raising and lowering indices

**bleplast** · August 18th, 2012, 11:57 AM

I am studying tensors, primarily to understand general relativity.

One thing I cannot understand about tensors is the ease with which indicies can be raised or lowered.

I thought I understood that indicies referred to either covariant or contravariant tensors. If so, then how can one switch between the two systems so easily without any consequences?

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**river_rat** · August 18th, 2012, 03:30 PM

Hi bleplast, remember its the metric tensor that allows you to raise and lower indices and that is kind of by construction.

**Markus Hanke** · August 19th, 2012, 02:44 AM

As river_rat has pointed out already, the covariant and contravariant elements are not completely independent, but related to each other via the metric tensor. Thus you can transform one into the other ( raisin/lowering indices ) like so :

and likewise

This of course only works if the entity E is actually a tensor.

**Guitarist** · August 20th, 2012, 10:21 AM

Yeah well, it looks like magic, doesn't it? It does if you write tensors in component form (I'll show what that is is a bit). But if it is does in all verbosity, we will see inside the magician's hat, and where the rabbit came from.

Suppose

a vector space and then define the space

as being the space of maps

So that, for all

and all

will have that

.

Now define the vector space

. One calls this the space of type (0,2) tensors.

Consider

, so that for all

then

.

If and only if we can think of

as an inner product then

1.

is to be called the metric tensor;

2. We may fix some

and form

.

Notice from the definition of the dual space that now

, so that it is a dual vector or covector or covariant vector or 1-form, your choice of terminology.

We can bring this into register with Markus's post by doing what's called "expanding vectors on orthonormal bases". So one may write

where the set

are basis vectors, and likewise

again the set

are basis vectors.

, and since we on orthonormal bases this is

, which becomes

since we are just taking sums of products (of real numbers)

The

are called "components" and it is usual (for a very good reason) to write tensors in this form

The metric tensor, for historical reasons only, follows a sightly different convention;