Tuesday, September 12, 2023

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 uβx, uβy and uβz correspondingly.
At time t=0 this object is at the origin of β-frame.

Our task is to determine the components uαx, uαy and uαz 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) = uβx·t
y(t) = uβy·t
z(t) = uβz·t

dx(t)/dt = uβx.
dy(t)/dt = uβy.
dz(t)/dt = uβz.

According to Lorentz transformation,
T =
t + v·x/c²
X =
x + v·t
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

All calculations we did in the previous lecture, when motion of an abject was restricted to X-direction, are good for getting uαx, 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
X =
uβx·t + v·t
Y = uβy·t
Z = uβz·t

dX/dt = (dX/dT)·(dT/dt),
we can express the uαx=dX/dT as
uαx = dX/dT = (dX/dt)/(dT/dt)
and we can easily resolve our problem as follows
dX/dt = (v+uβx )/1−(v/c)² 
dT/dt = (1+v·uβx /c²)/1−(v/c)² 
dX/dT = (v+uβx )/(1+v·uβ /c²)

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

uαx =
v + uβx
1 + v·uβx /c²

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

uαy =
(1+v·uβx /c²)

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

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

uαz =
(1+v·uβx /c²)

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

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