##
**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 |

### Keywords:

Einstein-scalar field equations; asymptotic behavior; generalized solutions; black hole; event horizon
PDF
BibTeX
XML
Cite

\textit{D. Christodoulou}, Commun. Math. Phys. 109, 613--647 (1987; Zbl 0613.53049)

Full Text:
DOI

### 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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.