Re: Is charge conserved between frames?

From: Tom Roberts (tjroberts_at_lucent.com)
Date: 10/31/04 

Date: Sun, 31 Oct 2004 23:59:22 GMT

J.J. Simplicio wrote:
> E_a is associated with a particular observer S and it is defined only
> along the world line of S. (It would have been nice if Wald had added
> some sort of notation on E_a to denote the particular observer chosen.)
> As Tom emphasized, it is not to be considered as a 4-vector field on the
> manifold. So, when Wald refers to E_a as 'the electric field measured by
> that observer' he must be using a pun (to borrow from Tom) on the word
> 'field'.
>
> At any point of the world line of the observer S we have this 4-vector
> E_a. In what sense is E_a the 'electric field measured by S'? As far as
> I can tell, the meaning is that in the coordinate frame of S (and only
> in this frame), the spatial components E_k are numerically equal to
> F_k0. The tensor components F_k0 are often referred to as the 'electric
> field' components of the tensor field F_ab. Thus, the use of the phrase
> 'electric field' in connection with E_a is not to denote that it is a
> field on the manifold but simply that it is associated with the
> 'electric field components' of F_ab in the frame of the observer S.
>
> Am I sort of on track?

Yes.

The original problem is that in ancient times (i.e. >100 years ago), the
electric field was a field on 3-space. In fact, mathematicians defined
"field" based on physicists' use of electric and magnetic fields on 3-space.

        Note the mathematicians also have a pun, and "field" has two
        distinct meanings; this is not the algebaric meaning (the
        field of real numbers...) but rather the geometric meaning
        (a field on a manifold...).

But the electric field is NOT a field on the spacetime manifold; the
actual field is the Maxwell 2-form.

Tom Roberts tjroberts@lucent.com

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