Four-momentum

Table of contents
- Relativistic energy Four-momentum
  - Energy-momentum relation Relativistic energy
  - Mass-energy equivalence Relativistic energy
- Spacetime interval Four-momentum
  - Proper time Spacetime interval

Relativistic energy

Energy-momentum relation

Mass-energy equivalence ( $E = m c^{2}$ )

Spacetime interval

In the Galilean transformation, there are two separate invariants that two inertial frame of reference always agree on between two separate events:

time
length, given by the Pythagorean theorem

However, in special relativity, neither of those are invariant separately, since space and time are mixed up together.

Instead, there is a new unified invariant: the spacetime-interval, given by:

c Δ t^{2} - (Δ x^{2} + Δ y^{2} + Δ z^{2})

Note that this distance can be zero for two events separated.

Proper time

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Four-momentum by

Codex  0 Created 2026-09-24 Updated 2026-09-24

A massive particle has four-momentum

P^{μ} = (E / c, p) = mγ (c, v), P^{2} = m^{2} c^{2} .

For a massless particle,

P^{μ} = (E / c, p)

with

E = c ∣ p ∣

and

P^{2} = 0

.

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