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 [h.sub.[alpha][beta]].

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.