Tensors and Relativity: Chapter 3

Index ``raising'' and ``lowering''

Index ``raising'' and ``lowering''

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Next: General properties of tensors Up: More general tensors Previous: Tensors of type M/N

In the same way that the metric maps a vector tex2html_wrap_inline1774 into a one- form, it maps a M/N tensor into a M-1/N+1 tensor, i.e. it lowers an index. Similarly the inverse metric maps a M/N tensor into a M+1/N-1 tensor, i.e. it raises an index. So for example 

eqnarray1497

In Special Relativity, raising or lowering a 0 component changes the sign of the component; raising or lowering 1, 2, or 3 components has no effect.

We can operate the inverse metric on the metric to get the Kroneker delta [ Assignment 3 ]:

equation1499

So far we have confined our attention to Lorentz frames [ i.e. inertial frames ]. We can also allow more general coordinate transformations in a more general space i.e. tex2html_wrap_inline1908 . We then define

equation1503

Tensors will then transform as before, for example

equation1507

Old fashioned texts regard the above as the definition of a tensor. Raised indices are called contravariant  because they transform ``contrary'' to basis vectors:

equation1513

Lowered indices are called covariant :

equation1517

In particular one- forms are sometimes called covariant vectors , while ordinary vectors are called contravariant vectors .