Utah State University
Physics Colloquium
The Howard L. Blood
Scholarship
RecipientsÕ Research, Section
II
Creating a Database of Einstein Exact Solution in Maple Sam Barkat
I will review the historical development of ideas which
led to Einstein's theory of gravity. In Einstein's theory, gravitational fields
are determined by matter and energy distributions according to an
extremely complicated system of non-linear partial differential equations known
as the Einstein equations. Despite their complexity, a large number
of solutions are known to these equations, corresponding to a wide variety of
physical situations. I will report on on-going research to use computer
algebra methods to analyze these solutions and develop a comprehensive database
of them.
A
Wave Equation in Biconformal Space Juan T. Trujillo
Just as the symmetry group of Newtonian mechanics, which is the
group of translations and rotations, and the symmetry group of relativity,
which comprises the PoincarŽ group (the group of Lorentz transformations and
translations), the conformal group seems to be an accurate and more complete
symmetry group of the universe (comprising not only the subgroups already
mentioned, but also the group of special conformal transformations and the
group of dilatations). Noting that
Dirac was able to write the Dirac equation, notably by taking the square root
of the Klein-Gordon equation, we attempted, first of all, to find a wave equation
based on the conformal group by forming a Casimir operator—an operator
that commutes with every generator (infinitesimal operator) of the conformal
group—by choosing a
representation of the generators that included not only generalized
coordinates, but also generalized momenta. In effect, by ÒsquaringÓ these generators, we were able to
arrive at one such equation that treated the generalized coordinates and the
generalized momenta on an equal basis.
Our next approach, was to form the generalized Klein-Gordon equation by
noticing that biconformal space—the conformal group gauged by the Lorentz
transformations—has an induced metric formed by the dual forms of the
generators of translations and special conformal transformations as given by
the Maurer-Cartan equations. From
this particular choice of the induced metric, we were able to form an
eight-dimensional Laplacian using differential form operators. Since both of these operators, or
equations, only had second order terms, we attempted to find the Òsquare rootÓ
of them by trying to find two linear operators such that when applied twice,
yielded the original operator. In
doing so, we had to attempt to satisfy simultaneously six equations. We were able to come up with a solution
for an orthonormal induced metric.
SER BLDG, Room 244
SER BLDG, Room 244