B Understanding Spacetime Diagrams | Basics Explained

[Grasshopper]

Spacetime diagrams seem to be the most used explanation for relativity weirdness, so I’d like some clarification in how to make them, it anyone wants to help.

(1) Light’s worldline is 45 degrees, obviously. No issues there, I don’t think.

(2) How do I determine the angles of the moving frame? I would imagine it is related to v/c. Maybe c/v.

(3) How do I determine the size of the tick marks on the moving frame? This, I assume, must be related to the Lorentz factor.

I’m sure there are brainless ways to do it. And I could probably find it through google. But I trust this community more than a random internet search.

Thanks to all replies.

 

[Ibix]

Write down a point an event on the $t'$ axis and inverse Lorentz transform it to get its $x,t$ coordinates. Ditto the $x'$ axis. Repeat for a few points events (or just write it generally in the first place) to confirm the axes are straight lines.

Useful thing to consider: as you vary $v$ what path does $x,t$ trace out?

 

[Grasshopper]

Write down a point on the $t'$ axis and inverse Lorentz transform it to get its $x,t$ coordinates. Ditto the $x'$ axis. Repeat for a few points (or just write it generally in the first place) to confirm the axes are straight lines.

Useful thing to consider: as you vary $v$ what path does $x,t$ trace out?

My guess is that it’s a hyperbola, although this is based only on the algebra of the Lorentz factor. I will have to test this when I get my calculator though.

But you’re answer actually seems kind of obvious in hindsight. Lorentz transforming would clearly give the correct values.

 

[Ibix]

Your guess is correct.

Minkowski diagrams are, indeed, incredibly simple. Once you get that "events move along a hyperbola when boosted" thing into your head, and remember which direction they move when boosted by $\pm v$, they're a very good way of visualising how a Lorentz boost is going to come out before you do the maths.

If you've got a device with a mouse rather than a touchscreen, my old animated Minkowski diagram tool (ibises.org.uk/Minkowski.html) let's you sketch scenarios and then animate boosts. If you only have a touchscreen the UI doesn't work so well, but you can still see canned diagrams (click buttons near the bottom of the page) and boost them. There's a diagram of hyperbolae, which you can watch being invariant (analogous to concentric circles under rotation in Euclidean geometry).

 

[vanhees71]

For more details, you may also have a look here:

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[robphy]

It turns out that you don't need to explicitly draw a hyperbola.
Instead, with one corner fixed at the center of the hyperbola, draw equal-area "[causal] diamonds", which are parallelograms with edges parallel to the light cone.

The timelike diagonal is along the worldline with a chosen velocity.
The spacelike diagonal is along the line of simultaneity for that worldline.
The stretching and shrinking factor of the reshaped diamond is the Doppler factor, k.
The diamonds are the spacetime-diagrams of light clocks.

Visit my PF Insights:
https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/
https://www.physicsforums.com/insights/relativity-rotated-graph-paper/

This links to an early draft of my paper and a link to the published version:
https://arxiv.org/abs/1111.7254

Interactive visualizations are useful to get an intuition.
Here are some I made:

• Relativity on Rotated Graph Paper (robphy) - MAA2016
  https://www.geogebra.org/m/HYD7hB9v
  (GeoGebra allows one to download the .ggb file. Navigate to the upper-right corner there.)
• Relativity: Velocity, Doppler-Bondi-k, and Rapidity (robphy)
  https://www.geogebra.org/m/bFxv6tja
  (PF Insight: https://www.physicsforums.com/insights/relativity-variables-velocity-doppler-bondi-k-rapidity/ )
• Relativity-LightClock-MichelsonMorley-2018 (robphy)
  https://www.geogebra.org/m/XFXzXGTq

For more emphasis on hyperbolas (and "circles"), here are some I did in Desmos (where you can more easily see the underlying equations):
Use the E-slider to the geometric analogies from a circle (in Euclidean geometry), a Galilean-circle (in Galilean relativity), and a hyperbola (in Special Relativity)

• robphy's spacetime diagrammer v.2 (2017)
  https://www.desmos.com/calculator/awgqxtkqcc
• Here's the latest version (9e-2021):
  robphy spacetime diagrammer for relativity (v9e-TwinParadox-Doppler)
  https://www.desmos.com/calculator/8kr9uc9zwu