Lecture: Lorentz Transformation (Geometric)

Theoretical Mechanics 2020
In this lecture, students see a geometric derivation of the Lorentz Transformation on a spacetime diagram.
What students learn
  • the Lorentz Transformation can be formulated in terms of hyperbola trig
  • how hyperbola trig can be used on spacetime diagrams
  • Media
    • 2381/space_4.png
    • 2381/space_3.png
    • 2381/space_2.png
    • 2381/space_1.png
    • 2381/time_5.png
    • 2381/time_4.png
    • 2381/time_3.png
    • 2381/time_2.png
    • 2381/time_1.png
    • 2381/length.png
    • 2381/time.png

Here is a geometric derivation of the Lorentz Transformation using hyperbola geometry: \begin{align*} \left[\begin{array}{c} x'\\ ct' \end{array}\right] = \left[\begin{array}{c c} \cosh\alpha & -\sinh\alpha\\ -\sinh\alpha & \cosh\alpha \end{array}\right] \left[\begin{array}{c} x\\ ct \end{array}\right] \end{align*}

On the spacetime diagrams, the large black dot is the event we're trying to describe. The small black dot indicates a right angle for a hyperbolic triangle.

Starting with the time coordinate:

Now thinking about the spatial coordinate:

Keywords
Special Relativity Lorentz Transformation Hyperbola Trig
Learning Outcomes