Special Relativity

Composition of Velocities

The relativistic relation between velocities measured in inertial reference frames.

This essay derives the relativistic composition of velocities from the Lorentz transformation for a general boost. The resulting relation respects the invariant speed of light and reduces to Galilean velocity addition in the low-speed limit.

Prerequisites

Lorentz Transformations
Vector calculus
Differentiation of vector-valued functions

I. Introduction
II. Derivation of Composition of Velocities
III. Limiting Cases
1. A Particle Moving at the Speed of Light in $F$
2. A Particle Moving at the Speed of Light in $F^{'}$
3. Relative Motion Approaching the Speed of Light

I. Introduction

The preceding essay Lorentz Transformations derives the transformation between two inertial reference frames

F

and

F^{'}

, where

F^{'}

moves with velocity

v = v e_{1}

relative to

F

. For the derivation below, we need the Lorentz transformation written explicitly in terms of this relative velocity:

Eq:Lorentz_transformation_in_velocity_form

Λ (v) = \frac{γ}{v ^{2}} ∣ v, 0 ⟩ ⟨ v, 0 ∣ - \frac{γ}{c} ∣ v, 0 ⟩ ⟨ 0, 1 ∣ + ∣ e_{2}, 0 ⟩ ⟨ e_{2}, 0 ∣ + ∣ e_{3}, 0 ⟩ ⟨ e_{3}, 0 ∣ - \frac{γ}{c} ∣ 0, 1 ⟩ ⟨ v, 0 ∣ + γ ∣ 0, 1 ⟩ ⟨ 0, 1 ∣,

where

γ = \frac{1}{1 - \frac{v ^{2}}{c ^{2}}} .

The time-coordinate component of this transformation is

Eq:Lorentz_time_coordinate

c t^{'} = γ (- \frac{v}{c} ⟨ e_{1} ∣ x ⟩ + c t) .

II. Derivation of Composition of Velocities

In classical mechanics, a velocity

u^{'}

measured in the inertial frame

F^{'}

is measured in the inertial frame

F

u = v + u^{'},

which is the familiar Galilean law of velocity addition. In relativity, however, velocities must be composed according to the Lorentz transformation, with the Galilean law recovered in the limit in which the relevant velocities are much smaller than the speed of light. We consider the inertial reference frames

F

and

F^{'}

defined in the introduction and the general motion of a body in

F^{'}

described by the spacetime trajectory

∣ x^{'}, 0 ⟩ + c t^{'} ∣ 0, 1 ⟩,

where we assume that

x^{'} = x^{'} (t^{'})

. The motion of the body observed in

F

is related to the motion observed in

F^{'}

through the Lorentz transformation

Λ (v) (∣ x, 0 ⟩ + c t ∣ 0, 1 ⟩) = ∣ x^{'}, 0 ⟩ + c t^{'} ∣ 0, 1 ⟩,

where

Λ (v)

is given by Lorentz transformation in velocity form and

x = x (t)

. Differentiating the spacetime trajectory with respect to the time

t

measured in

F

yields

Λ (v) \frac{d}{d t} (∣ x, 0 ⟩ + c t ∣ 0, 1 ⟩) = \frac{d}{d t} (∣ x^{'}, 0 ⟩ + c t^{'} ∣ 0, 1 ⟩) .

Differentiating the time-coordinate transformation Lorentz time coordinate , we obtain

Eq:Time_derivative_velocity_composition

\frac{d t ^{'}}{d t} = γ (1 - \frac{⟨ v ∣ u ⟩}{c ^{2}}) .

Since

\frac{d x ^{'}}{d t} = \frac{d x ^{'}}{d t ^{'}} \frac{d t ^{'}}{d t} = u^{'} \frac{d t ^{'}}{d t},

the differentiated Lorentz transformation becomes

Λ (v) (∣ u, 0 ⟩ + c ∣ 0, 1 ⟩) = γ (1 - \frac{⟨ v ∣ u ⟩}{c ^{2}}) (∣ u^{'}, 0 ⟩ + c ∣ 0, 1 ⟩) .

