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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
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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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