| 摘要: | 研究期間：10108~10201;We propose to continue our investigations of recent years into dynamic spacetime geometry, mainly for gauge theories of gravity based on local spacetime symmetry (independent metric & connection, especially the Poincar´e Gauge Theory (PG), the metric-affine (MA) theory, and teleparallel theories (including the new f(T ) gravity) as well as Einstein’s GR). Our major interest is quasi-local gravitational energy. The main approach is via our covariant Hamiltonian formalism. The principal tools are differential geometry, the variational principle, the canonical Hamiltonian analysis, spinor techniques, symbolic and numerical computer calculations. We expect some more new results re propagating mode dynamics, energy-momentum (also angular momentum & center-of-mass) and its quasi-localization, energy positivity, and long range effects. We are interested in alternative gravity theories which can account for the accelerating universe and “dark energy” a/o account for the galactic scale observations without large amounts of “dark matter”. Our latest investigations include parity violating effects. Using symbolic computer algebra calculations we could seek new solutions for the PGT, MA and teleparallel theories. With numerical calculations we could dynamically evolve cosmological models or even the 4D covariant general equations, identifying the energy and radiation. We are also interested in electrodynamic radiation reaction. We plan to further develop our general covariant canonical analysis and our quasi-local energy-momentum expressions, examining criteria, comparing with other expressions, refining their application to black hole thermodynamics, probing more their relationship with pseudotensors, and especially the choice of reference and boundary values. We expect to look deeper into the Bondi limit, covariance, quasi-local angular momentum, center-of mass, and energy flux, as well as the scope and limitations of spinor formulations. We will continue developing our manifestly covariant canonical formalism and further refine our study of the under-appreciated small region limit. Applications of our positive energy test to alternative theories may be pursued. This essential requirement (roughly, notwithstanding dark energy, gravity is innately purely attractive) has already shown that a large class of alternate theories cannot account for the dynamics at the galactic scale and that most PGT parameter values are not viable. The constraint analysis of special PG and MA cases reveals their gauge structure. Hidden gauge symmetries can pose serious problems. We found that most PG models have the curious conditional constraint bifurcation phenomena, intimately connected to another (fatal) problem: tachyonic propagation. Both arise from non-linear constraints; they provide a strong alternate gravity test, imposing severe conditions on the PG & MA parameters. We found that the scalar dynamic torsion modes are ok; and in fact they could account for the accelerating universe. Now we have expanded our investigation to include parity violating terms. For promising theories we could seek a positive energy proof using techniques which worked for GR: spinor fields and our special orthonormal frame (SOF) gauge conditions (now we can show existence and uniqueness). New developments in spinor and geometric (Clifford) algebra enabled a geometric spacetime algebra formulation for dynamic geometry and a new gravitational gauge theory. We want to further study this theory and other applications of these techniques. We found an important class of spinor-curvature identities. One permitted a new positive energy proof, another provided a quadratic spinor Lagrangian for GR, thereby links between self dual gravity, Ashtekar variables, teleparallel theory, special orthonormal frames, positivity proofs and localizations. But these beautiful spinor formulations apparently have severe limitations re angular momentum and the center-of-mass. We are investigating some new ideas that might get around this. The utility and efficacy of these Clifford algebra/geometric calculus/spinor methods and our SOF gauge conditions will be explored. Our main concern is the proper formulation of quasi-local energy for gravitating systems. Recently we found, surprisingly, that many cosmological models have negative energy, in sharp contrast with our positivity expectations. This will be explored further. |
