Graduate Thesis Or Dissertation
 

Parametric manifolds

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  • A standard tool in general relativity is the 3+1 or ADM point of view, namely slicing spacetime into spacelike hypersurfaces of constant time and then describing physics in terms of time-dependent quantities on a typical such hypersurface. Much less well-known is the 1+3 point of view, in which one foliates spacetime with timelike curves, then describes physics in terms of the surfaces "locally orthogonal" to the given foliation. This is precisely the description of physics as seen by a single observer. However, in many instances there do not exist such orthogonal hypersurfaces. One may instead attempt to describe physics on the manifold of orbits defined by the timelike curves, but one must then develop a parametric theory to handle the time dependent objects defined on the manifold of orbits. I will present two equivalent descriptions of parametric manifolds. The first is based on a generalized Gauss-Codazzi formalism which involves projection to a lower-dimensional "surface". The second is an intrinsic description which involves redefining the action of vector fields on functions. In either description one is lead to generalized notions of connections, Lie bracket, and exterior differentiation. Unique to a parametric theory of geometry is the deficiency. Although independent of the torsion, the deficiency behaves like torsion in the parametric direction. We will show how the deficiency emerges as a result of the above generalizations. The 3+1 formalism arises naturally in considering initial-value formulations both for fields on a fixed background spacetime and for the spacetime itself. The applicability of parametric manifolds to such problems will be discussed.
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