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The Lorentz Transformation for Velocity

"Great spirits have always encountered violent oppositions from mediocre
minds." - A. Einstein

It is recognized that Special Relativity provides the Lorentz
Transformations for Length and Time and Mass between reference frames having
a relative velocity. What is too often not recognized that, in so doing, it
provides the transformations for all other physical quantities! These
remaining transformations may be determined by applying the conventional
Lorentz Transformations to accepted physical equations.

Since a velocity is equal to a length divided by the time required to
traverse that length, the Lorentz Transformation for Velocity must equal the
Lorentz Transformation for Length [1/(1-V^2/C^2)^0.5] divided by the Lorentz
Transformation for Time [(1-V^2/C^2)^0.5]. The Lorentz Transformation for
Velocity is therefore equal to 1/(1-V^2/C^2).

The length contraction equation
L = L' / sqrt( 1- v^2/c^2 )
or
L' = L sqrt( 1- v^2/c^2 )
is only valid for two events satisfying T' = 0, as can be seen
from the real Lorentz transformation. It is valid for a rod a rest
in the unprimed frame with proper length L, and therefore the
endpoints of the rod must be measured simultaneously in the
primed frame, hence T' = 0, giving the measured length L'.

The time dilation equation
T = T' sqrt( 1 - v^2/c^2 )
or
T' = T / sqrt( 1 - v^2/c^2 )
is only valid for events satisfying L = 0, as can be seen again
from the real transformation equations. It is valid for a clock
at rest in the unprimed frame which means that the tickpoints
are colocal in this unprimed frame, hence L = 0. You can
check this again with the Lorentz transformation.

So when are expressions valid together?
The first expression implies T' = 0, so from the second
expression we also have T = 0.
The second expression implies L = 0, so from the first
expression we also have L' = 0.
So, excluding the trivial case where v=0, the expressions are
valid together if and only if
L = L' = T = T' = 0

Dividing your expressions to arrive at
L/T = L'/T' / ( 1-v^2/c^2)
is therefore not allowed.

Besides, the fraction L/T has no physical meaning at all. It is the
length of a rod at rest with respect to you, divided by the time
between ticks of a clock on your wrist. This is meaningless, if
not silly.

Start wondering what the products LT and L'T' might mean.
Someone calls it Spacetime Area and insists that it is invariant,
since clearly L T = L' T'.
Right, it is the invariance of 0. Brilliant.

Calling the equations
L = L' / sqrt( 1- v^2/c^2 )
T = T' sqrt( 1 - v^2/c^2 )
L/T = L'/T' / ( 1-v^2/c^2)
Lorentz Transformations for Length, Time and Velocity indicates
that you really have no idea what you are talking about.
You are making the classic mistake of not understanding what
the variables mean. You don't understand coordinates, so you
shouldn't talk about coordinate transformations.


This transformation yields a conclusion which is in conflict with the
current understanding of Special Relativity.

No, your mistakes yield a conclusion which is in conflict with your
chronic (and apparently deliberate) misunderstanding of Special Relativity.

Dirk Vdm