Substituting Lorentz transformation in velocity form , we obtain

γ (\frac{⟨ v ∣ u ⟩}{v ^{2}} - 1) ∣ v, 0 ⟩ + ∣ e_{2}, 0 ⟩ ⟨ e_{2} ∣ u ⟩ + ∣ e_{3}, 0 ⟩ ⟨ e_{3} ∣ u ⟩ + ∣ 0, 1 ⟩ γ c (1 - \frac{⟨ v ∣ u ⟩}{c ^{2}}) = γ (1 - \frac{⟨ v ∣ u ⟩}{c ^{2}}) (∣ u^{'}, 0 ⟩ + c ∣ 0, 1 ⟩) .

The time components on both sides are identical and therefore cancel, leaving

γ (\frac{⟨ v ∣ u ⟩}{v ^{2}} - 1) ∣ v, 0 ⟩ + ∣ e_{2}, 0 ⟩ ⟨ e_{2} ∣ u ⟩ + ∣ e_{3}, 0 ⟩ ⟨ e_{3} ∣ u ⟩ = γ (1 - \frac{⟨ v ∣ u ⟩}{c ^{2}}) ∣ u^{'}, 0 ⟩ .

Using the decomposition

∣ u, 0 ⟩ = \frac{⟨ v ∣ u ⟩}{v ^{2}} ∣ v, 0 ⟩ + ∣ e_{2}, 0 ⟩ ⟨ e_{2} ∣ u ⟩ + ∣ e_{3}, 0 ⟩ ⟨ e_{3} ∣ u ⟩,

the previous equation can be written more compactly as

γ \frac{⟨ v ∣ u ⟩}{v ^{2}} ∣ v, 0 ⟩ - γ ∣ v, 0 ⟩ - \frac{⟨ v ∣ u ⟩}{v ^{2}} ∣ v, 0 ⟩ + ∣ u, 0 ⟩ = γ (1 - \frac{⟨ v ∣ u ⟩}{c ^{2}}) ∣ u^{'}, 0 ⟩,

Eq:Velocities_relationship

(γ - 1) \frac{⟨ v ∣ u ⟩}{v ^{2}} ∣ v, 0 ⟩ - γ ∣ v, 0 ⟩ + ∣ u, 0 ⟩ = γ (1 - \frac{⟨ v ∣ u ⟩}{c ^{2}}) ∣ u^{'}, 0 ⟩ .

Solving for

∣ u^{'}, 0 ⟩

gives

Eq:Inverse_velocity_composition

∣ u^{'}, 0 ⟩ = \frac{1}{1 - \frac{⟨ v ∣ u ⟩}{c ^{2}}} (\frac{γ - 1}{γ} \frac{⟨ v ∣ u ⟩}{v ^{2}} ∣ v, 0 ⟩ - ∣ v, 0 ⟩ + \frac{∣ u , 0 ⟩}{γ}) .

The velocity

∣ u ⟩

can be easily expressed in terms of the velocity

∣ u^{'} ⟩

by exchanging the roles of

∣ u ⟩

and

∣ u^{'} ⟩

in the equation above, and changing the sign of the relative velocity

v

. As an exercise, we derive the same result by directly inverting velocities relationship . We start by factoring the velocity

∣ u ⟩

in velocities relationship and express the equation as an operator on the velocity

∣ u ⟩

(\frac{γ}{v ^{2}} ∣ v ⟩ ⟨ v ∣ + ∣ e_{2}, 0 ⟩ ⟨ e_{2}, 0 ∣ + ∣ e_{3}, 0 ⟩ ⟨ e_{3}, 0 ∣ + \frac{γ}{c ^{2}} ∣ u^{'} ⟩ ⟨ v ∣) ∣ u ⟩ = γ (∣ u^{'} ⟩ + ∣ v ⟩) .

