C. Hinsley
26 October 2023
Landau & Lifshitz [1] cover most of this material but most people these days seem to be afraid of (i) reading textbooks, (ii) 20th century Russian physicists, and (iii) getting a proper diet and frequent exercise, so I’ll do some pre-digestion for the common denominator by recapitulating the basics of special relativity before going into the topic I wanted to write about. The point of this article is really just that you can use elementary symmetry methods to rigorously extract the Lorentz transformation.
Schoolchildren know that light takes time to travel across space, having a constant speed. By experiment, we have determined the speed of light to be $c \overset{\text{def}}{\approx} 2.998 \times 10^8$ meters per second. It seems reasonable, then, to use $(ct, x, y, z)$ as a coordinate system for describing events in spacetime (as opposed to $(t, x, y, z)$), where $t$ is expressed in seconds and $x, y, z$ are expressed in meters, chosen in the usual fashion one proceeds in while working in non-relativistic mechanics.
It’s hard to see at first how constancy of light speed qualitatively changes the nature of mechanics. Luckily, there is a mathematical consequence that is easy to see with a little thought.
Suppose you’re floating in space with no forces gravitational or otherwise being applied to you; as an observer, you exhibit an inertial reference frame. Establish coordinates $ct, x, y, z$ with the usual units for spacetime. Let there be a lightbulb sitting stationary in space at $(1, 0, 0)$ (that is, $x = 1, y = z = 0$), and I’ll be sitting still (from your point of view) over at $(2, 0, 0)$ myself.
The light bulb turns on, broadcasting light in all directions at $t = 0$; that is, the bulb-on event is $(0, 1, 0, 0)$. The light is first observed at a location $(x, y, z)$ in space according to a light-like interval, when
$$ (ct)^2 - (x-1)^2 - y^2 - z^2 = 0. $$
For me, that’s when $ct = 1$ so that $t = \frac1c$. Likewise, for you, it’s at $t = \frac1c$. One would expect this; we are equidistant from the bulb and the light travels with constant velocity, so it reaches each of us simultaneously.
What you would observe if the bulb and I were at rest. Clearly the light reaches each of us at the same time, due to the constancy of light speed.
However, suppose instead that you had been moving to the right and starting from further away from the bulb so at time $t = \frac1c$ you end up right where you were in the first scenario; namely, that the lightbulb is at $(1, 1, 0, 0)$ at the time you receive the signal, and I am at $(1, 2, 0, 0)$ at that time. Then at $t = 0$ you would have observed the lightbulb at $(0, 1 + V, 0, 0)$ and observed me at $(0, 2+V, 0, 0)$ where $V = \frac{\textrm{d}x}{\textrm{d}ct}$ is your velocity from my reference frame (draw a picture, if it helps; this is just elementary kinematics; you are going to the right if $V > 0$ so, to end up at the same place at $ct=1$, the lightbulb and I would have had to have been further to the right to accommodate your head start).
But wait just a second — when do you see the light emitted? The light emits at the event $(0, 1+V, 0, 0)$, and so it will reach you at the event $(ct_{\text{you}}, 0, 0, 0)$ according to a light-like interval
$$ (ct_{\text{you}})^2 - (1+V)^2 = 0. $$
Hence $t_{\text{you}} = \frac{|1+V|}{c}$ is when you observe the light. When do you see me see the light? That should happen at the event $(ct_{\text{me}}, 2+V(1-ct_{\text{me}}), 0, 0)$. Again we get a light-like interval
$$ (ct_{\text{me}})^2 - (2+V(1-ct_{\text{me}})-2)^2 = 0. $$
In fact, $t_{\text{me}} = \frac{|V(1-ct_{\text{me}})|}{c}$. Since from your perspective I am moving towards where the light came from (even though the lightbulb is no longer there), clearly the light did not have to travel a full meter to reach me: $1 > ct_{\text{me}}$ so we can drop the absolute value. Thus $t_{\text{me}} = \frac{V}{c(1+V)} < \frac1c = t_{\text{you}}$. We have shown that the light reaches me before it reaches you, even though I didn’t change how the world “looks to you” (i.e., how it is described mathematically in your reference frame; you have to be careful when making such claims because there are strange optical phenomena that occur even if spatial configurations of rigid bodies are held constant) the very moment the light reaches you!