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Relativity and gravitation
Gauge theories of space-time symmetries, their variational principles,
energy and canonical Hamiltonian formulation. The ultimate goal: unified
field theory.
Of particular interest are the Poincare gauge theories (PGT) and the
Metric Affine (MA) theory which are generalizations of and promising
alternatives to Einstein's theory of gravity. For this class of theories
mass and spin together make curvature, torsion and, in the MA theory,
non-metricity. We look for exact solutions, investigate theoretical
properties and seek observable effects. Can torsion produce long range
detectable effects? Could galaxy dynamics anomalies be explained by an
alternate gravity theory? [1]
We have developed a covariant Hamiltonian formulation for general
gravitational theories. The first dividend was new expressions [4] for
the conserved quantities: total energy, momentum and angular momentum.
With the aid of the symplectic ideas of Tulczyjew and Kijowski, we have
now found good covariant Hamiltonian boundary expressions for the
corresponding quasi-local quantities [3]. We have already used
them to find a general formulation for black hole thermodynamics and are
presently exploring the connection with pseudotensors.
Dirac's constraint algorithm is being used to obtain the canonical
Hamiltonian formulation of the PGT and MA theory. This reveals all of
the constraints and any hidden gauge transformations. For many values of
the parameters we have recently found that these theories also suffer
from the ``constraint bifurcation'' phenomenon which we had previously
discovered in the teleparallel theory. We are now investigating the
relationship of this curious phenomenon with the problem of tachyonic
propagation modes and the need for a well posed initial value problem
[5,6].
The fundamental theoretical requirement of positive total energy
is being used as an effective test of alternate theories of gravity.
Positive total energy simply means that gravity is universally purely
attractive. Any theory which permits a repulsive solution, i.e.,
``antigravity'', can be rejected. Hence we try to find such ``bad''
solutions in a proposed alternate theory. Many otherwise viable theories
are failing this test.
New rotational gauge conditions have been found; the conditions select
certain preferred special orthonormal frames. These
frames are related to solutions of Dirac's equation and are promising
variables for a good description of gravity; they permit a new proof of
positive energy for Einstein's theory and a physically sensible
``quasi-localization'' of gravitational energy. They also mesh well with
the New Variables of Ashtekar, producing the most succinct known
positive energy proof/localization [7]. Further applications of these
special orthonormal frames are being sought.
Our investigations have been greatly enhanced by means of symbolic
computer calculations, currently we are using REDUCE with EXCALC and
Zhytnikov's GRG; MAPLE, MATHEMATICA and MACSYMA may also be used. We are
now planning to do numerical calculations for the full non-linear
general dynamical Einstein equations on a supercomputer.
We have found some new spinor-curvature identities; special cases are
used in the Witten positive energy proof and in our new 3-spinor
positive energy proof. Certain special cases have led to a promising new
class of quadratic spinor Lagrangians [2] for Einstein's
theory. We are looking into further applications of these identities as
well as other spinor and Clifford algebra techniques. In particular, the
Geometric Algebra formalism of Hestenes and the spacetime gauge theory
of Doran, Lasenby and Gull are being investigated.
We are also investigating some new ideas of Kijowski concerning
electromagnetic radiation reaction. Some of this work is being done in
collaboration with Prof. D.C. Chern.
Some recent research works are available at
the arXiv.org e-print database.
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