B Very simple exercises in General Relativity

[DaTario]

Hi All

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?

If so, what examples of exercises would be appropriate for the typical maturity of students in mathematics and physics?

Best wishes,
DaTario

 

[ergospherical]

I think Wald might have a few?

 

[DaTario]

I think Wald might have a few?

You mean Robert M. Wald's book?

 

[Frabjous]

There might be stuff in Exploring Black Holes by Taylor and Wheeler.
https://www.eftaylor.com/exploringblackholes/

also Gravity by Schutz
https://www.amazon.com/gp/product/0521455065/?tag=pfamazon01-20

 

[PeroK]

Hi All

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?

If so, what examples of exercises would be appropriate for the typical maturity of students in mathematics and physics?

Best wishes,
DaTario

What is the point of this? If the students have studied (and mastered) SR, Newtonian Gravity and classical EM, then they may be mature enough in terms of physics and maths to make something of GR. If not, then why try to teach something that is likely to far beyond their comprehension? Would the time and effort not be better spent on something more manageable?

 

[DaTario]

What is the point of this? If the students have studied (and mastered) SR, Newtonian Gravity and classical EM, then they may be mature enough in terms of physics and maths to make something of GR. If not, then why try to teach something that is likely to far beyond their comprehension? Would the time and effort not be better spent on something more manageable?

But being mature enough to learn a given physics content is not the same thing as learning that same physics content. It is about doing what many university physics courses seem not to do, namely, teaching GR concepts, GR methods and how to solve certain GR exercises.

 

[Vanadium 50]

Are you an actual teacher? I doubt it, because most teachers understand that if you add something to the curriculum you need to remove something. And you will need to remove a lot. This will do your students a serious disservice.

 

[PeroK]

But being mature enough to learn a given physics content is not the same thing as learning that same physics content. It is about doing what many university physics courses seem not to do, namely, teaching GR concepts, GR methods and how to solve certain GR exercises.

Yeah, okay, so you'll have high school students outdoing graduate physics students. They are not even going to know vector calculus, SR or four-vectors, let alone differential geometry - so you'll have to spoon-feed them some trivial stuff and give them the illusion of knowldege.

 

[DaTario]

Ok, Vanadium 50 and Perok. I agree with your points -- including GR notions may be troublesome. But you must concede that we are living in a time where the crisis of including modern physics in curriculum is on. My interest is not necessarily focused in high school. College physics seems to have the same problem.

It seems to me somewhat symptomatic that the contributions to this thread do not come as statements of simple problems, but as general and somewhat insecure references to books (for which I fraternally appreciate). It bothers me to notice in conversations with several colleagues who have degrees in physics that they do not seem to have done any exercises, however simple, in general relativity. I did not do any during my physics course.

 

[Frabjous]

I am not aware of too many physicists who are happy with the generic high school physics curriculum. I have doubts about adding GR, but one should not fear experimentation.

 

[Orodruin]

It bothers me to notice in conversations with several colleagues who have degrees in physics that they do not seem to have done any exercises, however simple, in general relativity. I did not do any during my physics course.

There is a simple reason for this: Most physics graduates do not need to know GR, even if the continue within academia. Not even if they do theory that does not have a strong connection to GR. People specializing in things like cosmology, string theory, black hole astrophysics, etc, obviously need GR, but that will typically be taken as specialization courses or as part of a PhD focused on those subjects.

 

[DaTario]

There is a simple reason for this: Most physics graduates do not need to know GR, even if the continue within academia. Not even if they do theory that does not have a strong connection to GR. People specializing in things like cosmology, string theory, black hole astrophysics, etc, obviously need GR, but that will typically be taken as specialization courses or as part of a PhD focused on those subjects.

First of all, I would like to thank everyone for participating so far.

Orodruim, I kind of understand your argument. But a physicist's education should give him or her the condition to have at least good introductory notions about the various topics of physics. Especially if we take into account the number of times that topics of general relativity are appearing in the media and social networks.

