What does it mean for the Lorentz Transform to preserve areas?

Question

If we look at the Lorentz Transform from some frame $S$ to some frame $S'$ moving relative to $S$...

$$\begin{bmatrix} \gamma & -\gamma v\\ -\frac{\gamma v}{c^2}& \gamma \end{bmatrix}\begin{bmatrix} x\\ t \end{bmatrix}=\begin{bmatrix} x'\\ t' \end{bmatrix}$$

...we can see that the transformation has a determinant of $1$...

$$\gamma^2 - \gamma^2 \frac{v^2}{c^2}=\gamma^2(1-\frac{v^2}{c^2})=\frac{\gamma^2}{\gamma^2}=1$$

...which means that if we have some shape in the $S$ frame, where the outline of the shape corresponds to a whole bunch of events happening some time $(x,t)$ after $(0,0)$, and we rewrite the space time coordinates of all those events from the point of view of the $S'$ frame, the area within the transformed shape is the same as the area within the original shape.

I feel like this has some significance, but, I have no idea what it means.

Any insights?

Answer 1

One meaning (very mundane, I am sure there are better ones) is that $L't'=L_0 \tau$, where $L_0$ is the rest length of an arbitrary object and $\tau$ an arbitrary amount of proper time. $t'$ is the time it takes in S' to see a stationary clock in S advance $\tau$. Basically $L't'=L_0/\gamma*\tau \gamma=L_0\tau$

Answer 2

The key idea is
in (1+1)-Minkowski spacetime,
the "causal diamond" of two timelike-related events, O and Q (with $O\ll Q$),
formed by the intersection of the future of O and the past of Q has
area proportional to the square-interval between O and Q.
(If one counts in "units of light-clock-diamonds" (described below) , the area is equal to square-interval.)

While the Euclidean rotation and Galilean boost transformations also have determinant 1,
I have used this determinant=1 property for the Lorentz boost
to formulate a graphical method for calculations in Special Relativity
("Relativity on Rotated Graph Paper", American Journal of Physics 84, 344 (2016); https://doi.org/10.1119/1.4943251 ; see also this earlier draft https://arxiv.org/abs/1111.7254 ).
This builds on Mermin's light-rectangles idea mentioned in the stackexchange link in the comments: Motivation for preservation of spacetime volume by Lorentz transformation? )

So, the Lorentz boost preserves this diamond area (since its determinant is equal to 1) and preserves the diamond's lightlike sides (since its eigenvectors are lightlike, with eigenvalue equal to the Doppler factor and its reciprocal).

We can use this idea to consider timelike events from O that are "one tick along inertial worldlines" to draw causal diamonds corresponding to the observer light-clocks ticking once. (See below for a diagram, essentially using light-cone coordinates [an eigenbasis of the boost].)

So, this gives a visualization of "1 tick" along an inertial worldline [called a "light-clock diamond"]... which is easy to visualize and construct on rotated graph paper. Calculations in special relativity can be visualized by counting diamonds, suitably interpreted.
Refer to the paper referenced for textbook problems treated using this "relativity on rotated graph paper" approach.