Date of Award
Doctor of Philosophy (PhD)
Dr. Edwin C. Thrig
This thesis investigates the problem of motion for extended bodies from the viewpoint of classical field theory, where the classical field is the body's energy-momentum or matter tensor.
In Special Relativity a symmetric and divergence-free matter tensor combined with inertial frames is used to generate a kinematics for extended bodies; but I have shown that if the matter tensor also obeys the weak energy condition, then both types of massless spin radiation must have an infinite spatial extent in all Lorentz frames. This does not agree with the observation that finite light beams put a torque on crystals as they change their polarization while traversing the crystal.
In General Relativity I have suggested a kinematics analogous to that accepted in Special Relativity and applied it to the simplest non-trivial example of static, spherical stars. In essence one looks for special sets of vector fields whose matter currents are conserved. Such a set of ten vector fields defines a special frame and integrals of the conserved matter currents define ten momenta which give the kinematics a simple application of de Rham cohomology theory shows that the conserved matter currents for isolated bodies will have mechanical potentials which enable the momenta to be found from flux integrals evaluated in the vacuum region surrounding the body. These potentials contain the full Riemann curvature allowing a body's General Relativistic momenta to be determined by its vacuum gravitational field.
This approach has several important differences with previous attempts at a General Relativistic kinematics. By working directly with the matter tensor employed in Einstein's equations, it seems unnecessary to invent energy pseudo-tensors or other secondary objects to define momenta. By integrating matter currents which vanish in the vacuum, the momenta receive no contribution from vacuum regions. In this way one avoids the problem of motion without matter, which arises if the vacuum is endowed with momenta. By integrating divergence-free matter currents, one obtains conserved momenta for isolated bodies. The existence of divergence-free vector fields is a very weak condition, quite unlike the existence of metric symmetries, so that conserved momenta can be obtained in the absence of metric symmetries, as is explicitly done for six of the ten momenta for static spherical stars.
Although an example of this kinematics has been given for static spherical stars, much remains to be done. I have shown that there are many conserved matter currents for arbitrary bodies, but it is not yet known how to use Einstein's equations to single out the physically interesting ones for arbitrary space-times. Related to this problem is the question of the final form of the mechanical potential. Work in this area will shed some light on the special frames. Are they directly analogous to the inertial frames of flat space, but determined by the matter distribution? What is the group of transformations which links the special frames in arbitrary space-times? If it is not the Poincaré group, then would the group provide a richer physical structure?
Woodside, Robert William MacLaren, "Kinematics in Special and General Relativity" (1979). Open Access Dissertations and Theses. Paper 639.