_d_τ′ = _d_
from this it follows, remembering the law of Jacobi,
(19)
$$ | \frac{\partial x'_{\sigma}}{dx_{\mu}} | = 1 $$
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
transformations 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.