Physics › 🕳️ Relativity

Mass–Energy Equivalence

A sliver of missing mass inside a star is the same thing as the sunlight on your face — E=mc² is the exchange rate between them.

Uni Year 1
💡
The big idea: Mass and energy are not two different things that convert into each other like currencies — they're the same physical quantity, measured in different units, with c² as the conversion factor. Because c² is enormous, even a tiny amount of mass corresponds to a staggering amount of energy, and pushing a massive object toward the speed of light demands energy that grows without bound.
🎯 By the end, you'll be able to
  • Explain what E=mc² actually means physically, not just recite it
  • Compute rest energy and total relativistic energy (E=γmc²) for a given mass and speed
  • Explain, using the Lorentz factor, why no object with mass can ever reach the speed of light
  • Connect mass-energy equivalence to how nuclear reactions release energy via a measurable 'mass defect'
📎 You should already know
  • Special relativity basics: time dilation and length contraction
  • The Lorentz factor \(\gamma\)
  • Basic kinematics: velocity, momentum, and kinetic energy

The energy that was once mass

Every second, the Sun's core fuses hydrogen into helium, and the resulting helium nuclei weigh very slightly less than the hydrogen that made them. That missing mass doesn't vanish — it reappears, moments later, as the light and heat radiating outward. Einstein's insight was that this isn't a special nuclear trick; it's a universal fact about nature. Mass is a form of energy, sitting there even when an object is perfectly still. \(E=mc^2\) tells you exactly how much.

🔑 Mass is stored energy

Every object at rest carries an energy content \(E_0 = mc^2\) simply because it has mass \(m\). This is called its rest energy. Nothing needs to happen to the object for this energy to exist — it's baked into having mass at all. Because \(c\) is about \(3\times10^8\) m/s, \(c^2\) is roughly \(9\times10^{16}\,\text{m}^2/\text{s}^2\) — an almost absurdly large conversion factor. That's why even a gram of matter corresponds to a huge amount of energy.

\[ E_0 = mc^2 \]
Rest energy: the energy an object has purely by virtue of its mass, even at rest.
\[ E = \gamma m c^2, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \]
Total relativistic energy of a moving object. At v = 0, this reduces to E = mc².
\[ KE = (\gamma - 1)mc^2 \]
Kinetic energy is whatever total energy exceeds the rest energy — the 'extra' from motion.
🎮 Interactive: The Light Clock LIVE
Watch how the Lorentz factor γ grows as speed approaches c — the same γ that multiplies mc² and makes the energy needed to go faster balloon toward infinity.

Why nothing with mass can catch a light beam

Look again at \(E=\gamma mc^2\). As \(v\) creeps toward \(c\), the term \(v^2/c^2\) creeps toward 1, so \(1-v^2/c^2\) shrinks toward zero — and \(\gamma\), sitting under a square root in the denominator, shoots toward infinity. That means the kinetic energy \((\gamma-1)mc^2\) needed to keep accelerating a massive object also shoots toward infinity. There's no finite amount of energy — not all the fuel in the universe — that gets a massive object all the way to \(c\). Photons, by contrast, have zero rest mass, so their energy comes entirely from motion (\(E=pc\)) with no contradiction: massless is precisely what lets them travel at \(c\).

📝 Worked example: A proton has a rest mass of about \(1.67\times10^{-27}\) kg. How much energy is locked in its mass alone, and what is that in mega-electronvolts (MeV)?
  1. Rest energy: \(E_0 = mc^2\).
  2. \(E_0 = (1.67\times10^{-27}\,\text{kg})(3.00\times10^{8}\,\text{m/s})^2\)
  3. \(c^2 = 9.00\times10^{16}\,\text{m}^2/\text{s}^2\), so \(E_0 = 1.67\times10^{-27} \times 9.00\times10^{16} \approx 1.50\times10^{-10}\,\text{J}\)
  4. Convert to eV by dividing by \(1.60\times10^{-19}\,\text{J/eV}\): \(E_0 \approx \dfrac{1.50\times10^{-10}}{1.60\times10^{-19}} \approx 9.4\times10^{8}\,\text{eV} = 938\,\text{MeV}\)
✓ About \(1.50\times10^{-10}\) J, or roughly 938 MeV — matching the proton's well-known rest energy.
📝 Worked example: An electron (rest energy 0.511 MeV) moves at \(v = 0.8c\). Find its total relativistic energy and its kinetic energy.
  1. Lorentz factor: \(\gamma = \dfrac{1}{\sqrt{1-v^2/c^2}} = \dfrac{1}{\sqrt{1-0.64}} = \dfrac{1}{\sqrt{0.36}} = \dfrac{1}{0.6} \approx 1.67\)
  2. Total energy: \(E = \gamma mc^2 \approx 1.67 \times 0.511\,\text{MeV} \approx 0.852\,\text{MeV}\)
  3. Kinetic energy is everything beyond rest energy: \(KE = (\gamma-1)mc^2 \approx 0.667 \times 0.511\,\text{MeV} \approx 0.341\,\text{MeV}\)
✓ Total energy ≈ 0.852 MeV; kinetic energy ≈ 0.341 MeV. Notice the electron needed kinetic energy comparable to its entire rest energy just to reach 80% of light speed — a preview of why the last bit toward c gets so expensive.
⚠️ Common trap: 'mass turns into energy and disappears'

It's tempting to picture mass vanishing and energy appearing out of nowhere, but total energy — including whatever is stored as rest mass — is always conserved. In a nuclear reaction, the total rest mass of the products is measurably less than that of the reactants; this difference, the mass defect \(\Delta m\), reappears as kinetic energy and radiation according to \(\Delta E = \Delta m\,c^2\). Nothing is created or destroyed — energy just changes which form it's stored in. Also remember that \(E=mc^2\) itself is the rest-energy special case; a moving object's full energy is \(E=\gamma mc^2\).

Check your understanding

1. What does the equation E=mc² fundamentally state?
E=mc² says rest mass IS a form of energy, related by the constant c². It's not a rule limited to nuclear physics — it's a general statement about what mass is.
2. A block has a rest mass of 0.50 kg. Using c² ≈ 9×10^16 m²/s², what is its rest energy?
E₀ = mc² = 0.50 kg × 9×10^16 m²/s² = 4.5×10^16 J — enough energy to power a large city for a long time, from half a kilogram of mass.
3. An electron travels at v = 0.8c. What is its approximate Lorentz factor γ?
γ = 1/√(1 − v²/c²) = 1/√(1 − 0.64) = 1/√0.36 = 1/0.6 ≈ 1.67.
4. Why can't a particle with nonzero rest mass ever reach the speed of light?
As v approaches c, γ diverges to infinity, so the kinetic energy (γ−1)mc² needed to push the object faster also diverges. No finite energy source can supply that, which is why massive objects only ever approach — never reach — c.
✅ Key takeaways
  • E=mc² says mass and energy are the same physical quantity, just measured in different units — c² is the huge conversion factor between them.
  • Total relativistic energy is E=γmc²; kinetic energy is the extra (γ−1)mc² an object gains from motion, on top of its rest energy.
  • Because γ→∞ as v→c, the energy needed to accelerate a massive object toward light speed grows without bound — that's the real reason nothing with mass ever reaches c.
  • Nuclear reactions release energy by converting a measurable 'mass defect' (Δm) into kinetic energy and radiation via ΔE=Δmc² — energy is conserved, not created from nothing.