We now introduce the idea that the metric acts as a mapping between vectors and one- forms. To see how this works, consider and a single vector . Since the metric requires two vectorial arguments, the expression still lacks one: when another one is supplied, it becomes a number. Therefore can be considered as a linear function of vectors producing real numbers: a one- form. We call it .
So we have
Then is a one- form whose value on a vector is :
What are the components of . They are
This can be summarized as follows: If
The components of are obtained from those of by changing the sign of the time component.
Since [ i.e. non zero ], there exists an inverse metric which we can write as [ In Special Relativity the components of are the same as , but this will not be true in the curved spacetime of General Relativity ].
The inverse metric defines a map from one- forms to vectors
In particular, we can map the gradient one- form into a vector gradient:
We can regard a vector as a 1/0 tensor i.e. a map from one- forms into the reals, so
The inverse metric can be used to define the magnitude of a one- form:
and the scalar product of two one- forms:
These are identical to the corresponding quantities for vectors, i.e.
Thus one- forms are timelike/spacelike/null if the corresponding vectors are.