B Very simple exercises in General Relativity

[PeterDonis]

Some small gems, beautiful problems, simple ones, that can shine some light on this subject.

I'm not sure why you would expect this to work. In order to solve problems using a scientific theory, you need to learn it. Solving problems with the theory is not where the learning process starts.

If you want an idea that can at least shine some light on a basic principle of GR, the best one I know of is the observation that Einstein called "the happiest thought of my life", which set him on the road to GR: if a person falls freely, they will not feel their own weight. But that's still just one idea, and there's a lot that still needs to be learned before one can say one has learned GR. It took Einstein eight years after having that thought to come up with the right field equation for GR.

I usually ask my fellow physicists (like me) about simple GR problems they have seen or solved during their careers. The answer is always the same: I have not solved any GR-related problems.

Did any of those physicists ever take a course in GR?

 

[PeroK]

I usually ask my fellow physicists (like me) about simple GR problems they have seen or solved during their careers. The answer is always the same: I have not solved any GR-related problems.

I'm just a retired amateur with an interest in these things, but I did one:

https://www.physicsforums.com/threa...n-in-a-black-hole.1012103/page-3#post-6599762

On a constructive note, Hartle's Book Gravity is probably the most accessible introduction to GR - although it's still advanced undergraduate. He has many good exercises and problems throughout the book.

IMO, however, if you haven't mastered SR, there is little or no point in studying GR.

 

[Orodruin]

If you want an idea that can at least shine some light on a basic principle of GR, the best one I know of is the observation that Einstein called "the happiest thought of my life", which set him on the road to GR: if a person falls freely, they will not feel their own weight. But that's still just one idea, and there's a lot that still needs to be learned before one can say one has learned GR. It took Einstein eight years after having that thought to come up with the right field equation for GR.

And arguably the very same holds in Newtonian physics … although that is not compatible with special relativity.

 

[DaTario]

Some universities offer introductory GR as an undergraduate course. I don't think "easy" would be an apt description.

I'm not sure what you mean by "some elements". Either you teach a scientific theory or you don't. You can't just teach "some elements" of it. Scientific theories aren't a la carte menus.

I can give you an example. The syllabus for the Modern Physics 3 course, an undergraduate course offered to future high school teachers, contains introductory notions of molecular orbitals, laser theory, solid physics, nuclear physics, particle physics and cosmology. All of this in one semester. It is one of the last courses in their training. In this course, the teacher must select topics and try to work on them in the classroom, seeking to both clarify some (few) concepts and encourage students to further study these topics in greater depth.

 

[Vanadium 50]

What you need is introductory calculus on manifolds/

Why npt simply teach that to 8 year olds? Then they will be ready.

 

[DaTario]

I'm not sure why you would expect this to work. In order to solve problems using a scientific theory, you need to learn it. Solving problems with the theory is not where the learning process starts.

My investigation and curiosity here has to do with knowing first if typical introductory exercises in GR exist (with or without simple formulas to plug numbers into). I am not working now on to build an entire course. I am not here in a serious reflexion about mounting a course -- how it would be from the begining to the end.
This OP is humble in this sense. It basically asks: what was the simplest GR exercise you did in your student carreer?

 

[PeroK]

It basically asks: what was the simplest GR exercise you did in your student carreer?

Things are only truly simple once you have the prerequisites. There may be several exercise in Hartle and Carroll (the two GR books I've studied) that I found simple. But, that was after having studied the material. And, having the right prerequisites to start the books themselves. I would say I was studying physics for six months a year on average for 3-4 years before I was ready for GR - and it made a big difference when I was locked down and studying full-time during Covid.

This is what it takes. I'm not a genius, but I'm fairly good at picking stuff up.

That said, if someone gave me an hour's lecture on GR, I know what I would present. Whether an audience of high-school students would understand the implications of what I was talking about is another matter. Even with SR, which is somewhat trivial once you've fully digested it, you can see how much students struggle with the basics.

If someone wanted to learn SR, I'd advise them to set aside 1-2 days a week study time for 3-6 months. That's the reality of these subjects. There's no royal road or magic bullet.

And GR is a different ball-game altogether.

 

[DaTario]

I'm just a retired amateur with an interest in these things, but I did one:

https://www.physicsforums.com/threa...n-in-a-black-hole.1012103/page-3#post-6599762

On a constructive note, Hartle's Book Gravity is probably the most accessible introduction to GR - although it's still advanced undergraduate. He has many good exercises and problems throughout the book.

IMO, however, if you haven't mastered SR, there is little or no point in studying GR.

Thank you very much, Perok (Are all the posts in that thread related to the interesting problem posed in the OP?).
Forgive me, but you are not just a retired amateur. You are a recognized member/collaborator in one of the most prestigious forums of physics in the world, working here for years. I say this with gratitude and true admiration. But when you say that you did just one exercise on GR my point gains in relevance. IMHO, we should have been exposed to more basic exercises, to the art of solving problems, and even to formulas we are not able to fully demonstrate, concerning GR.

Thank you again for sharing the problem you solved. Beautiful this one.

 

[DaTario]

I'm not sure why you would expect this to work. In order to solve problems using a scientific theory, you need to learn it. Solving problems with the theory is not where the learning process starts.

If you want an idea that can at least shine some light on a basic principle of GR, the best one I know of is the observation that Einstein called "the happiest thought of my life", which set him on the road to GR: if a person falls freely, they will not feel their own weight. But that's still just one idea, and there's a lot that still needs to be learned before one can say one has learned GR. It took Einstein eight years after having that thought to come up with the right field equation for GR.


