E = mc²
Mass is energy
A paperclip holds more energy than the bomb that destroyed Hiroshima. Choose something, convert it, and see where in nature mass really does become energy.
This is the object’s full rest energy, E = mc². No real process can release all of it from ordinary matter. Fuel, bombs and stars free only a small fraction (see below). Letting the object annihilate with the same mass of antimatter would release all of it, plus the antimatter’s equal share: twice the number shown.
What the equation really says
E = mc² says that anything with mass has a rest energy: energy it holds just by existing, even when it’s sitting still. Mass isn’t a separate thing from energy. It is energy, measured in kilograms.
E = m c² · c² ≈ 8.99 × 10¹⁶ J per kg
The conversion factor is the speed of light squared, an enormous number. That’s why one gram is worth 89.9 TJ, about 1.4 times the energy of the Hiroshima bomb (about 15 kilotons of TNT, or 63 TJ). For a moving object the full relation is E² = (mc²)² + (pc)², where p is momentum. E = mc² is the special case at rest.
Everything weighs its energy
The rule works both ways and applies to all energy, not just nuclear. Anything that stores energy gets heavier, and anything that releases energy gets lighter.
- Burning. Forming 18 g of water from hydrogen and oxygen releases 285.8 kJ of heat. The water ends up about 3.2 nanograms lighter than the gases it came from, a fraction of 1.8 × 10⁻¹⁰.
- Heating. Warm 1 kg of water by 80 °C (about 335 kJ) and it gets heavier by about 3.7 nanograms.
Those changes are real but far too small for any scale to notice. That’s why chemists can say mass is conserved and still be right to about ten decimal places. In nuclear reactions the energies per atom are millions of times larger, so the mass changes become measurable.
How much mass becomes energy?
- Mass before
- Mass after
- Converted to energy
- Energy released
- Per kg of fuel
Fraction of mass converted, on a log scale. Each step to the right is ten times more.
The Sun loses 4 million tonnes a second
The Sun radiates 3.83 × 10²⁶ watts. Divide by c² and it is getting lighter by 4.26 million tonnes every second, just from the light it shines. Since fusion converts only 0.71% of the hydrogen’s mass, it has to fuse about 600 million tonnes of hydrogen per second to keep that up.
That sounds catastrophic, but the Sun is vast. Even at today’s brightness for all 4.6 billion years of its life, it would have lost only about 0.03% of its mass this way. The young Sun was a little dimmer, so the real figure is slightly less.
The mass defect tab shows where the energy comes from. A helium nucleus weighs 0.75% less than two free protons and two free neutrons. That missing mass, 28.3 MeV, is the energy that binds the nucleus together, and it’s released as the nucleus forms.
Why we can’t just convert a paperclip
Nature keeps strict accounts. Reactions can rearrange protons and neutrons, but the total number of them (physicists call it baryon number) never changes. So the most any nuclear reaction can release is the difference in binding energy. Even fusing hydrogen all the way to iron, the most tightly bound nucleus there is, would free only about 0.9% of the mass.
Releasing 100% needs antimatter. There’s no natural supply of it, and making it in particle accelerators takes vastly more energy than it ever gives back. So antimatter could only ever be a way to store energy, never a source of it, and an absurdly inefficient one at that.
Why nothing with mass reaches c
Energy without limit
Newton’s formula, ½mv², says reaching light speed would take a finite amount of energy: half of mc². Relativity says the kinetic energy is (γ − 1)mc², and γ grows without limit as you approach c. At low speeds the two agree almost perfectly. At 0.99c, pushing a single kilogram takes 547 PJ, about 8,719 Hiroshima bombs. Reaching c itself would take infinite energy.
The Large Hadron Collider shows this every day. It gives protons 6.8 TeV, about 7,247 times their rest energy, yet they travel only about 2.9 metres per second slower than light. Almost all that extra energy shows up as more energy and momentum, not more speed.
From 1905 to the lab
Einstein published the idea in 1905, in a three-page follow-up to his special relativity paper. Its title asked, “Does the inertia of a body depend upon its energy content?” His answer: a body that gives off energy E as light loses mass E/c². He suggested radioactive materials might one day test it.
In 1932 John Cockcroft and Ernest Walton split lithium nuclei with protons, and the energy of the flying helium nuclei matched the measured loss of mass. Since then the equation has run through all of nuclear and particle physics. In 2005 a team led by Simon Rainville and David Pritchard tested it directly. They weighed silicon and sulfur atoms before and after a neutron capture, using extremely precise atom traps, and measured the energy of the gamma rays given off. The two matched to within about four parts in ten million (Nature, 2005).