B Very simple exercises in General Relativity

[Orodruin]

Sorry if the next paragraph is completely wrong. I am just trying to see if it is a possible and reasonable way to work an exercise on subjects related to GR in a introductory level.

One idea that came into my mind is to show to the students two points in a Cartesian plane, (0,0) and (3,4), and show that before determining the distance between them, I have to ask an authority for the metric (which will be given by a symmetric 2x2 matrix $A$ different from the identity_2x2). Then I would apply a procedure similar to Pythagoras theorem, something possibly like $ds^2 = (dx,dy) A (dx,dy)^T$, using that matrix to conclude that the distance is another number, not the expected 5. Perhaps it would be possible for us to easily make some inferences on the angles of the triangle (0,0), (3,0), (3,4).

Are your students proficient in variational calculus?

 

[Nugatory]

Sorry if the next paragraph is completely wrong. I am just trying to see if it is a possible and reasonable way to work an exercise on subjects related to GR in a introductory level.

One idea that came into my mind is to show to the students two points in a Cartesian plane, (0,0) and (3,4), and show that before determining the distance between them, I have to ask an authority for the metric (which will be given by a symmetric 2x2 matrix $A$ different from the identity_2x2). Then I would apply a procedure similar to Pythagoras theorem, something possibly like $ds^2 = (dx,dy) A (dx,dy)^T$, using that matrix to conclude that the distance is another number, not the expected 5. Perhaps it would be possible for us to easily make some inferences on the angles of the triangle (0,0), (3,0), (3,4).

I am not seeing how this exercise helps with understanding general relativity. It provides an example of how spatial curvature can be handled mathematically.... but general relativity is not spatial curvature and pretty much everything that your students think they have learned from this exercise will have to be unlearned before they can advance.

 

[Orodruin]

I am not seeing how this exercise helps with understanding general relativity. It provides an example of how spatial curvature can be handled mathematically.... but general relativity is not spatial curvature and pretty much everything that your students think they have learned from this exercise will have to be unlearned before they can advance.

If they can even do the exercise properly, which includes finding the actual shortest path.

 

[DaTario]

If they can even do the exercise properly, which includes finding the actual shortest path.

Finding the shortest path would be very nice, and the (high school) students are not supposed to have mastered variational calculus. But college ones may have at the moment this exercise is put in front of them.

 

[DaTario]

I am not seeing how this exercise helps with understanding general relativity. It provides an example of how spatial curvature can be handled mathematically.... but general relativity is not spatial curvature and pretty much everything that your students think they have learned from this exercise will have to be unlearned before they can advance.

The notion that the curvature happens in the space-time is a very rich one.
I had a positive experience here on the forum related to the maturation of GR ideas in my head. That's when I asked about the transition from straight 'geodesics' to circular 'geodesics' that are presented in many animations. I was worried that I would never see the animation of the transition from open curves (straight lines) to closed curves (circles or ellipses). The answer I got here, and which was illuminating, was that geodesics are trajectories in space-time and therefore can never be closed in ordinary situations. Therefore the earth follows not a circular geodesic around the Sun but a helical geodesic. Many years ago I read an article in the American Journal of Physics (unhappily I don't remember the author or the title) in which there was the following comment. When we throw an object upwards obliquely in the x,z plane within the Earth's gravitational field, it travels a curved, parabolic trajectory. The curvature of this trajectory is evident, but this spatial curvature is not the curvature that the Earth's mass produces by its gravitational field. To see this GR curvature we must introduce the temporal axis properly multiplied by the speed of light, so that the x, z, ct diagram will show in this ordinary launch a line that has a very low curvature, which reveals the Earth's low power in curving space time.

 

[robphy]

Possibly interesting:
https://www.eftaylor.com/leastaction.html#forcingenergy
https://www.eftaylor.com/general.html
from the same site that @Frabjous mentioned early on in this thread.

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Here's an update on the Sector Model approach mentioned by @pervect

https://arxiv.org/abs/2406.02324
V-SeMo: a digital learning environment for teaching general relativity with sector models
S. Weissenborn, U. Kraus, C. Zahn

I learned about Sector Models from a conference I attended in 2019
https://www.physicsforums.com/threa...-as-a-challenge-for-physics-education.966053/
The above article describes some software they were demonstrating at the conference.
Here's a book that came out of the conference. I wrote Chapter 7.
Sector Models are in Chapter 12.
https://www.routledge.com/Teaching-...-Teachers/Kersting-Blair/p/book/9781003161721

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[DaTario]

Possibly interesting:
https://www.eftaylor.com/leastaction.html#forcingenergy
https://www.eftaylor.com/general.html
from the same site that @Frabjous mentioned early on in this thread.

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Here's an update on the Sector Model approach mentioned by @pervect

https://arxiv.org/abs/2406.02324
V-SeMo: a digital learning environment for teaching general relativity with sector models
S. Weissenborn, U. Kraus, C. Zahn

I learned about Sector Models from a conference I attended in 2019
https://www.physicsforums.com/threa...-as-a-challenge-for-physics-education.966053/
The above article describes some software they were demonstrating at the conference.
Here's a book that came out of the conference. I wrote Chapter 7.
Sector Models are in Chapter 12.
https://www.routledge.com/Teaching-...-Teachers/Kersting-Blair/p/book/9781003161721

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This approach seems to be really interesting.

 

[haushofer]

If I'd expose high school students to GR, I'd use classical physics to explain relativistic phenomena, and let them think about why some explanations turn out to give the right answer, and some don't. Examples:

* The classical derivation of the Schwarzschild-radius. Why does this derivation give the right answer? (based on units, one expects $R \sim GM/c^2$, so the question is whether it's such a coincidence that a factor of 2 turns out to be right)
* Explain the principle of equivalence by applying Newton's laws on a falling elevator with a rope hanging on the ceiling and a ball attached to it. Can they explain why the tension of the rope vanishes in free fall?
* Explain tidal effects and using the binomial formula to give an expression for it.
* Calculate the gravitational field of a black hole using Newton, compare to the relativistic expression, and explain whether they prefer to sit next to a small black hole or a large one. How can they reconcile the idea that a large black hole seems to have a small gravitational field just outside of the horizon?
* Perform computer simulations with the bending of light using Newtonian physics. What are the assumptions you make in treating light with Newton's laws? How does the expression deviate from the relativistic formula for the angle of bending? Can you invoke units to explain this formula?

And indeed, Schutz' "Gravity from the ground up" is a great book for this project.