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Register a new userA robust proof of the instability of naked singularities of a scalar field in spherical symmetry
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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 another related paper, we are able to prove instability theorems of the spherical symmetric naked singularities under certain isotropic gravitational perturbations without symmetries. The argument given in this paper plays a central role.
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In this paper, we initiate the study of the instability of naked singularities without symmetries. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simplest case: a characteristic initial value problem of this coupled system with the initial data given on two intersecting null cones, the incoming one of which is assumed to be spherically symmetric and singular at its vertex, and the outgoing one of which has no symmetries. It is shown that, arbitrarily fixing the initial scalar field, the set of the initial conformal metrics on the outgoing null cone such that the maximal future development does not have any sequences of closed trapped surfaces approaching the singularity, is of first category in the whole space in which the shear tensors are continuous. Such a set can then be viewed as exceptional, although the exceptionality is weaker than the at least $1$ co-dimensionality in spherical symmetry. Almost equivalently, it is also proved that, arbitrarily fixing an incoming null cone $underline{C}_varepsilon$ to the future of the initial incoming null cone, the set of the initial conformal metrics such that the maximal future development has at least one closed trapped surface before $underline{C}_varepsilon$, contains an open and dense subset of the whole space. Since the initial scalar field can be chosen such that the singularity is naked if the initial shear is set to be zero, we may say that the spherical naked singularities of a self-gravitating scalar field are not stable under gravitational perturbations. This in particular gives new families of non-spherically symmetric gravitational perturbations different from the original spherically symmetric scalar perturbations given by Christodoulou.
We show that the spherically symmetric Einstein-scalar-field equations for wave-like decaying initial data at null infinity have unique global solutions in (0, infty) and unique generalized solutions on [0, infty) in the sense of Christodoulou. We emphasize that this decaying condition is sharp.
In previous work, we analyzed the linear and nonlinear stability of static, spherically symmetric wormhole solutions to Einsteins field equations coupled to a massless ghost scalar field. Our analysis revealed that all these solutions are unstable with respect to linear and nonlinear spherically symmetric perturbations and showed that the perturbation causes the wormholes to either decay to a Schwarzschild black hole or undergo a rapid expansion. Here, we consider charged generalization of the previous models by adding to the gravitational and ghost scalar field an electromagnetic one. We first derive the most general static, spherically symmetric wormholes in this theory and show that they give rise to a four-parameter family of solutions. This family can be naturally divided into subcritical, critical and supercritical solutions depending on the sign of the sum of the asymptotic masses. Then, we analyze the linear stability of these solutions. We prove that all subcritical and all critical solutions possess one exponentially in time growing mode. It follows that all subcritical and critical wormholes are linearly unstable. In the supercritical case we provide numerical evidence for the existence of a similar unstable mode.
In this paper, we construct a class of collapsing spacetimes in vacuum without any symmetries. The spacetime contains a black hole region which is bounded from the past by the future event horizon. It possesses a Cauchy hypersurface with trivial topology which is located outside the black hole region. Based on existing techniques in the literature, the spacetime can in principle be constructed to be past geodesically complete and asymptotic to Minkowski space. The construction is based on a semi-global existence result of the vacuum Einstein equations built on a modified version of the a priori estimates that were originally established by Christodoulou in his work on the formation of trapped surface, and a gluing construction carried out inside the black hole. In particular, the full detail of the a priori estimates needed for the existence is provided, which can be regarded as a simplification of Christodoulous original argument.
We simulate the behaviour of a Higgs-like field in the vicinity of a Schwarzschild black hole using a highly accurate numerical framework. We consider both the limit of the zero-temperature Higgs potential, and a toy model for the time-dependent evolution of the potential when immersed in a slowly cooling radiation bath. Through these numerical investigations, we aim to improve our understanding of the non-equilibrium dynamics of a symmetry breaking field (such as the Higgs) in the vicinity of a compact object such as a black hole. Understanding this dynamics may suggest new approaches for studying properties of scalar fields using black holes as a laboratory.
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