General Relativity: A Geometric Approach

Geometric Analysis and General Relativity

Geometric analysis can be said to originate in the 19’th century work of Weierstrass, Riemann, Schwarz and others on minimal surfaces, a problem whose history can be traced at least as far back as the work of Meusnier and Lagrange in the 18’th century. The experiments performed by Plateau in the mid-19’th century, on soap films spanning wire contours, served as an important inspiration for this...

Geometric Bounds in Spherically Symmetric General Relativity

We exploit an arbitrary extrinsic time foliation of spacetime to solve the constraints in spherically symmetric general relativity. Among such foliations there is a one parameter family, linear and homogeneous in the extrinsic curvature, which permit the momentum constraint to be solved exactly. This family includes, as special cases, the extrinsic time gauges that have been exploited in the pa...

A Geometric Theory of Zero Area Singularities in General Relativity

The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such “zero area singularities” in Riemannian manifolds, generalizing far beyond the Schwarzschild case (for example, allowing the singularit...

A Spinor Approach to General Relativity

A calculus for general relativity is developed in which the basic role of tensors is taken over by spinors. The Riemann-Christoffel tensor is written in a spinor form according to a scheme of Witten. It is shown that the curvature of empty space can be uniquely characterized by a totally symmetric four-index spinor which satisfies a first order equation formally identical with one for a zero re...

Reflections on the Application of the Geometric Approach to Quantum Mechanics to General Relativity