## A mathematical theory of gravitational collapse.(English)Zbl 0613.53049

This paper supplements three previous papers of the author [ibid. 105, 337-362 (1986; Zbl 0608.35039), ibid. 106, 587-622 (1986), and ibid. 109, 591-611 (1987); see the preceding reviews)]. In this paper the author investigates the asymptotic behavior of the generalized solutions as the retarded time u tends to infinity. It is shown, when the final Bondi mass $$M_ 1\neq 0$$ as $$u\to \infty$$, a black hole forms of mass $$M_ 1$$ surrounded by vacuum. Further it is shown that in the region exterior to the Schwarzschild sphere, $$r=ZM_ 1$$, the solution tends to stationary as $$u\to \infty$$ and the mass remaining outside this sphere tends to zero as $$u\to \infty$$. Finally, it asserts the formation of an event horizon as $$u\to \infty$$, which is the part of the limiting hypersurface $$u=\infty$$ interior to this sphere. The rate of decay of the metric function and the asymptotic behaviour of the incoming light rays are obtained.
Reviewer: N.Sengupta

### MSC:

 53C80 Applications of global differential geometry to the sciences 35L70 Second-order nonlinear hyperbolic equations 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 83C30 Asymptotic procedures (radiation, news functions, $$\mathcal{H}$$-spaces, etc.) in general relativity and gravitational theory 83C40 Gravitational energy and conservation laws; groups of motions

### Citations:

Zbl 0613.53048; Zbl 0613.53047; Zbl 0608.35039
Full Text:

### References:

 [1] Christodoulou, D.: The problem of a self-gravitating scalar field. Commun. Math. Phys.105, 337 (1986) · Zbl 0608.35039 [2] Christodoulou, D.: Global existence of generalized solutions of the spherically symmetric Einstein-scalar equations in the large. Commun. Math. Phys.106, 587 (1986) · Zbl 0613.53047 [3] Christodoulou, D.: The structure and uniqueness of generalized solutions of the spherically symmetric Einstein-scalar equations. Commun. Math. Phys.109, 591 (1987) · Zbl 0613.53048
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