Relativistic energy and momentum

Energy-momentum relations for an object in motion are among the most famous results of Einstein's special relativity (1905). For a particle with rest mass m moving at velocity v, total energy, momentum and relativistic kinetic energy are:

E = gamma * m * c^2 (total energy)

p = gamma * m * v (relativistic momentum)

T = (gamma - 1) * m * c^2 (kinetic energy)

where gamma = 1/sqrt(1 - v^2/c^2) is Lorentz factor and c = 299 792 458 m/s is light speed.

Rest energy E_0 = m c^2

Perhaps most famous physics equation, derived by Einstein in 'Does the Inertia of a Body Depend Upon Its Energy Content?' (1905). Every object with mass has rest energy:

E_0 = m * c^2

Enormity: 1 kg of mass = 9 * 10^16 J = 25 billion kWh = energy of ~20 Mt (megatons) of TNT. Basis of nuclear fission and fusion, where small mass changes release huge energies.

Relativistic invariant

Most fundamental quantity:

E^2 = p^2 c^2 + m^2 c^4

Invariant under Lorentz transformations: all inertial observers compute same value. Four-momentum P = (E/c, p) transforms as 4-vector, and |P|^2 = m^2 c^2 is invariant.

Photons and massless particles

For m = 0 particles (photons, gluons):

E = p c (linear dispersion relation)

Photons always travel at c, regardless of frame. Photon energy linked to frequency by E = h nu (Planck) or wavelength by E = h c / lambda (de Broglie).

Five modes

1. Energy and momentum: compute E, p, T for given mass and velocity.
2. Invariant: verify E^2 - p^2 c^2 = (m c^2)^2.
3. Photons: m=0 particles, E = pc relation, frequency, wavelength.
4. Binding energy: nuclear mass defect Delta m * c^2 in MeV.
5. Kinetic: comparison of relativistic T vs classical (1/2) m v^2.

Nuclear binding energy

Most concrete application of E = m c^2 in nuclear physics. Nucleus mass always less than sum of free proton and neutron masses. 'Mass defect' Delta m converts to binding energy:

B = Delta m * c^2

Expressed in MeV/nucleon:

• H-2 (deuterium): 1.11 MeV/A
• He-4: 7.07 MeV/A
• Fe-56: 8.79 MeV/A (MAXIMUM, most stable)
• U-235: 7.59 MeV/A (fissile)

Classical vs relativistic limit

For v << c, gamma ~ 1 + v^2/(2 c^2) + ..., so T ~ (1/2) m v^2. Relative error v^2/(2c^2):

• v = 0.01 c: 0.005% error
• v = 0.1 c: 0.5% error
• v = 0.5 c: 15% error
• v = 0.9 c: 125% error
• v = 0.99 c: 1300%+ error

Applications

• Particle physics: LHC collision kinematics with protons at gamma ~ 7000.
• Nuclear physics: fission (U-235 ~200 MeV per fission), fusion (D-T gives 17.6 MeV).
• Astrophysics: supernova energetics, black holes, ultra-energetic cosmic rays.
• Medicine: PET scanner uses e+ e- annihilation -> 2 photons 511 keV.
• Cosmology: dark matter-energy, universe expansion, CMB 2.7 K.
• Technology: nuclear reactors, accelerators, ultrashort-pulse lasers.

The four-momentum and conservation in collisions

Energy and momentum are not separate quantities in relativity: they merge into a single four-dimensional object, the four-momentum P = (E/c, p_x, p_y, p_z), which transforms from one frame to another like spacetime coordinates. Its magnitude, E^2/c^2 - |p|^2 = m^2 c^2, is a Lorentz invariant: all inertial observers agree on the rest mass m, even though they measure different energies and momenta.

In particle collisions the total four-momentum is conserved component by component: the sum of the energies and the sum of the momenta before the collision equal those after. From this arises the concept of the invariant mass of a system, which can exceed the sum of the constituent masses because it includes kinetic and binding energy. It is precisely this conservation that lets one compute the threshold energies for creating new particles in high-energy collisions.

Relativistic energies in accelerators

In particle accelerators relativistic effects are not corrections but the dominant regime. The protons of the Large Hadron Collider reach a gamma factor of order 7500, travelling at over 99.9999% of the speed of light: their total energy is almost entirely kinetic, thousands of times the rest-mass energy. This is why energies are measured in electronvolts (eV) and multiples (MeV, GeV, TeV), the natural units for the subatomic world.

The choice between colliders (two beams meeting head-on) and fixed targets depends on the useful centre-of-mass energy, the only energy actually available to create new particles. In colliders nearly all the energy is usable, whereas with a fixed target much of it is wasted in the bulk motion of the products. The famous equation E = mc^2 becomes operational here: converting kinetic energy into mass is the mechanism by which heavy particles such as the Higgs boson are produced.

Updated July 2026