Tensors and Relativity: Chapter 3

Tensors of type 0/2

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The product of two one- forms, written as tex2html_wrap_inline1820 defines a linear map which takes two vectors into the reals:

equation1391

It is therefore a 0/2 tensor. tex2html_wrap_inline1824 denotes the   outer product. It is the formal notation to show how the 0/2 tensor is formed from two one- forms.

Note that this product is non- commutitative since tex2html_wrap_inline1828 gives a different result [ Assignment 3 ] i.e.

equation1405

The most general 0/2 tensor is a linear sum of such outer products. So

eqnarray1407

where tex2html_wrap_inline1832 are the components of the map f and we have used linearity.

If we take a basis  for f as tex2html_wrap_inline1838 [ 16 components ], then

equation1413

But

eqnarray1415

so we have

equation1417

Under a Lorentz transformation  , the components of f become:

equation1419

A 0/2 tensor is symmetric  if

equation1424

and anti- symmetric  if

equation1434

It follows that any 0/2 tensor can be uniquely decomposed into a symmetric  and anti- symmetric part .

equation1444

with the symmetric and anti- symmetric parts given by

equation1446

and

equation1448