These are tensors of type 0/1 and they map four- vectors into the reals. They are denoted by , so is a real number. One- forms form a vector space since
It is called the dual vector space to distinguish it from the space of four- vectors. The components of are . We can write
Generally we have
Note that the signs are positive [ c.f ]. We can think of vectors as columns and one- forms as rows:
is more fundamental than since the latter is defined only if there is a metric i.e. .
Now let us look at how one- forms transform:
Figure 3.1: is the number of surfaces the vector pierces.
So one- forms transform like basis vectors, not like vector components.
so is frame independent.
We can introduce a one- form basis , so that
so we must have
This gives the basis for one- forms. It is said to be dual to .
One can show that
so the basis one- forms transform like vector components [ as required notationally ] .
Both vectors and one- forms have four components but they have different geometrical interpretation. Vectors are like arrows but one- forms can be thought of as like three dimensional surfaces with the spacing between the surfaces defining the magnitude of [ see Figure 3.1 ].