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