Note that we all have elementary exercises on quantum mechanics (although some physicist may want to specialize in acoustics):

1) what is the De Broglie wavelength of a 55-gram tennis ball?

2) what is the probability that the spin measurement of an electron is measured 'up' if its quantum state is given by $\vert \psi > = (2/3) \vert up > + (\sqrt{5}/3) \vert down >$ ?

OBS.: The only exercise of GR I can think of here has to do with using the concept of gravitational escape velocity to determine the largest radius of a sphere of mass m that would prevent light from escaping.

 

[Frabjous]

OBS.: The only exercise of GR I can think of here has to do with using the concept of gravitational escape velocity to determine the largest radius of a sphere of mass m that would prevent light from escaping.

Amusingly I just read this
https://www.realclearscience.com/bl...ght_up_black_holes_241_years_ago_1065979.html

 

[DaTario]

Amusingly I just read this
https://www.realclearscience.com/bl...ght_up_black_holes_241_years_ago_1065979.html

Very interesting. The history of science is likely to hide several such 'secrets' or 'more accurate versions'.

 

[Vanadium 50]

1. You are not a physicist.
2. You are not a teacher.
3. Yoi are unable to come up with a single example on your own of the sort described in the title.

Yet you ask us to believe that high school physics is being taight incorrectly and that you (and you alone) know how to fic it.

I'm sorry, but few will find this a convincing argument. You will need to try harder,

 

[DaTario]

1. You are not a physicist.
2. You are not a teacher.
3. Yoi are unable to come up with a single example on your own of the sort described in the title.

Yet you ask us to believe that high school physics is being taight incorrectly and that you (and you alone) know how to fic it.

I'm sorry, but few will find this a convincing argument. You will need to try harder,

• "You are not a physicist."
  is it a document that decides if you are correct of not?
  
  
• "You are not a teacher."
  is it a document that decides if you are correct of not?
  
  
• "Yoi are unable to come up with a single example on your own of the sort described in the title."
  I could try something related to what was discussed in my post of 2017 (PF has brought it to me I think that automatically). It would be like this:
  
  Given a matrix A 2x2 with elements such and such, consider it a space time metric (1 + 1) and calculate the distance ds between points P and Q.
  
  I am trying here because may be here there are someone with more imagination than me.
  
  But you have tried your inferences. Now it is my turn.
  
  1. You should be more delicate to the people that bring doubts to this forum
  
  2. This forum in not yours for you to decide be rude when you want.
  
  3. You are not obliged to work here as a responder. Therefore, if you did not like the question, just do not answer it. Let those who have animation to debate do it.
  

 

[Orodruin]

But a physicist's education should give him or her the condition to have at least good introductory notions about the various topics of physics.

Yes, but GR is not introductory. Quite the opposite. Many things in GR are so far removed from regular intuition built from our everyday experience and the mathematics so involved that it is not really possible to be both accurate and give a good introduction at the level you seem to want.

Note that we all have elementary exercises on quantum mechanics (although some physicist may want to specialize in acoustics):

1) what is the De Broglie wavelength of a 55-gram tennis ball?

2) what is the probability that the spin measurement of an electron is measured 'up' if its quantum state is given by |ψ>=(2/3)|up>+(5/3)|down> ?

You havd those because they are extremely easy and elementary, requiring at most very basic linear algebra. You will not find similar things in GR that are not in essence the Newtonian limit, inaccurate, or both. Take your example:

OBS.: The only exercise of GR I can think of here has to do with using the concept of gravitational escape velocity to determine the largest radius of a sphere of mass m that would prevent light from escaping

This is both inaccurate and relies on the Newtonian limit. As such, it could be argued that it is is making things worse when it comes to having an actual understanding of basic GR.

 

[pervect]

Hi All

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?

If so, what examples of exercises would be appropriate for the typical maturity of students in mathematics and physics?

Best wishes,
DaTario

I think some of the sector model approach could be introduced at a high school level. See for instance https://www.physicsforums.com/threa...s-space-in-the-real-case.1050537/post-6972834. I was particularly impressed by Zahn and Kraus, I cite a couple of their papers in their article.

