Tensors and Relativity: Chapter 3

One- forms

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 tex2html_wrap_inline1710 , so tex2html_wrap_inline1712 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 tex2html_wrap_inline1710 are tex2html_wrap_inline1716 . We can write


Generally we have


Note that the signs are positive [ c.f tex2html_wrap_inline1718 ]. We can think of vectors as columns and one- forms as rows:


tex2html_wrap_inline1712 is more fundamental than tex2html_wrap_inline1690 since the latter is defined only if there is a metric i.e. tex2html_wrap_inline1724 .

Now let us look at how one- forms transform:


Figure 3.1: tex2html_wrap_inline1726 is the number of surfaces the vector tex2html_wrap_inline1692 pierces.

So one- forms transform like basis vectors, not like vector components.

Now since


we have


so tex2html_wrap_inline1712 is frame independent.

We can introduce a one- form basis  tex2html_wrap_inline1732 , so that




so we must have


This gives the basis for one- forms. It is said to be dual to tex2html_wrap_inline1734 .

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 tex2html_wrap_inline1710 [ see Figure 3.1 ].