The Final Fate of Spherical Inhomogeneous Dust Collapse

We examine the role of the initial density and velocity distribution in the gravitational collapse of a spherical inhomogeneous dust cloud. Such a collapse is described by the Tolman-Bondi metric which has two free functions: the `mass-function' and the `energy function', which are determined by the initial density and velocity profile of the cloud. The collapse can end in a black-hole or a naked singularity, depending on the initial parameters characterizing these profiles. In the marginally bound case, we find that the collapse ends in a naked singularity if the leading non-vanishing derivative of the density at the center is either the first one or the second one. If the first two derivatives are zero, and the third derivative non-zero, the singularity could either be naked or covered, depending on a quantity determined by the third derivative and the central density. If the first three derivatives are zero, the collapse ends in a black hole. In particular, the classic result of Oppenheimer and Snyder, that homogeneous dust collapse leads to a black hole, is recovered as a special case. Analogous results are found when the cloud is not marginally bound, and also for the case of a cloud starting from rest. We also show how the strength of the naked singularity depends on the density and velocity distribution. Our analysis generalizes and simplifies the earlier work of Christodoulou and Newman [4,5] by dropping the assumption of evenness of density functions. It turns out that relaxing this assumption allows for a smooth transition from the naked singularity phase to the black-hole phase, and also allows for the occurrence of strong curvature naked singularities.Comment: 23 pages; Plain Tex; TIFR-TAP preprin

Genericity aspects in gravitational collapse to black holes and naked singularities

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.Comment: 24 pages, 6 figures, some changes in text and figures to match the version accepted for publication by IJMP

On the Role of Initial Data in the Gravitational Collapse of Inhomogeneous Dust

We consider here the gravitational collapse of a spherically symmetric inhomogeneous dust cloud described by the Tolman-Bondi models. By studying a general class of these models, we find that the end state of the collapse is either a black hole or a naked singularity, depending on the parameters of the initial density distribution, which are $\rho_{c}$, the initial central density of the massive body, and $R_0$, the initial boundary. The collapse ends in a black hole if the dimensionless quantity $\beta$ constructed out of this initial data is greater than 0.0113, and it ends in a naked singularity if $\beta$ is less than this number. A simple interpretation of this result can be given in terms of the strength of the gravitational potential at the starting epoch of the collapse.Comment: Original title changed, numerical range of naked singularity corrected. Plain Tex File. 14 pages. To appear in Physical Review

Stability of Naked Singularity arising in gravitational collapse of Type I matter fields

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 $\nu(t,r)$), 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.Comment: 16 pages, no figure, Latex, submitted to Praman

Connections between normalized Wright functions with families of analytic functions with negative coefficients

In this article we present sufficient conditions that ensures that normalized Wright functions belong to certain subclasses of analytic univalent functions with negative coefficients in the unit disc U. We also provide some geometric properties of integral transforms involving normalized Wright functions

Probing large distance higher dimensional gravity from lensing data

The modifications induced in the standard weak-lensing formula if Newtonian gravity differs from inverse square law at large distances are studied. The possibility of putting bounds on the mass of gravitons from lensing data is explored. A bound on graviton mass, esitmated to be about 100 Mpc$^{-1}$ is obtained from analysis of some recent data on gravitational lensing.Comment: 6 pages, 1 figure, added reference

Initial data and the end state of spherically symmetric gravitational collapse

Generalizing earlier results on the initial data and the final fate of dust collapse, we study here the relevance of the initial state of a spherically symmetric matter cloud towards determining its end state in the course of a continuing gravitational collapse. It is shown that given an arbitrary regular distribution of matter at the initial epoch, there always exists an evolution from this initial data which would result either in a black hole or a naked singularity depending on the allowed choice of free functions available in the solution. It follows that given any initial density and pressure profiles for the cloud, there is a non-zero measure set of configurations leading either to black holes or naked singularities, subject to the usual energy conditions ensuring the positivity of energy density. We also characterize here wide new families of black hole solutions resulting from spherically symmetric collapse without requiring the cosmic censorship assumption.Comment: Ordinary Tex file, 31 pages no figure

Gravitational collapse of an isentropic perfect fluid with a linear equation of state

We investigate here the gravitational collapse end states for a spherically symmetric perfect fluid with an equation of state $p=k\rho$. It is shown that given a regular initial data in terms of the density and pressure profiles at the initial epoch from which the collapse develops, the black hole or naked singularity outcomes depend on the choice of rest of the free functions available, such as the velocities of the collapsing shells, and the dynamical evolutions as allowed by Einstein equations. This clarifies the role that equation of state and initial data play towards determining the final fate of gravitational collapse.Comment: 7 Pages, Revtex4, To appear in Classical and Quantum Gravit

Analysis of the B→K2∗(→Kπ)l+l−B \to K^*_{2} (\to K \pi) l^+ l^- decay

In this paper we study the angular distribution of the rare B decay $B \to K^*_2 (\to K \pi) l^+ l^-$, which is expected to be observed soon. We use the standard effective Hamiltonian approach, and use the form factors that have already been estimated for the corresponding radiative decay $B \to K^*_2 \gamma$. The additional form factors that come into play for the dileptonic channel are estimated using the large energy effective theory (LEET), which enables one to relate the additional form factors to the form factors for the radiative mode. Our results provide, just like in the case of the $K^*(892)$ resonance, an opportunity for a straightforward comparison of the basic theory with experimental results which may be expected in the near future for this channel.Comment: 14 pages, 5 figures; as accepted for Phys. Rev.