Relative Velocity and Lorentz Factors

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Two future-directed unit timelike vectors at the same event determine an invariant relative Lorentz factor. They also determine a relative-velocity vector, but that vector lives in the measuring observer’s rest space. The speed is reciprocal; the two velocity vectors are not literally negatives because they belong to different three-spaces.

Let $u^\mu$ and $v^\mu$ be the four-velocities of two observers at the same event. Both satisfy $u^2=v^2=-1$ .

Required background. Events, Intervals, and Minkowski Geometry begins the Special Relativity sequence and supplies causal geometry; Four-Vectors and Lorentz-Invariant Decomposition supplies four-vector algebra; Observers, Rest Spaces, and Measured Components supplies the observer projector. Helpful background. Calculus and Linear Algebra supplies inner products and projections; Classical and Variational Mechanics supplies the classical relative-velocity limit.

Invariant Lorentz factor

Define

\gamma_{\rm rel}=-u_\mu v^\mu.

For future-directed unit timelike vectors, $\gamma_{\rm rel}\ge1$ , with equality only when $u^\mu=v^\mu$ . This scalar is manifestly symmetric:

\gamma(u,v)=\gamma(v,u).

The relative rapidity $\chi_{\rm rel}\ge0$ and relative speed $w$ are

\gamma_{\rm rel}=\cosh\chi_{\rm rel}, \qquad w=\tanh\chi_{\rm rel} =\sqrt{1-\frac{1}{\gamma_{\rm rel}^2}}.

Thus both observers assign the same relative speed. Rapidity is the hyperbolic distance between their velocity vectors on the future unit hyperboloid.

The velocity vector measured by one observer

Define the rest-space projector of $u$ by

h^\mu{}_{\nu}(u)=\delta^\mu{}_{\nu}+u^\mu u_\nu.

Decompose $v^\mu$ relative to $u^\mu$ :

v^\mu=\gamma_{\rm rel}(u^\mu+w^\mu_{(u)}),

where

w^\mu_{(u)} =\frac{h^\mu{}_\nu(u)v^\nu}{\gamma_{\rm rel}} =\frac{v^\mu}{\gamma_{\rm rel}}-u^\mu.

It obeys

u_\mu w^\mu_{(u)}=0, \qquad w^\mu_{(u)}w^{(u)}_\mu=w^2\lt1.

The reversed construction $w^\mu_{(v)}$ is orthogonal to $v^\mu$ , not to $u^\mu$ . At the same event, the two are directly comparable as spacetime vectors because both belong to the same tangent space. What is not automatic is an identification of the Euclidean three-spaces $u^\perp$ and $v^\perp$ . The standard boost in the $u$ – $v$ plane maps one rest space to the other and sends $w_{(u)}$ to $-w_{(v)}$ ; without choosing such a Lorentz map, only their equal magnitudes are immediately comparable.

This invariant construction and the identity $w^2=1-\gamma_{\rm rel}^{-2}$ agree with Gourgoulhon 2007, §5.3.1, printed pp. 76–77, Eqs. (5.28)–(5.36), which uses the same metric signature and $c=1$ .

Coordinate formula and velocity addition

In one inertial chart, write

u^\mu=\gamma_u(1,\mathbf u), \qquad v^\mu=\gamma_v(1,\mathbf v).

Then

\gamma_{\rm rel} =\gamma_u\gamma_v(1-\mathbf u\cdot\mathbf v).

This formula is often the shortest way to find relative speed even for noncollinear motion. For collinear velocities $u$ and $v$ along the same axis, it yields the signed velocity component measured in the $u$ rest frame,

w_x=\frac{v-u}{1-uv}.

The denominator enforces $|w_x|\lt1$ whenever $|u|,|v|\lt1$ . In the low-speed limit, $w_x$ becomes the Galilean difference $v-u$ . Its magnitude is the invariant speed $w$ defined above.

For the noncollinear formula, choose the $x$ -axis along the coordinate velocity $\mathbf u=u\,\hat{\mathbf x}$ , where $u=|\mathbf u|$ . Decompose the second coordinate velocity as

\mathbf v=v_\parallel\hat{\mathbf x}+\mathbf v_\perp, \qquad \hat{\mathbf x}\mathbin{\cdot}\mathbf v_\perp=0.

The components measured in the $u$ rest frame are then

w_\parallel=\frac{v_\parallel-u}{1-uv_\parallel}, \qquad \mathbf w_\perp =\frac{\mathbf v_\perp}{\gamma_u(1-uv_\parallel)}.

The product of two noncollinear pure boosts is generically a boost composed with a spatial Thomas–Wigner rotation. The boost–rotation ordering and rotation sign depend on active/passive and ordering conventions; the convention-sensitive construction is shown in Tong, Topics in Quantum Mechanics, Chapter 3, §3.1.3, printed pp. 71–72. Treating three-velocity addition as ordinary vector addition misses this geometry.

Worked example: crossing observers

Two observers have equal coordinate speeds $0\le q\lt1$ along perpendicular axes in an inertial frame:

\mathbf u=(q,0,0), \qquad \mathbf v=(0,q,0).

Their invariant factor is

\gamma_{\rm rel} =\gamma_q^2, \qquad \gamma_q=\frac{1}{\sqrt{1-q^2}},

w^2 =1-\frac{1}{\gamma_q^4} =2q^2-q^4.

For small $q$ , $w\simeq\sqrt2\,q$ , the Euclidean result. As $q\to1$ , $w\to1$ without exceeding it. Simply subtracting the coordinate velocity vectors would give magnitude $\sqrt2\,q\gt1$ for sufficiently large $q$ and is therefore not a physical relative speed.

Locality and limits

The invariant definition is local. Minkowski spacetime’s affine structure canonically identifies tangent spaces by translation, so inertial four-velocities at separated events can be compared algebraically. Turning that comparison into an operational distant velocity still requires a simultaneity, radar, or other measurement prescription; in curved spacetime, even tangent-space identification requires a transport rule.
A photon has a null tangent and no rest frame. A massive observer always measures its local speed as one; frequency and direction contain the nontrivial information.
Coordinate recession speeds in cosmology or noninertial charts can exceed one because they are not this local relative speed.
The scalar $-u\cdot v$ fixes speed but not spatial direction. Direction requires the projected vector and a spatial triad.
If either vector is not normalized, its inner product is not a Lorentz factor. Normalize physical four-velocities first.

The coordinate rapidity and velocity-transformation checks are derived in Siklos 2011, §6.5, printed pp. 13–14, Eq. (6.14), and §6.8.6, printed pp. 24–25, with signs translated from the source’s $(+,-,-,-)$ metric.

Exercises

Recover the speed from the projector

Starting from $w_{(u)}^\mu=v^\mu/\gamma_{\rm rel}-u^\mu$ , derive $w_{(u)}^2=1-\gamma_{\rm rel}^{-2}$ .

Solution

Use $u^2=v^2=-1$ and $u\mathbin{\cdot}v=-\gamma_{\rm rel}$ :

\begin{aligned} w_{(u)}^2 &=\frac{v^2}{\gamma_{\rm rel}^2} -\frac{2u\mathbin{\cdot}v}{\gamma_{\rm rel}}+u^2 \\ &=-\frac{1}{\gamma_{\rm rel}^2}+2-1 =1-\frac{1}{\gamma_{\rm rel}^2}. \end{aligned}

Because $w_{(u)}$ lies in the positive-definite rest space of $u$ , this squared norm is the measured speed squared.

Convert Lorentz factor to speed and rapidity

One observer measures another with $\gamma_{\rm rel}=5/3$ . Find the relative speed and rapidity.