This question is primarily for publius, because it's a pretty gnarly general relativity issue, but anyone with insight on the issue is encouraged to chime in.

General relativity is a nonlinear theory, so you cannot say that the gravitational effect of 10⁵⁷ protons will be the same as 10⁵⁷ times the effect of one proton (as Newton would have it). But in simpler versions of GR, you can still get to the 10⁵⁷ proton gravity by adding up all the protons but modifying them for their potential interaction with each other, like subtracting their binding energy. This sounds rather like a "correspondence principle", where the theory that we would imagine applying to just one or a few protons would simply accumulate to a large number of them, in bottom-up fashion even if it wouldn't be the sensible way to do it.

But what if we have a rapidly spinning neutron star? My impression is, things are so complicated there that you'd have a hard time building that up one atom at a time, but rather you need to do it "holistically" with the full mass and all the required information in there right from the get-go to use the exact formalism of GR. But I don't know if that's true-- the stress-energy tensor is built up additively, so if it only depends on that, it has a correspondence principle. Similarly with the angular momentum. But does the exact result depend only on these additive entities? A black hole has "no hair", so is built from additive entities, but what about a neutron star? I guess the way to say this is, let's say you know the current neutron star gravity and spin, and so forth. Now add a single particle with a known rest mass, energy, and angular momentum. Can you calculate the new gravity of the neutron star exactly? Or do you need to specify more information than you can track from the particle you are adding-- in other words, is there an exact correspondence principle in general relativity?

__________________
If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

Ken,

Forgive me as I didn't see this until now. Chris Hillman joined BAUT here a few days ago and was posting a bit, and this would be a good one for him to answer indeed. Don't take this as definitive, just my gut ramblings:

Consider Schwarzschild. Our magic metric factor that's at the base of everything is 1 - R/r, where R = 2GM/c^2. Note that "M" there is "the mass you've got" there, the "actual gravitational charge" (rest mass less the binding energy and whatever else goes on), and that's for the "external field" of that M, if it is perfectly spherically symmetrical. For a black hole, where the thing has collapsed all the way, that's what you've got. No hair, M is all you have.

If you double M, you double the R, but as you noted, due to the non-linear nature of the geodesic equations, the effect of doubling M on some test particle at some 'r' will not merely double, although it will converge to that for large enough r.

For example, we might ask what is the force required to hold the test particle stationary at r. In the strong field that would not be linear in M, but another thing would happen as well. As we increase M, R increases, and when R = r, our test particle is at the horizon and cannot remain stationary! Our original question would no longer be valid if we let M get large enough. The change due to increasing M can blow everything away at some point, depending.

And note how drastic this is, much more than a simple non-linear relation. Up to some M, we can imagine a test particle sitting at 'r' at times on our external Schwarzschild clock. But for R >= r, that makes no sense. A test particle can't "be there" at any time on our clock.

And that illustrates the rub. Adding more particles, more mass, to something is going to ultimately be a full GR dynamic process. Rather than a fixed stress-energy tensor you specify and solve for, you're going to have a maddeningly complex situation where the stress-energy tensor dynamically evolves due to its own field.

Consider a spherically symmetric ball of fluid. The external field is Schwarzschild, but the internal field, below the surface is something else, the Schwarzschild interior. The stress-energy tensor lives in that interior. Externally, we can speak of big 'M' for the regular Schwarzschild solution, but there is more to that M that you'd think at first blush, some density*volume. It would reduce to that in the weak field limit of course, but in the strong field, it's complex.

And it's more than just the binding energy. The pressure contributes to the "gravitational charge", M, as well in a complex way. So the M we get there is going to depend on more than just the density. It will depend on the equation of state of the matter as well. So technically, a 10lb ball of foam makes a slightly different gravity field than a 10lb ball of lead of the same size. {ETA: depends on what you mean by mass of 10lbs. If it's 10lbs of Schwarzschild M externally, it's the same. However, if it's 10lbs defined as some volume integral of the density term in the stress-energy tensor, then it gets fun}

So if you imagine adding more and more particles, you've got all that going on, and at some point, depending on the equation of state, the thing would collapse to a black hole, which is the equivalent of the R >= r limit above. You've can't have more than M packed into R.

