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Tensors and Relativity: Chapter 3

# The metric as a mapping of vectors onto one- forms

Next: More general tensors Up: One- forms Previous: Gradients

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

Thus

and

This can be summarized as follows: If

then

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

and

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   .