E = mc²

Mass is a form of energy. A body at rest contains energy equal to its mass multiplied by the square of the speed of light:

$E = mc^2$

Since c ≈ 3×10⁸ m/s, c² ≈ 9×10¹⁶ m²/s² — an enormous number. Even a small mass contains a staggering amount of energy. One kilogram of matter, fully converted to energy, releases about 9×10¹⁶ joules — equivalent to roughly 21 megatons of TNT.

The equation is the most famous result of Einstein's special theory of relativity (1905), but its significance is often misrepresented. It does not say "matter can be converted to energy" in the sense of a one-way transaction. It says mass and energy are the same thing. They can flow between different forms. Chemical reactions release energy by converting a tiny fraction of mass. Nuclear fission and fusion release energy by converting a larger fraction. In principle, matter-antimatter annihilation converts all the mass to energy.

What It Changed

Before 1905, conservation of mass and conservation of energy were separate laws. You couldn't create or destroy mass; you couldn't create or destroy energy. Einstein showed they are one law: conservation of mass-energy. The separate conservation was an approximation valid when the energies involved are small compared to mc². In chemistry, the mass deficit from reactions is so tiny it was undetectable before relativistic theory; in nuclear reactions, it is large enough to matter enormously.

Kinetic Energy at High Speed

For a moving object, the full relativistic energy is:

$E = \gamma mc^2, \quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$

At low speeds (v ≪ c), this reduces to the familiar Newtonian kinetic energy plus the rest-mass energy: E ≈ mc² + ½mv². The "extra" rest-mass energy mc² was always there; Newtonian mechanics just never needed to account for it because it never changes in non-nuclear processes.

At speeds approaching c, γ → ∞: it would take infinite energy to accelerate a massive object to the speed of light. This is why the speed of light is an absolute barrier for massive objects.

Connections

• conservation-of-energy — E = mc² unifies mass and energy conservation into one law
• stellar-nucleosynthesis — stars shine because nuclear fusion converts mass to energy
• four-fundamental-forces — mass-energy equivalence connects to gravitation (energy has mass)
• general-relativity-gravity — light bends near massive objects because it has energy, and energy has mass

Sources

• six-easy-pieces-feynman