arXiv subjects
Publications and source records attributed to R. V. Saraykar.
In this article, we review and discuss different aspects of stability and genericity of some properties of space-times which occur in various contexts in the General Theory of Relativity. We also give argument supporting the conclusion that Linearization Stability is a generic property if we restrict space-times to the class of those which admit compact spacelike constant mean curvature hypersurfaces.
In this paper, we prove that Linearization Stability of Einstein Field Equations is a Generic Property in the sense that within the class $\mathcal{V}$ of space-times which admit a compact Cauchy hypersurface of constant mean curvature, the subclass $\mathcal{V}_K$ of space-times which are linearization stable forms an open and dense subset of $\mathcal{V}$ under $C^\infty$ - topology. The work is based upon the work of Ebin $[1]$, Fischer, Marsden and Moncrief $[2,3]$ and Beig, Chrusciel and Schoen $[4]$.
In this paper, we prove that the set of solutions of constraint equations for coupled Einstein and scalar fields in classical general relativity possesses Hilbert manifold structure. We follow the work of R. Bartnik [2] and use weighted Sobolev spaces and Implicit Function Theorem to prove our results.
We investigate here gravitational collapse of a perfect fluid with a linear isentropic equation of state $p = k ρ$. A class of collapse models is given which is a family of solutions to Einstein equations and the final fate of collapse is analyzed in terms of the formation of black holes and naked singularities. The collapse evolves from a regular initial data and the positivity of energy conditions and other physical regularity conditions are satisfied. As we provided here an explicit class, this gives useful insights into the endstates of collapse with a physically reasonable and relevant equation of state and for the cosmic censorship hypothesis.
We study stability of occurrence of black holes and naked singularities that arise as a final state for a complete gravitational collapse of type I matter field in a spherically symmetric $N$ dimensional spacetime with equation of state $p = k ρ$, $0 \leq k \leq 1$. We prove that for a regular initial data comprising of pressure (or density) profiles at an initial surface $t = t_{i}$, from which the collapse evolves, there exists a large class of the velocity functions and classes of solutions of Einstein equations, such that the spacetime evolution goes to a final state which is either a black hole or a naked singularity. We further prove that in an infinite dimensional separable Banach space, the set of regular initial data leading the collapse to a black hole or a naked singularity, forms an open subset of the set of all regular initial data. In this sense, the gravitational collapse leading to either a black hole or a naked singularity is stable. These results are discussed and analyzed in the light of the cosmic censorship hypothesis in black hole physics.
This is a short review article in which we discuss and summarize the works of various researchers over past four decades on Zeeman topology and Zeeman-like topologies, which occur in special and general theory of relativity. We also discuss various properties and inter-relationship of these topologies.
We present a short review of geometric and algebraic approach to causal cones and describe cone preserving transformations and their relationship with causal structure related to special and general theory of relativity. We describe Lie groups, especially matrix Lie groups, homogeneous and symmetric spaces and causal cones and certain implications of these concepts in special and general theory of relativity related to causal structure and topology of space-time. We compare and contrast the results on causal relations with those in the literature for general space-times and compare these relations with K-causal maps. We also describe causal orientations and their implications for space-time topology and discuss some more topologies on space-time which arise as an application of domain theory.
Using K-causal relation introduced by Sorkin and Woolgar [26], we gener- alize results of Garcia-Parrado and Senovilla [8, 9] 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 [3] 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 [11, 19] and it has been used by Penrose et.al [20] 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.
We study the gravitational collapse of Type I matter field in $N$ dimensional spacetime with radial pressure $p_{r}$ as a function of $r $. We find that for a given smooth initial data set satisfying physical requirements, naked singularities exist for spacetime dimensions N=4 and 5 while for $N \geq 6$ Cosmic Censorship Conjecture withholds its ground. We, also, study the collapse with linear equation of state and find that, similar to dust collapse with appropriate choice of initial data naked singularities occur in all dimensions.
We study five dimensional(5D) spherically symmetric self-similar perfect fluid space-time with adiabatic equation of state, considering all the families of future directed non-spacelike geodesics. The space-time admits globally strong curvature naked singularities in the sense of Tipler and thus violates the cosmic censorship conjecture provided a certain algebraic equation has real positive roots. We further show that it is the weak energy condition (WEC) that is necessary for visibility of singularities for a finite period of time and for singularities to be gravitationally strong. We, also, match the solution to 5D Schwarzschild solution using the junction conditions.
Considering gravitational collapse of Type I matter fields, we prove that, given an arbitrary $C^{2}$- mass function $\textit{M}(r,v)$ and a $C^{1}$- function $h(r,v)$ (through the corresponding $C^{1}$- metric function $ν(t,r)$), there exist infinitely many choices of energy distribution function $b(r)$ such that the `true' initial data ($\textit{M},h(r,v)$) 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 $p_{r}(r)$ 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.
In this paper, we find all the Conformal Killing Vectors (CKVs) and their Lie Algebra for the recently reported [cqg1] spherically symmetric, shear-free separable metric spacetimes with non-vanishing energy or heat flux. We also solve the geodesic equatios of motion for the spacetime under consideration.
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 [cqg1]. 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.
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's sense and thus violating cosmic censorship conjecture.