Derive the Lorentz transformation in Minkowski four-dimensional spacetime 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.

 

[FactChecker]

Derivation of the Lorentz transformation in Minkowski spacetime diagrams:

This is now my favorite video explanation of the Minkowski diagram. It is great that a school like Stanford makes this available on the internet!

 

[robphy]

Given a line, to draw a "perpendicular [to it]" in Minkowski spacetime...

from my answer https://www.physicsforums.com/threa...me-meaning-visualization.1016318/post-6644678 ,
use Minkowski's construction

From Minkowski's "Space and Time"... $$
We decompose any vector, such as that from O to x, y, z, t into four
components x, y, z, t. If the directions of two vectors are, respectively, that
of a radius vector OR from O to one of the surfaces \mp F=1, and that of
a tangent RS at the point R on the same surface, the vectors are called
normal to each other. Accordingly,
c^2tt_1 − xx_1 − yy_1 − zz_1 = 0
is the condition for the vectors with components x, y, z, t and x_1, y_1, z_1, and t_1 to be normal to each other.

In other words, draw a "circle"
(in Minkowski spacetime, [the future-branch of] a hyperbola--- the locus of events (say) one-tick-later along inertial-worldlines from event O ) centered at the origin.
To construct a "perpendicular" to a line, find the intersection of your line and the "circle".
Construct the tangent at that intersection point.
(Key idea: The tangent is "perpendicular" to the radius.)
Through the origin, draw a parallel to that tangent line.... That is the x'-axis for the observer along the t'-axis.

desmos.com/calculator/dlsljxj8n2
(Try tuning the E-value: E=1 is Minkowski, E=-1 is Euclidean, E=0 is Galilean)



See also my spacetime diagrammer www.desmos.com/calculator/4765550a89
To see a variation of @Ibix 's construction of the "perpendicular",
click on the circle for "----ticks---lightclockDiamonds"... but only for the E=1 case.
Further down, to see "grids"
you can click on the circles for
coordinateSystem BOB (2)
----ticks---A's rotated graph paper
----ticks---B's rotated graph paper

 

[robphy]

Along the lines of @Ibix 's construction of the "perpendicular",
the essential part of the construction is
1. constructing the light-cones of two events on an inertial worldline, then
2. constructing the hyperplane that contains the intersection of those light-cones
3. constructing the parallel-hyperplane through the origin

Here's an extension of my earlier desmos file
desmos.com/calculator/hfaxw4bgwq


(The light-cone is a hyperbola (hyperboloid) of "radius zero".)

 

[robphy]

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.

First, I acknowledge what you say about spacetime diagrams being confusing.
So, some of my efforts have tried to develop additional geometric intuition, not found in typical references.

Second, I do wish to point out that a "PHY-101 position-vs-time graph"
is also a diagram with a non-euclidean geometry
since it also true (but not well-known or appreciated) that,
as you said,
"You cannot apply a Euclidean interpretation of lengths and angles. Pythagoras's theorem does not apply!"

PHY-101 students who understand position-vs-time graphs have been able
interpret aspects of the geometry of a Galilean spacetime diagram,
although they are likely unaware of its underlying geometry.

My spacetime diagrammer www.desmos.com/calculator/4765550a89 (by tuning the E-slider)
is an attempt to help connect Euclidean geometry
and the spacetime geometries associated with Galilean relativity and Special Relativity.