In this lecture, students see a geometric derivation of the Lorentz Transformation on a spacetime diagram.
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:
assignment Homework
Which pairs of events (if any) are simultaneous in the unprimed frame?
Which pairs of events (if any) are simultaneous in the primed frame?
Which pairs of events (if any) are colocated in the unprimed frame?
Which pairs of events (if any) are colocated in the primed frame?
Which event occurs first in the unprimed frame?
Which event occurs first in the primed frame?
group Small Group Activity
5 min.
Special Relativity Spacetime Diagrams Simultaneity Colocation
Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.group Small Group Activity
30 min.
assignment_ind Small White Board Question
10 min.
assignment_ind Small White Board Question
10 min.
group Small Group Activity
30 min.
group Small Group Activity
10 min.
group Small Group Activity
30 min.
accessibility_new Kinesthetic
10 min.
assignment_ind Small White Board Question
10 min.
Special Relativity Spacetime Diagrams Worldlines Postulates of Relativity
Student consider several curves on a spacetime diagram and have to judge which curves could be worldlines for an object.