E = mc²

One of the most celebrated consequences of special relativity is that mass itself is a form of energy. Einstein showed that the total energy of a free particle of mass $m$ is not just its kinetic energy, but includes a term depending on mass alone:

$\ev{E_\text{tot} = \gamma\,m\,c^2}$

The most interesting case is the particle at rest. For $v = 0$ we have $\gamma = 1$, and what remains is the rest energy:

$\ev{E_0 = m\,c^2}$

It is the most famous formula in physics. It says something surprising: even a small mass hides an enormous amount of energy, because the conversion factor $c^2$ is gigantic. A mass of just $1\;\text{kg}$ contains an energy of $9\cdot 10^{16}\;\text{J}$, equivalent to millions of tonnes of TNT. Mass and energy are not two separate quantities linked by a formula: they are the same thing, measured in different units.

From Hiroshima to nuclear power plants

The formula $E = mc^2$ received its most dramatic (and tragic) confirmation in 1945 with the atomic bombs of Hiroshima and Nagasaki: less than a gram of uranium converted into energy razes a city to the ground. The same formula, however, also explains the Sun (nuclear fusion: every second $4\cdot 10^{9}\;\text{kg}$ of hydrogen are converted into energy), nuclear power plants (fission) and medical PET scans (electron-positron annihilation). Always the same principle: a pinch of mass that disappears, a river of energy that appears.

▶ The Real Meaning of E=mc² — PBS Space Time

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Topics: Special relativity Concepts: Mass-energy equivalence

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