A proof
gulsen said:
Why is charge invariant under Lorentz transformation. Is there a fundamental and theoretical answer to this question?
Yes, both charge and its Lorentz invariance can in fact already be
derived from the classical wave equation:
<br />
\mbox{Classical Wave equation:} \qquad \frac{\partial^2 <br />
\psi}{\partial t^2}\ =\ v^2 \frac{\partial^2 \psi}{\partial x^2} \qquad \qquad (1)<br />
The derivatives in time and space are proportional by a constant
which stems from the characteristic speed of the medium. This simply
means that the equation is satisfied by any arbitrary function which
shifts along with a speed v (or -v). We can expand the equation to
three dimensions, for instance for the electric potential field V:
<br />
\mbox{Electric Potential:} \qquad \frac{\partial^2 V}{\partial t^2}\ <br />
=\ c^2 \frac{\partial^2 V}{\partial x^2} + c^2 \frac{\partial^2 <br />
V}{\partial y^2} + c^2 \frac{\partial^2 V}{\partial z^2}\qquad \qquad (2) <br />
Where c is the speed of light. The same expression holds for the three
components of the magnetic vector potential. Again these equations
are satisfied by any arbitrary function which shifts along with the
characteristic speed c: The electro magnetic waves. In our world however we also see things which are stationary or move at
other speeds than the speed of light. If we go to three (or more) space
dimensions then such solutions become possible. A stable solution which
shifts along with an arbitrary speed v in the x direction will satisfy both
(1) with a speed of v and (2). We can use this to eliminate the time
dependency by substitution:
<br />
\left(1-\frac{v^2}{c^2}\right)c^2 \frac{\partial^2 V}{\partial x^2}\ +\ c^2 \frac{\partial^2 <br />
V}{\partial y^2}\ +\ c^2 \frac{\partial^2 V}{\partial z^2}\ =\ 0 <br />This shows that the solutions are Lorentz contracted in the direction
of v by a factor \gamma, The first order derivatives are higher by a factor
\gamma and the second order by a factor \gamma^2. Velocities higher then c are
not possible. The solution for v=0 is:
<br />
\frac{\partial^2 V}{\partial x^2}\ +\ \frac{\partial^2 V}{\partial <br />
y^2}\ +\ \frac{\partial^2 V}{\partial z^2}\ =\ 0, \qquad <br />
\Rightarrow \qquad V\ =\ \frac{1}{r}<br />
Which is the electro static potential. The equation is satisfied at
all points except for r=0 where we have a singularity. This
singularity is now associated with the classical (point)charge.
Without it there would be no solutions at sub-luminal speeds.
The charge is defined by what we measure, the fields. Since charge
is conserved (does not change in time) and the fields are real
scalars it is sufficient to use the Lorentz contraction as a prove
for Lorentz invariance.
The total solution is an arbitrary superposition of 1/r functions.
This includes the Quantum Mechanical fields where charge is spread
out over the wavefunction. Regards, Hans.
