Relativistic Mass and Potential Energy

Question

In the chapter 16–4 of the Feynman Lectures, Feynman employs a thought experiment to explain why "if two particles come together and produce potential or any other form of energy; if the pieces are slowed down by climbing hills, doing work against internal forces, or whatever; then it is still true that the mass is the total energy that has been put in."

It is clear in the case of kinetic energy and heat. After all it is expressed directly in the formula for mass:

\begin{equation} \label{Eq:I:16:10} m_u=\frac{m_0}{\sqrt{1-u^2/c^2}}. \end{equation}

If two bodies come together in an inelastic collision, then we can see how the particles that make up the new body, increase its mass due to their increased kinetic energy (heat).

But suppose the two masses are two electrons and as they approach each other they slow down to a halt due to repulsion. If we imagine them at the moment in time in which their velocity is 0, their combined mass is still more than twice the rest mass of an electron. But where is it?

Is the Special Relativity formula for mass incomplete? is the extra mass hiding in $m_0$? Does it hide somewhere else?

Answer 1

Energy is stored in electric and magnetic fields. There's a little bit stored at each point in space. It's proportional to the square of the electric field at that point. The total energy stored that way is just the integral (i.e. sum) of that the energy at each point.

As you move charges around, you change the field values. If you push two like charges together (i.e. electrons), the field between them goes up; that corresponds to an increase of energy in the field.

Summary: The energy you put into pushing them together was pushing against their repulsion, and went into increased energy of their electric field.