Some consequences of a covariant differential coordinate system

Download
Author
Comstock, Craig
Date
1961Advisor
Torrance, Charles C.
Metadata
Show full item recordAbstract
The assumption is made that the concept of a covariant coordinate system is meaningful and that such a system must satisfy the differential equations
a) dx(I) = g(ij)dx (j)
where the g(ij) are the components of the metric tensor in the corresponding contravarient system. Using this system it is shown that the derivatives of tensors by these new variables are just those quantities that would result from the raising of indices in the derivatives by the corresponding contravariant variables. The requirement that the equations a) be solvable for a set of covariant variables x(I), namely
b) (handwritten equations) produces a radical change in the nature of the Christoffel symbols, making them triply symmetric. The equations b) also permit a new physical interpretation of the symbols of the second kind as a direct dual of the symbols of the first kind, c) (handwritten equations) and d) (handwritten equations). This dualism is applied to the usual expression for the LaPlacean of a function and the resulting symmetry noted.. Several approaches are made to apply the above results to the Riemann-Christoffel tensor. The expression for this tensor is simplified in form by the use of c and d. It is noted that the Riemann-Christoffel tensor is now a first order set of nonlinear differential equation in the g(ij); however the solution appears to be just as difficult. In particular, the method of solution of the Einstein Field Equations by assuming a linear approximation for the g(ij) can no longer be applied with any degree of simplicity. The simplification of the form of the Riemann-Christoffel tensor does not alter the number of independent components of this tensor, nor does it change the nature of the various contractions of this tensor. These conclusions are reached by direct calculation of the twenty components and by the application of some elements of group representation theory following a technique used by D. E. Littlewood.
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.Collections
Related items
Showing items related by title, author, creator and subject.
-
Riemannian Geometry as a Field Over Another Geometry
Connor, George Henry, Jr. (Monterey, California. Naval Postgraduate School, 1968-09);The basic tensors of a Riemannian geometry are found in terms of tensor components by considering the geometry as a field over another arbitrary Riemannian geometry. The approach exhibits symmetries not previously noted. ... -
A Relativistic Mass Tensor with Geometric Interpretation
Rockower, Edward B. (American Association of Physics Teachers, 1987-01);We derive a relativistic mass tensor (dyadic or matrix) whose origin and properties have a direct geometric interpretation in terms of projection operators related to the particle's world line and local inertial frame in ... -
High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model
Giraldo. Francis X.; Restelli, M. (2009);We extend the explicit in time high-order triangular discontinuous Galerkin (DG) method to semi-implicit (SI) and then apply the algorithm to the two-dimensional oceanic shallow water equations; we implement high-order SI ...