Did any of those physicists ever take a course in GR?

Thank you for the contribution with Einstein's happiest thought.

"Did any of those physicists ever take a course in GR?"
All of them did physics, I am afraid no one has done this course. In the Federal University of Rio de Janeiro, for instance, the GR course is just an option to the students, differently from courses like calculus or quantum mechanics.

 

[Orodruin]

the GR course is just an option to the students, differently from courses like calculus or quantum mechanics.

There are several reasons for this. One being that GR is a more advanced topic than either of those. Another being that, unlike GR, both of those have widespread applications in a very broad range while GR does not.

 

[DaTario]

There are several reasons for this. One being that GR is a more advanced topic than either of those. Another being that, unlike GR, both of those have widespread applications in a very broad range while GR does not.

Yes, it reasonable. I agree with this. At the end, I guess my worries with respect to this OP have more to do with a marketing question in the physics realm. It is like: "How so you physicists don't know simple applications of this cute subject professor Tysson and professor Sagan (among others...) are talking every day in social networks?"

I would say more. If the majority of the physicists is ok with not knowing how to solve one basic formal question in GR, I am prone to think it's better to do as they do. Note that the majority of the physicist do not work with nuclear physics, but almost all of them can manipulate formulas like $E = m c^2$ and have a neat discussion about the bonding energy of $H_2$.

 

[jedishrfu]

Why npt simply teach that to 8 year olds? Then they will be ready.

Someday, in the near future likely beyond this generation that may well happen.

Looking at school courses over the years, things have been dropped from math, and more modern topics taught previously in college have been added. Calculus and calculus-based phyiscs programs are prime examples where Calculus was previously an advanced subject taught only in college.

Some others are geometric proofs was pretty much all of geometry when I was in school but now proofs are taught in one chapter and then they move onto other geometric concepts. Cursive writing was a big thing when I was a kid and now it hardly matters as people print their letters except for their name.

Lets get back to the thread and stop sidetracking it.

 

[DaTario]

Summarizing the contributions made in this thread, for which I am very grateful. The contributions are books where some member saw a good chance of finding basic exercises in General Relativity.

Gravity - An Introduction to Einstein's General Relativity
James B. Hartle

Spacetime and Geometry - An Introduction to General Relativity
Sean Carroll

General Relativity
Robert M. Wald

Exploring Black Holes_ Introduction to General Relativity
Edwin F. Taylor, John Archibald Wheeler

Gravity from the Ground Up: An Introduction Guide to Gravity and General Relativity
Bernard Schutz

Plus the problem posed and solved in the OP of the following PF thread:
https://www.physicsforums.com/threa...n-in-a-black-hole.1012103/page-3#post-6599762

Plus the problem: find the altitude where orbital speed kinematic time dilation cancels gravitational time dilation.

Thank you all, for the contributions and for the debate.

 

[Frabjous]

I have not looked at it, but Susskind came out with a Theoretical Minimum book on GR last year.

In case you are unaware, Carroll and Wald are graduate texts.

 

[Bosko]

Would it be possible to introduce some exercises in general relativity at the high school level?

It is better to start, step by step, with the Special Relativity ( at the high school level)

 

[Nugatory]

Would it be possible to introduce some exercises in general relativity at the high school level? Exercises that would at least help to raise students' awareness of the aspects that will be studied in greater depth at university or in post-doctorate studies?

The posts above have made it pretty clear that if we're thinking in terms of exercises, as in "go work this problem to gain some insight", the answer to this question is "no".

However it may be possible to teach a reasonable qualitative notion of what GR is about and how it differs from the Newtonian theory that presumably the students are studying. Start with the classical position-time graph that high school students should learn anyway, the one in which we demonstrate that the slope/first derivative is velocity and the second derivative is acceleration. Point out that Newton's laws say that the path of an object not subject to forces is a straight line on this graph. So far this is just a different way of looking at the Newtonian physics the students are learning. But now we can introduce the idea that curving/stretching the graph causes these straight lines to converge or diverge in ways that produce the effect of a force acting on the objects.... And that's Newton vs Einstein at the high school level. From there a bit of handwaving will get us to a qualitative explanation of why gravitational and inertial mass are the same, why objects fall at the same speed regardless of weight.

Of course this is all handwaving, but I don't see how we can do better at the high school level. There's no intermediate step between this level and Hartle, and Hartle is a heavy lift even for the last year of an undergraduate degree.

(This line of thought is of course inspired by
edit: and with a belated thanks and attribution to member @A.T.)

 

[DaTario]

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).

 

[robphy]

Perhaps it would be possible for us to easily make some inferences on the angles of the triangle (0,0), (3,0), (3,4).

Some care is needed.
An "angle" measures the separation of two rays.
With a Lorentzian-signature metric, it's not clear (to me)
how to define the "angle" between a timelike-vector and a spacelike-vector, in general.

 

[DaTario]

Some care is needed.
An "angle" measures the separation of two rays.
With a Lorentzian-signature metric, it's not clear (to me)
how to define the "angle" between a timelike-vector and a spacelike-vector, in general.

Are you taking this exercise I am proposing as consisting of one spatial dimensional and one time dimension (1+1), aren't you?

In this case, sorry. I was proposing a purely spatial question (x and y being spatial dimensions with no time dimension included).

 

[robphy]

I was proposing a purely spatial question

So, your A is not just symmetric, but also positive-definite.

 

[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.

---

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

---

 

[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.

---

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

---

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.