New Approaches to Variational Principles and Gauge Theories in General Relativity.
Abstract
We develop new variational techniques, acting on classes of Lagrangians with the same functional dependence but arbitrary functional form, for the derivation of general, strongly conserved quantities, supplementing the usual procedure for deriving weak conservation laws via Noether's theorem. Using these new techniques we generate and generalize virtually all energymomentum complexes currently known. In the process we discover and understand the reason for the difficulties associated with energymomentum complexes in general relativity. We study a Palatini variation of a novel Lagrangian due to Nissani. We find that Nissani's principal claim, that his Lagrangian specifies Riemannian geometry in the presence of a generalized matter tensor, is not in fact justifiable, and prove that his Lagrangian is not unique. We speculate on the possibility of deriving a generalrelativistic analog of Maxwell's current equation, a matter current equation, yielding an entirely new approach to the idea of energymomentum in general relativity. We develop the SL(2,C) times U(1) spinor formalism naturally combining the gravitational and electromagnetic potentials in a single objectthe spinor connection. Variably charged matter is rigourously introduced, through the use of spin densities, in the unified potential theories we develop. We generate both the EinsteinMaxwell equations and new equations. The latter generalize both the Maxwell equation and the Einstein equation which includes a new "gravitational stressenergy tensor". This new tensor exactly mimicks the electromagnetic stressenergy tensor with Riemann tensor contractions replacing Maxwell tensor contractions. We briefly consider the introduction of matter. A Lagrangian generalizing the two spinor Dirac equations has no gravitational currents and the electromagnetic currents must be on the light cone. A Lagrangian generalizing the Pauli equations has both gravitational and electromagnetic currents. The equations of both Lagrangians demonstrate beautifully how the divergence of the total stressenergy tensor vanishes in this formalism. In the theory of the generalized EinsteinMaxwell and Pauli equations we succeed in deriving an equation describing a generalized matter charge current density.
 Publication:

Ph.D. Thesis
 Pub Date:
 1990
 Bibcode:
 1990PhDT.......251C
 Keywords:

 Physics: General