23 research outputs found
N-dimensional geometries and Einstein equations from systems of PDE's
The aim of the present work is twofold: first, we show how all the
-dimensional Riemannian and Lorentzian metrics can be constructed from a
certain class of systems of second-order PDE's which are in duality to the
Hamilton-Jacobi equation and second we impose the Einstein equations to these
PDE's
Tensorial Spin-s Harmonics
We show how to define and go from the spin-s spherical harmonics to the
tensorial spin-s harmonics. These quantities, which are functions on the sphere
taking values as Euclidean tensors, turn out to be extremely useful for many
calculations in General Relativity. In the calculations, products of these
functions, with their needed decompositions which are given here, often arise
naturally
Twisting Null Geodesic Congruences, Scri, H-Space and Spin-Angular Momentum
The purpose of this work is to return, with a new observation and rather
unconventional point of view, to the study of asymptotically flat solutions of
Einstein equations. The essential observation is that from a given
asymptotically flat space-time with a given Bondi shear, one can find (by
integrating a partial differential equation) a class of asymptotically
shear-free (but, in general, twistiing) null geodesic congruences. The class is
uniquely given up to the arbitrary choice of a complex analytic world-line in a
four-parameter complex space. Surprisingly this parameter space turns out to be
the H-space that is associated with the real physical space-time under
consideration. The main development in this work is the demonstration of how
this complex world-line can be made both unique and also given a physical
meaning. More specifically by forcing or requiring a certain term in the
asymptotic Weyl tensor to vanish, the world-line is uniquely determined and
becomes (by several arguments) identified as the `complex center-of-mass'.
Roughly, its imaginary part becomes identified with the intrinsic spin-angular
momentum while the real part yields the orbital angular momentum.Comment: 26 pages, authors were relisted alphabeticall
Electrodynamic Radiation Reaction and General Relativity
We argue that the well-known problem of the instabilities associated with the
self-forces (radiation reaction forces) in classical electrodynamics are
possibly stabilized by the introduction of gravitational forces via general
relativity
Noether symmetries in the phase space
The constants of motion of a mechanical system with a finite number of degrees of freedom are related to the variational symmetries of a Lagrangian constructed from the Hamiltonian of the original system. The configuration space for this Lagrangian is the phase space of the original system. The symmetries considered in this manner include transformations of the time and may not be canonical in the standard sense.The authors acknowledge the support from CONACyT, SNI and VIEP-BUAP
Twisting Null Geodesic Congruences and the Einstein-Maxwell Equations
The purpose of the present work is to extend the earlier results for
asymptotically flat vacuum space-times to asymptotically flat solutions of the
Einstein-Maxwell equations. Once again, in this case, we get a class of
asymptotically shear-free null geodesic congruences depending on a complex
world-line in the same four-dimensional complex space. However in this case
there will be, in general, two distinct but uniquely chosen world-lines. One of
which can be assigned as the complex center-of- charge while the other could be
called the complex center of mass. Rather than investigating the situation
where there are two distinct complex world-lines, we study instead the special
degenerate case where the two world-lines coincide, i.e., where there is a
single unique world-line. This mimics the case of algebraically special
Einstein-Maxwell fields where the degenerate principle null vector of the Weyl
tensor coincides with a Maxwell principle null vector. Again we obtain
equations of motion for this world-line - but explicitly found here only in an
approximation. Though there are ambiguities in assigning physical meaning to
different terms it appears as if reliance on the Kerr and charged Kerr metrics
and classical electromagnetic radiation theory helps considerably in this
identification. In addition, the resulting equations of motion appear to have
many of the properties of a particle with intrinsic spin and an intrinsic
magnetic dipole moment. At first order there is even the classical
radiation-reaction term 2/3{q^{2}}{c^{-3}}ddot{v}, now obtained without any use
of the Lorentz force law but obtained directly from the asymptotic fields
themselves. One even sees the possible suppression, via the Bondi mass loss, of
the classical runaway solutions due to the radiation reaction force
The Universal Cut Function and Type II Metrics
In analogy with classical electromagnetic theory, where one determines the
total charge and both electric and magnetic multipole moments of a source from
certain surface integrals of the asymptotic (or far) fields, it has been known
for many years - from the work of Hermann Bondi - that energy and momentum of
gravitational sources could be determined by similar integrals of the
asymptotic Weyl tensor. Recently we observed that there were certain overlooked
structures, {defined at future null infinity,} that allowed one to determine
(or define) further properties of both electromagnetic and gravitating sources.
These structures, families of {complex} `slices' or `cuts' of Penrose's null
infinity, are referred to as Universal Cut Functions, (UCF). In particular, one
can define from these structures a (complex) center of mass (and center of
charge) and its equations of motion - with rather surprising consequences. It
appears as if these asymptotic structures contain in their imaginary part, a
well defined total spin-angular momentum of the source. We apply these ideas to
the type II algebraically special metrics, both twisting and twist-free.Comment: 32 page