Re: currents or charges densities

Lets look at the expression j(x) = e | v \delta^4[x-y(\tau)] d\tau.
This one may be innocent but it looks odd to me for two reasons.
1. Where does it live? j(x) look to be an object on spacetime since that is
where x is. The \delta^4[x-y(\tau)] is a distributional function of x
and so lives on spacetime but d\tau is a 1-form on the line. How would you
unravel the functional analytic definition of this distribution? Go back to
first principles. There should be no prefered volume element (see below)

2. According to the modern view the spacetime metric only enters when the
constitutive equations relating E and D and H and B are invoked. Maxwells
equations make sense prior to that (see Foundations of Classical
Electrodynamics (Progress in Mathematical Physics) by F. W. Hehl, Yuri N.
Obukhov ).
Now without a metric we have no prefered volume element on spacetime so
expressions like
\int_M\delta(x-a)dV are out.

You wrote

""But I can give
you an expression that operationally makes sense as a current density
concentrated on a 2-surface in space-time. Suppose the surface S is
parametrized by s and t. Let vol(s,t) ds dt be the pull back of the area
form from the space-time to S. Let v(s,t) be the 4-velocity vector and e
the electric charge. Then I can define a current density as

j(x) = int_S vol(s,t) delta^4(x-S(s,t)) ev ds dt.""

You pullback the "area element" on spacetime? I suppose you mean the area on
S. What is the area element on S if there is no volume form on
M? Again, we shouldn't have to use any metric notions at this level. Another
thing: Current density is supposed to always be a 3-form! If you pull back a
3-form to a 2-surface you get 0. On the other hand, in DeRham's theory of
"currents" a 3-form can have support on a 2-surface so all is well.

Also, couldn't the notation be better? The left hand side, j(x), looks like
it lives on
M while the righthand side look like it lives in s,t parameter space since
it ends in dsdt. If j(x) is a current density how would I integrate it over
a 3-volume? Current density is still a 3-form even if it have singular
support. It really looks funny to me. Think about a Dirac delta function: It
has support on a point but is integrated over a volume. The dirac function
\delta(x-a) above should be integrated over a 4-volume. So what does \int_V
j(x)dV mean? But j should be a 3-form. What is the dV here?
Since x is in M, I assume that we must integrate over a 4-volume. But again
no volume element and besides j should be a 3-form.
There may be a way to make sense of you expression but as it stand I find it
confusing.
It seems to me that if one tries to make sense of dirac delta function type
(distributional) objects on a manifold with no metric then one ends up with
deRham currents.

"Igor Khavkine" <igor.kh@xxxxxxxxx> wrote in message
news:7a6be$42bcd4ad$cf700abf$6969@xxxxxxxxxxxx
> On 2005-06-24, cyberkatru <cyberkatru@xxxxxxxxx> wrote:
>> I am aware or the Swartz theory of distributions. It is iruotine but in a
>> geometric setting one has to be careful. Th analogue of distributions for
>> differential forms are the DeRham currents as explained in his book on
>> differentiable manifolds. Currents are dual to smooth forms of compact
>> support.
>> There are some subtle issues when one wants to concentrate onto a
>> submaifold
>> and do so in a geometrically meaningful way. In this setting I have seen
>> what I think are abuses of the dirac delta function idea that doesn't pan
>> out from the functional analytic point of view. This is what I meant by
>> willy nilly.
>> How would you define a 3 form with support on a 2-surface? This can't be
>> done as far as I know with ordinary dirac delta functions.
>
> Unfortunately, I don't know much about de Rham currents. But I can give
> you an expression that operationally makes sense as a current density
> concentrated on a 2-surface in space-time. Suppose the surface S is
> parametrized by s and t. Let vol(s,t) ds dt be the pull back of the area
> form from the space-time to S. Let v(s,t) be the 4-velocity vector and e
> the electric charge. Then I can define a current density as
>
> j(x) = int_S vol(s,t) delta^4(x-S(s,t)) ev ds dt.
>
> This integral is S-reparametrization invariant. It gives you a
> distributional vector field vector field supported only on the surface
> S.
>
> The expresson for j(x) is really a current density. To make sense of it,
> you must integrate over a 3-volume, which will give you the amount of
> charge that it contains (in the case of a space-like volume) or the
> amount that has passed through a 2-dimensional spacial slice (if the
> 3-volume extends in a time-like direction).
>
> Q = int_V n.j(x) dx,
>
> where n is the unit normal vector to V. To perform this integral, the
> order of integration over V and S must be switched. Again, looks like
> this integral is also V-reparametrization invariant. Would you say that
> what I've done is "willy nilly"? If so, why?
>
> Igor
>

.

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