Existence of Negative mass

Question

While deriving potential energy stored in space due to two stationary opposite charges we end up with negative value of energy which upon on dividing by '$c$ square' provides us negative value of mass. What is the significance of this mass other than reducing the total mass of system of charges. ($c$ stands for speed of light in vacuum).

Answer 1

Let's call the two particles a proton and electron for convenience.

Suppose we start with the proton and electron at rest at some distance $r$ apart. Then to separate the particles to infinity we have to do some work on them. This work is, of course, just (minus) the potential energy $U(r)$.

If we now measure the total mass of the separated particles we find it is just the mass of the proton plus the mass of the electron:

$$ M_\infty = m_p + m_e $$

So far so good, but hang on we had to put in an energy $U(r)$ to separate the two particles and that corresponds to a mass $m = U/c^2$. So if the final mass is $M_\infty$ that must mean that the initial mass was lower:

$$\begin{align} M_r &= M_\infty - U/c^2 \\ &= m_p + m_e - U/c^2 \end{align}$$

And this is quite correct. What you've discovered is that the mass of a bound state is always less that the total mass of its component parts. The difference is the binding energy.

But this doesn't mean there is a negative mass hiding somewhere. It just means that for composite systems mass is a more complicated concept than you thought. The mass of a composite system is not located at any specific point or points - it is a property of the system as a whole.

If you want to pursue this further there are aleady a lot of questions and answers dealing with it.