Examples of naked singularity formation in the gravitational collapse of a scalar field
type: Item
instance of: scholarly article
title: Examples of naked singularity formation in the gravitational collapse of a scalar field
zbMATH Open document ID: 0822.53066
DOI: 10.2307/2118619
author: Demetrios Christodoulou
published in: Annals of Mathematics. Second Series
publication date: 1995-10-05T00:00:00Z
review text: The author studies a special class of spherically symmetric spacetimes whose energy momentum tensor describes a massless scalar field. He proves the existence of asymptotically flat solutions with naked singularities and announces a further paper in which these solutions are shown to be unstable. This (and the announced) paper have been part of the folklore for years [cf. \textit{P. T. Chruściel}, On uniqueness in the large of solutions of Einstein's equations (``Strong cosmic censorship''), Canberra: Australian National University, Proc. Cent. Math. Appl. 27, 130 p. (1991; Zbl 0747.53050) and \textit{C. J. S. Clarke}, Classical Quantum Gravity 11, No. 6, 1375-1386 (1994; Zbl 0808.53067)]. The author chooses coordinates \(r\), \(u\) such that the surfaces orthogonal to the spheres of symmetry have the metric \(g = -e^{2\nu} du^ 2 - 2e^{\nu + \lambda} udr\). He assumes self-similarity, i.e. he demands that there exists a conformal Killing field \(S\) such that \({\mathcal L}_{S}g = -2g\), \(dr(S) = r\), \(d\phi(S) = -k\), where \(\phi(u,r)\) is the scalar field and \(k\) a constant. Then Einstein's equations are invariant under the flow of \(S\). In particular, \(\lambda\) and \(\nu\) depend only on a single function \(x = -r/u\). Einstein's equations reduce to a dynamical system in the plane which he discusses extensively. In a final chapter, he shows that these self-similar solutions can be used to construct examples with naked singularities. Because of self-similarity these solutions cannot be asymptotically flat, but the author sketches how the part of them containing the naked singularity can be smoothly matched to an asymptotically flat outer solution. It should be noted that in an important paper \textit{A. Ori} and \textit{T. Piran} [Phys. Rev. D 42, No. 4, 1068-1090 (1990)] have studied the same question in the more general context of perfect fluid spacetimes with linear equation of state. While their approach is less mathematical and (partly) relies on comp…
Mathematics Subject Classification ID: 53Z05
Mathematics Subject Classification ID: 83C75
Mathematics Subject Classification ID: 83C20
zbMATH DE Number: 750883
zbMATH Keywords: cosmic censorship
zbMATH Keywords: spherically symmetric spacetimes
zbMATH Keywords: massless scalar field
zbMATH Keywords: naked singularities
zbMATH Keywords: folklore
zbMATH Keywords: self-similarity
zbMATH Keywords: asymptotically flat
reviewed by: Marcus Kriele
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118619
OpenAlex ID: W2020184849
Wikidata QID: Q56058262
description: scientific article; zbMATH DE number 750883