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Thread: raising and lowering indices

  1. #1 raising and lowering indices 
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    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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  3. #2  
    Forum Professor river_rat's Avatar
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    Hi bleplast, remember its the metric tensor that allows you to raise and lower indices and that is kind of by construction.


    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  4. #3  
    Moderator Moderator Markus Hanke's Avatar
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    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.
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  5. #4  
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    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; when written in component form

    Putting all together I finally have that which we may just as well write as the identity



    Phew! Why does it take me so long to explain things?
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  6. #5  
    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by Guitarist View Post
    Phew! Why does it take me so long to explain things?
    Because you're a mathematician
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