Differential geometry of a space with a two-point differential metric
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis we have generalized the Riemannian line element […] to the case where […] is a function of two points, x1, x2, and we consider the differential geometry of the line elem...
| Summary: | NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
In this thesis we have generalized the Riemannian line element […] to the case where […] is a function of two points, x1, x2, and we consider the differential geometry of the line element […].
The extremalizing of L = […] leads to a pair of curves […], called dyodesics, these curves being obvious generalizations of the geodesics of Riemannian geometry. A projective geometry of these paths is then investigated.
We then introduce a concept of parallel displacement of vectors relative to two paths […] which is directly analagous to parallel displacement in a Riemannian space. Parallel displacement is found to depend in a very natural way on six fundamental two-point tensors, the vanishing of these tensors implying that the space is flat, and for this case the dyodesics take the simple forms […] for special coordinate systems.
From the definition of parallel displacement arises a method for generating new two-point tensor fields by a process equivalent to covariant differentiation in Riemannian geometry. Parallel vector fields and ennuples of vectors are then introduced. It is shown that the ennuples […], […], form parallel vector fields for the metric space […]. We then define parallel displacement in sub-spaces and introduce a generalized covariant differentiation process, this last enabling us to develop second fundamental forms for hyper-surfaces. It is found that special and important types of coordinate systems may be set up independently at the points M1, and M2. These coordinates enable us to generate new tensors by a method of extension. An equivalence problem is then studied.
Finally, a line element […] is introduced for two masses at M1, M2, the […] satisfying […], the T's corresponding to the Ricci tensor of Riemannian geometry. The dyodesics obtained for this space approximate the Einstein solution for the one body problem when the mass of the particle at M, is small compared with that at M2. The motion for two equal masses differs from that obtained by Robertson in his solution of the equations of motion obtained by Einstein, Infeld, and Hoffman. The difference lies in the yet undetermined periastron effect for double stars.
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