The co-variant fundamental tensor—In the invariant expression of the square of the linear element
ds² = gμν dxμ dxν
dxμ plays the rôle of any arbitrarily chosen contravariant vector, since further gμν = gνμ, it follows from the considerations of the last paragraph that gμν 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 | gμν | the minors of gμν and divide them by the determinant g = | gμν | we get certain quantities gμν = gνμ, which as we shall prove generates a contravariant tensor.
According to the well-known law of Determinants
(16) gμσ gνσ = δμν
where δμν is 1, or 0, according as μ = ν or not. Instead of the above expression for ds², we can also write
gμσ δνσ dxμ dxν
or according to (16) also in the form