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MATHEMATICAL PROCEDURE by which Albert Einstein derived Lorentz transformation is incorrect. The transformation is an imaginary "solution" to a set of equations which evaluate to zero throughout the derivation process. Author derives Lorentz transformation the way Einstein did, and shows the places where errors were made.
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Mathematical Invalidity of the
Lorentz Transformation in Relativity Theory
Aleksandar Vukelja
aleksandar@masstheory.org
http://www.masstheory.org
April 2005

LEGAL:
This work is released in public domain.
Abstract
MATHEMATICAL PROCEDURE by which Albert Einstein derived Lorentz
transformation is incorrect. The transformation is an imaginary "solution" to a set of
equations which evaluate to zero throughout the derivation process.
Author derives Lorentz transformation the way Einstein did, and shows the places
where errors were made.
NOTE: This document is found through search engines more often than
Triangle of Velocities”, which offers complete analysis of Lorentz transformation
and mathematical proof that the transformation is based on error.
Reader is encouraged to read http://www.masstheory.org/triangle_of_velocities.pdf
2

Critical Analysis of Einstein's paper
“Simple Derivation of the Lorentz Transformation”
Readers will find in the appendix B unmodified copy of Einstein's derivation, as published in “Relativity:
The Special and General Theory”, 1920.
We have two frames of reference, K which is still, and K' which is moving along the
positive side of x axes of K with speed v.
A light signal, which is proceeding along the positive axis of x, is transmitted
according to the equation
x=ct or xct=0
(1)
where c is the speed of light.
(Postulate 1) Assuming that the speed of light is independent of the speed of observer
v, in K' for the same front of the light signal we write
x ' =ct ' or x ' ct ' =0
(2)
To find out how to transform coordinates between K and K',
(Error 1) (Postulate 2) We assume that there is a proportionality quotient λ,
x ' ct ' =  xct
(3)
(Error 2) If we apply quite similar considerations to light rays which are being
transmitted along the negative x axis,
(Postulate 3) then we have another quotient µ
x ' ct ' =  xct
(4)
By adding (3) and (4) we have
x ' =a xb ct
(5a)
ct ' =actbx
(5b)
 
 −
where for convenience we introduced a=
and b=
.
2
2
If we would know quotients a and b, our derivation would be complete. For the base
of K' we always have x' = 0,
(Error 3) then from (5a) we have
bc
x bc
x=
t or v= =
(6)
a
t
a
where v is the speed of K' system relative to K.
3

Observed from K, relation of x' and x recorded when t = 0, from (5a) is
x ' =ax
(7)
Two points of the x' axis which are separated by the distance x' = 1 measured in K' are
thus separated at the same moment in K by the distance
1
x=
(8)
a
But if the snapshot be taken from K' at the moment t' = 0,
(Error 4) Observed from K', relation between x' and x recorded when t' = 0, from (5b)
we have
bx
act=bx or t=
(9)
ac
(Error 5) and by using (9) in the same equation (5a) from which it was derived, we get
b2
x ' =a 1−  x
(10)
a2
Using expression (6) for speed v, expression (10) becomes:
v2
x ' =a 1−  x
(11)
c2
From this we conclude that two points on the x axis and separated by the distance 1
(relative to K) will be represented on our snapshot by the distance:
v2
x '=a1− 
(12)
c2
But from what has been said, the two snapshots must be identical; hence  x in (8)
must be equal to  x ' in (12), so we obtain:
1
a2=
v2
(13)
1− c2
The equations (6) and (13) determine the constants a and b. By inserting the values of
these constants in (5a), we obtain the Lorentz Transformation:
v
t
x
xvt
c2
x ' =  ,t'=
(14)
v2
v2
1−
1−
c2
c2
4

Explanations of Errors
Error 1 Expression (3) is useless. We have
xct=0
(1)
x ' ct ' =0
(2)
In (3) Einstein writes
x ' ct ' =  xct
(3)
Because of (1) and (2) we can write (3) as
0= 0
One can postulate that meaningful values, which are at least sometimes both different
from zero, are somehow related. Introducing proportionality quotient between nothing
and nothing has no meaning.
Error 2 Explanation is inapplicable. If rays of light are traveling in both positive and
negative directions of the x axes, then one cannot combine their positive and negative
x coordinates as they represent different events (addition of apples and oranges).
Equation (4) can be used in a valid context, and that is postulation of new relationship
between the same coordinates on the positive side, besides existing postulate (3).
However this is not what he wrote.
Error 3 Expression (6) is a cardinal error. Derivation begins with (1) x=ct for the
front of the beam of light.
Few lines below, in (6) Einstein now presents relative speed of two coordinate
systems as
x
v=
or x=vt in the same context.
t
This can only be valid so long as vt=ct or v=c . As that was the point of
inclusion of relative speed v into equations, all that follows is meaningless.
Error 4 Expression (9) is valid only when x' = 0, which according to (2) implies also
that t' = 0. This can happen only when also x = 0 and t = 0, rendering (9) useless.
Otherwise, because (5a) and (5b) are the same equation, when x' = 0 then t' = 0 and we
have
x bc
=
from (5a)
t
a
x
ac
=
from (5b)
t
b
Which would it be if x≠0 ?
5

