dx_{μ} plays the rôle of any arbitarily chosen contravariant vector, since further g_{μν} = g_{νμ}, it follows from the considerations of the last paragraph that g_{μν} is a symmetrical co-variant tensor of the second rank. We call it the "fundamental tensor." Afterwards we shall deduce some properties of this tensor, which will also be true for any tensor of the second rank. But the special rôle of the fundamental tensor in our Theory, which has its physical basis on the particularly exceptional character of gravitation makes it clear that those relations are to be developed which will be required only in the case of the fundamental tensor. The co-variant fundamental tensor.
If we form from the determinant scheme | g_{μν} | the
minors of g_{μν} and divide them by the determinat g = | g_{μν} |
we get certain quantities g^{μν} = g^{νμ}, which as we shall
prove generates a contravariant tensor.
According to the well-known law of Determinants
(16) g_{μσ} g^{νσ} = δ_{μ}^{ν}
where δ_{μ}^{ν} is 1, or 0, according as μ = ν or not. Instead of the above expression for ds^2, we can also write
g_{μσ} δ_{ν}^{σ} dx_{μ} dx_{ν}
or according to (16) also in the form
g_{μσ} g_{ντ} g^{στ} dx_{μ dx_{ν}
