The Initial Value Problem Using Metric and Extrinsic Curvature

The initial value problem is introduced after a thorough review of the essential geometry. The initial value equations are put into elliptic form using both conformal transformations and a treatment of the extrinsic curvature introduced recently. This use of the metric and the extrinsic curvature is manifestly equivalent to the author's conformal thin sandwich formulation. Therefore, the reformulation of the constraints as an elliptic system by use of conformal techniques is complete.Comment: 10 pages, to appear in the Proceedings of the Tenth Marcel Grossmann Meeting on General Relativit

Application of quasi-homogeneous anisotropic laminates in grid-stiffened panel design

Composite laminates are derived for standard configurations with quasi-homogeneous anisotropic properties, whereby in-plane and out-of-plane stiffness properties are concomitant. Dimensionless parameters, and their relationship to the well-known ply- orientation-dependent lamination parameters, are also developed from which the elements of the extensional and bending stiffness matrices are readily calculated for any fiber/resin properties. The definitive list of laminate configurations for up to 21 plies is presented, together with graphical representations of the lamination parameter design space for standard ply orientations +45, -45, 0 and 90 degrees. Finally, the potential of quasi-homogeneous anisotropic laminates as an optimum design solution for anisogid structures is explored for cases where buckling and strength constraints are both active

Path Integral Over Black Hole Fluctuations

Evaluating a functional integral exactly over a subset of metrics that represent the quantum fluctuations of the horizon of a black hole, we obtain a Schroedinger equation in null coordinate time for the key component of the metric. The equation yields a current that preserves probability if we use the most natural choice of functional measure. This establishes the existence of blurred horizons and a thermal atmosphere. It has been argued previously that the existence of a thermal atmosphere is a direct concomitant of the thermal radiation of black holes when the temperature of the hole is greater than that of its larger environment, which we take as zero.Comment: 5 pages, added a couple of clarification

Conformal ``thin sandwich'' data for the initial-value problem of general relativity

The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The resulting equations have the {\it same} elliptic form as does the one-hypersurface formulation. The metrical roots of this form are revealed by a conformal ``thin sandwich'' viewpoint coupled with the transformation properties of the lapse function.Comment: 7 pages, RevTe

Fixing Einstein's equations

Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique solutions that depend smoothly on the initial data. To remedy this failing, there has been widespread interest recently in reformulating Einstein's theory as a hyperbolic system of differential equations. The physical and geometrical content of the original theory remain unchanged, but dynamical evolution is made sound. Here we present a new hyperbolic formulation in terms of $g_{ij}$, $K_{ij}$, and \bGam_{kij} that is strikingly close to the space-plus-time (``3+1'') form of Einstein's original equations. Indeed, the familiarity of its constituents make the existence of this formulation all the more unexpected. This is the most economical first-order symmetrizable hyperbolic formulation presently known to us that has only physical characteristic speeds, either zero or the speed of light, for all (non-matter) variables. This system clarifies the relationships between Einstein's original equations and the Einstein-Ricci and Frittelli-Reula hyperbolic formulations of general relativity and establishes links to other hyperbolic formulations.Comment: 8 pages, revte