Tensors and Relativity: Chapter 3

One- forms

next up previous index
Next: Gradients Up: Title page Previous: Metrics and forms

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

equation1196

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

equation1208

Generally we have

equation1211

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

eqnarray1219

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:

eqnarray1233

  figure260
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

eqnarray1243

we have

equation1249

so tex2html_wrap_inline1712 is frame independent.

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

equation1255

Then

equation1257

so we must have

equation1263

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

One can show that

equation1265

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 ].