future directed light cone with vertex at the central geodesic. Suppose there exist spheres \(S_{1,0},S_{2,0}\subset C^+_ 0\) \((S_{2,0}\) to the future of \(S_{1,0})\) which are orthogonal to the null geodesic generators of \(C^+_ 0\) such that the (Hawking) mass function m (that coincides with the usual mass function for spherically symmetric spacetimes) satisfies \[ 2(m(S_{2,0})-m(S_{1,0})/r_{2,0}\geq E(r_{2,0}/r_{1,0})-1), \] where \(r_{i,0}\) denotes the radius of \(S_{i,0}\) and E is a certain explicitly given increasing function with \(E(0)=0\). Then (M,g) contains a singularity. But the author gives a far more detailed account of what is happening.

Reviewer: M.Kriele (Berlin)

MSC:

53C80	Applications of global differential geometry to the sciences
83C75	Space-time singularities, cosmic censorship, etc.
83C57	Black holes

Keywords:

gravitational groups; scalar field; spherically symmetric; black hole; Einstein’s equation; curvature singularity; mass function

References:

[1]	Christodoulou, Commun. Math. Phys. 105 pp 337– (1986)
[2]	Christodoulou, Commun. Math. Phys. 106 pp 587– (1986)
[3]	Christodoulou, Commun. Math. Phys. 109 pp 591– (1987)
[4]	Christodoulou, Commun. Math. Phys. 109 pp 613– (1987)
[5]	and , The global nonlinear stability of Minkowski space, to appear.