Note that I haven't reviewed this particular approach extensively, I got interested in it from a recommendation by another poster and after a small amount of reading about it I found these particular authors. I was impressed enough to remember it and I occasionally tout it.

To recite them here, https://arxiv.org/pdf/1405.0323.pdf "Sector Models – A Toolkit for Teaching General Relativity. Part 1: Curved Spaces and Spacetimes" and https://iopscience.iop.org/article/10.1088/1742-6596/1286/1/012025/pdf.

Basically, the idea is to introduce the idea of curvature of space by cutting and pasting pieces of paper together and making simple curved surfaces. And one can generalize this to a similar "cut and paste" process in three dimensions. The fundamental idea is that one can make an approximation to a curved 2d surface (such as a sphere) by the process of cutting and pasting. And this can be generalized to higher dimensions, though anything above three is probably not going to be that easy to visualize.

This could help get across the idea of curvature, in a way that can be extended beyond two dimensions as Zahn , et al, does. I'm not going to get into that extension process here. One of my previous ideas, before the sector model approachy, was using spherical trignometry to talk about curvature. But references are important, and I don't think I've seen anyone attempt to talk about curvature in approach based on spherical trignometry.

There is another important limiation in what I've talked about so far, though. To really talk about GR, one needs to talk about curved space-time. This entails understanding both the concepts of space-time and the concept of curvature first, and we've only talked about the curvature part. If one doesn't understand space-time in a way other than as word salad, I can't see much as being accomplished :(. I am found of Taylor & Wheeler's "Space-time physics", in particular their "parable of the surveyor" as a way of understanding the idea of space-time, but I'm not sure if it would be a good fit for high school.

I'll digress enough to say that my take on Taylor & Wheeler's "Parable of the surveyor" is to talk about why we consider a plane to be a 2 dimensional surface, rather than treating "north-south" and "east west" as two separate dimensions. The answer to this question sheds light on why we consider space-time to be a unified entity , rather than seprate it into space and time. My interpretation of how "the parable of the surveyor" answers to say this is that the answer is symmetry. On the plane, we have rotational symmetry, which can turn north-south distances into east-west distances,. In space-time, we have a "boost" symmetry.

This is getting long, but I'm going to push it further and go one step further to actually talk about curved space-time cause that's where I wanted to end up.

As I mentioned in the previously cited PF thread

From: https://arxiv.org/pdf/1405.0323.pdf

In a spatial sector model a sector is rotated in order to lay it alongside the neighbouring sector. In the spatiotemporal case the rotation is replaced by a Lorentz transformation.

Lorentz transformations aren't going to be familiar at the high school level. It's possible to re-write the basic idea in common language, with the usual ambiguties that entails, but I'm not sure the idea is going to get across well. The issue, as I see it , is that needs a sold background in both curvature and space-time before one can even start to think about what curved space-time is.

But here goes my "common language" descriptoion of what the above means. A Lorentz boost is nothing but a moving frame of reference, so one basicaly replaces the idea of cutting and pasting pieces of paper together to form (i curved 2d surfaces with the idea of cuts and pastes together stationary frames of reference to moving frames of reference to generate a curved-space time. (It's not smooth, alas, because it's cut and pasted together from flat parts). I have never seen anyone try this approach, and I have some doubts about how well it would really work, but it could work in principle.

I'll talk about another idea I had of how to talk about GR at a very elementary level. First one introduces space-time diagrams as a way about talking about space-time. One needs to get across the idea that a space-time diagram represents time and space. One can potentially skip the whole discussion of why we want to unify the two. The next step is to talks about drawing these space-time diagrams on curved surfaces (like a sphere), rather than on flat pieces of paper. A weakness here is that in this super-simplified approach I mostly omit talking about why a sphere is curved and a plane is flat, and I don't really talk about curvature in general, just hoping the reader will agree that planes are flat and spheres are not.

