Integrability of the Minimal Strain Equations for the Lapse and Shift in 3+1 Numerical Relativity

Brady, Creighton and Thorne have argued that, in numerical relativity simulations of the inspiral of binary black holes, if one uses lapse and shift functions satisfying the ``minimal strain equations'' (MSE), then the coordinates might be kept co-rotating, the metric components would then evolve on the very slow inspiral timescale, and the computational demands would thus be far smaller than for more conventional slicing choices. In this paper, we derive simple, testable criteria for the MSE to be strongly elliptic, thereby guaranteeing the existence and uniqueness of the solution to the Dirichlet boundary value problem. We show that these criteria are satisfied in a test-bed metric for inspiraling binaries, and we argue that they should be satisfied quite generally for inspiraling binaries. If the local existence and uniqueness that we have proved holds globally, then, for appropriate boundary values, the solution of the MSE exhibited by Brady et. al. (which tracks the inspiral and keeps the metric evolving slowly) will be the unique solution and thus should be reproduced by (sufficiently accurate and stable) numerical integrations.Comment: 6 pages; RevTeX; submitted to Phys. Rev. D15. Technical issue of the uniqueness of the solution to the Dirichlet problem clarified. New subsection on the nature of the boundary dat

Cosmic Censorship: As Strong As Ever

Spacetimes which have been considered counter-examples to strong cosmic censorship are revisited. We demonstrate the classical instability of the Cauchy horizon inside charged black holes embedded in de Sitter spacetime for all values of the physical parameters. The relevant modes which maintain the instability, in the regime which was previously considered stable, originate as outgoing modes near to the black hole event horizon. This same mechanism is also relevant for the instability of Cauchy horizons in other proposed counter-examples of strong cosmic censorship.Comment: 4 pages RevTeX style, 1 figure included using epsfi

Breast Cancer: Modelling and Detection

This paper reviews a number of the mathematical models used in cancer modelling and then chooses a specific cancer, breast carcinoma, to illustrate how the modelling can be used in aiding detection. We then discuss mathematical models that underpin mammographic image analysis, which complements models of tumour growth and facilitates diagnosis and treatment of cancer. Mammographic images are notoriously difficult to interpret, and we give an overview of the primary image enhancement technologies that have been introduced, before focusing on a more detailed description of some of our own recent work on the use of physics-based modelling in mammography. This theoretical approach to image analysis yields a wealth of information that could be incorporated into the mathematical models, and we conclude by describing how current mathematical models might be enhanced by use of this information, and how these models in turn will help to meet some of the major challenges in cancer detection

Odd-parity perturbations of self-similar Vaidya spacetime

We carry out an analytic study of odd-parity perturbations of the self-similar Vaidya space-times that admit a naked singularity. It is found that an initially finite perturbation remains finite at the Cauchy horizon. This holds not only for the gauge invariant metric and matter perturbation, but also for all the gauge invariant perturbed Weyl curvature scalars, including the gravitational radiation scalars. In each case, `finiteness' refers to Sobolev norms of scalar quantities on naturally occurring spacelike hypersurfaces, as well as pointwise values of these quantities.Comment: 28 page

Phases of massive scalar field collapse

We study critical behavior in the collapse of massive spherically symmetric scalar fields. We observe two distinct types of phase transition at the threshold of black hole formation. Type II phase transitions occur when the radial extent $(\lambda)$ of the initial pulse is less than the Compton wavelength ($\mu^{-1}$) of the scalar field. The critical solution is that found by Choptuik in the collapse of massless scalar fields. Type I phase transitions, where the black hole formation turns on at finite mass, occur when $\lambda \mu \gg 1$. The critical solutions are unstable soliton stars with masses \alt 0.6 \mu^{-1}. Our results in combination with those obtained for the collapse of a Yang-Mills field~{[M.~W. Choptuik, T. Chmaj, and P. Bizon, Phys. Rev. Lett. 77, 424 (1996)]} suggest that unstable, confined solutions to the Einstein-matter equations may be relevant to the critical point of other matter models.Comment: 5 pages, RevTex, 4 postscript figures included using psfi

Stability of degenerate Cauchy horizons in black hole spacetimes

In the multihorizon black hole spacetimes, it is possible that there are degenerate Cauchy horizons with vanishing surface gravities. We investigate the stability of the degenerate Cauchy horizon in black hole spacetimes. Despite the asymptotic behavior of spacetimes (flat, anti-de Sitter, or de Sitter), we find that the Cauchy horizon is stable against the classical perturbations, but unstable quantum mechanically.Comment: Revtex, 4 pages, no figures, references adde