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Tensors and Relativity: Chapter 3

The metric as a mapping of vectors onto one- forms

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We now introduce the idea that the metric acts as a mapping between vectors and one- forms. To see how this works, consider tex2html_wrap_inline1772 and a single vector tex2html_wrap_inline1774 . Since the metric tex2html_wrap_inline1772 requires two vectorial arguments, the expression tex2html_wrap_inline1778 still lacks one: when another one is supplied, it becomes a number. Therefore tex2html_wrap_inline1778 can be considered as a linear function of vectors producing real numbers: a one- form. We call it tex2html_wrap_inline1782 .

So we have

equation1318

Then tex2html_wrap_inline1774 is a one- form whose value on a vector tex2html_wrap_inline1692 is tex2html_wrap_inline1788 :

equation1332

What are the components of tex2html_wrap_inline1790 . They are

eqnarray1344

Thus

equation1350

and

equation1352

This can be summarized as follows: If

equation1354

then

equation1359

The components of tex2html_wrap_inline1790 are obtained from those of tex2html_wrap_inline1774 by changing the sign of the time component.

Since tex2html_wrap_inline1796 [ i.e. non zero ], there exists an inverse metric which we can write as tex2html_wrap_inline1798 [ In Special Relativity the components of tex2html_wrap_inline1800 are the same as tex2html_wrap_inline1798 , but this will not be true in the curved spacetime of General Relativity ].

The inverse metric  defines a map from one- forms to vectors

equation1364

In particular, we can map the gradient one- form tex2html_wrap_inline1762 into a vector gradient :

equation1366

We can regard a vector as a 1/0 tensor i.e. a map from one- forms into the reals, so

equation1371

and

equation1377

The inverse metric tex2html_wrap_inline1798 can be used to define the magnitude of a one- form :

equation1381

and the scalar product of two one- forms :

equation1383

These are identical to the corresponding quantities for vectors, i.e.

equation1385

Thus one- forms are timelike/spacelike/null if the corresponding vectors are   .