J. M. Nester (聶斯特) J. M. Nester (聶斯特)
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 {\it 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.
References