## 1. Minkowski

trell.org interactive Minkowski spacetime diagram

## 2. Lorentz factor γ (gamma)

The single most important *meta*physical fact (e.g., beyond the straight mathematical description of the physical facts) about the Lorentz factor is this: it is a *premise. *Harvey Brown has put a huge amount of work into re-popularizing this idea in the early 2000s; it was widely known prior to Einstein’s 1905 work, and a major reason that Lorentz was metpahysically unimpressed with his own factor prior to 1905. Physics went through a long period of blandly accepting that the Lorentz factor is a premise, but Brown and others have become increasingly dissatisfied with the metaphysical state of affairs.

What I mean by *premise* is this: we lack a *physical explanation *of why it should be the case that that the Lorentz factor is so useful in explaining how the physical world looks to those moving through it. It is not derived from any physical facts. That it empirically explains the world is *not *an explanation — as is the case for any mere mathematical equation. That’s why Newton said *hypotheses non fingo* (“I frame no hypotheses”) about his own gravitational equations, which he was keenly aware implied instantaneous movement of information through spacetime, which was physically ridiculous. That an equation describes the world is *not *equivalent to having an explanation of why the world is as it is, any more than a picture of your friend in a red shirt explains why he is wearing red today rather than blue or purple.

The premise that drives the Lorentz equation is simply that *c *is identical in all frames of reference. Thus we know that

### 1.1 Derivations of γ

Here are multiple thoughtful derivations of the Lorentz factor (gamma):

- Rindler
- Mermin: Derived from the second postulate, the invariance of the speed of light. Contrast with Yakovenko. Very conversational and aimed at the intelligent high school student or college freshman. Very thoughtful and deep, however. You don’t need much more than this presentation.
- Shankar (Yale)
- Levy
- Hashemi website, PDF
- Yakovenko: Derived from first postulate, the equivalence of all reference frames; contrast with Mermin.
- Stack Exchange
- Deriving Lorentz factor for high-school students

## 2. Online Resources

### 2.1 Harvey Brown

Martinez review of Brown’s book

### 2.2 David Mermin’s Excellent Course Notes

Here are the notes in PDF form.

### 2.3 Rindler’s Scholarpedia Resource

Key Quotes:

- “Later it was found (Ignatowski 1910 and, e.g., Rindler 1979) that Einstein’s light postulate is, in fact, an almost inevitable consequence of the relativity principle. The latter, by itself, has the interesting mathematical implication of leaving exactly one velocity invariant among inertial frames, which can be chosen at will.”
- “Newton’s theory is at the limiting edge of possible relativistic theories; in it, the invariant velocity is infinity. In it, therefore, a particle (or just an ideal point) moving with infinite velocity in one inertial frame moves with infinite velocity in all other inertial frames. Let us stand back for a moment to see what this implies. A particle moving along a line with infinite velocity is everywhere at once. It thus passes all the clocks along the way at the same reading, say at noon. And in Newton’s theory it then does the same thing in every other inertial frame as well. So when the clocks of one inertial frame all read noon, those of another such frame will also all read the same time, which might as well be noon. This leads to Newton’s ‘absolute’ (or universal) time. Not so in special relativity. For various reasons — mainly because of Maxwell’s theory — the invariant speed is here chosen to be c . The infinite speed is no longer invariant. Time is no longer absolute.”