Relativistic Momentum and Mass-Energy Equivalence

In special relativity the momentum of a particle of rest mass m is p = γmv and its total energy is E = γmc2, which includes a rest energy E0 = mc2; the kinetic energy is KE = (γ − 1)mc2, and energy and momentum are related by E2 = (pc)2 + (mc2)2. In brief: as v approaches c, γ and the momentum and energy become infinite, so no object with mass can reach c. Mass and energy are equivalent: a mass change Δm corresponds to an energy change Δm c2, as seen in nuclear reactions, in particle creation and annihilation, and in the Sun. For a photon (m = 0), E = pc. At low speeds, KE → ½mv2.

Relativity changes the laws of mechanics at high speed, and reveals that mass is a form of energy. This guide derives the relativistic expressions for momentum and energy, explains E = mc2 and the energy-momentum relation, compares Newtonian and relativistic formulas in a table, works through numerical examples with electrons and protons and corrects common mistakes.

What is relativistic momentum?

If momentum were still mv, the law of conservation of momentum would not hold in all inertial frames when the velocities are measured with the relativistic transformations. The quantity that is conserved in all frames is p = γmv, with m the rest (invariant) mass and γ = 1 ÷ √(1 − v2/c2). For small speeds, γ ≈ 1 and p ≈ mv. As v → c, p increases without limit, so a force acting on a particle for a long time cannot accelerate it to the speed of light. In particle accelerators, protons reach 99.999999 percent of c, but each increase in energy produces only a minuscule increase in speed. Newton’s second law holds in the form F = dp/dt with the relativistic momentum; the acceleration is no longer in the direction of the force in general, and a = F/(γ3m) along the direction of motion, so the “inertia” increases with speed.

What is relativistic energy?

The total energy of a free particle is E = γmc2. At rest (γ = 1), it has the rest energy E0 = mc2. The kinetic energy is the difference: KE = (γ − 1)mc2, which for small v expands to ½mv2 + (3/8)mv4/c2 + …. As v approaches c, KE grows without limit, and the work needed to reach c is infinite. The energy and momentum are related by the invariant combination

E2 = (pc)2 + (mc2)2.

For a particle of zero mass (the photon), m = 0 and E = pc, which agrees with E = hf and p = h/λ. It can be shown that v = pc2/E. In particle physics, masses and momenta are usually quoted in energy units: MeV/c2 and MeV/c; for example, the electron mass is 0.511 MeV/c2, and the proton mass is 938.3 MeV/c2. The relation also shows that the energy gained by a charge e accelerated through a potential difference V is eV, so an electron accelerated through 1.0 MV gains 1.0 MeV of kinetic energy.

0.90 0 1 2 3 0.5 v/c KE/mc2 1.29 0.41

What does mass-energy equivalence mean?

Einstein’s 1905 conclusion was that the inertia of a body depends on its energy content: ΔE = Δm c2. A body that emits energy E in the form of radiation loses mass E/c2; a hot object is slightly more massive than a cold one; a compressed spring has a larger mass than a relaxed one. In everyday chemistry, the changes are tiny: burning 1 kg of coal releases 3 × 107 J, a mass change of 3.3 × 10−10 kg (not measurable). In nuclear reactions, the effect is a thousand to a million times bigger (0.1 percent of the mass in fission, 0.7 percent in fusion). In particle physics, the kinetic energy of colliding particles can create new particles (electron-positron pairs from a photon with E ≥ 1.022 MeV), and the annihilation of matter and antimatter converts all the mass into energy. The mass of a nucleus or a proton is mostly due to the binding energy of its components (the proton’s mass is largely the energy of the strong-force field of the quarks, not the sum of the quark masses). Mass, in this modern view, is the energy of an object in its rest frame, divided by c2.

