Photon redshift and the appearance of a naked singularity

In this paper we analyze the redshift as observed by an external observer receiving photons which terminate in the past at the naked singularity formed in a Tolman-Bondi dust collapse. Within the context of models considered here it is shown that photons emitted from a weak curvature naked singularity are always finitely redshifted to an external observer. Certain cases of strong curvature naked singularities, including the self-similar one, where the photons are infinitely redshifted are also pointed out.Comment: Latex file, 14 pages, no figures, one change in the reference. Accepted for publication in Phys. Rev.

Event-related brain potential correlates of human auditory sensory memory-trace formation

The event-related potential (ERP) component mismatch negativity (MMN) is a neural marker of human echoic memory. MMN is elicited by deviant sounds embedded in a stream of frequent standards, reflecting the deviation from an inferred memory trace of the standard stimulus. The strength of this memory trace is thought to be proportional to the number of repetitions of the standard tone, visible as the progressive enhancement of MMN with number of repetitions (MMN memory-trace effect). However, no direct ERP correlates of the formation of echoic memory traces are currently known. This study set out to investigate changes in ERPs to different numbers of repetitions of standards, delivered in a roving-stimulus paradigm in which the frequency of the standard stimulus changed randomly between stimulus trains. Normal healthy volunteers (n = 40) were engaged in two experimental conditions: during passive listening and while actively discriminating changes in tone frequency. As predicted, MMN increased with increasing number of standards. However, this MMN memory-trace effect was caused mainly by enhancement with stimulus repetition of a slow positive wave from 50 to 250 ms poststimulus in the standard ERP, which is termed here "repetition positivity" (RP). This RP was recorded from frontocentral electrodes when participants were passively listening to or actively discriminating changes in tone frequency. RP may represent a human ERP correlate of rapid and stimulus-specific adaptation, a candidate neuronal mechanism underlying sensory memory formation in the auditory cortex

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

Null Geodesic Expansion in Spherical Gravitational Collapse

We derive an expression for the expansion of outgoing null geodesics in spherical dust collapse and compute the limiting value of the expansion in the approach to singularity formation. An analogous expression is derived for the spherical collapse of a general form of matter. We argue on the basis of these results that the covered as well as the naked singularity solutions arising in spherical dust collapse are stable under small changes in the equation of state.Comment: 10 pages, Latex File, No figure

The final fate of spherical inhomogeneous dust collapse II: Initial data and causal structure of singularity

Further to results in [9], pointing out the role of initial density and velocity distributions towards determining the final outcome of spherical dust collapse, the causal structure of singularity is examined here in terms of evolution of the apparent horizon. We also bring out several related features which throw some useful light towards understanding the nature of this singularity, including the behaviour of geodesic families coming out and some aspects related to the stability of singularity.Comment: Latex file, uses epsf.sty, 15 pages and 3 eps figures. Paragraph on role of smooth functions rewritten. Four references added. To appear in Classical & Quantum Gravit

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

Stability constants of some lanthanide mixed ligand complexes

722-725Stability constants of the mixed ligand complexes of some lanthanides, viz., La(III), Pr(III), Nd(III), Gd(III) and Dy(III) with oxydiacetic acid (ODA), tartaric acid (TRA), malic acid (MEA), iminodiacetic acid (IDA) and glycine (GLY) as ligands have been determined pH-metrically at 25 ± 1°C and at an ionic strength of 0.1mol dm-3 (KNO3). A comparison of the values of stability constants of the complexes reveals the order, La(III) K values are negative for all the ternary and quaternary systems; however, DD log K values are significantly positive. These have been explained in terms of intramolecular hydrophobic ligand-ligand interactions. The stability constant data have been used to study the electrostatic factors involved in the formation of these mixed complexes

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

Physical nature of the central singularity in spherical collapse

We examine here the nature of the central singularity forming in the spherically symmetric collapse of a dust cloud and it is shown that this is always a strong curvature singularity where gravitational tidal forces diverge powerfully. An important consequence is that the nature of the naked singularity forming in the dust collapse turns out to be stable against the perturbations in the initial data from which the collapse commences.Comment: Latex file, 11 pages, 2 figures, Updated version to match the published version in PR