## Activity: Lorentz Transformation (Geometric)

Theoretical Mechanics (3 years)
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

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)
1. Which pairs of events (if any) are simultaneous in the unprimed frame?

2. Which pairs of events (if any) are simultaneous in the primed frame?

3. Which pairs of events (if any) are colocated in the unprimed frame?

4. Which pairs of events (if any) are colocated in the primed frame?

1. For each of the figures, answer the following questions:
1. Which event occurs first in the unprimed frame?

2. Which event occurs first in the primed frame?

• group Events on Spacetime Diagrams

group Small Group Activity

5 min.

##### Events on Spacetime Diagrams
Theoretical Mechanics 2021

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
Theoretical Mechanics (4 years)

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.
• assignment_ind Time Dilation

assignment_ind Small White Board Question

10 min.

##### Time Dilation
Theoretical Mechanics (4 years)

• assignment_ind Dot Product Review

assignment_ind Small White Board Question

10 min.

##### Dot Product Review
Static Fields 2023 (7 years)

This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.
• group Applying the equipartition theorem

group Small Group Activity

30 min.

##### Applying the equipartition theorem
Contemporary Challenges 2021 (4 years)

Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.
• group Sequential Stern-Gerlach Experiments

group Small Group Activity

10 min.

##### Sequential Stern-Gerlach Experiments
Quantum Fundamentals 2023 (3 years)
• group Visualization of Divergence

group Small Group Activity

30 min.

##### Visualization of Divergence
Vector Calculus II 23 (9 years) Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the definition of the divergence of a vector field at a point: The divergence of a vector field at a point is flux per unit volume through an infinitesimal box surrounding that point. Optionally, students can use a Mathematica notebook to verify their predictions.
• accessibility_new Curvilinear Basis Vectors

accessibility_new Kinesthetic

10 min.

##### Curvilinear Basis Vectors
Static Fields 2023 (9 years)

Curvilinear Coordinate Sequence

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
• assignment_ind Possible Worldlines

assignment_ind Small White Board Question

10 min.

##### Possible Worldlines
Theoretical Mechanics (4 years)

Student consider several curves on a spacetime diagram and have to judge which curves could be worldlines for an object.

Learning Outcomes