12,596 research outputs found

### 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

### The topology of deformation spaces of Kleinian groups

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible
boundary and let AH(\pi_1(M)) denote the space of (conjugacy classes of)
discrete faithful representations of \pi_1(M) into PSL 2 (C). The components of
the interior MP(\pi_1(M)) of AH(\pi_1(M)) (as a subset of the appropriate
representation variety) are enumerated by the space A(M) of marked
homeomorphism types of oriented, compact, irreducible 3-manifolds homotopy
equivalent to M. In this paper, we give a topological enumeration of the
components of the closure of MP(\pi_1(M)) and hence a conjectural topological
enumeration of the components of AH(\pi_1(M)). We do so by characterizing
exactly which changes of marked homeomorphism type can occur in the algebraic
limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use
this enumeration to exhibit manifolds M for which AH(\pi_1(M)) has infinitely
many components.Comment: 49 pages, published versio

### Uniformly exponential growth and mapping class groups of surfaces

We show that the mapping class group of an orientable finite type surface has
uniformly exponential growth, as well as various closely related groups. This
provides further evidence that mapping class groups may be linear.Comment: 6 pages, no figure

### Einstein-Bianchi Hyperbolic System for General Relativity

By employing the Bianchi identities for the Riemann tensor in conjunction
with the Einstein equations, we construct a first order symmetric hyperbolic
system for the evolution part of the Cauchy problem of general relativity. In
this system, the metric evolves at zero speed with respect to observers at rest
in a foliation of spacetime by spacelike hypersurfaces while the curvature and
connection propagate at the speed of light. The system has no unphysical
characteristics, and matter sources can be included.Comment: 25 pp., Latex, to appear in Topol. Methods in Nonlinear Analysis,
typos corrected and further citations adde

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