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

Notes to a video lecture on UNIZOR.COM

Adding Velocities
along Y-, Z-axis

Let's complete our formulas for adding velocities with those related to Y- and Z-axis, while assuming that the movement of one reference frame relatively to another is along the X-axis.

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 some direction with X-, Y- and Z-components of its velocity vector u_βx, u_βy and u_βz correspondingly.
At time t=0 this object is at the origin of β-frame.

Our task is to determine the components u_αx, u_αy and u_αz of the velocity of this object in α-frame, using Lorentz Transformation.

In the previous lecture we discussed the analogous task but restricted only to an object moving in the β-frame along X-axis.
This lecture is about a more general movement of the object along any direction.
Obviously, the formula derived in the previous lecture for a movement along X-axis is fully applicable to the X-component of the general velocity vector, and out task is restricted to add Y- and Z-components of the velocity vector.

We will use exactly the same methodology for Y- and Z-components as we used for X-component.

The object in β-frame moves according to this formulas for its coordinates
x(t) = u_βx·t
y(t) = u_βy·t
z(t) = u_βz·t

Obviously,
dx(t)/dt = u_βx.
dy(t)/dt = u_βy.
dz(t)/dt = u_βz.

According to Lorentz transformation,

T =	t + v·x/c² √1−(v/c)²
X =	x + v·t √1−(v/c)²
Y =	y
Z =	z

Our task is to find all the components of the velocity of an object in α-frame, that is to find
u_αx=dX/dT
u_αy=dY/dT
u_αz=dZ/dT

All calculations we did in the previous lecture, when motion of an abject was restricted to X-direction, are good for getting u_αx, but we will repeat them here.

Using an explicit form of functions x(t), y(t) and z(t), the above formulas of Lorentz Transformation are

T =	t + v·uβx·t/c² √1−(v/c)²
X =	uβx·t + v·t √1−(v/c)²
Y =	uβy·t
Z =	uβz·t

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

The final formula for X-component of the object's velocity u_αx in α-frame is

uαx =

v + uβx 1 + v·uβx /c²

Let's analyze the Y-component of the object's velocity.
Since
dY/dt = (dY/dT)·(dT/dt),
we can express the u_αy=dY/dT as
u_αy = dY/dT = (dY/dt)/(dT/dt)
and we can easily resolve our problem as follows
dY/dt = dy/dt = u_βy
As before,
dT/dt = (1+v·u_βx /c²)/√1−(v/c)² 
Therefore,

uαy =

uβy·√1−(v/c)²  (1+v·uβx /c²)

In particular, if an object moves along y-axis in β-frame (u_βx=0), its Y-axis speed in α-frame is
u_αy = u_βy·√1−(v/c)² 

The Z-component of the velocity is analogous to Y-component.
Since
dZ/dt = (dZ/dT)·(dT/dt),
we can express the u_αz=dZ/dT as
u_αz = dZ/dT = (dZ/dt)/(dT/dt)
and we can easily resolve our problem as follows
dZ/dt = dz/dt = u_βz
As before,
dT/dt = (1+v·u_βx /c²)/√1−(v/c)² 
Therefore,

uαz =

uβz·√1−(v/c)²  (1+v·uβx /c²)

In particular, if an object moves along z-axis in β-frame (u_βx=0), its Z-axis speed in α-frame is
u_αz = u_βz·√1−(v/c)²