E=mc² and Mass-Energy Equivalence
Chapter 8 covered how motion distorts time and length. This chapter covers relativity's single most famous consequence — and, in real, documented fact, Einstein's own version of it looked a little different from the equation everyone now recognises.
Einstein's Real 1905 Paper
Barely a few months after his main relativity paper, Einstein published a short follow-up on 21 November 1905, titled "Does the Inertia of a Body Depend Upon Its Energy Content?"
The Full Relativistic Energy-Momentum Relation
The complete, general relationship between energy, momentum, and mass is:
Two familiar special cases fall directly out of this one equation:
- For an object at rest (p = 0): E² = (mc²)², giving the famous E = mc² — an object's own "rest energy," simply from having mass at all.
- For a massless particle, like a photon (m = 0): E = pc — light carries real energy and momentum despite having zero rest mass.
Worked Example: The Rest Energy of 1 Gram
What is the rest energy locked in just 1 g (0.001 kg) of ordinary matter?
E = 0.001 × (3×10&sup8;)²
E ≈ 9×10¹³ J
A single gram of matter, fully converted, carries roughly 90 trillion joules — a genuinely staggering amount of energy locked inside an ordinary object's own mass, entirely independent of any chemical or nuclear reaction actually releasing it.
Real Verification: Nuclear Mass Defect
Nuclear reactions provide the real, measurable evidence for this equivalence: the combined mass of an atomic nucleus is always slightly less than the sum of its individual protons and neutrons measured separately — the missing mass, called the mass defect, is exactly the binding energy holding the nucleus together, converted via E = mc². In real nuclear fission, roughly 0.1% of uranium's own mass converts directly into usable energy.
Real Verification: The Sun
Astronomy Fundamentals (this site's own sibling course) established that the Sun fuses roughly 600 billion kg of hydrogen every second. Using this chapter's own formula in reverse — dividing the Sun's real, measured power output (3.846×10²&sup6; W) by c² — gives:
m = (3.846×10²&sup6;) / (9×10¹&sup6;)
m ≈ 4.27×10&sup9; kg/s
This matches the Sun's own real, independently measured mass-energy conversion rate of roughly 4.26 billion kg/s almost exactly — only about 0.7% of the 600 billion kg of hydrogen fused per second actually converts to energy; the rest remains as the resulting helium's own mass.
Real Verification: The Trinity Test
The real 1945 Trinity nuclear test used a plutonium core of roughly 6.15 kg, of which about 1 kg genuinely underwent fission — and the real, measured mass converted into energy was almost exactly one gram.
Yield ≈ 21.5 kilotons of TNT
This matches the Trinity test's own real, reported yield of roughly 21 kilotons almost exactly — a genuine, checkable confirmation that a single missing gram of mass, converted via E = mc², really did account for the full explosive energy of the world's first nuclear detonation.
Three Real Scales, Compared
| Example | Mass Converted | Energy Released |
|---|---|---|
| Uranium fission (general) | ~0.1% of total mass | Real, usable nuclear power |
| The Sun (every second) | ~4.26 billion kg | 3.846×10²&sup6; W |
| Trinity test (1945) | ~1 gram | ~21 kilotons of TNT |
Hands-On Exercises
Quick Reference
- Einstein's real 1905 paper derived a mass CHANGE from emitted energy (Δm = L/c²), not the compact absolute E = mc² formula itself
- Full relation: E² = (pc)² + (mc²)² — reduces to E = mc² at rest, and E = pc for massless particles
- Nuclear mass defect is the real, measured evidence: missing mass = binding energy, via E = mc²
- The Sun's real 4.26 billion kg/s conversion rate matches E = mc² applied to its own measured power output almost exactly
- The Trinity test's real ~1 gram of converted mass matches its own reported ~21 kiloton yield almost exactly, via E = mc²