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

On-lattice agent-based simulation of populations of cells within the open-source chaste framework

Over the years, agent-based models have been developed that combine cell division and reinforced random walks of cells on a regular lattice, reaction-diffusion equations for nutrients and growth factors and ordinary differential equations (ODEs) for the subcellular networks regulating the cell cycle. When linked to a vascular layer, this multiple scale model framework has been applied to tumour growth and therapy. Here we report on the creation of an agent-based multiscale environment amalgamating the characteristics of these models within a Virtual Pysiological Human (VPH) Exemplar Project. This project enables re-use, integration, expansion and sharing of the model and relevant data. The agent-based and reactiondiffusion parts of the multiscale model have been implemented and are available for download as part of the latest public release of Chaste (“Cancer, Heart and Soft Tissue Environment”), (http://www.cs.ox.ac.uk/chaste/) version 3.1, part of the VPH Toolkit (http://toolkit.vph-noe.eu/). The environment functionalities are verified against the original models, in addition to extra validation of all aspects of the code. In this work, we present the details of the implementation of the agent-based environment, including the system description, the conceptual model, the development of the simulation model and the processes of verification and validation of the simulation results. We explore the potential use of the environment by presenting exemplar applications of the “what if” scenarios that can easily be studied in the environment. These examples relate to tumour growth, cellular competition for resources and tumour responses to hypoxia. We conclude our work by summarising the future steps for the expansion of the current system

Phase properties of hypergeometric states and negative hypergeometric states

We show that the three quantum states (P$\acute{o}$lya states, the generalized non-classical states related to Hahn polynomials and negative hypergeometric states) introduced recently as intermediates states which interpolate between the binomial states and negative binomial states are essentially identical. By using the Hermitial-phase-operator formalism, the phase properties of the hypergeometric states and negative hypergeometric states are studied in detail. We find that the number of peaks of phase probability distribution is one for the hypergeometric states and $M$ for the negative hypergeometric states.Comment: 7 pages, 4 figure

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

Instability of black hole formation under small pressure perturbations

We investigate here the spectrum of gravitational collapse endstates when arbitrarily small perfect fluid pressures are introduced in the classic black hole formation scenario as described by Oppenheimer, Snyder and Datt (OSD) [1]. This extends a previous result on tangential pressures [2] to the more physically realistic scenario of perfect fluid collapse. The existence of classes of pressure perturbations is shown explicitly, which has the property that injecting any smallest pressure changes the final fate of the dynamical collapse from a black hole to a naked singularity. It is therefore seen that any smallest neighborhood of the OSD model, in the space of initial data, contains collapse evolutions that go to a naked singularity outcome. This gives an intriguing insight on the nature of naked singularity formation in gravitational collapse.Comment: 7 pages, 1 figure, several modifications to match published version on GR

Why do naked singularities form in gravitational collapse?

We investigate what are the key physical features that cause the development of a naked singularity, rather than a black hole, as the end-state of spherical gravitational collapse. We show that sufficiently strong shearing effects near the singularity delay the formation of the apparent horizon. This exposes the singularity to an external observer, in contrast to a black hole, which is hidden behind an event horizon due to the early formation of an apparent horizon.Comment: revised for clarity, new figure included; version accepted by Phys. Rev. D (RC

Fair Equality of Opportunity Critically Reexamined: The Family and the Sustainability of Health Care Systems

A complex interaction of ideological, financial, social, and moral factors makes the financial sustainability of health care systems a challenge across the world. One difficulty is that some of the moral commitments of some health care systems collide with reality. In particular, commitments to equality in access to health care and to fair equality of opportunity undergird an unachievable promise, namely, to provide all with the best of basic health care. In addition, commitments to fair equality of opportunity are in tension with the existence of families, because families are aimed at advantaging their own members in preference to others. Because the social-democratic state is committed to fair equality of opportunity, it offers a web of publicly funded entitlements that make it easier for persons to exit the family and to have children outside of marriage. In the United States, in 2008, 41% of children were born outside of wedlock, whereas, in 1940, the percentage was only 3.8%, and in 1960, 5%, with the further consequence that the social and financial capital generated through families, which aids in supporting health care in families, is diminished. In order to explore the challenge of creating a sustainable health care system that also supports the traditional family, the claims made for fair equality of opportunity in health care are critically reconsidered. This is done by engaging the expository device of John Rawls's original position, but with a thin theory of the good that is substantively different from that of Rawls, one that supports a health care system built around significant copayments, financial counseling, and compulsory savings, with a special focus on enhancing the financial and social capital of the family. This radical recasting of Rawls, which draws inspiration from Singapore, is undertaken as a heuristic to aid in articulating an approach to health care allocation that can lead past the difficulties of social-democratic policy

Role of Initial Data in Higher Dimensional Quasi-Spherical Gravitational Collapse

We study the gravitational collapse in ($n+2$)-D quasi-spherical Szekeres space-time (which possess no killing vectors) with dust as the matter distribution. Instead of choosing the radial coordinate `$r$' as the initial value for the scale factor $R$, we consider a power function of $r$ as the initial scale for the radius $R$. We examine the influence of initial data on the formation of singularity in gravitational collapse.Comment: 7 Latex Pages, RevTex Style, 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

Divergence of the Quantum Stress Tensor on the Cauchy Horizon in 2-d Dust Collapse

We prove that the quantum stress tensor for a massless scalar field in two dimensional non-selfsimilar Tolman Bondi dust collapse and Vaidya radiation collapse models diverges on the Cauchy horizon, if the latter exists. The two dimensional model is obtained by suppressing angular co-ordinates in the corresponding four dimensional spherical model.Comment: 16 pages, no figures, LaTeX fil