J.E.R. O'Connor and G.E. Prince School of Mathematics
Kimura investigated static spherically symmetric metrics and found
several to have quadratic first integrals.
We use REDUCE and the package Dimsym to seek collineations for these
metrics.
For one metric we find that three proper projective collineations
exist, two of which are
associated with the two irreducible quadratic first integrals found by Kimura.
The third
projective collineation is found to have a reducible quadratic first integral.
We also
find that this metric admits two conformal motions and that the
resulting
reducible conformal Killing tensors also lead to Kimura's quadratic
integrals. We demonstrate that when a Killing
tensor is known for a metric we can seek an associated collineation by solving
first order
equations that give the Killing tensor in terms of the collineation rather than
the second
order determining equations for collineations. We report less
interesting results for other Kimura metrics.
Key words and phrases: General relativity, geodesic
equations, Killing tensors, Kimura metrics, collineations, symmetry.