The Einstein mass-energy relation

In 1905 Albert Einstein published his Special theory of relativity. One of the conclusions of this theory is that matter and energy are equivalent (see below). In fact the history of the Universe is a good example of this. In the early stages of development of the Universe there was a great deal more energy in the Universe and this has been converted into matter as time has passed.

Matter and energy are related by the famous equation proposed by Einstein:

The quantity E is the amount of energy produced when a mass m is totally converted to energy and c is the speed of light (3x10⁸ ms^-1).

For example if a mass of say 1 kg of any matter could be converted into energy a huge amount of energy could be produced. By Einstein's equation:

E = 1 x [3x10⁸]² = 9x10¹⁶ J

This amount of energy (equivalent to 2.5x10¹⁰ kWh) would heat each of the 10 000 houses in a medium sized town with a 1 kW fire for nearly 300 years! Just think of how much useful energy might be produced just by being able to convert household waste into energy.

Unfortunately no means has yet been devised for doing this on a large scale. However the conversion of mass to energy and energy to mass can be carried out at a nuclear level. Pair production (energy to matter) and annihilation (matter to energy) have both been observed. (See: Antimatter)

If we write m_o as the rest mass of the object then E = m_ogc² where g= (1 - v²/c²)^1/2

Relativity and the equivalence of mass and energy

The Einstein equation (E = mc²) can be deduced from special relativity as follows:

m = m_o/[1 - v²/c²]^1/2 therefore m²c² – m²v² – m_oc² = 0 where m_o is the rest mass of the object.

Differentiating with respect to time gives: c² 2m[dm/dt] – 2mv[d(mv)/dt] = 0

Rearranging this equation gives: c² 2m[dm/dt] = 2mv[d(mv)/dt]
and therefore c²[dm/dt] = v[d(mv)/dt]

However dE/dt = Fv and so dE/dt = Fv = v[d(mv)/dt] = c²dm/dt

and so

E = mc²