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:
After performing some minor modifications, the aforementioned system is converted into:
|x' = ax-bct|
|ct' = act-bx|
At a first sight, it seems that
is not better than
: the searched relationship (i.e.,
) has been artificially provoked by relying on two unknown constants (i.e.,
); 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
) occurring during a given time period (from
). 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
always equals 2 would certainly be helpful; unlikely what would happen with
only being 2 in some of the treated cases.
After the aforementioned equation
, 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
) may not be zero. Otherwise, no velocity might have been considered; or, alternatively, the associated velocity (
) 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
(equivalently to what happens with
) through the constant
, what avoids these variables to be independent upon each other (i.e.,
might take any value above zero, but only as far as
would also be equal to
). 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
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:
Such a conversion occurs by creating a new variable (the velocity
) from the fraction
. 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
(spatial coordinate) and
(variation between two spatial coordinates; what the
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:
|x-ct = 0|
|x'-ct' = 0|
Even by ignoring the new
(where could a new variable come from?), it should have been clear that, without accounting for additional information, the proposed system cannot be solved.