Relativistic unit circle

Visualization of equidistant points in four-dimensional Minkowski space.

According to special relativity, every point of the 4D sphere in the animation, including the ones in the far future, is exactly one year away from the origin due to time dilation and length contraction. The sphere's radius represents the physical distance we can travel in one year, and animation time is physical time, both measured in a stationary frame of reference. The sphere is expanding according to a hyperbola, derived from the proper time formula:

$$\begin{aligned} \tau^2 = t^2-r(t)^2 \quad\quad\quad \rightarrow \quad\quad\quad r(t) = \pm \sqrt{t^2-\tau^2} \end{aligned}$$

where $r$ is the radius of the sphere (for example in light-years), $t$ is the time measured in a stationary frame, and $\tau$ is the proper time (held constant, for example one year) between the origin and points at the sphere's surface.

This shows that the popular interpretation of special relativity as "the faster one moves through space, the slower they move through time" is either meaningless (to define how fast one moves through time, we should define a universal time), or, if we use proper time as the baseline, one actually travels faster in both time and space with increasing velocity.

Note:

The slope of the hyperbola $r(t)$ at $t = \tau$ is vertical, so — as expected — this effect is negligible for speeds much lower than the speed of light.

Note 2:

I took some liberties by calling it a unit circle, as proper time is not a distance function in the strict mathematical sense, but it captures a very similar concept.

Requirements

• Python 3.8+
• numpy, matplotlib

Quick start

Just run the main.py script, really.