With this choice of co-ordinates, only substitutions with determinant 1, are allowable.
It would however be erroneous to think that this step signifies a partial renunciation of the general relativity postulate. We do not seek those laws of nature which are co-variants with regard to the tranformations having the determinant 1, but we ask: what are the general co-variant laws of nature? First we get the law, and then we simplify its expression by a special choice of the system of reference.
Building up of new tensors with the help of the fundamental tensor.
Through inner, outer and mixed multiplications of a
tensor with the fundamental tensor, tensors of other
kinds and of other ranks can be formed.
Example:—
A^μ = g^{μσ} A_{σ}
A = g_{μν} A^{μν}
We would point out specially the following combinations:
A^{μν} = g^{μα} g^{νβ} A_{αβ}
A_{μν} = g_{μα} g_{νβ} A^{αβ}
(complement to the, co-variant or contravariant tensors)
and, B_{μν} = g_{μν} g^{αβ} A_{αβ}
We can call B_{μν} the reduced tensor related to A_{μν}.
