Mass–Energy Equivalence

The Paper

In September 1905, three months after the relativity paper, Einstein sent the Annalen der Physik a short follow-up with a question for a title: "Does the Inertia of a Body Depend Upon Its Energy Content?" His answer was yes.

The famous formula does not appear in it. Einstein stated the result in words: if a body gives off an amount of energy $L$ as radiation, its mass decreases by $L/c^2$. In modern symbols:

Read the other way, a body at rest with mass $m$ holds an energy

For anything in motion, the complete relation includes momentum $p$:

What It Means

Mass and energy are not two separate things that happen to be exchangeable. Mass is a measure of energy content. Because $c^2$ is an enormous number, a tiny mass corresponds to a vast energy: one gram is equivalent to about 90 trillion joules, roughly the energy released by 21,000 tons of TNT.

This is true everywhere, not only in nuclear reactions. A hot cup of coffee weighs more than a cold one, and a charged battery more than a flat one. The differences are far too small to measure.

How It Was Confirmed

In 1932 John Cockcroft and Ernest Walton split lithium nuclei with accelerated protons and found that the energy released matched the mass that disappeared. Every nuclear reaction measured since has agreed. A direct test published in 2005, a century after the paper, confirmed the relation to better than one part in a million.

What People Get Wrong

• It is not "the formula for the atomic bomb." It explains where the energy of fission comes from. It says nothing about how to start a chain reaction, which required discoveries made in the late 1930s by others.
• Einstein did not work on the bomb. See Nuclear Energy for what he did and did not do.
• The old idea that a moving object's "mass increases" is a dated way of speaking. Physicists today treat mass as fixed and let energy and momentum carry the dependence on speed.

Where It Stands

The equation is the daily accounting rule of nuclear and particle physics. Stars shine by turning mass into energy; the Sun loses about four million tons every second. In a PET scanner the same conversion happens in miniature, as an electron and a positron vanish into two photons.