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

Gluon confinement criterion in QCD

We fix exactly and uniquely the infrared structure of the full gluon propagator in QCD, not solving explicitly the corresponding dynamical equation of motion. By construction, this structure is an infinite sum over all possible severe (i.e., more singular than $1/q^2$) infrared singularities. It reflects the zero momentum modes enhancement effect in the true QCD vacuum, which is due to the self-interaction of massless gluons. It existence automatically exhibits a characteristic mass (the so-called mass gap). It is responsible for the scale of nonperturbative dynamics in the true QCD ground state. The theory of distributions, complemented by the dimensional regularization method, allows one to put the severe infrared singularities under the firm mathematical control. By an infrared renormalization of a mass gap only, the infrared structure of the full gluon propagator is exactly reduced to the simplest severe infrared singularity, the famous $(q^2)^{-2}$. Thus we have exactly established the interaction between quarks (concerning its pure gluon (i.e., nonlinear) contribution) up to its unimportant perturbative part. This also makes it possible for the first time to formulate the gluon confinement criterion and intrinsically nonperturbative phase in QCD in a manifestly gauge-invariant ways.Comment: 10 pages, no figures, no tables. Typos corrected and the clarification is intoduced. Shorten version to appear in Phys. Lett.

The asymptotic regimes of tilted Bianchi II cosmologies

In this paper we give, for the first time, a complete description of the dynamics of tilted spatially homogeneous cosmologies of Bianchi type II. The source is assumed to be a perfect fluid with equation of state $p = (\gamma -1) \mu$, where $\gamma$ is a constant. We show that unless the perfect fluid is stiff, the tilt destabilizes the Kasner solutions, leading to a Mixmaster-like initial singularity, with the tilt being dynamically significant. At late times the tilt becomes dynamically negligible unless the equation of state parameter satisfies $\gamma > {10/7}$. We also find that the tilt does not destabilize the flat FL model, with the result that the presence of tilt increases the likelihood of intermediate isotropization