Re: Is charge conserved between frames?

From: sal (pragmatist_at_nospam.org)
Date: 10/31/04 

Date: Sat, 30 Oct 2004 22:05:42 -0400

On Sat, 30 Oct 2004 04:21:04 +0000, Tom Roberts wrote:

> sal wrote:
>> So, please, Tom, admit you're human -- repeat after me: "I, Tom Roberts,
>> don't think it's a good idea to call it an electric field 4-vector"
>>
>> That's a far more defensible statement than, "It's _NOT_ an electric
>> field 4-vector"
>
> Well, while it is the components of a 4-vector, it is quite clearly not
> THE electric field, and is not even a field at all.

I can't resist :-) though I may get flamed for this...

Because, you see, it actually _is_ a field, or at any rate it can be
extended to one quite easily.

What is a 4-vector field, after all? It's a slice through the tangent
bundle, right? In other words, it's a mapping from the manifold to the
tangent space at each point:

   F:X -> T_X

If we have a particular frame of reference, S, with an associated
coordinate system which I'll also refer to as S, then at each point where
S is defined, we can find the so-called electric field 4-vector
associated with frame S at that point.

Now, you may object that S may not cover the manifold, in which case we
still haven't got a vector field. Call what we've got so far a "partial
field". We certainly can find a _set_ of coordinate systems that will
cover the manifold. Choose an appropriate collection that forms a locally
finite cover of the manifold, and which includes S. Find the electric
field 4-vector "partial field" for each of them, and then stitch them all
together with a partition of unity.

And that's a field, at least as I understand the term.

What it means is left as an exercise for the reader. ;-)

> But yes, it is most definitely not a good idea to call it that.
>
>
> Tom Roberts tjroberts@lucent.com 

-- 
I can be contacted through http://www.physicsinsights.org

Relevant Pages

  • Re: Is charge conserved between frames?
    ... > the manifold. ... > vector to the observer's worldline, and is defined only on that worldline. ... E_a is associated with a particular observer S and it is defined only along ... In what sense is E_a the 'electric field measured by S'? ...
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  • Re: Is charge conserved between frames?
    ... > the manifold. ... > vector to the observer's worldline, and is defined only on that worldline. ... E_a is associated with a particular observer S and it is defined only along ... In what sense is E_a the 'electric field measured by S'? ...
    (sci.physics.relativity)