Instability of spherical naked singularities of a scalar field under gravitational perturbations
Junbin Li, Jue Liu
备注：
88 pages, 2 figures. In this new version, the main result is replaced by a statement on all (not only isotropic) perturbations with exceptionality being of first category, weaker than being of co-dimension at least 1. The title changes according to this
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.
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 in…
In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in $3+1$ dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Connected to this is a new geometric twisting phenomenon which plays the leading role in singularity formation. Prior to this work, the only known examples of naked singularities were the solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar-field syst…
In a previous work [I. Rodnianski and Y. Shlapentokh-Rothman, Naked Singularities for the Einstein Vacuum Equations: The Exterior Solution, arXiv:1912.08478] we constructed solutions to the Einstein vacuum equations in 3+1 dimensions which corresponded to the exterior region of a naked singularity. In this work we construct solutions which correspond to the interior region and show that the two solutions may be glued together to produce a naked singularity. Fundamental to our construction is the novel type of self-similarity for the Einstein vacuum equations that we introduced in our previous …
Proceedings of the second edition of the international Workshop "Dynamics and Thermodynamics of Blackholes and Naked Singularities" (Department of Mathematics of the Politecnico of Milano from May 10-12, 2007.
In this paper, we derive general conditions for a shadow to occur, without a photon sphere in a spacetime, caused by central nulllike or timelike naked singularities. Using these conditions, we propose classes of spacetimes which have nulllike and timelike naked singularities at the center, and that fulfill these `shadow without photon sphere' conditions. Considering additional asymptotically flat conditions, we show that, for a fixed Schwarzschild mass, the timelike naked singularities can cast a shadow of a size that is equal to, greater or smaller, than the size of a black hole shadow. On t…
We investigate the gravitational field of a static mass with quadrupole moment in empty space. It is shown that in general this configuration is characterized by the presence of curvature singularities without a surrounding event horizon. These naked singularities generate an effective field of repulsive gravity which, in turn, drastically changes the behavior of test particles. As a possible consequence, the accretion disk around a naked singularity presents a particular discontinuous structure.
Within the framework of the scalar-tensor models of gravitation and by relying on analytical and numerical techniques, we establish the existence of a class of spherically symmetric spacetimes containing a naked singularity. Our result relies on and extends a work by Christodoulou on the existence of naked singularities for the Einstein-scalar field equations. We establish that a key parameter in Christodoulou's construction couples to the Brans-Dicke field and becomes a dynamical variable, which enlarges and modifies the phase space of solutions. We recover analytically many properties first …
In order to show that there exists a relation between the validity of the Weyl entropy and the cosmic censorship hypothesis, we discuss the relation between naked singularities and the gravitational entropy estimator using the Weyl invariant. We look at the examples of Vaidya and Tolman-Bondi spacetimes in asymptotically flat and de Sitter spacetime and look at Weyl entropy as a possible indicator of the prohibition of naked singularities.
Naked singularities are hypothetical astrophysical objects, characterized by a gravitational singularity without an event horizon. Penrose has proposed a conjecture, according to which there exists a cosmic censor who forbids the occurrence of naked singularities. Distinguishing between astrophysical black holes and naked singularities is a major challenge for present day observational astronomy. A possibility of differentiating naked singularities from black holes is through the comparative study of thin accretion disks properties around rotating naked singularities and Kerr-type black holes,…
In this paper, We prove a conjecture which states that if M is a nakedly singular future boundary or nakedly singular past boundary spacetime, then the space of null geodesics, N, is non-Hausdorff. Also, we show that every two-dimensional strongly causal spacetime M is causally simple if and only if it is null pseudoconvex. As a result, it implies the converse of the conjecture for two-dimension but there are examples that refute it for more dimensions.
Static spherically symmetric solutions of 4d Brans-Dicke theory include a set of naked singularity solutions. Dilatonic effects near the naked singularities result in either a shielding or an antishielding effect from intruding massive test particles. One result is that for a portion of the solution parameter space, no communication between the singularity and a distant observer is possible via massive particle exchanges. Kaluza-Klein gravity is considered as a special case.
