Mathematics

We investigate here the genericity and stability aspects for naked singularities and black holes that arise as the final states for a complete gravitational collapse of a spherical massive matter cloud. The form of the matter considered is a general Type I matter field, which includes most of the physically reasonable matter fields such as dust, perfect fluids and such other physically interesting forms of matter widely used in gravitation theory. We first study here in some detail the effects of small pressure perturbations in an otherwise pressure-free collapse scenario, and examine how a collapse evolution that was going to the black hole endstate would be modified and go to a naked singularity, once small pressures are introduced in the initial data. This allows us to understand the distribution of black holes and naked singularities in the initial data space. Collapse is examined in terms of the evolutions allowed by Einstein equations, under suitable physical conditions and as evolving from a regular initial data. We then show that both black holes and naked singularities are generic outcomes of a complete collapse, when genericity is defined in a suitable sense in an appropriate space.

Chaotic behavior in a charged three-body self-gravitating system J. Math. Phys. 50, 082703 (2009) Anisotropic cosmological models with spinor and scalar fields and viscous fluid in presence of a Λ term: Qualitative solutions A diffusion process in curved space-time J. Math. Phys. 45, 2744 (2004) Lie aspects of the dynamics of self-gravitating ellipsoids in n dimensions

On a wave-particle in closed and open isotropic universes J. Math. Phys. 52, 013508 On a class of global characteristic problems for the Einstein vacuum equations with small initial data Following Ashtekar's new Hamiltonian formulation of general relativity ["Anew Hamiltonian formulation of general relativity," Syracuse University preprint, 1986; Phys. Rev. ] on the structure of the space of solutions of vacuum Einstein equations in the case of space-times admitting a compact Cauchy hypersurface are rederived and extended.

An important issue in the study of continual gravitational collapse of a massive matter cloud in general relativity is whether shell-crossing singularities develop as the collapse evolves. We examine this here to show that for any spherically symmetric collapse in general, till arbitrarily close to the final singularity, there is always a finite neighborhood of the center in which there are no shell-crossings taking place. It follows that in order to study the visibility or otherwise of the ultra-dense region close to the final singularity of collapse where physical radius of the matter cloud shrinks to an arbitrarily small value, we can always consider without loss of generality a collapsing ball of finite comoving radius in which there are no shell-crossings.

11, 10] proved many interesting results. Especially after good deal of effort, Minguzzi proved that K -causality condition is equivalent to stably causal condition. More recently, K.Martin and Panangaden making use of domain theory, a branch of theoretical computer science, proved fascinating results in the causal structure theory of space-time. The remarkable fact about their work is that only order is needed to develop the theory and topology is an outcome of the order. In addition to this consequence, there are abstract approaches, algebraic as well as geometric to the theory of cones and cone preserving mappings. Use of quasi-order (a relation which is reflexive and transitive) and partial order is made in defining the cone structure. Such structures and partial orderings are used in the optimization problems [13], game theory and decision making etc . The interplay between ideas from theoretical computer science and causal structure of space-time is becoming more evident in the recent works . Keeping these developments in view , in this article, we present a review of geometric and algebraic approach to causal cones and describe cone preserving transformations and their relationship with causal structure. We also describe certain implications of these concepts in special and general theory of relativity related to causal structure and topology of space-time. Thus in section 2, we begin with describing Lie groups, especially matrix Lie groups, homogeneous spaces and then causal cones. We give an algebraic description of cones by using quasi-order. Furthermore, we describe cone preserving transformations. These maps are generalizations of causal maps related to causal structure of spacetime which we shall describe in section 3. We then describe explicitly Minkowski space as an illustration of these concepts and note that some of the space-time models in general theory of relativity can be described as homogeneous spaces. In section 3, we describe causal structure of space time, causality conditions, Kcausality and hierarchy among these conditions in the light of recent work of S. Janardhan and R.V. Saraykar and E.Minguzzi and M.Sanchez [8,. We also describe geometric structure of causal group, a group of transformations preserving causal structures or a group of causal maps on a space-time.

In this paper we prove a stability theorem for large solutions of three dimensional incompressible Magnetohydrodynamic equations under suitable initial & boundary conditions and appropriate boundedness assumptions on the initial data. We use Sobolev spaces as function space for various quantities.