Error 5 Expression (11) is yet another cardinal error. It is accumulation of all previous
errors, and adds additional nonsense:
One cannot assume x' = 0 in (5b) and then include (9) which was derived from that
assumption, back into the same equation (5a) and then pretend that x' is now different
from zero. This is how (11) was derived.
It takes exceptionally strong illusions and lack of math skills to make five such errors
on a single sheet of paper.
By making such errors, one can derive nearly anything, Lorentz transformation
included. Unfortunately for science, this is one thing that his many followers learned
very well from him, and happily applied.
References
Albert Einstein
Relativity: The Special and General Theory. (a copy of examined text is included in
appendix)
This article was inspired by work of
Milan Pavlović
“Ajnštajnova teorija relativnosti – naučna teorija ili OBMANA”, downloaded from
http://users.net.yu/~mrp
6

Appendix – copy of Einstein's original derivation
Albert Einstein (1879–1955). Relativity: The Special and General Theory. 1920.
Simple Derivation of the Lorentz Transformation
[SUPPLEMENTARY TO SECTION XI]
FOR the relative orientation of the co-ordinate systems indicated in Fig. 2, the x-axes of both systems permanently
coincide. In the present case we can divide the problem into parts by considering first only events which are
localized on the x-axis. Any such event is represented with respect to the co-ordinate system K by the abscissa x
and the time t, and with respect to the system k' by the abscissa x' and the time t'. when x and t are given.
A light-signal, which is proceeding along the positive axis of x, is transmitted according to the equation
x=ct or xct=0
(1)
Since the same light-signal has to be transmitted relative to k' with the velocity c, the propagation relative to the
system k' will be represented by the analogous formula
x ' ct ' =0
(2)
Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the
relation
x 'ct ' =  xct
(3)
is fulfilled in general, where l indicates a constant; for, according to (3), the disappearance of (x – ct) involves the
disappearance of (x' – ct').
If we apply quite similar considerations to light rays which are being transmitted along the negative x-axis, we obtain
the condition
x 'ct ' =  xct
(4)
By adding (or subtracting) equations (3) and (4), and introducing for convenience the constants a and b in place of
the constants l and m where
 
a= 2
and
 −
b= 2
we obtain the equations
x ' =axbct
(5)
ct ' =actbx
We should thus have the solution of our problem, if the constants a and b were known. These result from the
following discussion.
For the origin of k' we have permanently x' = 0, and hence according to the first of the equations (5)
bc
x=
t
a
If we call v the velocity with which the origin of k' is moving relative to K, we then have
bc
v=
(6)
a
The same value v can be obtained from equation (5), if we calculate the velocity of another point of k' relative to K,
7

or the velocity (directed towards the negative x-axis) of a point of K with respect to K'. In short, we can designate v
as the relative velocity of the two systems.
Furthermore, the principle of relativity teaches us that, as judged from K, the length of a unit measuring-rod which is
at rest with reference to k' must be exactly the same as the length, as judged from K', of a unit measuring-rod which
is at rest relative to K. In order to see how the points of the x'-axis appear as viewed from K, we only require to take
a “snapshot” of k' from K; this means that we have to insert a particular value of t (time of K), e.g. t = 0. For this
value of t we then obtain from the first of the equations (5)
x ' =ax
Two points of the x'-axis which are separated by the distance x'=1 when measured in the k' system are thus
separated in our instantaneous photograph by the distance
1
x=
(7)
a
But if the snapshot be taken from K'(t' = 0), and if we eliminate t from the equations (5), taking into account the
expression (6), we obtain
v2
x ' =a 1−  x
c2
From this we conclude that two points on the x-axis and separated by the distance 1 (relative to K) will be
represented on our snapshot by the distance
v2
x '=a 1− 
(7a)
c2
But from what has been said, the two snapshots must be identical; hence ∆x in (7) must be equal to ∆x' in (7a), so
that we obtain
1
a2=
v2
(7b)
1− c2
The equations (6) and (7b) determine the constants a and b. By inserting the values of these constants in (5), we
obtain the first and the fourth of the equations given in Section XI.
xvt
x ' =  v2
1− c2
v
t
x
(8)
c2
t ' =  v2
1− c2
Thus we have obtained the Lorentz transformation...
8

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