Adding Velocities: UNIZOR.COM - Relativity 4 All - Einstein View

Notes to a video lecture on UNIZOR.COM

Adding Velocities

In classic Mechanics that uses Galilean transformation, if an object moves in reference frame β with velocity u_β, and its β-frame moves relative to another frame α with velocity v, the composition of movements of an object and its frame can be represented in α-frame as a simple vector sum of velocities of an object:
u_α = u_β + v

In Theory of Relativity that uses Lorentz Transformation the situation is a bit more complex.

Consider, as before, two inertial reference frames:
α-frame with coordinates {X,Y,Z} and time T and
β-frame with coordinates {x,y,z} and time t.

At time T=t=0 both reference frames coincide.
Assume that β-frame uniformly moves relatively to α-frame with speed v along α-frame's X-axis, maintaining parallelism of all corresponding axes.

Assume an object moves uniformly in β-frame along its x-axis with speed u_β.
At time t=0 this object is at the origin of β-frame.

Our task is to determine the speed of this object in α-frame u_α, using Lorentz Transformation.

IMPORTANT NOTE:
We deliberately restricted movements only along the X- and x-axes to simplify the calculations.
To expand the results to all three space dimensions is trivial.

The object in β-frame moves according to a simple law
x(t) = u_β·t
Obviously,
dx(t)/dt = u_β.

According to Lorentz transformation,

T =	t + v·x/c² √1−(v/c)²

X =	x + v·t √1−(v/c)²

Our task is to find the speed of an object in α-frame, that is to find u_α=dX/dT.

Using an explicit form of function x(t), the above formulas of Lorentz Transformation are

T =	t + v·uβ·t/c² √1−(v/c)²

X =	uβ·t + v·t √1−(v/c)²

The above expressions are linear functions of time t and their derivative by t would result in constants.

Since
dX/dt = (dX/dT)·(dT/dt),
we can express the u_α as
u_α = dX/dT = (dX/dt)/(dT/dt)
and we can easily resolve our problem as follows
dX/dt = (v+u_β )/√1−(v/c)² 
dT/dt = (1+v·u_β /c²)/√1−(v/c)² 
dX/dT = (v+u_β )/(1+v·u_β /c²)

The final formula is

uα =	v + uβ 1 + v·uβ /c²

The above formula represents the law of addition of velocities in Theory of Relativity.

Obviously, if speeds of an object u_β and of β-frame v are low relative to speed of light c, which is the case under most normal conditions, the above transformation is approximated by the Galilean one quite well.