Updated by Terence Tao 1997.

Original by Philip Gibbs 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*?

v u -------> A -------> B C w ----------------->In non-relativistic mechanics the velocities are simply added and the answer is that

u + v w = --------- 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

w - u v = --------- 1 - wu/cNotice that the only case with velocities less than or equal to^{2}

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):

xTo go from the frame of_{B}= γ(v)( x_{A}- v t_{A}) t_{B}= γ(v)( t_{A}- v/c^{2}x_{A}) γ(v) = 1/sqrt(1-v^{2}/c^{2})

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

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

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

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 + sThis follows from the identity of hyperbolic tangents

tanh x + tanh y tanh(x+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.

Consider now 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

xSince C is moving along the line_{B}= γ(v_{x})( x_{A}- v_{x}t_{A}) y_{B}= y_{A}z_{B}= z_{A}t_{B}= γ(v_{x})( t_{A}- v_{x}/c^{2}x_{A})

(xwe see, after some computation, that in_{B},y_{B},z_{B},t_{B}) = (u_{x}t, u_{y}t, u_{z}t, t) (t real),

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

u_{x}+ v_{x}w_{x}= ------------ 1 + u_{x}v_{x}/c^{2}u_{y}w_{y}= ------------------- (1 + u_{x}v_{x}/c^{2}) γ(v_{x}) u_{z}w_{z}= ------------------- (1 + u_{x}v_{x}/c^{2}) γ(v_{x}) γ(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 1977.

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

wHowever, in special relativity the relative speed is instead given by the formula^{2}= (u-v).(u-v) = (u_{x}- v_{x})^{2}+ (u_{y}- v_{y})^{2}+ (u_{z}- v_{z})^{2}.

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

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

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

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

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