Updated by Terence Tao 22-February-1997

Original by Philip Gibbs 6-October-1996

Suppose an object *A* is moving with a velocity *v* relative
to an object *B* and *B* is moving with a velocity *u*
(in the same direction) relative to an object *C*. What is the
velocity of *A* relative to *C*?

vu------->A------->BCw----------------->

In non-relativistic mechanics the velocities are simply added
and the answer is that *A* is moving with a velocity *w = u+v* relative
to *C*. But in special relativity the velocities must be combined
using the formula

w = (u + v)/(1 + uv/c^{2})

If *u* and *v* are both small compared to the speed
of light *c*, then the answer is approximately the same as the
non-relativistic theory. In the limit where *u* is equal to
*c* (because *C* is a massless particle moving to the
left at the speed of light), the sum gives *c*. This
confirms that anything going at the speed of light does so in all
reference frames.

This change in the velocity addition formula is not due to making measurements without taking into account time it takes light to travel or the Doppler effect. It is what is observed after such effects have been accounted for and is an effect of special relativity which cannot be accounted for with Newtonian mechanics.

The formula can also be applied to velocities in opposite directions
by simply changing signs of velocity values or by rearranging the
formula and solving for *v*. In other words, If *B* is
moving with velocity *u* relative to *C* and *A*
is moving with velocity *w* relative to *C* then the
velocity of *A* relative to *B* is given by,

v = (w - u)/(1 - wu/c^{2})

Notice that the only case with velocities less than or equal to
*c* which is singular is *w = u = c* which gives the
indeterminate zero divided by zero. In other words it is meaningless
to ask the relative velocity of two photons going in the same
direction.

Naively the relativistic formula for adding velocities does not
seem to make sense. This is due to a misunderstanding of the question
which can easily be confused with the following one: Suppose the
object *B* above is an experimenter who has set up a reference
frame consisting of
a marked ruler with clocks positioned at measured intervals along it.
He has synchronised the clocks carefully by sending light signals
along the line taking into account the time taken for the signals
to travel the measured distances. He now observes the objects *A*
and *C* which he sees coming towards him from opposite
directions. By watching the times they pass the clocks at measured
distances he can calculate the speeds they are moving towards him.
Sure enough he finds that *A* is moving at a speed *v*
and *C* is moving at speed *u*. What will *B*
observe as the speed at which the two objects are coming together?
It is not difficult to see that the answer must be *u+v*
whether or not the problem is treated relativistically. In this
sense velocities add according to ordinary vector addition.

But that was a different question from the one asked before.
Originally we wanted to know the speed of *C* as measured
relative to *A* not the speed at which *B* observes
them moving together. This is different because the rulers and
clocks set up by *B* do not measure distances and times
correctly in the reference from of *A* where the clocks
do not even show the same time. To go from the reference
frame of *A* to the reference frame of *B* you need to
apply a Lorentz transformation on co-ordinates as follows
(taking the x-axis parallel to the direction of travel).

x_{B}= gamma(v)( x_{A}- v t_{A}) t_{B}= gamma(v)( t_{A}- v/c^{2}x_{A}) gamma(v) = 1/sqrt(1-v^{2}/c^{2})

To go from the frame of *B* to the frame of *C*
you must apply a similar transformation

x_{C}= gamma(u)( x_{B}- u t_{B}) t_{C}= gamma(u)( t_{B}- u/c^{2}x_{B})

These two transformations can be combined to give a transformation which simplifies to

x_{C}= gamma(w)( x_{A}- w t_{A}) t_{C}= gamma(w)( t_{A}- w/c^{2}x_{A}) w = (u + v)/(1 + uv/c^{2})

This gives the correct formula for combining parallel velocities in
special relativity. A feature of the formula is that if you
combine two velocities less than the speed of light you always
get a result which is still less than the speed of light. Therefore
no amount of combining velocities can take you beyond light speed.
Sometimes physicists find it more convenient to talk about the
**rapidity** *r* which is defined by the relation,

v = c tanh(r/c)

The hyperbolic tangent function *tanh* maps the real line
from minus infinity to plus infinity onto the interval -1 to +1.
So while velocity *v* can only vary between *-c* and *c*,
the rapidity *r* varies over all real values. At small
speeds rapidity and velocity are approximately equal. If *s*
is also the rapidity corresponding to velocity *u* then
the combined rapidity *t* is given by simple addition.

t = r+s

This follows from the identity of hyperbolic tangents

tanh(x+y) = ( tanh(x) + tanh(y) )/( 1 + tanh(x)tanh(y) )

Rapidity is therefore useful when dealing with combined velocities in the same direction and also for problems of linear acceleration

For example, if we combine the speed *v* *n*
times, the result is,

w = c tanh( n tanh^{-1}(v/c) )

The previous discussion only concerned itself with the case
when both velocities *v* and *u* were aligned along
the *x*-axis;
the *y* and *z* directions were ignored.