Relativistic and Newtonian kinetic energy against speed 0.0 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5 Speed v / c Kinetic energy / mc2 relativistic Newtonian ½mv2
Kinetic energy in units of the rest energy mc²: relativistic (γ − 1) (solid) and Newtonian ½mv² (dashed). They agree at low speed but the relativistic energy grows without limit as v approaches c. At 0.9c (dotted line) the relativistic value is 1.29 mc², about three times the Newtonian one; for an electron that is 0.661 MeV against 0.207 MeV.

Newtonian and relativistic formulas compared

Newtonian and relativistic mechanics
Quantity Newtonian Relativistic
Momentummvγmv
Kinetic energy½mv2(γ − 1)mc2
Total energy½mv2 (plus a constant)γmc2 = KE + mc2
Limit on speedNonev < c for m > 0
PhotonNot definedE = pc

Worked example: energies and speeds

  1. Rest energy of an electron: mc2 = 9.109 × 10−31 × (2.998 × 108)2 = 8.19 × 10−14 J = 0.511 MeV. For a proton: 938.3 MeV.
  2. Electron at 0.90c. γ = 2.294. KE = (γ − 1) × 0.511 = 0.661 MeV. Total energy E = 2.294 × 0.511 = 1.172 MeV. The momentum: pc = γmvc = 2.294 × 0.90 × 0.511 = 1.055 MeV (so p = 1.055 MeV/c). Check: √(1.0552 + 0.5112) = √(1.113 + 0.261) = 1.172 MeV ✓. The Newtonian KE, ½mv2 = 0.5 × 0.511 × 0.81 = 0.207 MeV, is three times too small.
  3. Speed from accelerating voltage. An electron is accelerated through 1.0 MV: KE = 1.0 MeV, γ = 1 + 1.0 ÷ 0.511 = 2.957. v = c√(1 − 1/γ2) = c√(1 − 0.1144) = 0.941c. The Newtonian formula would give v = √(2KE/m) = √(2 × 1.0 ÷ 0.511) c = 1.98c, impossible.
  4. Large Hadron Collider proton. A proton with total energy 6.5 TeV: γ = 6.5 × 1012 ÷ 938.3 × 106 = 6,930. The speed differs from c by a fraction 1/(2γ2) = 1.0 × 10−8: v = 0.99999999c.
  5. Mass-energy. The conversion of 1.0 g of mass: E = 1.0 × 10−3 × (2.998 × 108)2 = 9.0 × 1013 J, equivalent to about 21 kilotons of TNT (the Hiroshima bomb converted about 0.7 g).
  6. Pair production. The minimum photon energy to create an electron-positron pair: 2 × 0.511 = 1.022 MeV, with the wavelength λ = 1,240 eV·nm ÷ 1.022 × 106 eV = 1.2 × 10−3 nm. Annihilation of an electron and a positron at rest produces two 0.511 MeV photons.
pc = 1.055 MeV mc2 = 0.511 MeV E = 1.172 MeV

How was E = mc² confirmed?

Einstein published the mass-energy relation in a short paper in September 1905. Cockcroft and Walton verified it in 1932 by splitting lithium nuclei with accelerated protons and comparing the energy released with the mass difference. Carl Anderson discovered the positron in 1932, confirming Dirac’s prediction, and pair production and annihilation became standard processes. Precise measurements of the masses of nuclei and the energies of nuclear reactions (to better than one part in a million) have confirmed E = mc2. Particle accelerators operate every day on the basis of relativistic mechanics.

Which mistakes are common?

Mistake 1: Saying that mass increases with speed

Modern physics treats the (rest) mass as invariant. It is the momentum and energy that increase with γ.

Mistake 2: Using ½mv² at high speed

At 0.5c, the error is already about 19 percent; at 0.9c, a factor of three.

Mistake 3: Confusing total energy with kinetic energy

E = γmc2 includes the rest energy. KE = E − mc2.

Mistake 4: Thinking that E = mc² applies only to nuclear reactions

It applies to all energy changes; they are simply too small to measure in chemical reactions.