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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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Aleksandar Vukelja

aleksandar@masstheory.org

http://www.masstheory.org

April 2005

LEGAL:

This work is released in public domain.

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.

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

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,

positive side of

A light signal, which is proceeding along the positive axis of x, is transmitted

according to the equation

(1)

where

(Postulate 1) Assuming that the speed of light is independent of the speed of observer

(2)

To find out how to transform coordinates between

(Error 1) (Postulate 2) We assume that there is a proportionality quotient λ,

(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

(4)

By adding (3) and (4) we have

(5a)

(5b)

−

where for convenience we introduced

and

.

2

2

If we would know quotients

of

(Error 3) then from (5a) we have

(6)

where

3

Observed from

(7)

Two points of the

thus separated at the same moment in

1

(8)

But if the snapshot be taken from

(Error 4) Observed from

we have

(9)

(Error 5) and by using (9) in the same equation (5a) from which it was derived, we get

(10)

Using expression (6) for speed

(11)

From this we conclude that two points on the x axis and separated by the distance 1

(relative to

(12)

But from what has been said, the two snapshots must be identical; hence

must be equal to

1

(13)

1−

The equations (6) and (13) determine the constants

these constants in (5a), we obtain the Lorentz Transformation:

(14)

1−

1−

4

Error 1 Expression (3) is useless. We have

(1)

(2)

In (3) Einstein writes

(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

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)

front of the beam of light.

Few lines below, in (6) Einstein now presents relative speed of two coordinate

systems as

or

This can only be valid so long as

inclusion of relative speed

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

=

from (5a)

=

from (5b)

Which would it be if

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.

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

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

(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

(2)

Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the

relation

(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

(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

and

−

we obtain the equations

(5)

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)

If we call v the velocity with which the origin of k' is moving relative to K, we then have

(6)

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)

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

(7)

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

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

(7a)

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

(7b)

1−

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.

1−

(8)

1−

Thus we have obtained the Lorentz transformation...

8

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