1,232 research outputs found
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
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 , , 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
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
New Minimal Distortion Shift Gauge
Based on the recent understanding of the role of the densitized lapse
function in Einstein's equations and of the proper way to pose the thin
sandwich problem, a slight readjustment of the minimal distortion shift gauge
in the 3+1 approach to the dynamics of general relativity allows this shift
vector to serve as the vector potential for the longitudinal part of the
extrinsic curvature tensor in the new approach to the initial value problem,
thus extending the initial value decomposition of gravitational variables to
play a role in the evolution as well. The new shift vector globally minimizes
the changes in the conformal 3-metric with respect to the spacetime measure
rather than the spatial measure on the time coordinate hypersurfaces, as the
old shift vector did.Comment: 5 page ReVTeX4 twocolumn latex file, no figures; slight revision:
last sentence of section 2 deleted and replaced, citations reordered,
additonal paragraph added to introduction with short explanation of the
initial value problem and its thin sandwich variation, Yvonne Choquet-Bruhat
reference added and acknowledgment expanded to include he
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