Completed (14 days)
Minkowski space-time >
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).
All the red-coloured numeric references are directly taken from that book.
The current section is mainly focused on Appendix II MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD") (SUPPLEMENTARY TO SECTION 17).
The analysis of the Minkowski theory is started in equation
(11a), which is the last one in the previous Appendix I (where the Lorentz transformation is discussed). As proven in Lorentz transformation > General analysis, this equation is already wrong. Nevertheless, some of the next steps are certainly worth noting and that's why I will continue the analysis anyway.
In the next step, the previous equation is converted into
(12)by performing these substitutions:
After performing this conversion, Einstein writes the following:
We see from (12) that the imaginary time co-ordinate x4, enters into the condition of transformation in exactly the same way as the space co-ordinates x1, x2, x3. It is due to this fact that, according to the theory of relativity, the "time" x4, enters into natural laws in the same form as the space co ordinates x1, x2, x3.
This text seems to indicate that a mere formal change in the variables is enough to convert space into time (?!). On the other hand, what is the exact point of some ambiguous remarks, like quoting time or referring to an imaginary term? Was the author trying to create a "super-variable" with a dual space-time nature?
Such an intention is certainly not communicated to the mathematical derivation and, more specifically, to the units which remain unaltered. In SI, the units of the new temporal variable are metres, exactly the same than what happened before performing the aforementioned replacement. An unarguable conclusion by bearing in mind that velocity (
c) is multiplied by time (logically, the fact of being constant does not make it unit-less. In any case, note that this constancy is also treated somewhere else). That is:
If you have a spatial variable (i.e., measured in spatial units), what would make you think that it is actually related to time? Or better: what would you do to let the mathematical derivation know about your intention? The answer is clear: there is nothing you can do to "convert" a spatial variable into a temporal variable (at least, by relying on Mathematics and/or Physics).
What is the right interpretation of
x4? It is just a spatial term which includes all the errors of the Lorentz transformation. That is: a wrongly-generated element which was created while trying to extract inexistent information from a system of equations describing 3D space/time relationships. Additionally, the essence of this (already-wrong) term was arbitrarily converted from spatial into temporal (or weird space-time mixture), apparently for no other reason than agreeing with the statement
[...]due to this fact that, according to the theory of relativity, the "time" x4, enters into natural laws in the same form as the space co ordinates[...].