77 research outputs found
Normal frames and the validity of the equivalence principle
We investigate the validity of the equivalence principle along paths in
gravitational theories based on derivations of the tensor algebra over a
differentiable manifold. We prove the existence of local bases, called normal,
in which the components of the derivations vanish along arbitrary paths. All
such bases are explicitly described. The holonomicity of the normal bases is
considered. The results obtained are applied to the important case of linear
connections and their relationship with the equivalence principle is described.
In particular, any gravitational theory based on tensor derivations which obeys
the equivalence principle along all paths, must be based on a linear
connection.Comment: 14 pages, LaTeX 2e, the package amsfonts is neede
Normal frames and the validity of the equivalence principle. I. Cases in a neighborhood and at a point
A treatment in a neighborhood and at a point of the equivalence principle on
the basis of derivations of the tensor algebra over a manifold is given.
Necessary and sufficient conditions are given for the existence of local bases,
called normal frames, in which the components of derivations vanish in a
neighborhood or at a point. These frames (bases), if any, are explicitly
described and the problem of their holonomicity is considered. In particular,
the obtained results concern symmetric as well as nonsymmetric linear
connections.Comment: LaTeX2e, 9 pages, to be published in Journal of Physics A:
Mathematical and Genera
Normal frames and the validity of the equivalence principle. III. The case along smooth maps with separable points of self-intersection
The equivalence principle is treated on a mathematically rigorous base on
sufficiently general subsets of a differentiable manifold. This is carried out
using the basis of derivations of the tensor algebra over that manifold.
Necessary and/or sufficient conditions of existence, uniqueness, and
holonomicity of these bases in which the components of the derivations of the
tensor algebra over it vanish on these subsets, are studied. The linear
connections are considered in this context. It is shown that the equivalence
principle is identically valid at any point, and along any path, in every
gravitational theory based on linear connections. On higher dimensional
submanifolds it may be valid only in certain exceptional cases.Comment: 15 standard LaTeX 2e (11pt, A4) pages. The package amsfonts is
require
Natural Diagonal Riemannian Almost Product and Para-Hermitian Cotangent Bundles
We obtain the natural diagonal almost product and locally product structures
on the total space of the cotangent bundle of a Riemannian manifold. We find
the Riemannian almost product (locally product) and the (almost) para-Hermitian
cotangent bundles of natural diagonal lift type. We prove the characterization
theorem for the natural diagonal (almost) para-K\"ahlerian structures on the
total spaces of the cotangent bundle.Comment: 10 pages, will appear in Czechoslovak Mathematical Journa
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