Expressing

∣ u^{'} ⟩

in the basis

{e_{1}, e_{2}, e_{3}}

, the operator becomes

(γ (1 + \frac{v}{c ^{2}} u_{1}^{'}) ∣ e_{1}, 0 ⟩ ⟨ e_{1}, 0 ∣ + γ \frac{v}{c ^{2}} u_{2}^{'} ∣ e_{2}, 0 ⟩ ⟨ e_{1}, 0 ∣ + γ \frac{v}{c ^{2}} u_{3}^{'} ∣ e_{3}, 0 ⟩ ⟨ e_{1}, 0 ∣ + ∣ e_{2}, 0 ⟩ ⟨ e_{2}, 0 ∣ + ∣ e_{3}, 0 ⟩ ⟨ e_{3}, 0 ∣) \cdot ∣ u, 0 ⟩ = γ (∣ u^{'} ⟩ + ∣ v ⟩) .

The operator

γ (1 + \frac{v}{c ^{2}} u_{1}^{'}) ∣ e_{1}, 0 ⟩ ⟨ e_{1}, 0 ∣ + γ \frac{v}{c ^{2}} u_{2}^{'} ∣ e_{2}, 0 ⟩ ⟨ e_{1}, 0 ∣ + γ \frac{v}{c ^{2}} u_{3}^{'} ∣ e_{3}, 0 ⟩ ⟨ e_{1}, 0 ∣ + ∣ e_{2}, 0 ⟩ ⟨ e_{2}, 0 ∣ + ∣ e_{3}, 0 ⟩ ⟨ e_{3}, 0 ∣

has the inverse

\frac{1}{γ ( 1 + \frac{v}{c ^{2}} u _{1}^{'} )} ∣ e_{1}, 0 ⟩ ⟨ e_{1}, 0 ∣ - \frac{v}{c ^{2}} \frac{u _{2}^{'}}{1 + \frac{v}{c ^{2}} u _{1}^{'}} ∣ e_{2}, 0 ⟩ ⟨ e_{1}, 0 ∣ - \frac{v}{c ^{2}} \frac{u _{3}^{'}}{1 + \frac{v}{c ^{2}} u _{1}^{'}} ∣ e_{3}, 0 ⟩ ⟨ e_{1}, 0 ∣ + ∣ e_{2}, 0 ⟩ ⟨ e_{2}, 0 ∣ + ∣ e_{3}, 0 ⟩ ⟨ e_{3}, 0 ∣ .

We can express the velocity

∣ u ⟩

∣ u, 0 ⟩ = \frac{1}{γ ( 1 + \frac{v}{c ^{2}} u _{1}^{'} )} ∣ e_{1}, 0 ⟩ ⟨ e_{1}, 0 ∣ - \frac{v}{c ^{2}} \frac{u _{2}^{'}}{1 + \frac{v}{c ^{2}} u _{1}^{'}} ∣ e_{2}, 0 ⟩ ⟨ e_{1}, 0 ∣ - \frac{v}{c ^{2}} \frac{u _{3}^{'}}{1 + \frac{v}{c ^{2}} u _{1}^{'}} ∣ e_{3}, 0 ⟩ ⟨ e_{1}, 0 ∣ + ∣ e_{2}, 0 ⟩ ⟨ e_{2}, 0 ∣ + ∣ e_{3}, 0 ⟩ ⟨ e_{3}, 0 ∣ \cdot (v γ ∣ e_{1}, 0 ⟩ + γ ∣ u^{'} ⟩),

which simplifies to

∣ u, 0 ⟩ = \frac{1}{1 + \frac{v}{c ^{2}} u _{1}^{'}} ((v + u_{1}^{'}) ∣ e_{1}, 0 ⟩ + 1 - \frac{v ^{2}}{c ^{2}} u_{2}^{'} ∣ e_{2}, 0 ⟩ + 1 - \frac{v ^{2}}{c ^{2}} u_{3}^{'} ∣ e_{3}, 0 ⟩) .

Suppressing the zero time component in the notation, we can write the previous equation in the more compact vector form

Eq:Velocity_composition

∣ u ⟩ = \frac{1}{1 + \frac{⟨ v ∣ u ^{'} ⟩}{c ^{2}}} (\frac{γ - 1}{γ} \frac{⟨ u ^{'} ∣ v ⟩}{v ^{2}} ∣ v ⟩ + ∣ v ⟩ + \frac{∣ u ^{'} ⟩}{γ}) .

