Adding Velocities: UNIZOR.COM - Relativity 4 All - Einstein View
Channel: Zor Shekhtman on UNIZOR Education
Date: 2023-07-04

UNIZOR.COM - Physics through a Looking Glass of Mathematician
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Adding Velocities

In classic Mechanics that uses Galilean transformation, if an object moves in reference frame β with velocity uβ, 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:
uα = uβ + 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 uβ.
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 uα, 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) = uβ·t
Obviously,
dx(t)/dt = uβ.

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 uα=dX/dT.

Using an explicit form of function x(t), 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 uα as
uα = dX/dT = (dX/dt)/(dT/dt)
and we can easily resolve our problem as follows
dX/dt = (v+uβ )/√1−(v/c)² 
dT/dt = (1+v·uβ /c²)/√1−(v/c)² 
dX/dT = (v+uβ )/(1+v·uβ /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 uβ 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.