Relativistic Addition of Velocities

​
u / c
0.75
α / 2π
0.125
v / c
0.75
β / 2π
0.25
This Demonstration considers the composition of velocities in accordance with the special theory of relativity. Consider a system moving with velocity represented by the red arrow, with magnitude
u
and direction
α
, with respect to a stationary frame of reference. The red disk recapitulates this magnitude, which has an upper limit extending to the red circle, corresponding to the speed of light
c
. The blue arrow represents a second velocity, which has a magnitude
v
and direction
β
, with respect to the moving frame of reference. The velocity with respect to the original stationary frame is then represented by
w=u⊕v
. A compact formulation gives the components of
w
parallel and perpendicular to
u
:
w

=
u+
v

1+u·v/
2
c
and
w
⊥
=
v
⊥
1+u·v/
2
c
1-
2
u
2
c
.
​The gray arrow represents the vector
w
.

Details

Snapshot 1: for
u,v≪c
, the Galilean result
w≈u+v
is a good approximation
Snapshot 2: if
u=c
or
v=c
, then
w=c
Snapshot 3: the collinear case reduces to Einstein's well-known formula
w=
u+v
1+uv/
2
c
Snapshots 4, 5: velocity addition is not commutative;
u⊕v≠v⊕u
Reference: J. D. Jackson, Classical Electrodynamics, 3rd ed., New York: John Wiley & Sons, 1998 p. 531.

External Links

Special Relativity (ScienceWorld)

Permanent Citation

S. M. Blinder
​
​"Relativistic Addition of Velocities"​
​http://demonstrations.wolfram.com/RelativisticAdditionOfVelocities/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011