Some Spinor-Curvature Identities

We describe a class of spinor-curvature identities which exist for Riemannian or Riemann-Cartan geometries. Each identity relates an expression quadratic in the covariant derivative of a spinor field with an expression linear in the curvature plus an exact differential. Certain special cases in 3 and 4 dimensions which have been or could be used in applications to General Relativity are noted.Comment: 5 pages Plain TeX, NCU-GR-93-SSC

A Quadratic Spinor Lagrangian for General Relativity

We present a new finite action for Einstein gravity in which the Lagrangian is quadratic in the covariant derivative of a spinor field. Via a new spinor-curvature identity, it is related to the standard Einstein-Hilbert Lagrangian by a total differential term. The corresponding Hamiltonian, like the one associated with the Witten positive energy proof is fully four-covariant. It defines quasi-local energy-momentum and can be reduced to the one in our recent positive energy proof. (Fourth Prize, 1994 Gravity Research Foundation Essay.)Comment: 5 pages (Plain TeX), NCU-GR-94-QSL

Another positivity proof and gravitational energy localizations

Two locally positive expressions for the gravitational Hamiltonian, one using 4-spinors the other special orthonormal frames, are reviewed. A new quadratic 3-spinor-curvature identity is used to obtain another positive expression for the Hamiltonian and thereby a localization of gravitational energy and positive energy proof. These new results provide a link between the other two methods. Localization and prospects for quasi-localization are discussed.Comment: 14 pages REVTe

Ashtekar's New Variables and Positive Energy

We discuss earlier unsuccessful attempts to formulate a positive gravitational energy proof in terms of the New Variables of Ashtekar. We also point out the difficulties of a Witten spinor type proof. We then use the special orthonormal frame gauge conditions to obtain a locally positive expression for the New Variables Hamiltonian and thereby a ``localization'' of gravitational energy as well as a positive energy proof.Comment: 12 pages Plain Te

Lagrange formulation of the symmetric teleparallel gravity

We develop a symmetric teleparallel gravity model in a space-time with only the non-metricity is nonzero, in terms of a Lagrangian quadratic in the non-metricity tensor. We present a detailed discussion of the variations that may be used for any gravitational formulation. We seek Schwarzschild-type solutions because of its observational significance and obtain a class of solutions that includes Schwarzschild-type, Schwarzschild-de Sitter-type and Reissner-Nordstr\"{o}m-type solutions for certain values of the parameters. We also discuss the physical relevance of these solutions.Comment: Corrected typos, Accepted for publication in IJMP-

Quasi-local Energy for Spherically Symmetric Spacetimes

We present two complementary approaches for determining the reference for the covariant Hamiltonian boundary term quasi-local energy and test them on spherically symmetric spacetimes. On the one hand, we isometrically match the 2-surface and extremize the energy. This can be done in two ways, which we call programs I (without constraint) and II (with additional constraints). On the other hand, we match the orthonormal 4-frames of the dynamic and the reference spacetimes. Then, if we further specify the observer by requiring the reference displacement to be the timelike Killing vector of the reference, the result is the same as program I, and the energy can be positive, zero, or even negative. If, instead, we require that the Lie derivatives of the two-area along the displacement vector in both the dynamic and reference spacetimes to be the same, the result is the same as program II, and it satisfies the usual criteria: the energies are non-negative and vanish only for Minkowski (or anti-de Sitter) spacetime.Comment: 16 pages, no figure

Ashtekar Variables in Classical General Realtivity

This paper contains an introduction into Ashtekar's reformulation of General Relativity in terms of connection variables. To appear in "Canonical Gravity - From Classical to Quantum", ed. by J. Ehlers and H. Friedrich, Springer Verlag (1994).Comment: 31 Pages, Plain-Tex; Further comments were added, minor grammatical changes made and typos correcte