4,545 research outputs found

### Monodromy transform and the integral equation method for solving the string gravity and supergravity equations in four and higher dimensions

The monodromy transform and corresponding integral equation method described
here give rise to a general systematic approach for solving integrable
reductions of field equations for gravity coupled bosonic dynamics in string
gravity and supergravity in four and higher dimensions. For different types of
fields in space-times of $D\ge 4$ dimensions with $d=D-2$ commuting isometries
-- stationary fields with spatial symmetries, interacting waves or partially
inhomogeneous cosmological models, the string gravity equations govern the
dynamics of interacting gravitational, dilaton, antisymmetric tensor and any
number $n\ge 0$ of Abelian vector gauge fields (all depending only on two
coordinates). The equivalent spectral problem constructed earlier allows to
parameterize the infinite-dimensional space of local solutions of these
equations by two pairs of \cal{arbitrary} coordinate-independent holomorphic
$d\times d$- and $d\times n$- matrix functions ${\mathbf{u}_\pm(w),
\mathbf{v}_\pm(w)}$ of a spectral parameter $w$ which constitute a complete set
of monodromy data for normalized fundamental solution of this spectral problem.
The "direct" and "inverse" problems of such monodromy transform --- calculating
the monodromy data for any local solution and constructing the field
configurations for any chosen monodromy data always admit unique solutions. We
construct the linear singular integral equations which solve the inverse
problem. For any \emph{rational} and \emph{analytically matched} (i.e.
$\mathbf{u}_+(w)\equiv\mathbf{u}_-(w)$ and
$\mathbf{v}_+(w)\equiv\mathbf{v}_-(w)$) monodromy data the solution for string
gravity equations can be found explicitly. Simple reductions of the space of
monodromy data leads to the similar constructions for solving of other
integrable symmetry reduced gravity models, e.g. 5D minimal supergravity or
vacuum gravity in $D\ge 4$ dimensions.Comment: RevTex 7 pages, 1 figur

### Infinite hierarchies of exact solutions of the Einstein and Einstein-Maxwell equations for interacting waves and inhomogeneous cosmologies

For space-times with two spacelike isometries, we present infinite
hierarchies of exact solutions of the Einstein and Einstein--Maxwell equations
as represented by their Ernst potentials. This hierarchy contains three
arbitrary rational functions of an auxiliary complex parameter. They are
constructed using the so called `monodromy transform' approach and our new
method for the solution of the linear singular integral equation form of the
reduced Einstein equations. The solutions presented, which describe
inhomogeneous cosmological models or gravitational and electromagnetic waves
and their interactions, include a number of important known solutions as
particular cases.Comment: 7 pages, minor correction and reduction to conform with published
versio

### 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

### Equilibrium configurations of two charged masses in General Relativity

An asymptotically flat static solution of Einstein-Maxwell equations which
describes the field of two non-extreme Reissner - Nordstr\"om sources in
equilibrium is presented. It is expressed in terms of physical parameters of
the sources (their masses, charges and separating distance). Very simple
analytical forms were found for the solution as well as for the equilibrium
condition which guarantees the absence of any struts on the symmetry axis. This
condition shows that the equilibrium is not possible for two black holes or for
two naked singularities. However, in the case when one of the sources is a
black hole and another one is a naked singularity, the equilibrium is possible
at some distance separating the sources. It is interesting that for
appropriately chosen parameters even a Schwarzschild black hole together with a
naked singularity can be "suspended" freely in the superposition of their
fields.Comment: 4 pages; accepted for publication in Phys. Rev.

### 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

### Integrability of generalized (matrix) Ernst equations in string theory

The integrability structures of the matrix generalizations of the Ernst
equation for Hermitian or complex symmetric $d\times d$-matrix Ernst potentials
are elucidated. These equations arise in the string theory as the equations of
motion for a truncated bosonic parts of the low-energy effective action
respectively for a dilaton and $d\times d$ - matrix of moduli fields or for a
string gravity model with a scalar (dilaton) field, U(1) gauge vector field and
an antisymmetric 3-form field, all depending on two space-time coordinates
only. We construct the corresponding spectral problems based on the
overdetermined $2d\times 2d$-linear systems with a spectral parameter and the
universal (i.e. solution independent) structures of the canonical Jordan forms
of their matrix coefficients. The additionally imposed conditions of existence
for each of these systems of two matrix integrals with appropriate symmetries
provide a specific (coset) structures of the related matrix variables. An
equivalence of these spectral problems to the original field equations is
proved and some approach for construction of multiparametric families of their
solutions is envisaged.Comment: 15 pages, no figures, LaTeX; based on the talk given at the Workshop
``Nonlinear Physics: Theory and Experiment. III'', 24 June - 3 July 2004,
Gallipoli (Lecce), Italy. Minor typos, language and references corrections.
To be published in the proceedings in Theor. Math. Phy

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