Understanding and using Minkowski diagrams (spacetime diagrams)

(Originally posted on Thursday, 26 March 2026; updated on 7 April 2026)

If anybody is interested in Minkowski diagrams (spacetime diagrams), I've made a video on this topic. Enjoy!



If anybody is interested I prepared day-by-day RELATIVISTIC calculations for acceleration of 1G:



Part 1 (from 0:00 to 2:28)

Minkowski diagrams (spacetime diagrams) are rendered with straight lines, so they assume INSTANT acceleration which is simply not possible. I was curious how good an approximation they are, so I prepared day-by-day RELATIVISTIC calculations for acceleration of 1G (acceleration on the surface of Earth due to Earth's gravity).

In my video “Understanding and using Minkowski diagrams (spacetime diagrams)” in the step 9 I described a classic case of twins, where the time on Earth is 8 years, while the time on a spaceship is 6.92 years (ship's time). At that case the ship was moving with the speed of 0.5c, so in my day-by-day calculations I “stopped the acceleration” after reaching the speed of 0.5c. Please notice that I use the notation “v” which means velocity (speed as a vector).

The precise day-by-day result is 7.10 years (ship's time), which is much closer to the result from the Minkowski diagram that I thought it would be.

Part 2 (from 2:29 to 3:22)

By the way I discovered that with acceleration of 1G a spaceship could reach almost the speed of light in around 354 days ship's time. ONLY 354 days ship's time!!! At that moment ANY distance would shrink to almost zero, so the ship could reach ANY place in a universe almost instantly (from the point of view of the ship). From the point of view of Earth the ship would still need millions of years to reach another galaxy, but I think some people would like to take part in such a “one-way journey” (of a spaceship into the future) anyway.

The problem is that the amount of fuel needed to accelerate with 1G for almost a year and then to decelerate with 1G for almost a year would be insane, so the problem is practical (not theoretical).

Part 3 (from 3:23 to 4:24)

At the end of the video I changed the starting speed to 0.5c and the acceleration to ZERO because I wanted to verify my formulas by comparing the result to the Minkowski diagram (spacetime diagram). I ended up with 6.9289 years (ship's time), so the result was the same when rounded down, which means that my formulas were correct. The result had to be rounded DOWN because in the Minkowski diagram (spacetime diagram) I used a rounded value of the Lorentz factor (1.155 instead of 1.1547).

At the speed of 0.5c the relativistic factor (Lorentz factor) is precisely 1.1547, so one simultaneity plane is equal to precisely 1.73205 years (1.1547 * (1 + 0.5)), which gives the precise result 6.9282 for the Minkowski diagram (spacetime diagram). There is still a slight difference (6.9289 vs. 6.9282), but it's because in the day-by-day calculations I used the Lorentz factor from day TWO. When starting with the speed of 0 it gives almost perfect result, but when starting with the speed of 0.5c the difference is already detectable. So, the correction to the day-by-day calculations with instant acceleration to 0.5c would have to be:
(1264.5311 - 1 + 0.8660) / 365 * 2 = 6.9282

PERFECT!