Some consequences of a covariant differential coordinate system
Torrance, Charles C.
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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.
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