14,183 research outputs found
Proof of a generalized Geroch conjecture for the hyperbolic Ernst equation
We enunciate and prove here a generalization of Geroch's famous conjecture
concerning analytic solutions of the elliptic Ernst equation. Our
generalization is stated for solutions of the hyperbolic Ernst equation that
are not necessarily analytic, although it can be formulated also for solutions
of the elliptic Ernst equation that are nowhere axis-accessible.Comment: 75 pages (plus optional table of contents). Sign errors in elliptic
case equations (1A.13), (1A.15) and (1A.25) are corrected. Not relevant to
proof contained in pape
Solving the characteristic initial value problem for colliding plane gravitational and electromagnetic waves
A method is presented for solving the characteristic initial value problem
for the collision and subsequent nonlinear interaction of plane gravitational
or gravitational and electromagnetic waves in a Minkowski background. This
method generalizes the monodromy transform approach to fields with nonanalytic
behaviour on the characteristics inherent to waves with distinct wave fronts.
The crux of the method is in a reformulation of the main nonlinear symmetry
reduced field equations as linear integral equations whose solutions are
determined by generalized (``dynamical'') monodromy data which evolve from data
specified on the initial characteristics (the wavefronts).Comment: 4 pages, RevTe
Collision of plane gravitational and electromagnetic waves in a Minkowski background: solution of the characteristic initial value problem
We consider the collisions of plane gravitational and electromagnetic waves
with distinct wavefronts and of arbitrary polarizations in a Minkowski
background. We first present a new, completely geometric formulation of the
characteristic initial value problem for solutions in the wave interaction
region for which initial data are those associated with the approaching waves.
We present also a general approach to the solution of this problem which
enables us in principle to construct solutions in terms of the specified
initial data. This is achieved by re-formulating the nonlinear dynamical
equations for waves in terms of an associated linear problem on the spectral
plane. A system of linear integral ``evolution'' equations which solve this
spectral problem for specified initial data is constructed. It is then
demonstrated explicitly how various colliding plane wave space-times can be
constructed from given characteristic initial data.Comment: 33 pages, 3 figures, LaTeX. Accepted for publication in Classical and
Quantum Gravit
Observables for spacetimes with two Killing field symmetries
The Einstein equations for spacetimes with two commuting spacelike Killing
field symmetries are studied from a Hamiltonian point of view. The complexified
Ashtekar canonical variables are used, and the symmetry reduction is performed
directly in the Hamiltonian theory. The reduced system corresponds to the field
equations of the SL(2,R) chiral model with additional constraints.
On the classical phase space, a method of obtaining an infinite number of
constants of the motion, or observables, is given. The procedure involves
writing the Hamiltonian evolution equations as a single `zero curvature'
equation, and then employing techniques used in the study of two dimensional
integrable models. Two infinite sets of observables are obtained explicitly as
functionals of the phase space variables. One set carries sl(2,R) Lie algebra
indices and forms an infinite dimensional Poisson algebra, while the other is
formed from traces of SL(2,R) holonomies that commute with one another. The
restriction of the (complex) observables to the Euclidean and Lorentzian
sectors is discussed.
It is also shown that the sl(2,R) observables can be associated with a
solution generating technique which is linked to that given by Geroch.Comment: 23 pages (LateX-RevTeX), Alberta-Thy-55-9
Monodromy-data parameterization of spaces of local solutions of integrable reductions of Einstein's field equations
For the fields depending on two of the four space-time coordinates only, the
spaces of local solutions of various integrable reductions of Einstein's field
equations are shown to be the subspaces of the spaces of local solutions of the
``null-curvature'' equations constricted by a requirement of a universal (i.e.
solution independent) structures of the canonical Jordan forms of the unknown
matrix variables. These spaces of solutions of the ``null-curvature'' equations
can be parametrized by a finite sets of free functional parameters -- arbitrary
holomorphic (in some local domains) functions of the spectral parameter which
can be interpreted as the monodromy data on the spectral plane of the
fundamental solutions of associated linear systems. Direct and inverse problems
of such mapping (``monodromy transform''), i.e. the problem of finding of the
monodromy data for any local solution of the ``null-curvature'' equations with
given canonical forms, as well as the existence and uniqueness of such solution
for arbitrarily chosen monodromy data are shown to be solvable unambiguously.
The linear singular integral equations solving the inverse problems and the
explicit forms of the monodromy data corresponding to the spaces of solutions
of the symmetry reduced Einstein's field equations are derived.Comment: LaTeX, 33 pages, 1 figure. Typos, language and reference correction
No New Symmetries of the Vacuum Einstein Equations
In this note we examine some recently proposed solutions of the linearized
vacuum Einstein equations. We show that such solutions are {\it not} symmetries
of the Einstein equations, because of a crucial integrability condition.Comment: 9 pages, Te
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