Activity: Right Angles on Spacetime Diagrams

Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.
What students learn
  • The spacetime axes are orthogonal
  • Doing an inner product requires using the correct metric
  • A Lorentz transformation does not change the vector - it is a change of coordinates.
  • In special relativity, right angles don't look “square” - they look symmetric around the line x=ct.
  • Media
    • activity_media/spacetime_axes.png
    • activity_media/spacetime_vectors.png

Right Angles on a Spacetime Diagram

  1. Draw two spacetime vectors, one on the \(ct'\) axis and one on the \(x'\) axis.
  2. Write these vectors are column matrices with the primed coordinates. (You can choose numbers or do it generally with variables.)
  3. Take the dot product between these two vectors. What is the result? What does it mean?
  4. How would an observer in the unprimed frame write these columns? (Hint: Lorentz Transformation.)
  5. Take the dot product between these two vectors. What is the result? What does it mean?


Learning Outcomes