We study the steady state Magnetohydrodynamic (MHD) equations in the whole space Following the work of C. Bjorland and M. Schonbek [4] on Navier -Stokes equations in the whole space, we prove the existence of at least one solution with finite Dirichlet Integral to steady state Magnetohydrodynamic equations in the whole space. Further, we show that these solutions are unique among all solutions with finite energy and finite Dirichlet Integral .

In this paper, following recent results on generalized Vaidya solutions by Wang, we prove that under certain conditions on generalized mass function, strong curvature naked singularities exist in radiation collapse in monopole-Vaidya space-times and also in charged-Vaidya space-times. We thus unify and generalize results of Dwivedi-Joshi and Lake-Zannias. The general case also covers de Sitter-Vaidya space-time recently treated by Wagh-Maharaj with a view to study existence of naked singularities.

Abstract :  Following the work of researchers spread over more than four decades, we describe a number of topologies on the space of Lorentz metrics on a space-time manifold and using some of these topologies, especially Whitney Cr - topology, we discuss stable and unstable global properties of space-times. We also discuss a new topology introduced by Noldus on this space.

Considering gravitational collapse of Type I matter fields, we prove that, given an arbitrary C2- mass function M(r, v) and a C1- function h(r, v) (through the corresponding C1-metric function (t, r)), there exist infinitely many choices of energy distribution function b(r) such that the ‘true’ initial data (M, h(r, v)) leads the collapse to the formation of naked singularity. We further prove that the occurrence of such a naked singularity is stable with respect to small changes in the initial data. We remark that though the initial data leading to both black hole and naked singularity form a “big” subset of the true initial data set, their occurrence is not generic. The terms ‘stability’ and ‘genericity’ are appropriately
defined following the theory of dynamical systems. The particular case of radial pressure pr(r) has been illustrated in details to get clear picture of how naked singularity is formed and how, it is stable with respect to initial data.

We study the occurrence of naked singularities in the spherically symmetric collapse of radiation shells in a higher dimensional spacetime. The necessary conditions for the formation of a naked singularity or a black hole are obtained. The naked singularities are found to be strong in the Tipler sense and thus violate the cosmic censorship conjecture.

We investigate the occurrence and nature of naked singularities in the gravitational collapse of a self-similar adiabatic perfect fluid in a five dimensional space-time. The naked singularities are found to be gravitationally strong in the sense of Tipler and thus violate the cosmic censorship conjecture.

Using the technique of the conformal deformation of initial data, we study the problem of mapping Hat data onto solutions of Ashtekar's constraints. The procedure followed is that of York and we show that it is useful even for complex general relativity.

Using K-causal relation introduced by Sorkin and Woolgar [1], we generalize results of Garcia-Parrado and Senovilla [2,3] on causal maps. We also introduce causality conditions with respect to K-causality which are analogous to those in classical causality theory and prove their inter-relationships. We introduce a new causality condition following the work of Bombelli and Noldus [4] and show that this condition lies in between global hyperbolicity and causal simplicity. This approach is simpler and more general as compared to traditional causal approach and it has been used by Penrose et al in giving a new proof of positivity of mass theorem. C 0 -space-time structures arise in many mathematical and physical situations like conical singularities, discontinuous matter distributions, phenomena of topology-change in quantum field theory etc.

Using semigroup theory for linear operators and a bootstrap argument,we prove that linear instability implies nonlinear instability for solutions of incompressible magnetohydrodynamic equations in three dimensions. We also prove a stability theorem for these solutions under different assumptions on the spectrum of a certain linear operator.

Gravitational collapse of radiation shells in a non self-similar higher dimensional
spherically symmetric spacetime is studied. Strong curvature naked singularities form
for a highly inhomogeneous collapse, violating the cosmic censorship conjecture. As
a special case, self similar models can be constructed.

In this paper, we investigate the nature of the singularity in the spherically
symmetrical, shear-free, gravitational collapse of a star with heat flux using a
separable metric [1]. For any non-singular, regular, radial density profile for
a star described by this metric, eq. (2.1), the singularity of the gravitational
collapse is not naked locally. Our results here unequivocally support the
Strong Cosmic Censorship Hypothesis.