Repulsive gravity is a well known characteristic of naked singularities. In this work, we explore light surfaces and find new effects of repulsive gravity. We compare Kerr naked singularities with the corresponding black hole counterparts and find certain structures that are identified as horizon remnants. We argue that these features might be significant for the comprehension of processes that lead to the formation or eventually destruction of black hole Killing horizons. These features can be detected by observing photon orbits, particularly close to the rotation axis, which can be used to d…
We investigate the occurrence of naked singularities, local and global, in the incoming Vaidya spacetimes with zero initial mass. While it is well-known that these spacetimes admit locally and globally naked singularities, we demonstrate that globally naked singularities are significantly more common than is stressed in the literature, being generic in a natural topology on this collection of spacetimes. A heuristic consequence of the results is that the slow accumulation of mass is both necessary and sufficient for both types of naked singularities in these spacetimes. We demonstrate that the…
Following recent literature on dS instability in presence of interactions, we study the decay of massive particles in general FRW models and the emission from naked singularities either associated with 4D charged black holes or with 2D shock waves, by means of the Hamilton--Jacobi tunneling method. It is shown that the two-dimensional semi-classical tunneling amplitude from a naked singularity computed in that way is the same as the one-loop result of quantum field theory.
We comment here on the results in Ref [4] that showed naked singularities in dynamical gravitational collapse of inhomogeneous dust to be stable but non-generic. The definition of genericity used there is reconsidered. We point out that genericity in terms of an open set, with a positive measure defined suitably on the space of initial data, is physically more appropriate compared to the dynamical systems theory definition used in [4] which makes both black holes and naked singularities non-generic as collapse outcomes.
In this paper, we will investigate the consequences of loop quantum gravitational modifications on the formation of naked singularities. The loop quantum gravitational effects will be incorporated in the collapsing system using gravity's rainbow. This will be done by using rainbow functions, which are constructed from loop quantum gravitational modifications to the energy momentum dispersion. It will be observed that this modification will prevent the formation of naked singularity. Thus, such a modification can ensures that the weak cosmic censorship hypothesis will hold for any collapsing sy…
We investigate the existence of repulsive gravitational acceleration near naked singularities. The investigation is carried out by means of the acceleration tensor, which is a coordinate invariant object. We find that the gravitational acceleration is repulsive in the vicinity of the origin in the Reissner-Nordstrøm and in the Kerr space-times, and attractive at large distances in the expected Newtonian way. We further address the space-time of a point massive particle, which also exhibits repulsive effects near the origin.
We discuss the existence of maximum forces in 3+1 dimensional spacetimes and show that the existence of a mass-independent maximum force does not occur in general relativity in spaces of more than three dimensions. Instead, the maximum force increases with the masses, M, of merging objects as M to the power (N-3)/(N-2) and allows unbounded gravitational forces to occur. This suggests that naked singularities can arise in more than three dimensions because they are unprotected by a maximum force at the horizon surface. This creates a new perspective on the expectation of naked singularities in …
We present a quite simple analytical study on the appearance or absence of naked singularities in binary systems. As an example we consider the double Reissner-Nordström solution and fix the conditions it should satisfy in order to avoid or develop singular surfaces off the axis. The proof shows that singular surfaces appear as a consequence of the presence of negative masses in the solution.
In this work we study collapse of a general matter in a most general spacetime i.e., a spacetime with any matter and without (assuming) any symmetry. We show that the energy is completely trapped inside the null singularity and therefore they cannot be experimentally observed. This most general result implies, there is no physical significance of the null naked singularities irrespective of their existence. This conclusion strongly supports the essence of cosmic censorship hypothesis.
In the present paper, we study the thermal properties of the geometrically thin accretion disks surrounding the null naked singularity (NNS) and the first type of Joshi-Malafarina-Narayan (JMN1) spacetimes and compare the results with the accretion disk around equally massive Schwarzschild black hole. First, we examine the properties of the circular orbits in these spacetimes. The emitted flux, radiation spectra, disk efficiency, and temperature distribution on the disk surface are then investigated. The efficiency of the conversion of the accreting matter into radiation is found to be substan…
