Metrics and forms

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Recall that the scalar product of two four- vectors is :

so

where

are the components of the metric tensor .

We can think of the metric tensor as a map  which takes two four- vectors and into the reals: .

The map is linear in the arguments, in the sense that

We can generalize this by defining a tensor of type 0/N as a map which takes N four- vectors into the reals which is linear in all its arguments, for example the metric tensor  is a type 0/2 tensor.