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 inertial reference frames.

This change in the velocity addition formula from the non-relativistic to the relativistic theory is not the result of making measurements that have neglected to take light-travel times or the Doppler effect into account. Rather, it is what is observed after such effects have been accounted for. It is an effect of special relativity which cannot be accounted for using 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

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 might not seem to make sense. But this is due to a
misunderstanding of the idea, 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 calculates the speeds with which 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 situation, the two velocities *do* add according to ordinary vector
addition.

But that was a different scenario and question to the first one asked above. Originally we asked for the
velocity of *C* as measured relative to *A*, and not the speed at which
*B* observes *A* and *C* to approach each other. This is different because the rulers
and clocks set up by *B* cannot be used to measure distances and times correctly by *A*, since
for *A* the clocks do not even show the same time. To go from the reference frame of *A* to the
reference frame of *B*, we must apply a Lorentz transformation on co-ordinates in the following way (taking
the x-axis parallel to the direction of travel and the spacetime origins to coincide):

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 velocity addition formula is that if you combine two velocities less than the speed of light, you always get a result that is still less than the speed of light. This means that 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 rapidity *t*
of the combined velocities is given by the 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 solving problems with 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. But note 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

wBut 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.