Let us now consider a more general case, where *B* is moving with
velocity *v = (v _{x},0,0)* in

There is one additional assumption we will need to make before we
can give the formula. Unlike the case of one spatial dimension, the
relative orientations of *B*'s frame of reference and *A*'s frame of
reference is now important. What *B* perceives as motion in the
*x*-direction (or *y*-direction, or
*z*-direction) may not agree with
what *A* perceives as motion in the *x*-direction (etc.), if *B* is facing
in a different direction from *A*.

We will thus make the simplifying assumption that *B* is oriented
in the standard way with respect to *A*, which means that the spatial
co-ordinates of their respective frames agree in all directions orthogonal
to their relative motion. In other words, we are assuming that

y_{B}= y_{A}z_{B}= z_{A}

In the technical jargon, we are requiring *B*'s frame of reference to be
obtained from *A*'s frame by a standard Lorentz transformation (also known
as a Lorentz boost).

In practice, this assumption is not a major obstacle,
because if *B* is not initially oriented in the
standard way with respect to *A*, it can be made
to be so oriented by a purely spatial rotation of axes.
However, it should be warned that if *B* is oriented
in the standard way with respect to *A*,
and *C* is oriented in the standard way with respect
to *B*, then it is not necessarily true that *C*
is oriented in the standard way with respect to *A*!
This phenomenon is known as **precession**. It's roughly analogous
to the three-dimensional fact that, if one rotates an object
around one horizontal axis and then about a second horizontal
axis, the net effect would be a rotation around an axis which
is not purely horizontal, but which will contain some vertical
components.

If *B* is oriented in the standard way with respect
to *A*, the Lorentz transformations are given by

x_{B}= gamma(v_{x})( x_{A}- v_{x}t_{A}) y_{B}= y_{A}z_{B}= z_{A}t_{B}= gamma(v_{x})( t_{A}- v_{x}/c^{2}x_{A})

Since C is moving along the line

{ (x_{B},y_{B},z_{B},t_{B}) = (u_{x}t, u_{y}t, u_{z}t, t): t real },

we see, after some computation, that in *A*'s frame of reference *C* is moving
along the line

{ (x_{A},y_{A},z_{A},t_{A}) = (w_{x}s, w_{y}s, w_{z}s, s): s real },

where

w_{x}= (u_{x}+ v_{x}) / (1 + u_{x}v_{x}/ c^{2}) w_{y}= u_{y}/ [(1 + u_{x}v_{x}/ c^{2}) gamma(v_{x})] w_{z}= u_{z}/ [(1 + u_{x}v_{x}/ c^{2}) gamma(v_{x})]. gamma(v_{x}) = 1/sqrt(1 - v_{x}^{2}/ c^{2}).

Thus the velocity *w = (w _{x}, w_{y}, w_{z})*
of

References: "Essential Relativity", W. Rindler, Second Edition. Springer-Verlag 1977.

If an observer *A* measures two objects *B* and *C*
to be travelling at velocities
*u = (u _{x}, u_{y}, u_{z})* and

*
w ^{2} = (u-v).(u-v) = (u_{x} - v_{x})^{2} + (u_{y} - v_{y})^{2} + (u_{z} - v_{z})^{2}.
*

However, in special relativity the relative speed is instead given by the formula

(u-v).(u-v) - (uXv)^{2}/c^{2}w^{2}= ---------------------- (1 - (u.v)/c^{2})^{2}

where
*u-v = (u _{x} - v_{x}, u_{y} - v_{y}, u_{z} - v_{z})*
is the vector difference of

When *u _{y} = u_{z} = v_{y} = v_{z} = 0*
the formula reduces to the more familiar

w = |u_{x}- v_{x}| / (1 - u_{x}v_{x}/c^{2}).

References:

N. M. J. Woodhouse, "Special Relativity", Lecture Notes in
Physics (m: 6), Springer Verlag, 1992.

J. D. Jackson, "Classical Electrodynamics", 2nd ed., 1975, ch 11.

P. Lounesto, "Clifford Algebras and Spinors", CUP, 1997