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Critical analysis of the main premises of special relativity: Lorentz & Minkowski

Completed on 14-Jun-2015 (14 days)

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).

The current section is mainly focused on Appendix I SIMPLE DERIVATION OF THE Lorentz transformation (SUPPLEMENTARY TO SECTION 11).

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 asv = x/t, its exact essence is transmitted better with the formulav = 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 (fromx1tox2) occurring during a given time period (fromt1tot2). 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 relationshipa = bmis 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 equationa = b+c, knowing thatcalways equals 2 would certainly be helpful; unlikely what would happen withconly being 2 in some of the treated cases.

(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) |

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.