Undergrad derive Lorentz transformation in Minkowski four-dimensional spacetime

[Xinze]

TL;DR

How to derive Lorentz transformation in Minkowski four-dimensional spacetime diagrams?

I attempted to draw a Minkowski four-dimensional spacetime diagram, using frame K as the reference frame. I identified a point A with spacetime coordinates (x, ct). My goal was to find its spacetime coordinates in frame r, which moves at a constant velocity v relative to K.

I tried drawing perpendicular lines from point A to the axes of frame r and attempted to determine the coordinates of these perpendicular points. However, I realized this approach might be incorrect because it doesn't yield the Lorentz transformation.

Could someone please help me with this? Thank you!

 

[Sagittarius A-Star]

The $x'$ axis rotates counter-clockwise (hyperbolic rotation of the coordinate system), see
https://www.geogebra.org/m/NnrRvA46

Also, the scales of the $ct'$ and $x'$ axes change with velocity.
This keeps the speed of light in vacuum (45° world-line) the same in both coordinate systems.

 

[Sagittarius A-Star]

Derivation of the Lorentz transformation in Minkowski spacetime diagrams:

 

[PeroK]

TL;DR: How to derive Lorentz transformation in Minkowski four-dimensional spacetime diagrams?

I attempted to draw a Minkowski four-dimensional spacetime diagram, using frame K as the reference frame. I identified a point A with spacetime coordinates (x, ct). My goal was to find its spacetime coordinates in frame r, which moves at a constant velocity v relative to K.

I tried drawing perpendicular lines from point A to the axes of frame r and attempted to determine the coordinates of these perpendicular points. However, I realized this approach might be incorrect because it doesn't yield the Lorentz transformation.

Could someone please help me with this? Thank you!

View attachment 368278

I'm one of the few students, it seems, that finds spacetime diagrams confusing. The problem is that the diagram you have drawn is in Minkowski spacetime, and is not Euclidean. You cannot apply a Euclidean interpretation of lengths and angles. Pythagoras's theorem does not apply!

You have to relearn how to interpret the geometry of a Minkowski spacetime diagram. If I'm honest, I never mastered this. For me, the Lorentz Transformation is too complex to visualise and I rely on algebra.

 

[Ibix]

Draw x-t axes with a scale chosen so that $c=1$, as you have done.

Draw the path of an object moving at constant speed $v$ passing through the origin. This object is at rest at the origin of the x'-t' coordinate system, so this is your t' axis.

Draw two more objects moving at speed $v$ equal distances ahead and behind the first. Have the middle object emit (at the same time) light pulses in both directions. The arrival of the pulses at the front and rear objects must be simultaneous in the x'-t' frame, so a line through these events must be parallel to the x' axis (as others have noted, the slant of the x' axis in your image is the wrong way).

Draw a light clock of period $T$ at rest in the x-t system. You know that the period measured by the x'-t' system is $\gamma T$. Drop linesfrom two consecutive ticks parallel to the x' axis and mark where they cross the t' axis. The distance between them is $\gamma T$ on the t' axis, so now you can draw the "lines of constant t'" part of the x'-t' grid.

Draw a light pulse emitted from the origin and mark where it crosses the t'=1, 2, 3,... lines. Drop lines from these parallel to the t' axis. These are the "lines of constant x'" part of the x'-t' grid.

With the grids it shouldn't be difficult to derive the transforms.