As a final remark, we should note two important cases. First, from equation velocity composition , we notice that the velocity of any object at rest in

F^{'}

is observed in

F

to move with velocity

∣ v ⟩

, as expected. Conversely, any object at rest in

F

is observed in

F^{'}

to move with velocity

- ∣ v ⟩

, as follows from equation inverse velocity composition . The important point is that, although the velocity vector changes sign when the roles of the two inertial frames are exchanged, the magnitude of their relative velocity does not. If

F^{'}

moves with velocity

v

relative to

F

, then

F

moves with velocity

- v

relative to

F^{'}

, and therefore both frames assign the same relative speed

v

to their motion. Consequently, they also associate the same Lorentz factor

γ (v)

with their relative motion. This reciprocity will play an important role in the discussion of the twin paradox.

III. Limiting Cases

In this section we derive the limiting formulas for the composition of velocities when one of the relevant velocities approaches the speed of light.

1. A Particle Moving at the Speed of Light in $F$

We start by considering the case where

∣ u ⟩

approaches the speed of light

c

and determine the value that

∣ u^{'} ⟩

reaches. To achieve this, we compute the square modulus of

∣ u^{'} ⟩

from inverse velocity composition , which reads

u^{'2} = \frac{1}{( 1 - \frac{⟨ v ∣ u ⟩}{c ^{2}} ) ^{2}} (\frac{⟨ u ∣ v ⟩ ^{2}}{c ^{2}} + v^{2} + \frac{u ^{2}}{γ ^{2}} - 2 ⟨ u ∣ v ⟩) .

When

∣ u ⟩ = c ∣ n ⟩

, where

∣ n ⟩

is a unit vector, the equation above simplifies to

u^{'2} = \frac{c ^{2}}{( 1 - \frac{⟨ v ∣ n ⟩}{c} ) ^{2}} (1 - \frac{⟨ v ∣ n ⟩}{c})^{2} = c^{2},

which implies that the velocity

∣ u^{'} ⟩

reaches the speed of light when

∣ u ⟩

approaches the speed of light.

2. A Particle Moving at the Speed of Light in $F^{'}$

Similarly, we can determine the limit of the velocity

∣ u ⟩

when the velocity

∣ u^{'} ⟩

approaches the speed of light. From the velocity composition formula velocity composition , the modulus of the velocity

∣ u ⟩

reads

u^{2} = \frac{1}{( 1 + \frac{⟨ v ∣ u ^{'} ⟩}{c ^{2}} ) ^{2}} (\frac{⟨ u ^{'} ∣ v ⟩ ^{2}}{c ^{2}} + v^{2} + \frac{u ^{'2}}{γ ^{2}} + 2 ⟨ u^{'} ∣ v ⟩) .

This differs from the previous expression only by the replacement

v \to - v

. When

∣ u^{'} ⟩ = c ∣ n ⟩

, where

∣ n ⟩

is a unit vector, the equation above simplifies to

u^{2} = \frac{c ^{2}}{( 1 + \frac{⟨ v ∣ n ⟩}{c} ) ^{2}} (1 + \frac{⟨ v ∣ n ⟩}{c})^{2} = c^{2},

which, also in this case, means that the velocity

∣ u ⟩

reaches the speed of light when

∣ u^{'} ⟩

approaches the speed of light.

3. Relative Motion Approaching the Speed of Light

Finally, consider the limit in which the magnitude of the relative velocity approaches the speed of light,

∣ v ⟩ ⟶ c ∣ n ⟩,

where

∣ n ⟩

is a unit vector. In this limit, the modulus of the velocity

∣ u ⟩

becomes

u^{2} ⟶ \frac{c ^{2}}{( 1 + \frac{⟨ u ^{'} ∣ n ⟩}{c} ) ^{2}} (1 + \frac{⟨ u ^{'} ∣ n ⟩}{c})^{2} = c^{2},

independently of the value of

∣ u^{'} ⟩

Contents

I. Introduction

II. Derivation of Composition of Velocities

III. Limiting Cases

1. A Particle Moving at the Speed of Light in F

2. A Particle Moving at the Speed of Light in F′

3. Relative Motion Approaching the Speed of Light

1. A Particle Moving at the Speed of Light in $F$

2. A Particle Moving at the Speed of Light in $F^{'}$