L. C. Shepley

Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant theories

We study spacetime diffeomorphisms in the Hamiltonian and Lagrangian formalisms of generally covariant systems. We show that the gauge group for such a system is characterized by having generators which are projectable under the Legendre map. The gauge group is found to be much larger than the original group of spacetime diffeomorphisms, since its generators must depend on the lapse function and shift vector of the spacetime metric in a given coordinate patch. Our results are generalizations of earlier results by Salisbury and Sundermeyer. They arise in a natural way from using the requirement of equivalence between Lagrangian and Hamiltonian formulations of the system, and they are new in that the symmetries are realized on the full set of phase space variables. The generators are displayed explicitly and are applied to the relativistic string and to general relativity. @S0556-2821~97!06202-4#

Gauge transformation in Einstein-Yang-Mills theories

We discuss the relation between space–time diffeomorphisms and gauge transformations in theories of the Yang–Mills type coupled with Einstein’s general relativity. We show that local symmetries of the Hamiltonian and Lagrangian formalisms of these generally covariant gauge systems are equivalent when gauge transformations are required to induce transformations which are projectable under the Legendre map. Although pure Yang–Mills gauge transformations are projectable by themselves, diffeomorphisms are not. Instead, the projectable symmetry group arises from infinitesimal diffeomorphism-inducing transformations which must depend on the lapse function and shift vector of the space–time metric plus associated gauge transformations. Our results are generalizations of earlier results by ourselves and by Salisbury and Sundermeyer.

Gauge group and reality conditions in Ashtekar’s complex formulation of canonical gravity

We discuss reality conditions and the relation between spacetime diffeomorphisms and gauge transformations in Ashtekar’s complex formulation of general relativity. We produce a general theoretical framework for the stabilization algorithm for the reality conditions, which is different from Dirac’s method of stabilization of constraints. We solve the problem of the projectability of the diffeomorphism transformations from configuration-velocity space to phase space, linking them to the reality conditions. We construct the complete set of canonical generators of the gauge group in the phase space which includes all the gauge variables. This result proves that the canonical formalism has all the gauge structure of the Lagrangian theory, including the time diffeomorphisms.

Reduced phase space: quotienting procedure for gauge theories

We present a reduction procedure for gauge theories based on quotienting out the kernel of the presymplectic form in configuration-velocity space. Local expressions for a basis of this kernel are obtained using phase-space procedures; the obstructions to the formulation of the dynamics in the reduced phase space are identified and circumvented. We show that this reduction procedure is equivalent to the standard Dirac method as long as the Dirac conjecture holds: that the Dirac Hamiltonian, containing the primary first-class constraints, with their Lagrange multipliers, can be enlarged to an extended Dirac Hamiltonian which includes all first-class constraints without any change of the dynamics. The quotienting procedure is always equivalent to the extended Dirac theory, even when it differs from the standard Dirac theory. The differences occur when there are ineffective constraints, and in these situations we conclude that the standard Dirac method is preferable - at least for classical theories. An example is given to illustrate these features, as well as the possibility of having phase-space formulations with an odd number of physical degrees of freedom.

Gauge symmetries in Ashtekar's formulation of general relativity

Abstract It might seem that a choice of a time coordinate in Hamiltonian formulations of general relativity breaks the full four-dimensional diffeomorphism covariance of the theory. This is not the case. We construct explicitly the complete set of gauge generators for Ashtekars formulation of canonical gravity. The requirement of projectability of the Legendre map from configuration-velocity space to phase space renders the symmetry group a gauge transformation group on configuration-velocity variables. Yet there is a sense in which the full four-dimensional diffeomorphism group survives. Symmetry generators serve as Hamiltonians on members of equivalence classes of solutions of Einsteins equations and are thus intimately related to the so-called “problem of time” in an eventual quantum theory of gravity.

Stability of two-dimensional quasisingular space-times

We study the stability of a class of two-dimensional cylindrical space-times with quasiregular singularities using massless scalar waves. The fact that the stress-energy scalarTμvTμv diverges indicates the instability of the singularity toward formation of a scalar curvature singularity. In special cases a nonscalar curvature singularity results.

Quasi-regular singularities based on null planes

We examine quasi-regular singularities that take the form of invariant two-dimensional null planes in Minkowski space-time, thus extending earlier studies of “conical” singularities based on timelike and spacelike planes. The result is described in terms of a deficit parameter. We also examine the form of the Riemann curvature tensor at the singularity.