Momentum has to be rebuilt
Conservation of momentum is too precious to lose, but the Newtonian form p = mv fails at high speed: analyze a collision in two frames using the old formula and momentum comes out conserved in one frame and not the other. The fix is to weave in the same Lorentz factor. Relativistic momentum keeps conservation exact in every inertial frame, and reduces to mv when v \ll c.
Relativistic momentum. As v → c, γ → ∞, so p grows without bound even though v cannot exceed c — a second sign that light-speed is an unreachable ceiling for matter.
Energy, and the most famous equation
Demanding that energy be conserved in every frame too forces a matching redefinition. The total relativistic energy of a free particle is E = \gamma mc^2. The startling part appears when the particle sits still: set v = 0, so \gamma = 1, and the energy does not go to zero. It settles at the rest energy E_0 = mc^2 — energy locked inside mass itself.
Total energy, rest energy, and kinetic energy as the surplus above rest. For v ≪ c, expanding γ gives K ≈ ½mv² — the old kinetic energy re-emerges as the first correction.
This is mass–energy equivalence: mass and energy are two names for one conserved thing, convertible at the ferocious exchange rate c^2. Because c^2 is enormous, a tiny mass hides a colossal energy. Convert just one gram entirely and you release E = (0.001)(3\times10^8)^2 = 9\times10^{13} J — roughly 20 kilotons of TNT, comparable to an early nuclear weapon. This is why nuclear reactions, which convert a fraction of a percent of their mass, dwarf every chemical fire.
It runs the sky, too. The Sun shines by fusing hydrogen into helium, converting about 4 million tonnes of mass into energy every second — yet with so much mass to spend it will burn for billions of years. E=mc² is the power source of the stars.
The master relation, and light itself
Energy and momentum knit together into one invariant equation, the relativistic counterpart of the spacetime interval. It holds in every frame, and it contains a surprise: a massless particle (m = 0) is not forbidden — it simply must travel at c and carries energy E = pc. That is exactly the photon, the quantum of light, which has energy and momentum yet no rest mass. Relativity and quantum theory meet here.
The energy–momentum relation. The rest energy mc² is the invariant 'length' of the energy–momentum vector; for light the mass term vanishes and E = pc.
A worked example, and where it leads
- Problem. An electron (rest energy mc² = 0.511 MeV) is accelerated to v = 0.99c. Find its kinetic energy relativistically, and compare with the Newtonian ½mv².
- Lorentz factor. γ = 1/√(1 − 0.99²) = 1/√(1 − 0.9801) = 1/√0.0199 ≈ 7.09.
- Relativistic KE. K = (γ − 1)mc² = (7.09 − 1)(0.511 MeV) ≈ 6.09 × 0.511 ≈ 3.1 MeV.
- Newtonian estimate. ½mv² = ½ mc² (v/c)² = ½ (0.511)(0.99²) ≈ 0.25 MeV — low by a factor of about twelve. At relativistic speeds the classical formula is not slightly off; it is badly wrong, and only K = (γ − 1)mc² is trustworthy.
You have now met the whole classical core of special relativity. Two roads lead onward. Include acceleration and gravity and you reach Einstein's general relativity, where mass curves spacetime — the dedicated Relativity domain. Combine relativity with quantum mechanics and E^2 = (pc)^2 + (mc^2)^2 becomes the license to create and destroy particles, opening particle physics and the Higgs story. Both begin exactly where this guide ends.