Digital Object Identiﬁer (DOI) https://doi.org/10.1007/s00220-018-3157-1
Commun. Math. Phys.
A Robust Proof of the Instability of Naked Singularities
of a Scalar Field in Spherical Symmetry
Jue Liu, Junbin Li
Department of Mathematics, Sun Yat-sen University, Guangzhou, China.
E-mail: firstname.lastname@example.org; email@example.com
Received: 12 October 2017 / Accepted: 9 March 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract: Published in 1999, Christodoulou proved that the naked singularities of a self-
gravitating scalar ﬁeld are not stable in spherical symmetry and therefore the cosmic
censorship conjecture is true in this context. The original proof is by contradiction and
sharp estimates are obtained strictly depending on spherical symmetry. In this paper,
appropriate a priori estimates for the solution are obtained. These estimates are more
relaxed but sufﬁcient for giving another robust argument in proving the instability, in
particular not by contradiction. In a companion paper, we are able to prove certain
instability theorems of the spherically symmetric naked singularities of a scalar ﬁeld
under gravitational perturbations without symmetries. The argument given in this paper
plays a central role.
In the paper , Christodoulou proved both the weak cosmic censorship conjecture
and the strong cosmic censorship conjecture for spherically symmetric solutions of the
Einstein equations coupled with a massless scalar ﬁeld. The coupled system reads
which we call the Einstein-scalar ﬁeld equations. The proof, which is by contradiction,
contains sharp estimates which may not be easily obtained beyond spherical symmetry.
In this paper, we will provide a robust proof which is not by contradiction, and contains
only relaxed estimates. The main advantage of this proof is that it has the potential to be
extended beyond spherical symmetry.