It is utilized to produce

anti-symmetric of last 4-bits (X (4 to 0)) when the MSB of X i.

where the anti-symmetric field tensor [PHI], given by

Finally, as a pure theory of gravitation, the results in the present work may be compared to those given in [8] and [9], wherein, based on the theory of chronometric invariants [7], a new geometric formulation of gravity (which is fully equivalent to the standard form of General Relativity) is presented in a way very similar to that of the electromagnetic field, based solely on a second-rank anti-symmetric field tensor.

ikl] are the components of the completely

anti-symmetric three-dimensional Levi-Civita permutation tensor density).

Then the covariant and contravariant components of the totally anti-symmetric permutation tensor are given by

which are actually anti-symmetric with respect to the first two indices iandj.

say W, with respect to the so-called Cartan basis as the totally anti-symmetric object

kl] = 0, the components of the anti-symmetric part of the stress tensor are then given by

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the covariant and contravariant components of the completely anti-symmetric Levi-Civita permutation tensor, respectively, with the ordinary permutation symbols being given as usual by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A spin frame is described by the anti-symmetric tensor product

x](M), we see that, astonishingly enough, the anti-symmetric product [A,B] is what defines the Lie (exterior) derivative of B with respect to A.

abcd] are the components of the completely anti-symmetric four-dimensional Levi-Civita permutation tensor and [psi] is a vector field normal to a three-dimensional space (hypersurface) [summation](t) defined as the time section ct = [x.