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**The instability of naked singularities in the gravitational collapse of a scalar field.**
*(English)*
Zbl 1126.83305

Introduction: One of the fundamental unanswered questions in the general theory of relativity is whether “naked” singularities, that is singular events which are visible from infinity, may form with positive probability in the process of gravitational collapse. The conjecture that the answer to this question is in the negative has been called “cosmic censorship”. The present paper addresses this question in the context of the spherical gravitational collapse of a scalar field.

The problem of a spherically symmetric self-gravitating scalar field is formulated in terms of a two-dimensional quotient space-time manifold \({\mathcal Q}\) with boundary [the author, Commun. Pure Appl. Math. 46, No. 8, 1131–1220 (1993; Zbl 0853.35122)]. The boundary of \({\mathcal Q}\) corresponds to the set of fixed points of the group action, the center of symmetry, which is a timelike geodesic \(\Gamma\). The manifold \({\mathcal Q}\) is endowed with a Lorentzian metric \(g_{ab}\), an area radius function \(r\) and a wave function \(\varphi\) satisfying the following nonlinear system of partial differential equations:

\[ \begin{aligned} r\nabla_a\nabla_br&= \tfrac12 g_{ab}(1-\partial^cr\partial_cr)- r^2T_{ab},\\ T_{ab}&= \partial_a\varphi\partial_b\varphi- \tfrac12 g_{ab}\partial^c \varphi\partial_c \varphi,\\ \nabla^a(r^2 \partial_a\varphi)&= 0. \end{aligned} \]

These imply the following equation for the Gauss curvature of \({\mathcal Q}\):

\[ K= r^{-2} (1-\partial^ar \partial_ar)+ \partial^a\varphi \partial_a\varphi. \]

The mass function \(m\) is defined by:

\[ 1- \frac{2m}{r}= g^{ab} \partial_ar \partial_br. \]

In [the author, Commun. Pure Appl. Math. 44, No. 3, 339–373 (1991; Zbl 0728.53061)] it was shown that given an initial future light cone with vertex at the center of symmetry and with a region bounded by two spheres such that the ratio of the mass contained in the region to the largest radius is large in comparison to the ratio of the radii minus 1, then a trapped region, namely a region where the future light cones have negative expansion, forms in the future terminating at a strictly space-like singular boundary. The trapped region contains a sphere whose mass is bounded from below by a positive number depending only on the two initial radii. The results shall be used in an essential way in the present paper.

In [the author, Commun. Math. Phys. 109, 613–647 (1987; Zbl 0613.53049)] it was shown that when the final Bondi mass, that is, the infimum of the mass at future null infinity, is different from zero, a black hole forms of mass equal to the final Bondi mass surrounded by vacuum. The rate of growth of the redshift of light seen by faraway observers was determined and the asymptotic wave behaviour at timelike infinity and along the event horizon, the boundary of the past of future null infinity, was analyzed.

In [the author, Ann. Math. (2) 140, No. 3, 607–653 (1994; Zbl 0822.53066)] we constructed examples of solutions corresponding to regular asymptotically flat initial data which develop singularities which are not preceeded by a trapped region but have future light cones expanding to infinity. Thus naked singularities do, in fact, occur in the spherical gravitational collapse of a scalar field.

The present paper nevertheless supports the cosmic censorship conjecture. For, we shall show in the following that in the space of initial conditions the subset of initial conditions leading to the formation of naked singularities has, in a certain sense, positive codimension, consequently the occurrence of naked singularities is an unstable phenomenon in the context of the spherical selfgravitating scalar field model.

The problem of a spherically symmetric self-gravitating scalar field is formulated in terms of a two-dimensional quotient space-time manifold \({\mathcal Q}\) with boundary [the author, Commun. Pure Appl. Math. 46, No. 8, 1131–1220 (1993; Zbl 0853.35122)]. The boundary of \({\mathcal Q}\) corresponds to the set of fixed points of the group action, the center of symmetry, which is a timelike geodesic \(\Gamma\). The manifold \({\mathcal Q}\) is endowed with a Lorentzian metric \(g_{ab}\), an area radius function \(r\) and a wave function \(\varphi\) satisfying the following nonlinear system of partial differential equations:

\[ \begin{aligned} r\nabla_a\nabla_br&= \tfrac12 g_{ab}(1-\partial^cr\partial_cr)- r^2T_{ab},\\ T_{ab}&= \partial_a\varphi\partial_b\varphi- \tfrac12 g_{ab}\partial^c \varphi\partial_c \varphi,\\ \nabla^a(r^2 \partial_a\varphi)&= 0. \end{aligned} \]

These imply the following equation for the Gauss curvature of \({\mathcal Q}\):

\[ K= r^{-2} (1-\partial^ar \partial_ar)+ \partial^a\varphi \partial_a\varphi. \]

The mass function \(m\) is defined by:

\[ 1- \frac{2m}{r}= g^{ab} \partial_ar \partial_br. \]

In [the author, Commun. Pure Appl. Math. 44, No. 3, 339–373 (1991; Zbl 0728.53061)] it was shown that given an initial future light cone with vertex at the center of symmetry and with a region bounded by two spheres such that the ratio of the mass contained in the region to the largest radius is large in comparison to the ratio of the radii minus 1, then a trapped region, namely a region where the future light cones have negative expansion, forms in the future terminating at a strictly space-like singular boundary. The trapped region contains a sphere whose mass is bounded from below by a positive number depending only on the two initial radii. The results shall be used in an essential way in the present paper.

In [the author, Commun. Math. Phys. 109, 613–647 (1987; Zbl 0613.53049)] it was shown that when the final Bondi mass, that is, the infimum of the mass at future null infinity, is different from zero, a black hole forms of mass equal to the final Bondi mass surrounded by vacuum. The rate of growth of the redshift of light seen by faraway observers was determined and the asymptotic wave behaviour at timelike infinity and along the event horizon, the boundary of the past of future null infinity, was analyzed.

In [the author, Ann. Math. (2) 140, No. 3, 607–653 (1994; Zbl 0822.53066)] we constructed examples of solutions corresponding to regular asymptotically flat initial data which develop singularities which are not preceeded by a trapped region but have future light cones expanding to infinity. Thus naked singularities do, in fact, occur in the spherical gravitational collapse of a scalar field.

The present paper nevertheless supports the cosmic censorship conjecture. For, we shall show in the following that in the space of initial conditions the subset of initial conditions leading to the formation of naked singularities has, in a certain sense, positive codimension, consequently the occurrence of naked singularities is an unstable phenomenon in the context of the spherical selfgravitating scalar field model.

### MSC:

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

35L67 | Shocks and singularities for hyperbolic equations |

35L75 | Higher-order nonlinear hyperbolic equations |

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |