arXiv subjects

J. D. Finley

Publications and source records attributed to J. D. Finley.

4 recordsLinked to original sources

arXiv1996

An Iterative Approach to Twisting and Diverging, Type N, Vacuum Einstein Equations: A (Third-Order) Resolution of Stephani's `Paradox'

In 1993, a proof was published, within ``Classical and Quantum Gravity,'' that there are no regular solutions to the {\it linearized} version of the twisting, type-N, vacuum solutions of the Einstein field equations. While this proof is certainly correct, we show that the conclusions drawn from that fact were unwarranted, namely that this irregularity caused such solutions not to be able to truly describe pure gravitational waves. In this article, we resolve the paradox---since such first-order solutions must always have singular lines in space for all sufficiently large values of $r$---by showing that if we perturbatively iterate the solution up to the third order in small quantities, there are acceptable regular solutions. That these solutions become flat before they become non-twisting tells us something interesting concerning the general behavior of solutions describing gravitational radiation from a bounded source.

gr-qc↗

arXiv1995

Estabrook-Wahlquist Prolongations and Infinite-Dimensional Algebras

Detailed mappings between zero-curvture equations for prolongation structures of nonlinear pde's and Estabrook-Wahlquist algorithms for same are given. The differences are exemplified by studies of the sine-Gordon equation. An example where the prolongation structure must be infinite-dimensional is given by the Robinson-Trautman equation, where the minimal algebra is $K_2$. In general these algorithms require integration of vector-field valued pde's; solutions of simultaneous flow equations are given. Applications to coupled systems of flow equations are given, where the result describes Lie algebras of vector fields vertical over fibers of pseudopotentials over a jet bundle appropriate for a given system of pde's; algebras invariant under sl(2,C) are of special interest.

solv-int↗

arXiv1995

Infinite-Dimensional Estabrook-Wahlquist Prolongations for the sine-Gordon Equation

We are looking for the universal covering algebra for all symmetries of a given pde, using the sine-Gordon equation as a typical example for a non-evolution equation. For non-evolution equations, Estabrook-Wahlquist prolongation structures for non-local symmetries depend on the choice of a specific sub-ideal, of the contact module, to define the pde. For each inequivalent such choice we determine the most general solution of the prolongation equations, as sub-algebras of the (infinite-dimensional) algebra of all vector fields over the space of non-local variables associated with the pde, in the style of Vinogradov covering spaces. We show explicitly how previously-known prolongation structures, known to lie within the Kac-Moody algebra, $A_1^{(1)}$, are special cases of these general solutions, although we are unable to identify the most general solutions with previously-studied algebras. We show the existence of gauge transformations between prolongation structures, viewed as determining connections over the solution space, and use these to relate (otherwise) distinct algebras. Faithful realizations of the universal algebra allow integral representations of the prolongation structure, opening up interesting connections with algebras of Toeplitz operators over Banach spaces, an area that has only begun to be explored.

solv-int↗

arXiv1995

The Robinson-Trautman Type III Prolongation Structure Contains K$_2$

The minimal prolongation structure for the Robinson-Trautman equations of Petrov type III is shown to always include the infinite-dimensional, contragredient algebra, K$_2$, which is of infinite growth. Knowledge of faithful representations of this algebra would allow the determination of Bäcklund transformations to evolve new solutions.

gr-qc↗