Question

What does the `E` represent in Einstein's equation, `E=mc^2`?

Answer 1

The E represents the energy that released in the annihilation of a mass m, and vice-versa.

Explanation:

If you measure the mass of a lump of coal and then burn it. Is that mass annihilated? No . If you gather the ashes, all the smoke and gasses given off and determine the mass of all that, you will find that the mass you measured before and the total of the masses you measured after were equal.

In nuclear reactions, like those in the sun or in an atomic energy plant, mass is annihilated. If you could have measured the mass of the all the mass of the original atoms and all the atoms and perhaps some smaller particles that result, you would find that some mass has disappeared. Call that discrepancy `m_"missing"`.

The thing is, there would also be a large amount of energy among the products of that nuclear reaction. That energy is the `E` of the equation
`E=mc^2`.

Assume that you measured the amount of energy released and then converted that equation into
`m = E/c^2`.
Now plug the speed of light and your measurement of the energy released into the revised version of Einstein's formula you would obtain a value for m. Compared that value of m to the discrepancy `m_"missing"`, you would find them to be equal.o

So Einstein is telling us that mass and energy are equivalent and his formula is how to calculate the relationship. Note: only unusual situations allow mass to convert to energy, or energy to convert to mass.

I hope this helps,
Steve

Answer 2

Just adding a quick relativistic perspective to Steve's answer:

Where special relativity is considered, the quantity `"mc"^2` represents the rest energy `E_0`. The rest energy is in effect the relativistic total energy of a particle measured in a frame of reference in which the particle is at rest. Sometimes `"m"` in this equation is referred to as the rest mass `m_0` to distinguish it from the relativistic mass.

The equation `"E"_0="mc"^2` suggests that mass can be expressed in units of energy divided by the speed of light squared `"c"^2`, such as `"MeV"//"c"^2`. For example, a proton has a rest energy of `938 "MeV"` and thus a mass of `938 "MeV"//"c"^2`.

The relativistic total energy is given as `E=K+E_0`, which is the sum of the particle's kinetic energy and rest energy. We can see that the rest energy plays a roll here analogous to the potential energy we are used to seeing in calculations of energy.