Relativity and gravitational theory Definition

Four-velocity

Builds on Four-vector World line Proper time Lorentz factor

Definition

Let $\sigma$ be a timelike world line in Minkowski space, parametrised by its proper time $\tau$ , so that $\tau \mapsto x(\tau)$ with $x(\tau)$ in inertial coordinates $(x^0, x^1, x^2, x^3)$ . The four-velocity at proper time $\tau$ is the four-vector $U(\tau) = \frac{dx}{d\tau}(\tau).$ If the world line is written as $t \mapsto (ct, \mathbf{x}(t))$ with velocity $\mathbf{u} = d\mathbf{x}/dt$ , then $U = \gamma(u) \, (c, \mathbf{u}),$ where $\gamma(u)$ is the Lorentz factor of $u = \abs{\mathbf{u}}$ .

Remarks

Example: for $\mathbf{u} = (0.6 c, 0, 0)$ , $\gamma = 1.25$ and $U = (1.25 c, 0.75 c, 0, 0)$ . Check: $\eta(U, U) = -(1.25)^2 c^2 + (0.75)^2 c^2 = -c^2$ . In general $\eta(U, U) = -c^2$ , because $d\tau = dt / \gamma$ and $\gamma^2 (u^2 - c^2) = -c^2$ . Non-example: the derivative $dx/dt = (c, \mathbf{u})$ with respect to coordinate time is not the four-velocity: it is not a four-vector, and its squared length $-c^2 + u^2$ depends on the frame. The four-momentum of a particle of mass $m$ is $P = m U$ .

Reading path

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Four-vector concept
Lorentz factor concept
Proper time concept
World line concept
Four-velocity definition

Tag	`VNP9RRYGVH`
URI	`m2://VNP9RRYGVH`
Kind	definition
Writing	W0 stub
MSC	83A05, 70B05
Defines	Four-velocity
Uses	Four-vector, World line, Proper time, Lorentz factor
Changed	3 November 2026

Four-velocity

Definition

Remarks

Reading path

Further reading

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