24 research outputs found
A Factorization Algorithm for G-Algebras and Applications
It has been recently discovered by Bell, Heinle and Levandovskyy that a large
class of algebras, including the ubiquitous -algebras, are finite
factorization domains (FFD for short).
Utilizing this result, we contribute an algorithm to find all distinct
factorizations of a given element , where is
any -algebra, with minor assumptions on the underlying field.
Moreover, the property of being an FFD, in combination with the factorization
algorithm, enables us to propose an analogous description of the factorized
Gr\"obner basis algorithm for -algebras. This algorithm is useful for
various applications, e.g. in analysis of solution spaces of systems of linear
partial functional equations with polynomial coefficients, coming from
. Additionally, it is possible to include inequality constraints
for ideals in the input
Geometric Interpretation of the Mixed Invariants of the Riemann Spinor
Mixed invariants are used to classify the Riemann spinor in the case of
Einstein-Maxwell fields and perfect fluids. In the Einstein-Maxwell case these
mixed invariants provide information as to the relative orientation of the
gravitational and electromagnetic principal null directions. Consideration of
the perfect fluid case leads to some results about the behaviour of the
Bel-Robinson tensor regarded as a quartic form on unit timelike vectors.Comment: 31 pages, AMS-LaTe
Obtaining a class of Type O pure radiation metrics with a cosmological constant, using invariant operators
Using the generalised invariant formalism we derive a class of conformally
flat spacetimes whose Ricci tensor has a pure radiation and a Ricci scalar
component. The method used is a development of the methods used earlier for
pure radiation spacetimes of Petrov types O and N respectively. In this paper
we demonstrate how to handle, in the generalised invariant formalism,
spacetimes with isotropy freedom and rich Killing vector structure. Once the
spacetimes have been constructed, it is straightforward to deduce their
Karlhede classification: the Karlhede algorithm terminates at the fourth
derivative order, and the spacetimes all have one degree of null isotropy and
three, four or five Killing vectors.Comment: 29 page
Type O pure radiation metrics with a cosmological constant
In this paper we complete the integration of the conformally flat pure
radiation spacetimes with a non-zero cosmological constant , and , by considering the case . This is a
further demonstration of the power and suitability of the generalised invariant
formalism (GIF) for spacetimes where only one null direction is picked out by
the Riemann tensor. For these spacetimes, the GIF picks out a second null
direction, (from the second derivative of the Riemann tensor) and once this
spinor has been identified the calculations are transferred to the simpler GHP
formalism, where the tetrad and metric are determined. The whole class of
conformally flat pure radiation spacetimes with a non-zero cosmological
constant (those found in this paper, together with those found earlier for the
case ) have a rich variety of subclasses with zero,
one, two, three, four or five Killing vectors
Complete solution of Hadamard\u27s problem for the scalar wave equation on Petrov type III space-times
Consequences of the validity of Huygens\u27 principle for the conformally invariant scalar wave equation, Weyl\u27s neutrino equation and Maxwell\u27s equations on Petrov type II space-times
GHP: a maple package for performing calculations in the Geroch-Held-Penrose formalism
We present a new symbolic algebra package, written for Maple, for performing computations in the Geroch-Held-Penrose formalism. We demonstrate the essential features and capabilities of our package by investigating Petrov-D vacuum solutions of Einstein\u27s field equations.<br /