Updated by Terence Tao, 1997.
Original by Philip Gibbs, 1996.

# How Do You Add Velocities in Special Relativity?

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 A is moving with a velocity w = u+v relative to C.  But in special relativity the velocities must be combined using the formula
u + v
w =  ---------
1 + uv/c2

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/c2
Notice that the only case with velocities less than or equal to c that is singular is w = u = c, which gives the indeterminate value zero divided by zero.  In other words, it's meaningless to ask for the relative velocity of two photons that are moving in the same direction.

## How can that be right?

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):

xB = γ(v)( xA - v tA )
tB = γ(v)( tA - v/c2 xA )

γ(v) = 1/sqrt(1-v2/c2)
To go from the frame of B to the frame of C you must apply a similar transformation
xC = γ(u)( xB - u tB )
tC = γ(u)( tB - u/c2 xB )

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

xC = γ(w)( xA - w tA )
tC = γ(w)( tA - w/c2 xA)

u + v
w =  ---------
1 + uv/c2
This 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 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 rapidity t of the combined velocities is given by the simple addition

t = r + s
This 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 velocity addition formula for non-parallel velocities

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 = (vx,0,0) in A's reference frame, and C is moving with velocity u = (ux, uy, uz) in B's reference frame.  The question is to find the velocity w = (wx, wy, wz) of C in A's reference frame.  This is still not quite the most general situation, since we are assuming B to be moving in the direction of A's x-axis, but it is a decent compromise, since the most general formula is somewhat messy.  In any event, one can always orient A's frame using Euclidean rotations so that B's direction of motion lies along the x-axis.

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

yB = yA
zB = zA

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

xB = γ(vx)( xA - vx tA )
yB = yA
zB = zA
tB = γ(vx)( tA - vx/c2 xA )
Since C is moving along the line
(xB,yB,zB,tB) = (ux t, uy t, uz t, t) (t real),
we see, after some computation, that in A's frame of reference C is moving along the line
(xA,yA,zA,tA) = (wx s, wy s, wz s, s) (s real),
where
ux + vx
wx = ------------
1 + uxvx/c2

uy
wy = -------------------
(1 + uxvx/c2) γ(vx)

uz
wz = -------------------
(1 + uxvx/c2) γ(vx)

γ(vx) = 1/sqrt(1 - vx2/c2)

Thus the velocity w = (wx, wy, wz) of C with respect to A is given by the above three formulae, assuming that B is orientated in the standard way with respect to A.  Note that if uy=uz=0 then this reduces to the simpler velocity addition formula given before.

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

## Relative speeds

If an observer A measures two objects B and C to be travelling at velocities u = (ux, uy, uz) and v = (vx, vy, vz) respectively, one may ask the question of what the relative speed between B and C are, or in other words at what speed w B would measure C to be travelling at, or vice versa.  In galileian relativity the relative speed would be given by

w2 = (u-v).(u-v) = (ux - vx)2 + (uy - vy)2 + (uz - vz)2.
But in special relativity the relative speed is instead given by the formula
(u-v).(u-v) - (u × v)2/c2
w2 =    -------------------------
(1 - (u.v)/c2)2

where u-v = (ux - vx, uy - vy, uz - vz) is the vector difference of u and v, u.v = ux vx + uy vy + uz vz is the inner product of u and v, and u×v is the vector product for which (u×v)2 = (u.u)(v.v) - (u.v)2.

When uy = uz = vy = vz = 0, the formula reduces to the more familiar

|ux - vx|
w =  -------------
1 - ux vx/c2

## 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.