In this second idea, I focus on geodesic deviation as the main consequence of curved space-time to make a specific example so that the discussion isn't so abstract that it's pointless. So I'll talk about how two geodesics on the sphere (great circle) appear to accelerate towards each other, and how this relates to GR's "geodesic deviation equation"..

It makes sense to me, but I don't think it actually gets across to the target audience very well (at least on PF). Probably it's not possible to actually cover that much material in a post, and I'm not patient enough to write a book. (Also, I'm not sure I still have the writing skills to do it well nowadays). I find references are needed, personal ideas aren't sufficient, and I don't have sufficient references for people doing things exactly this way.

I'll give an honorable mention to Baez & Bunn, https://arxiv.org/abs/gr-qc/0103044, "The Meaning of Einstein's equation", but I don't think it's high school level. In particular.

Before stating Einstein’s equation, we need a little preparation. We assume the
reader is somewhat familiar with special relativity — otherwise general relativity
will be too hard.

Unfortuantely, I don't think it's safe to assume that the high school student is familiar with special relativity, so I'd say Baez's & Bunn's approach is more college level. It's still intersting, though, IMO.

 

[Ibix]

OBS.: The only exercise of GR I can think of here has to do with using the concept of gravitational escape velocity to determine the largest radius of a sphere of mass m that would prevent light from escaping.

This sounds like Michell's "dark stars", which he proposed in 1784. Unfortunately, although a reasonable conjecture given the state of knowledge in the 18th century, the idea relies on an inaccurate theory of light and an inaccurate theory of gravity, and "dark stars" are nothing like black holes.

There are simple computations one can do with GR results. For example, you could find the altitude where orbital speed kinematic time dilation cancels gravitational time dilation. Or you could memorise the Schwarzschild metric, or the Kerr metric, and show that the latter reduces to the former when $a=0$. Or you could integrate $ds^2$ along a path, as you suggested. The question is, though, why bother? They're just party tricks if you don't understand where the formulas come from and how they fit together.

Just about the only useful thing I can think of to teach is Einstein's energy conservation argument for why gravitational redshift must exist. That gives some useful insight into physical thinking (it's a simple logical argument that shows you must either accept gravitational redshift or abandon energy conservation) and it's a step on the road. But where do you go from there? You run smack into covariant derivatives about two minutes later.

If you've got spare time in a physics curriculum, I'd teach a sanitised history of the development of GR. Maxwell, the ether, the failure of ether theory and attempts to address its failures, SR, GR. The insight into the scientific process (even the kind of just-so version of it I've sketched out above) is probably more useful to kids who are almost certainly not going to be working physicists.

 

[Vanadium 50]

If you've got spare time in a physics curriculum

...then something has gone wrong.

 

[DaTario]

Yes, but GR is not introductory. Quite the opposite. Many things in GR are so far removed from regular intuition built from our everyday experience and the mathematics so involved that it is not really possible to be both accurate and give a good introduction at the level you seem to want.

Just to add some levity to this discussion I cite the following work to respectfully (and with some good humor) confront your claim that the topic is complex from the start.

This is both inaccurate and relies on the Newtonian limit. As such, it could be argued that it is is making things worse when it comes to having an actual understanding of basic GR.

I was telling to a friend of mine (also a physicist) a couple of days ago, that I do not consider this exercise I mentioned as a genuine GR question. It seems to serve just as a motivation for teaching GR.


As exercises on GR I think we could find something that uses the linear algebra machinery so as to calculate distances or curvatures. Perhaps the use of matrices to provoke geometric distortions in space. There is a book to celebrate the aniversary of 70 years old of Einstein, in which several authors have written small texts. One of these texts talked about doing geometry with metal rules on a metal plane below which there were a couple of bunsen burners. The author says that the method of letting the rule get in thermal equilibrium with the local part of the plane will produce non uniform dilations in the rule which will correspond to distortions in the measurement of distances. So the sum of the internal angles of a triangle will not be 180 degrees anymore. I would like this idea to be a little more formalized. But perhaps this is difficult even in its simplest approach.

