Critical analysis of the main premises of special relativity: Lorentz & Minkowski
Completed on 14-Jun-2015 (14 days)

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Lorentz transformation >
General analysis

NOTE: the text below refers to what was written by A. Einstein in "relativity: The Special and General Theory" (you can download a copy from http://www.gutenberg.org/ebooks/30155).
All the red-coloured numeric references are directly taken from that book
.
The current section is mainly focused on Appendix I SIMPLE DERIVATION OF THE Lorentz transformation (SUPPLEMENTARY TO SECTION 11).


The derivation is started from the following system of equations:
x-ct = 0
(1)
x'-ct' = 0
(2)

After performing some minor modifications, the aforementioned system is converted into:
x' = ax-bct
(5)
ct' = act-bx

At a first sight, it seems that
(5)
is not better than
(1)
: the searched relationship (i.e.,
x'
&
t'
related to
x
&
t
) has been artificially provoked by relying on two unknown constants (i.e.,
a
&
b
); nevertheless, the original requirement of bringing further information into picture remains unaltered (i.e., previously, it had to explain the intended relationship; now, the meaning of the new constants).

Before analysing the next steps, I will clarify various issues whose misunderstanding is precisely the responsible for most of the subsequent errors:
  • Although velocity is conventionally defined as
    v = x/t
    , its exact essence is transmitted better with the formula
    v = Δx/Δt
    . This second version emphasises the required variation between two points (i.e.,
    Δx/Δt = x2-x1/t2-t1
    ). That is: velocity represents the spatial variation (from
    x1
    to
    x2
    ) occurring during a given time period (from
    t1
    to
    t2
    ). No velocity could be considered unless both variations would be present (i.e., greater than zero).
  • When performing any (mathematical) analysis, it is required to stick to the used assumptions throughout the whole process. That is: if at some point the relationship
    a = bm
    is verified (i.e., in a general way, without being expressly constrained to work under certain conditions), it would have to be true at any other point too.
  • When considering different scenarios to solve certain (mathematical) problem, the tested conditions would have to hold in all the considered situations. For example: with the equation
    a = b+c
    , knowing that
    c
    always equals 2 would certainly be helpful; unlikely what would happen with
    c
    only being 2 in some of the treated cases.
After the aforementioned equation
(5)
, the derivation follows with:
For the origin of K' we have permanently x' = 0

Such an assumption is wrong for various reasons. Firstly, it goes against the already-explained fact that
x'
(better:
Δx'
) may not be zero. Otherwise, no velocity might have been considered; or, alternatively, the associated velocity (
c
) would be zero, what is impossible on account of its essence (i.e., constant value much bigger than zero). In fact, this clarification denotes a second error: forgetting about the unbreakable relationship between
x'
&
t'
(equivalently to what happens with
x
&
t
) through the constant
c
, what avoids these variables to be independent upon each other (i.e.,
x'
might take any value above zero, but only as far as
t'
would also be equal to
x'/c
). There is a third error in the aforementioned statement: even in case that
x' = 0
would be valid, it would have been a very bad choice on account of its extremely limited applicability; that is: the conclusions outputted for
x' = 0
(e.g.,
x = bct/a
) wouldn't work when such a condition is not met (i.e., when
x' ≠ 0
,
x ≠ bct/a
).

After all the aforementioned errors, the resulting formula
x = bct/a
is converted into:
v = bc/a
(6)

Such a conversion occurs by creating a new variable (the velocity
v
) from the fraction
x/t
. That is: the author started from a fraction being equal to a constant and, after performing some formal replacements (i.e., not bringing any new information into picture), created a new variable defined by this same fraction. That is:
x/t = c
&
x/t = v
&
c ≠ v
(?!).

Afterwards, Einstein writes:
[...]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.

This "snapshot" represents a more intuitive way to understand the confusion between
x
(spatial coordinate) and
Δx
(variation between two spatial coordinates; what the
x
in
v = x/t
is actually referring to): it is impossible to take a snapshot of a spatial variation or a velocity, it would rather be a video.

I will stop my analysis of this document here, because of not seeing the point in continuing. The aforementioned errors are so clear and "unfixable" that I cannot think of a better way to transmit the intended ideas.

Lastly, I want to highlight an issue which, unlikely what some people seem to think, I consider very relevant: better making sure that everything works fine before bringing the more elegant/magical/cool ideas in. That is: any experienced person should have assumed that this development was faulty just after having quickly skimmed through it. More specifically, after having noticed the starting conditions (i.e., a system of two inter-independent equations, each of them inversely relating two variables to the same constant), the additional information being accounted (i.e., none) and the final results:
(starting point)
x-ct = 0
x'-ct' = 0
(final result)
Final1

Even by ignoring the new
v
(where could a new variable come from?), it should have been clear that, without accounting for additional information, the proposed system cannot be solved.