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Tensors and Relativity: Chapter 3

# One- forms

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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.

Now since

we have

so is frame independent.

We can introduce a one- form basis  , so that

Then

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 ].