Velocities [[??].sup.[mu]], [nabla][[omega].sup.[mu]] and projector [N.sub.[mu]v] transform like contravariant vectors and covariant tensor
, so the action is manifestly invariant under the general-coordinate transformations.
of dimension n, with the metric [h.sup.D] - 2[tau] + [gamma] - [theta][[rho].sup.D], where the function [theta] and covariant tensor
fields [tau], [[rho].sup.D], [h.sup.D], [gamma] on [summation], T*[summation] or V are identified with their pullbacks to (T*[summation] x V.
In view of definition (1), covariant tensor [A.sub.ij] of order-2 is bimetrically spherically symmetric if
Thus the bimetrically spherically symmetric covariant tensor [A.sub.ij] have the following non-vanishing components: [A.sub.11], [A.sub.14], [A.sub.41], [A.sub.44].
The bimetric plane symmetric covariant tensor
[K.sub.nm] can be expressed in the form of four functions, [I.sub.1] ...
In particular, we are interesting in the sub-module consisting of the covariant tensor
fields of degree 2.
Furthermore, we need an anti-symmetric, covariant tensor
which is non-degenerate.
This metric observable tensor, in real observations where the observer accompanies his references, is the same that the analogous built general covariant tensor
Besides, there is an obstacle related to definition of observable components of covariant tensors
(in which the indices are subscripts) and of mixed tensors, which have both subscripts and superscripts.