## Activity: 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.
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.

## 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?

• face Lorentz Transformation (Geometric)

face Lecture

30 min.

##### Lorentz Transformation (Geometric)
Theoretical Mechanics (3 years)

In this lecture, students see a geometric derivation of the Lorentz Transformation on a spacetime diagram.
• group Spherical Harmonics

group Small Group Activity

5 min.

##### Spherical Harmonics
Central Forces 2023 (4 years)
• 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.
• face Thermal radiation and Planck distribution

face Lecture

120 min.

##### Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
• keyboard Electrostatic potential of four point charges

keyboard Computational Activity

120 min.

##### Electrostatic potential of four point charges
Computational Physics Lab II 2023 (2 years)

Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.
• keyboard Position operator

keyboard Computational Activity

120 min.

##### Position operator
Computational Physics Lab II 2022

Students find matrix elements of the position operator $\hat x$ in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.
• face Introducing entropy

face Lecture

30 min.

##### Introducing entropy
Contemporary Challenges 2021 (4 years)

This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.
• assignment Extensive Internal Energy

assignment Homework

##### Extensive Internal Energy
Energy and Entropy 2021 (2 years)

Consider a system which has an internal energy $U$ defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where $\alpha$, $\beta$ and $\gamma$ are constants. The internal energy is an extensive quantity. What constraint does this place on the values $\alpha$ and $\beta$ may have?

• assignment Bottle in a Bottle 2

assignment Homework

##### Bottle in a Bottle 2
heat entropy ideal gas Energy and Entropy 2021 (2 years)

Consider the bottle in a bottle problem in a previous problem set, summarized here.

A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated.

The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was $p=106$ Pa and its temperature $T=300$ K. Approximate the helium gas as an ideal gas of equations of state $pV=Nk_BT$ and $U=\frac32 Nk_BT$.

1. How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

2. Compute the integral $\int \frac{{\mathit{\unicode{273}}} Q}{T}$ and the change of entropy $\Delta S$ between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).