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**Examples of naked singularity formation in the gravitational collapse of a scalar field.**
*(English)*
Zbl 0822.53066

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. 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 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 A. Ori and 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 computer simulations I believe that the main ideas are already present. It has not been quoted by the author.

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 A. Ori and 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 computer simulations I believe that the main ideas are already present. It has not been quoted by the author.

Reviewer: M.Kriele (Berlin)

### MSC:

53Z05 | Applications of differential geometry to physics |

83C75 | Space-time singularities, cosmic censorship, etc. |

83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |