Activity: Lorentz Transformation (Geometric)
In this lecture, students see a geometric derivation of the Lorentz Transformation on a spacetime diagram.
- face Lecture 30 min. Student handout (PDF)
- Search for related topics
Special Relativity Lorentz Transformation Hyperbola Trig
- the Lorentz Transformation can be formulated in terms of hyperbola trig
- how hyperbola trig can be used on spacetime diagrams
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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 Events on Spacetime Diagrams
assignment Homework
Events on Spacetime Diagrams
Special Relativity Spacetime Diagram Simultaneity Colocation Theoretical Mechanics (4 years)-
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?
- For each of the figures, answer the following questions:
Which event occurs first in the unprimed frame?
Which event occurs first in the primed frame?
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- group Events on Spacetime Diagrams
group Small Group Activity
5 min.
Events on Spacetime Diagrams
Theoretical Mechanics 2021Special 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 Right Angles on Spacetime Diagrams
group Small Group Activity
30 min.
Right Angles on Spacetime Diagrams
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assignment_ind Small White Board Question
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Time Dilation
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assignment_ind Small White Board Question
10 min.
Dot Product Review
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Applying the equipartition theorem
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Sequential Stern-Gerlach Experiments
Quantum Fundamentals 2023 (3 years) - group Visualization of Divergence
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Visualization of Divergence
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accessibility_new Kinesthetic
10 min.
Curvilinear Basis Vectors
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assignment_ind Small White Board Question
10 min.
Possible Worldlines
Theoretical Mechanics (4 years)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.
- Learning Outcomes
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