This section provides an introduction of Minkowski diagram, which represents the Lorentz Transformation overlaying a moving frame into a stationary frame.
What Is Minkowski Diagram?
Minkowski diagram is a spacetime diagram that overlays a moving frame into a stationary frame
to represent Lorentz Transformation in a geometric model.
Using the same thought experiment in the previous section as an example,
the Minkowski diagram from Bob's point of view can constructed as below:
Draw an orthogonal coordinate system of (x,ct) to represent Bob's frame.
Draw the ct' axis of Amy's frame along the worldline of Amy's location.
Draw the x' axis of Amy's frame by flipping ct' axis against the light cone line.
Scale ct' as (sqrt(1+(v/c)**2)/sqrt(1-(v/c)**2))*ct.
Scale x' as (sqrt(1+(v/c)**2)/sqrt(1-(v/c)**2))*x.
For a given event E=(X,cT), the (X',cT') coordinates calculated by the Lorentz Transformation
can be read from the Minkowski diagram geometrically:
Draw a point E at (X,cT) in Bob's frame.
Read X' value on x' axis by projecting E along ct' to x'
Read cT' value on ct' axis by projecting E along x' to ct'
For example, assuming the train is moving at the speed of 0.6*c,
an event E at (X,cT)=(5.5,6.5) on Bob's light cone would be observed by Bob
in Amy's frame as (X',cT')=(2,4) based on Lorentz transformation.
The following picture shows a Minkowski diagram of the above example
produced by the interactive Minkowski diagram tool at http://www.trell.org/div/minkowski.html: