Pedagogy: story lines for introducing GR?

[bcrowell]

There are various roads from SR to GR. In a couple of years I may get a chance to teach a semester-length class on relativity for liberal arts students. Any comments on what story line works best?

Some possibilities:

(1) Thought experiments with elevators suggest that the Newtonian distinction between inertial and noninertial frames rests on a shaky foundation. This leads to the equivalence principle, and more thought experiments with elevators show that there must be gravitational time dilation, as verified in Pound-Rebka and Hafele-Keating. Clocks and rulers change their behavior from point to point (i.e., the metric is not constant).

(2) Instantaneous propagation of signals is inconsistent with SR. Therefore Newtonian gravity can't be right, and we must have phenomena such as gravitational waves. Thought experiments such as Feynman's sticky bead and the "Atlas" argument in Taylor and Wheeler's spacetime physics show that gravitational waves carry energy. SR tells us that mass and energy are equivalent, so gravitational fields must themselves act as sources of gravitational fields. This is pretty much the Einstein field equations put into words.

(3) Maxwell's equations aren't invariant under Galilean boosts, so we're forced to use Lorentz transformations instead. But Newtonian gravity isn't invariant under Lorentz transformations, so again we need to make a new theory. This is summarized from the introduction to General Relativity from A to B by Geroch.

(4) The following is summarized from the introduction to Einstein's paper "The foundation of the general theory of relativity" (annotated translation at the end of the pdf version of this book http://www.lightandmatter.com/genrel/ ). Thought experiments such as the parable of the two planets lead us to Mach's principle. Applying Mach's principle to a rotating frame gives noneuclidean spatial geometry.

Of course more than one of these could be presented in a semester-length course -- maybe all of them could.

Geroch doesn't really spell out #3 very explicitly, and I'm not clear on exactly what he has in mind. When we apply the Lorentz transformation to electrical interactions, we're forced to invent magnetism. (I think Purcell was the first to do this at the undergrad level, and it can be done at a gen ed level, e.g., http://www.lightandmatter.com/html_books/7cp/ch06/ch06.html#Section6.2 .) So naively I guess this would suggest simply making Newtonian gravity into a twin of electromagnetism, and this is actually qualitatively a pretty decent picture, since it gives gravitational waves, although obviously it's wrong in detail (wrong polarization properties, ...) What is the crucial difference between the gravitational case and the EM case? Opposite sign of the coupling constant? The equivalence principle? The fact that mass, unlike charge, has special logical status in SR?

 

[zonde]

(1) Thought experiments with elevators suggest that the Newtonian distinction between inertial and noninertial frames rests on a shaky foundation. This leads to the equivalence principle, and more thought experiments with elevators show that there must be gravitational time dilation, as verified in Pound-Rebka and Hafele-Keating. Clocks and rulers change their behavior from point to point (i.e., the metric is not constant).

I would say this is rather clear. Probably therefore it's first on your list, right?
And speaking about equivalence principle have you seen this paper that PAllen posted in another thread http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf
And I would suggest to add Shapiro delay to the list of experiments. I personally like it very much and then I think you can't really have too much references to good experiments.

(2) Instantaneous propagation of signals is inconsistent with SR. Therefore Newtonian gravity can't be right, and we must have phenomena such as gravitational waves. Thought experiments such as Feynman's sticky bead and the "Atlas" argument in Taylor and Wheeler's spacetime physics show that gravitational waves carry energy. SR tells us that mass and energy are equivalent, so gravitational fields must themselves act as sources of gravitational fields. This is pretty much the Einstein field equations put into words.

As a non-expert I would say this way is much more advanced then (1) in particular bold part. But if you can lay it out in a clear manner then it seems quite interesting too.

(3) Maxwell's equations aren't invariant under Galilean boosts, so we're forced to use Lorentz transformations instead. But Newtonian gravity isn't invariant under Lorentz transformations, so again we need to make a new theory. This is summarized from the introduction to General Relativity from A to B by Geroch.

Are you sure that you can show in a straight forward way how GR solves the problem of Newtonian gravity not being invariant under Lorentz transformations?

(4) The following is summarized from the introduction to Einstein's paper "The foundation of the general theory of relativity" (annotated translation at the end of the pdf version of this book http://www.lightandmatter.com/genrel/ ). Thought experiments such as the parable of the two planets lead us to Mach's principle. Applying Mach's principle to a rotating frame gives noneuclidean spatial geometry.

I was reading that paper posted by PAllen and have to say that explanatory potential of Mach's principle does not seem very high for me even if it led to prediction of frame-dragging effect (and confirmation).

 

[cosmik debris]

I like the number 3 approach. Introducing symmetry and the corresponding conservation laws I think gives a good basis for understanding lots of physics.

 

[Mark M]

I prefer (1). I think the best approach is to introduce the equivalence principle, which then leads to gravitational redshift and then gravitational time dilation.

 

[atyy]

I like the make Newton compatible with SR road. So we make a field on flat spacetime, and this actually works as a coherent theory - that's Nordstrom's theory, which was the first consistent relativistic theory of gravitation.

Historically, Einstein (with Fokker) took Nordstrom's theory and cast it in geometrical form before he reached his field equations. Nordstrom's theory is ruled out on empirical grounds, for giving the wrong perihelion precession.

After Einstein got his geometrical field equations, the QFTers reflattened his theory as a spin-2 field on flat spacetime.

So both Nordstrom's and Einstein's theories have field on flat spacetime as well as gerometrical formulations - with some caveats about what happens near singularities.

An amazing claim I've read, but haven't understood, is that Weinberg showed that the quantum version of a spin-2 field in flat spacetime demands the equivalence principle - ie. the EP is derived!

 

[bcrowell]

So we make a field on flat spacetime, and this actually works as a coherent theory - that's Nordstrom's theory, which was the first consistent relativistic theory of gravitation.

Historically, Einstein (with Fokker) took Nordstrom's theory and cast it in geometrical form before he reached his field equations. Nordstrom's theory is ruled out on empirical grounds, for giving the wrong perihelion precession.

Interesting! But this seems to contradict Geroch's claim that you can't make a Lorentz-invariant theory of gravity in flat spacetime...? (He claims it as a theoretical no-go argument, not something that needs to be tested empirically.)

 

[pervect]

I'd say that the crucial difference between the gravitational case and the electromagnetic case can be traced back to what we call gravitational time dilation, which is really a symptom of space-time curvature.

We don't see any sort of electromagnetic time dilation, except insofar as the electromagnetic fields also act as gravitational ones.

This ultimately leads Einstein - and us - to the notion that what transforms covariantly is a rank-4 tensor, the Riemann, because that's how curvature transforms.

Electromagnetic fields can be represented by a rank two tensor. It's tempting to try and argue that the important difference is that you can't measure gravitational forces directly, you can only measure tidal forces. But taking this point of view seriously would lead one to a rank-3 tensor description of gravity, not the rank 4 tensor description. At least that's where it leads me.

A tidal tensor is simply the covariant derivative of a force, so we should be able to represent the tidal tensor of electromagnetism by taking the covariant derivative of the Faraday tensor.

The need to explain what we call gravitational time dilation, is what leads to the rank 4 tensor theory - IMO anyway.

 

[Ben Niehoff]

I'm not sure which approach is best. However,

(1) Thought experiments with elevators suggest that the Newtonian distinction between inertial and noninertial frames rests on a shaky foundation. This leads to the equivalence principle, and more thought experiments with elevators show that there must be gravitational time dilation, as verified in Pound-Rebka and Hafele-Keating. Clocks and rulers change their behavior from point to point (i.e., the metric is not constant).

I never liked the elevator thought experiments. However, you do need to introduce the equivalence principle somehow.

Also, clocks and rulers do not change their behavior from point to point. What changes is the relationship between a clock or ruler and the underlying coordinates.

(2) Instantaneous propagation of signals is inconsistent with SR. Therefore Newtonian gravity can't be right, and we must have phenomena such as gravitational waves. Thought experiments such as Feynman's sticky bead and the "Atlas" argument in Taylor and Wheeler's spacetime physics show that gravitational waves carry energy.

This might be a bit sophisticated for liberal arts majors. Also, I don't think "gravitational waves exist, and carry energy" is the most important thing to learn about GR. The equivalence principle is way more important. If you introduce gravitational waves too early, prepare for "speed of gravity" arguments.

SR tells us that mass and energy are equivalent, so gravitational fields must themselves act as sources of gravitational fields. This is pretty much the Einstein field equations put into words.

I really don't like this point of view. The EFE are nonlinear. But you can't move the nonlinear terms to the RHS and call them a source, because such a source is nontensorial. The real reason the EFE are nonlinear is because they are built out of curvature data, which are nonlinear. I would just leave it at that.

(3) Maxwell's equations aren't invariant under Galilean boosts, so we're forced to use Lorentz transformations instead.

Careful with that. It's not enough to say "The theory is prettier this way, so it must be true!" Maxwell's equations are Lorentz-invariant, but whether Lorentz- or Galilean-invariance holds in real life must be determined by experiment. You'll have to discuss the experiments that showed the aether theory to be untenable.

(4) The following is summarized from the introduction to Einstein's paper "The foundation of the general theory of relativity" (annotated translation at the end of the pdf version of this book http://www.lightandmatter.com/genrel/ ). Thought experiments such as the parable of the two planets lead us to Mach's principle. Applying Mach's principle to a rotating frame gives noneuclidean spatial geometry.

Haven't had time to read this. Is Mach's principle the one about motion "relative to distant stars"? I'm not sure I agree with such a thing. First of all, acceleration is a local property that can be measured without reference to distant objects. Second, rotation is not even uniform acceleration! Rotating frames are easy to detect. One does not need objects to rotate "with respect to"; an object can rotate with respect to itself. Indeed, the entire universe can be uniformly rotating.

 

[atyy]

Interesting! But this seems to contradict Geroch's claim that you can't make a Lorentz-invariant theory of gravity in flat spacetime...? (He claims it as a theoretical no-go argument, not something that needs to be tested empirically.)

Nordstrom gravity as a scalar field in flat spacetime is described by Ravndal and Giulini, and also mentioned by Weiss. Ravndal (section 2.4) also gives the Einstein-Fokker geometric reformulation of Nordstrom's theory.

Thorne does claim in his popular book about wormholes that GR can be considered a field in flat spacetime. I'm not sure if the physically relevant portion of the FLRW solution can be done this way, but I believe the physically relevant parts of the Schwarzschild solution can be.

Flat spacetime GR is described by Straumann, Deser, and Hinterbichler (p4-6 & p51-52).

While going back and forth between fields in inertial frames versus geometry, it might be nice to mention that Newton can be reformulated as geometry too.

Since the EP (minimal coupling) is key to GR, Weinberg's claim to derive the EP is given in this review by Bekaert, Boulanger and Sundell (Appendix A, p33-35). I'm still working on trying to understand it.

 

[bcrowell]

Interesting comments, Ben Niehoff -- thanks!

Careful with that. It's not enough to say "The theory is prettier this way, so it must be true!" Maxwell's equations are Lorentz-invariant, but whether Lorentz- or Galilean-invariance holds in real life must be determined by experiment. You'll have to discuss the experiments that showed the aether theory to be untenable.

Good point. This is an argument I'm paraphrasing from Geroch, and Geroch, who is a theorist, makes virtually no contact with experiment in that book. But note that what I'm really asking about in this thread is how to get from SR to GR, so we can assume that SR has already been developed appropriately.

Haven't had time to read this. Is Mach's principle the one about motion "relative to distant stars"? I'm not sure I agree with such a thing. First of all, acceleration is a local property that can be measured without reference to distant objects. Second, rotation is not even uniform acceleration! Rotating frames are easy to detect. One does not need objects to rotate "with respect to"; an object can rotate with respect to itself. Indeed, the entire universe can be uniformly rotating.

I have a different take on this. Einstein was only trying to write a heuristic motivation, and he didn't yet understand the implications of GR. Among the implications that he definitely did not understand at that stage were some of the facts you list, e.g., that the whole universe can be rotating. The job of a heuristic is to say, "Here's why it would be natural to go down this path," and if it does that, it's done its job. Furthermore, everything you're saying is a correct interpretation of GR, but basically none of it was empirically established until about 1975. What I mean by that is that, until ca. 1975, Brans-Dicke gravity was consistent with all observations, and in Brans-Dicke gravity Mach's principle is true and almost every statement you've made is false. Here's a longer explanation of my point of view: http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.3 If you have access to journals, the original paper is extremely interesting and readable: C. Brans and R. H. Dicke, “Mach's Principle and a Relativistic Theory of Gravitation,” Physical Review 124 (1961) 925

 

[bcrowell]

Nordstrom gravity as a scalar field in flat spacetime is described by Ravndal and Giulini, and also mentioned by Weiss. Ravndal (section 2.4) also gives the Einstein-Fokker geometric reformulation of Nordstrom's theory.

Cool, thanks for pointing me to those papers!

What I'm getting from the Giulini paper is that:
(1) Some of Einstein's criticisms of Nordstrom gravity were simply wrong.
(2) Nordstrom gravity disagrees with both prior and later empirical evidence. It predicts the wrong perihelion precession for Mercury, and predicts no deflection of free light rays by gravity. ("Free" is necessary because it does predict gravitational effects on EM waves that are confined in a box.)

I would guess that Geroch was either (a) wrongly influenced by some knowledge of #1 or (b) rightly influenced by the fact that #2 violates the expectation that all forms of mass-energy should couple to gravity.

 

[atyy]

I would guess that Geroch was either (a) wrongly influenced by some knowledge of #1 or (b) rightly influenced by the fact that #2 violates the expectation that all forms of mass-energy should couple to gravity.

Regarding #2, I'm not sure this is right, but I suspect that even though there is no global deflection of light by Nordstrom's theory, it would still pass all EP thought experiments like red shift, local deflection of light in the elevator. The reason for my guess is that http://relativity.livingreviews.org/Articles/lrr-2006-3/fulltext.html mentions that Nordstrom's theory satisfies SEP (last para in section 3.1).

The distinction between local and global light deflection is mentioned by Pössel.

 

[Cleonis]

There are various roads from SR to GR. In a couple of years I may get a chance to teach a semester-length class on relativity for liberal arts students. Any comments on what story line works best?

Like Ben Niehoff I feel the elevator cabin thought experiment is comparitively weak.
It is known that Einstein also used the thought experiment of a spinning disk as a heuristic.
Let's say you have a giant rotating wheel, so at all point not at the central axis the setup is pulling G's.
What happens if you set up a Pound-Rebka experiment somewhere along the rim? On the basis of SR you expect a frequency shift; the base of the tower is circumnavigating the central axis at a faster velocity than the top of the tower.
Einstein's first explorative theory of gravity had only gravitational time dilation, but no spatial effects.
I would make that the first step in the story line: an exploratory theory of gravitation, with only gravitational time dilation.

What is the crucial difference between the gravitational case and the EM case?

The crucial difference, in my opinion, is in how the mediator of the interaction is conceived.
The mediator of electromagnetic interaction is the electromagnetic field. A pair of particles in electromagnetic interaction is thought of as both coupling to each other's electromagnetic field.
The electromagnetic field is thought of as an occupant of spacetime. Like matter, the electromagnetic field is an inhabitant of spacetime.
The mediator of gravitational interaction is thought of as a deformation of spacetime itself. We have that around mass-energy spacetime is deformed, and we have equations to describe that. There is no notion of a gravitational field that is in some way an occupant of spacetime. The spacetime itself is the field.

A what if scenario:
What if a separate gravitational field exists, independently of a uniform Lorentz invariant background? Then you need the hypothesis that the time dilation and spatial curvature effects of the gravitational field are perfectly tuned to make the uniform Lorentz invariant background inaccessible to observation. It would exist, but always hidden. Such perfect hiding is acutely improbable. More likely there is no such thing as a separate-gravitational-field occupying a uniform-Lorentz-invariant-background. More likely the spacetime itself is the field.

The concept 'deformation of spacetime is the mediator of gravitational interaction' and the Principle of Equivalence are in effect the same concept. Each is a logical implication of the other.



Einstein pushed relentlessly for the principle of equivalence. He was the only one to do so.

If Einstein's effort would not have been there history would have unfolded differently, and in a very interesting way.
See John Norton's article Einstein, Nordström and the early Demise of Lorentz-covariant, Scalar Theories of Gravitation
John Norton describes how he thinks history would have unfolded without Einstein's input.

Nordström's theory of gravitation was a theory in which a separate gravitational field exists, in a uniform Lorentz invariant background. There would be several theories at the time, some of them maybe taylored to account for the anomalous Mercury precession.
Experimental results such as Shapiro delay would bring deficiencies of the Lorentz invariant theories into focus. To reproduce the experimental findings the theories would have to move to more and more compliance with the principle of equivalence.
Eventually the physics community would arrive at a theory with such complete incorporation of the principle of equivalence that the implicit uniform-Lorentz-invariant-background is in all circumstances inaccessible to observation.
And then the realization would come: "Hang on, a background that is always inaccessible to observation, we've been there before!"
And then the conclusion: the gravitational field and the background are one and the same thing.


Well, that's not how actual history unfolded. From the very beginning Einstein pushed hard for the principle of equivalence.
How history actually proceeded is a hard to understand story line for introduction to novices.
GR obsoleted the fledgling SR.
SR assumes the existence of a uniform Lorentz invariant background and then GR drops the 'uniform'. With GR we have a dynamic background.

 

[bcrowell]

The crucial difference, in my opinion, is in how the mediator of the interaction is conceived.
The mediator of electromagnetic interaction is the electromagnetic field. A pair of particles in electromagnetic interaction is thought of as both coupling to each other's electromagnetic field.
The electromagnetic field is thought of as an occupant of spacetime. Like matter, the electromagnetic field is an inhabitant of spacetime.
The mediator of gravitational interaction is thought of as a deformation of spacetime itself. We have that around mass-energy spacetime is deformed, and we have equations to describe that. There is no notion of a gravitational field that is in some way an occupant of spacetime. The spacetime itself is the field.

The problem with this as a story line is that it presupposes that we want a geometrical theory, but the story line is supposed to be an explanation of why we would even go looking for a geometrical theory. It doesn't answer the question of why we are content to have a nongeometrical theory of electromagnetism but want a geometrical theory of gravity, when they superficially appear very similar (both being 1/r^2 forces).

But I think the rest of your post pretty much hits the nail on the head for me. If you insist on the equivalence principle, a geometrical theory is very natural. If you don't, then you could very easily end up believing in the Nordstrom theory.

 

[Cleonis]

[...] the story line is supposed to be an explanation of why we would even go looking for a geometrical theory.

Einstein followed his intuition that told him that he had to go for a theory in which inertial and gravitational mass are equivalent as a matter of principle.
Einstein's contemporaries did not see it that way. At the time the case for a geometrical theory was not strong.
So, with only the clues of 1905/1915 available it cannot be argued that a geometrical theory is the way to go.

For a story line for the benefit of education the next best thing one can do is to provide plausibility arguments that a Nordström-type theory, when pushed by experimental results, will eventually end up as a theory in which an implicitly assumed uniform Lorentz-invariant background is inaccessible to observation.


Historically, Einstein abandoned looking for a Lorentz-invariant scalar theory because he expected that such theories cannot be made to uphold the conservation laws.
As it turned out Nordström's theory disproved that expectation.
Again, with only the 1905-1915 clues available the case for a geometrical theory can't be made.

 

[atyy]

But I think the rest of your post pretty much hits the nail on the head for me. If you insist on the equivalence principle, a geometrical theory is very natural. If you don't, then you could very easily end up believing in the Nordstrom theory.

Historically, Einstein abandoned looking for a Lorentz-invariant scalar theory because he expected that such theories cannot be made to uphold the conservation laws.
As it turned out Nordström's theory disproved that expectation.
Again, with only the 1905-1915 clues available the case for a geometrical theory can't be made.

But doesn't the Einstein-Fokker geometrical reformulation of Nordstrom's theory support the view that there isn't a distinction between geometrical and non-geometrical? If one restricts to geometrical theories, then isn't the distinction between Nordstrom's and Einstein's theories "background independence" or "no prior geometry"?

MTW point to a discussion by Anderson, which I've never read, but I believe PAllen has Anderson's text. Giulini has a discussion of background independence and the Einstein-Fokker geometric formulation of Nordstrom in section 2.4.1 of http://arxiv.org/abs/gr-qc/0603087.

 

[twofish-quant]

1) One thing that I would start out with are some demonstrations of what it means for space to be curved. There are some demonstrations with parallel transport with the idea that "space doesn't need to curve into anything."

2) Something else that might be useful is to use the "tetrad formalism." Something that makes initutive sense to me would be to think of curved space as pieces of flat space taped together, and I think this would be a good way of having students visualize what is going on.

3) One thing that you can show that would be a good motivator for why all of this is necessary is how Newtonian physics just falls apart when you put a time delay in propagation time. Showing that SR and Newtonian gravity are incompatible is pretty straightforward and shows that you need a new theory.

4) Finally, I got an excellent question over private mail from someone who had often heard it said that GR and QM were fundamentally incompatible, but wanted to know why they were incompatibility. I thought this was an excellent question because it was something that I knew could be explained simply but I couldn't off the top of my head explain it. Something that I would have liked to do was to have two columns (how QM sees the world | how GR sees the world) and then go into why those two clash.

5) Especially for non-science majors, I like to heavily emphasize the experimental background for the theories rather than talk about theoretical elegance. Go through the major experiment effects of GR with references to the experiments.

 

[Andy Resnick]

There are various roads from SR to GR. In a couple of years I may get a chance to teach a semester-length class on relativity for liberal arts students. Any comments on what story line works best?

<snip>

I use the story line #3 in class (health science students, algebra-based intro). Nobody has mentioned that liberal arts majors are going to be totally lost by all the jargon being thrown around here- presumably you realize that :)

Edit- I also introduce GR by analogy to the surface of the Earth- moving North-South vs. East-West introduces the notion of a 'fictitious force', combined with the paradox that force-free mechanics (motion in straight lines) appears violated for free body motion, because we can cancel gravity with a fictitious force. Thus gravity and curvature are interchangeable.

I don't know how much sticks, but I enjoy telling the story... :)

 

[piareround]

Thorne does claim in his popular book about wormholes that GR can be considered a field in flat spacetime. I'm not sure if the physically relevant portion of the FLRW solution can be done this way, but I believe the physically relevant parts of the Schwarzschild solution can be.

Correct! Actually I think that he publish a "lesson plan" of how to teach GR using wormholes. (http://www.physics.uofl.edu/wkomp/teaching/spring2006/589/final/wormholes.pdf )

I cite this as a way that you might attract liberal arts majors who might be more interested in the historical side of GR. :)

After all, we cannot simple come from a simple Piagetian based teaching style but must include social communication, scaffolding and interaction (Vygotisky) while appealing to twenty first century learning need to address different intellengences (music, art, kinestetic, etc.)