# Can you derive the Lorentz factor?

Anyone who claims to comprehend RT should certainly be able to make sense of the famous Lorentz factor $\gamma := 1 / \sqrt{1 - \beta^2}$, and to provide a derivation based (ideally strictly and exclusively) on foundational principles and definitions of RT relating to geometric or kinematic notions, as laid out by A.Einstein; instead, in particular, sticking to the derivation by H.A.Lorentz (or G.FitzGerald) with its appeal to dynamics such as to “classical electro-magnetic theory”, prior to Einstein’s (or H.Poincaré’s) contributions.

The actual derivations presented by Einstein, or by Poincaré, or any other, are themselves not beyond critique nor necessarily already ideal in the above sense. Rather, the goal of presenting a compreshensibe and convincing derivation of $" \sqrt{1 - \beta^2} "$ or various related expressions seems to invite some extend of “didactic-scientific creativity”. Different disseminators of RT, who well may each consider themselves careful educators, do take noticable liberties in “their style”.

Here, then, is my favorite derivation (for now …):

Consider the (hopefully familiar) cast of characters and the following (hopefully familiar) setup requirements (which I hope to illustrate eventually, too):

three distinct participants $A$, $B$, and $F$ who were

(1)

• at rest to each other, and
• straight to each other, i.e. such that the distance ratios between them satisfied $\frac{AB}{AF} + \frac{BF}{AF} = 1$, and

another three distinct participants $J$, $K$, and $Q$ who were likewise

(2)

• at rest to each other, and
• straight to each other, i.e. such that the distance ratios between them satisfied $\frac{JK}{JQ} + \frac{KQ}{JQ} = 1$, and such that further

(3)

• $J$, $K$, and/or $Q$ were not at rest to $A$, $B$, and/or $F$, but they were mutually moving uniformly “straight along, against” each other and satisfied the following further setup conditions:

(4)

• $F$ and $Q$ met each other (in passing) in coincidence with (or “just as”) both $F$ and $Q$ observed that $A$ and $K$ had met each other (in passing), i.e. the two corresponding events are (called) “light-like to each other”, i.e. $\mathcal{E}_{A \circ K} \nearrow \mathcal{E}_{F \circ Q}$ and

(5)

• $A$ and $K$ met each other (in passing), such that $A$‘s indication of having met (and being passed by) $K$ is simultaneous to $B$‘s indication of having met (and being passed by) $Q$, i.e. $A_{\circ K} \equiv A_{\circledS}^{B \oslash Q}$, and similarly
• $A$ and $J$ met each other (in passing), such that $J$‘s indication of having met (and being passed by) $A$ is simultaneous to $Q$‘s indication of having met (and being passed by) $F$, i.e. $J_{\circ A} \equiv J_{\circledS}^{Q \oslash F}$, and finally:

(6)

• that the two distance ratios $\frac{AB}{AF}$ and $\frac{JK}{KQ}$ are equal to each other (and, of course, non-zero) where their corresponding real number value is called $" \beta "$, $\frac{AB}{AF} = \frac{JK}{KQ} := \beta$.

(If it might be found that these requirements had not been satisfied for some particular set of participants $\{ A, B, F, J, K, Q \}$, in some particular trial, then this particular trial should be considered “invalid” and its data, if any, “to be discarded”. The following derivation of $" \sqrt{1 - \beta^2} "$ is meant only a concern of “valid” trials, in whith the stated requirements were found satisfied.

A point not to be missed is the question whether each and all of the stated requirements are even necessary and effective in distinguishing “valid” from “invalid” trials, or whether these “valid setup” requirements constitute an overconstraint. To address this question, however, the definitions and interdependencies must of course be considered of what is even meant at all by the notions “mutual rest”, “distance ratio”, and not least “simultaneity” at all.)

Now, given these requirements, can any conclusions be drawn (agreements reached, measurements obtained) regarding distance ratios $\frac{JQ}{AF}$ or $\frac{BF}{KQ}$, and so on? How can the given “two systems of participants” compare distances between each other at all, given that they are supposed to “move against each other” throughout the trial?

The two parts of requirement (5) suggest a particular symmetric relation, between the pairs $B$, $F$ and $K$, $Q$ on one hand, and in turn the pairs $J$, $Q$ and $A$, $F$ on the other. (It is possible to contruct other even more explicitly symmetric setup requirements, which however involve two systems of five participants each and are accordingly more lengthy to express.)

Therefore (and barring possible other arguments or finds related to “dynamics”) the two systems may agree, as an obvious convention, to compare distances between each other based on setting $\frac{JQ}{AF} = \frac{BF}{KQ}$.
Suitably combining the equations of requirements (1), (2) and (6):

$1 - \beta = 1 - \frac{AB}{AF} = \frac{BF}{AF}$

and

$1 + \beta = 1 + \frac{JK}{KQ} = \frac{JQ}{KQ}$

then allows to calculate

$\frac{JQ}{AF} = \frac{BF}{KQ} = \frac{BF}{AF} \frac{AF}{JQ} \frac{JQ}{KQ} = (1 - \beta) (1 + \beta) \frac{AF}{JQ} = (1 - \beta^2) \frac{AF}{JQ} = \sqrt{ (1 - \beta^2) }$.

There it is already — the inverse of the Lorentz factor $\gamma$ having been derived in the course of identifying a mutually agreeable convention for how to compare distances, i.e. to measure distance ratios, between systems of participants moving “against each other on straight lines”.

As a sequel, it follows how participants who are not at rest to each other can agree on comparisons of durations between each other; where the Lorentz factor will make another appearance.

The challenge remains of course to define the notions employed above (“mutual rest”, “distance ratio”, “simultaneity”) and corresponding measurement operations in the first place, based on foundational principles which will have to be introduced more broadly to begin with.

Supplement (June 23, 2015):

Only recently I noticed that the setup conditions required above also imply that

• $A$ and $J$ met each other (in passing) in coincidence with (or “just as”) both $A$ and $J$ observed that $B$ and $Q$ had met each other (in passing), i.e. that the two corresponding events had been “light-like to each other”; $\mathcal{E}_{B \circ Q} \nearrow \mathcal{E}_{A \circ J}$.

In consequénce, it can be shown that the four events $\mathcal{E}_{A \circ K}$, $\mathcal{E}_{A \circ J}$, $\mathcal{E}_{B \circ Q}$, and $\mathcal{E}_{F \circ Q}$ (which contain the entire observational content of the above description) are plane with respect to each other; i.e. their Cayley-Menger determinant (in terms of their (pairwise, symmetric) spacetime intervals) vanishes.

1. This is my first WP-post (aside from the initial quote) — thanks for this (free) opportunity; I hope to still improve the layout and (especially $\LaTeX$) redability a bit.