E = mc²
E² − (pc)² = (mc²)², residual zero — and a collision that CREATES mass.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
One particle, one speed, and the most famous equation ever written — shown here with the γ that the T-shirts leave off. Rest energy is the floor; motion builds on it; and the two together close a triangle every observer in the universe agrees about.
The triangle
E = γmc² and pc = γmβc² close a Pythagorean triangle with mc²: E² = (pc)² + (mc²)². Energy and momentum change from frame to frame, but that combination — the invariant — does not. The light clock had its triangle drawn in space; this one is drawn in energy–momentum, and it is the same postulate holding the pen.
The numbers
The numbers, and Newton’s alibi: at small β the relativistic kinetic energy hides inside ½mv², with a first correction of exactly ¾β² — a coefficient this page verifies against the site’s own classical arithmetic. Newton was not wrong; he was the low-speed chapter of a longer book.
The collision
Two identical lumps meet head-on and fuse. Momentum is zero before and after and energy is conserved, so the fused rest mass is 2γm: the incoming kinetic energy has become internal energy.
Audit
Audited: E²−(pc)²=(mc²)², the fused invariant mass is rebuilt from total energy and momentum, and the low-speed Taylor coefficient is checked independently.