 

[DaTario]

This sounds like Michell's "dark stars", which he proposed in 1784. Unfortunately, although a reasonable conjecture given the state of knowledge in the 18th century, the idea relies on an inaccurate theory of light and an inaccurate theory of gravity, and "dark stars" are nothing like black holes.

Yes, I agree with you. But it may serve as a motivation at least.

There are simple computations one can do with GR results. For example, you could find the altitude where orbital speed kinematic time dilation cancels gravitational time dilation.

This seems to be a beautiful exercise. I don't know how difficult it would be to solve.

 

[DaTario]

I think some of the sector model approach could be introduced at a high school level. See for instance https://www.physicsforums.com/threa...s-space-in-the-real-case.1050537/post-6972834. I was particularly impressed by Zahn and Kraus, I cite a couple of their papers in their article.

Note that I haven't reviewed this particular approach extensively, I got interested in it from a recommendation by another poster and after a small amount of reading about it I found these particular authors. I was impressed enough to remember it and I occasionally tout it.

To recite them here, https://arxiv.org/pdf/1405.0323.pdf "Sector Models – A Toolkit for Teaching General Relativity. Part 1: Curved Spaces and Spacetimes" and https://iopscience.iop.org/article/10.1088/1742-6596/1286/1/012025/pdf.

Basically, the idea is to introduce the idea of curvature of space by cutting and pasting pieces of paper together and making simple curved surfaces. And one can generalize this to a similar "cut and paste" process in three dimensions. The fundamental idea is that one can make an approximation to a curved 2d surface (such as a sphere) by the process of cutting and pasting. And this can be generalized to higher dimensions, though anything above three is probably not going to be that easy to visualize.

This could help get across the idea of curvature, in a way that can be extended beyond two dimensions as Zahn , et al, does. I'm not going to get into that extension process here. One of my previous ideas, before the sector model approachy, was using spherical trignometry to talk about curvature. But references are important, and I don't think I've seen anyone attempt to talk about curvature in approach based on spherical trignometry.

There is another important limiation in what I've talked about so far, though. To really talk about GR, one needs to talk about curved space-time. This entails understanding both the concepts of space-time and the concept of curvature first, and we've only talked about the curvature part. If one doesn't understand space-time in a way other than as word salad, I can't see much as being accomplished :(. I am found of Taylor & Wheeler's "Space-time physics", in particular their "parable of the surveyor" as a way of understanding the idea of space-time, but I'm not sure if it would be a good fit for high school.

I'll digress enough to say that my take on Taylor & Wheeler's "Parable of the surveyor" is to talk about why we consider a plane to be a 2 dimensional surface, rather than treating "north-south" and "east west" as two separate dimensions. The answer to this question sheds light on why we consider space-time to be a unified entity , rather than seprate it into space and time. My interpretation of how "the parable of the surveyor" answers to say this is that the answer is symmetry. On the plane, we have rotational symmetry, which can turn north-south distances into east-west distances,. In space-time, we have a "boost" symmetry.

This is getting long, but I'm going to push it further and go one step further to actually talk about curved space-time cause that's where I wanted to end up.

As I mentioned in the previously cited PF thread

From: https://arxiv.org/pdf/1405.0323.pdf



Lorentz transformations aren't going to be familiar at the high school level. It's possible to re-write the basic idea in common language, with the usual ambiguties that entails, but I'm not sure the idea is going to get across well. The issue, as I see it , is that needs a sold background in both curvature and space-time before one can even start to think about what curved space-time is.

But here goes my "common language" descriptoion of what the above means. A Lorentz boost is nothing but a moving frame of reference, so one basicaly replaces the idea of cutting and pasting pieces of paper together to form (i curved 2d surfaces with the idea of cuts and pastes together stationary frames of reference to moving frames of reference to generate a curved-space time. (It's not smooth, alas, because it's cut and pasted together from flat parts). I have never seen anyone try this approach, and I have some doubts about how well it would really work, but it could work in principle.

