Date sön 14 januari 2018

What is electromagnetic interaction, what is the nature of space, can we formulate ourself in a vivid mathematical language and hint on generalizartions for further developments. The following is a small observation but i find it interesting.

Consider a space of buckets \(V_t(r)\). We will consider channels in all directions e.g. for each point \(r\) and direction \(e\) we have a channel \(j(e,r)\). Then we will assume that the flow out of position \(r\) in direction \(e\) is

$$ A_t(r,e) = V_t(r) j(e,r) $$

Then in order to preserve the total quantity we have the continuity equation

$$ V_{t+dt}(r) = \int V_t(r + e dx) j(e,r + e dx)dS(e) $$

Now we will assume that \(j\) is linear, e.g. \(j=j_t^i(r)e_i\) we will get

$$ V_t(r)+\frac{\partial}{\partial t} V_t(r) = V_t(r) + \int (\nabla_r V_t(r))\cdot e \sum_i j^ie_i + V_t(r)(\sum_{i,k} \frac{\partial}{\partial_j}j^i_t(r)e_je_i) dS(e) $$

now \(\int e_i e_j dS(e) = c \delta_{i,j}\) e.g. we get

$$ \frac{\partial}{\partial t} V_t(r) = c \nabla_r \cdot(V_t j_t) $$

This is the classical transport equation where the potential \(V_t\) is transported by the field \(c j\).

Assume for all solution that are plane waves with speed c that we have a constant \(j\) so that the the flow in direction of the movement is constant constrained by the continuity equation.

Then \(V\) and \(A_i=V j^i_t\) represents the plane wave solution of maxwells equations with the Lorenz gauge.

Now an important point with these plane wave solutions is that we can view each channel flowing with the solution e.g. that channels flow as well as the things in the bucket and in ray like fashion. So by associating each quantitiy inside the bucket with a channell we still have all the plane wave solutions of the source free maxwells equations. This also means that any linear combination of the solutions, that comprises all solutions of Maxwells equations, but also that we can continue to use the association of rays with channels with these solution and maintain that interpretation.

This discussion ends with a question; We used a linear expression for \(j_t(r,e)\) what about other kinds of distributions.

Cheers


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