I'll talk about another idea I had of how to talk about GR at a very elementary level. First one introduces space-time diagrams as a way about talking about space-time. One needs to get across the idea that a space-time diagram represents time and space. One can potentially skip the whole discussion of why we want to unify the two. The next step is to talks about drawing these space-time diagrams on curved surfaces (like a sphere), rather than on flat pieces of paper. A weakness here is that in this super-simplified approach I mostly omit talking about why a sphere is curved and a plane is flat, and I don't really talk about curvature in general, just hoping the reader will agree that planes are flat and spheres are not.

In this second idea, I focus on geodesic deviation as the main consequence of curved space-time to make a specific example so that the discussion isn't so abstract that it's pointless. So I'll talk about how two geodesics on the sphere (great circle) appear to accelerate towards each other, and how this relates to GR's "geodesic deviation equation"..

It makes sense to me, but I don't think it actually gets across to the target audience very well (at least on PF). Probably it's not possible to actually cover that much material in a post, and I'm not patient enough to write a book. (Also, I'm not sure I still have the writing skills to do it well nowadays). I find references are needed, personal ideas aren't sufficient, and I don't have sufficient references for people doing things exactly this way.

I'll give an honorable mention to Baez & Bunn, https://arxiv.org/abs/gr-qc/0103044, "The Meaning of Einstein's equation", but I don't think it's high school level. In particular.



Unfortuantely, I don't think it's safe to assume that the high school student is familiar with special relativity, so I'd say Baez's & Bunn's approach is more college level. It's still intersting, though, IMO.

It seems to be an interesting idea to work with paper, scissors and glue. In fact these notions seems to fit well in physics classes but can also enter in geometry or linear algebra classes.

 

[PeterDonis]

I cite the following work to respectfully (and with some good humor) confront your claim that the topic is complex from the start.

Unfortunately, the "mass warps space" visualization you refer to is not GR--it's a common pop science misconception of GR, which would have to be unlearned by anyone who learned it and then tried to learn actual GR.

Trying to have useful ideas for teaching a subject that you don't understand yourself does not have a good track record of success.

 

[PeterDonis]

I do not consider this exercise I mentioned as a genuine GR question. It seems to serve just as a motivation for teaching GR.

Is this thread about how to motivate people to want to learn GR? Or about how to teach it? Your OP was about the latter.

 

[DaTario]

If you've got spare time in a physics curriculum, I'd teach a sanitised history of the development of GR. Maxwell, the ether, the failure of ether theory and attempts to address its failures, SR, GR. The insight into the scientific process (even the kind of just-so version of it I've sketched out above) is probably more useful to kids who are almost certainly not going to be working physicists.

Yes, I think you are right if it applies to high school. If it applies to college may be not so much.

 

[DaTario]

Is this thread about how to motivate people to want to learn GR? Or about how to teach it? Your OP was about the latter.

I confirm that the thread is about how to teach at least first notions, principles and methods of GR.

Incidentally, problems were mentioned that serve to motivate the theme, although they are not GR problems themselves.

 

[DaTario]

Trying to have useful ideas for teaching a subject that you don't understand yourself does not have a good track record of success.

Dear PeterDonis, I agree with you. But I am trying here to see if there are easy steps a teacher can take to teach some elements of GR to college students (I gave up with high school in this thread, although high school students in Brazil study introductory linear algebra).

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

 

[Orodruin]

although high school students in Brazil study introductory linear algebra

What you need is introductory calculus on manifolds/differential geometry, not introductory linear algebra. This is the main piece of language that is missing. Without it, any attempt to teach GR is instead a series of anecdotes and analogies.

Sure, you can do problems like what was suggested: to compute where gravitational and motional time dilation cancel out, but that again is either just plugging numbers into formulas - often using approximations and generally not giving any insight whatsoever as to where the formula comes from.

 

[PeterDonis]

I am trying here to see if there are easy steps a teacher can take to teach some elements of GR to college students

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.

 

[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.