All of that is in the full strong field, highly relativistic regimes. In the weak field, I'd say you can build things up atom by atom until the field gets too strong. Then you'd have a dynamic evolution where you had to specify the initial conditions of how you were going to add particles.

-Richard

Oh, and BTW, I'm going to be gone for about 2 weeks and doubt I'll have a chance to even log on BAUT. An aunt of mine in California died suddenly and I'm having to fly out there. Ironically, she was coming out of her doctor's office after some blood work and just fell down on the sidewalk going back to her car. Some blood vessels in her brain just burst, causing massive brain damage. She went into a coma and died the next day. Had she lived, she would've been a vegetable anyway.

-Richard

Gee, I just ranted in another thread about the low quality of knowledge exhibited by most of the posters in forums like BAUT in discussions of gtr, but I can see this thread might develop in interesting ways.

I'm new here, so I don't have solid impression of what math/physics background you two have, and this is pretty subtle stuff so I should probably try to come back when I have more time.

Ken, is it possible that you might want to ask about any or all of the following topics?

1. Newtonian limit (aka weak-field slow motion limit, giving the "correspondence" between gtr and Newtonian gravity),

2. Weak-field limit (aka "linearized gtr", often used for elementary theory of gravitational waves and early discussion of mass/momentum of isolated objects like rotating stars in gtr), or slightly more generally, linearized perturbations of more interesting exact solutions than Minkowski vacuum,

3. Ways in which nonlinearity manifests itself in gtr, e.g.

a. e.g. in the class of Weyl vacuums, all axisymmetric static vacuum solutions, it is not hard to work out a way to isolate the nonlinear contribution to the "superposition" of two Weyl vacuums to obtain a third Weyl vacuum,

b. symmetries of the field equations themselves, with or without conditions, e.g. for Weyl vacuum they reduce to a familar PDE, and then applying symmetries to Weyl vacuum solutions exhibits a notion of "mass" which varies from place to place,

c. etc.,

4. How can one try to build a mathematical model in gtr of an idealized situation in which two massive objects, or a massive object and a nongravitational field, "interact"? Given that this interaction will almost always create gravitational radiation? Given that the EFE provides a consistency check on a possible solution, i.e. a spacetime equipped with tensor fields representing various nongravitational forms of energy and momentum as appropriate, but by itself doesn't provide a prescription for finding such possible solutions?

5. When we write m and r in 1-2m/r in the most common way of writing down the Schwarzschild solution, what does this parameter m really mean and what does the coordinate r really mean? E.g. in comparision to another popular radial coordinate, the one which appears in so-called "isotropic" coordinates?

(Publius: sorry for the loss of your aunt.)

My condolences, publius. Life is so much about people; its tough to loose a family member.

__________________
Lighten up! This is a stellar board! Author: duh.

"The Sun, with all the planets revolving around it, and depending on it, can still ripen a bunch of grapes as though it had nothing else in the universe to do..." Author: Galileo supposedly.

You can always tell when you have asked an expert a question, because you never get the answer to your question, but you get two other things instead: (1) how you should have asked the question so that it would have an answer, and (2) what that answer is! In this case, I have a menu of options for how to ask a meaningful question that actually has an answer, so I have a little more work to figure out just what I'm asking. Ultimately, I suppose I'm really asking "everything that I am equipped to understand the answer to". My equipment is a Ph.D. in physics, but in the case of GR, that unfortunately is not much of a guarantee of anything.

First a little more context. The question emerges from a discussion I had where I mentioned that we retrofit the axioms of physics to the data we have, but it's not our place to assert what makes reality tick. The other person said that surely I imagine that the gravity of the Earth exerted on me comes from the gravitational interaction of all Earth's atoms on all my atoms, to which I responded that I'm not sure what he means by "comes from" because I haven't the vaguest idea what the gravitational interaction between two atoms really is.

I realize that the weak-field limit of GR recovers Newtonian gravity, that it unifies our understanding of why all objects fall at the same rate, and that it is more accurate for predicting planet and GPS orbits and weak deflection and redshifting of light by stars. It even works for strong enough deflections to produce gravitational lensing. But we don't really have a fundamental theory for it, because it hasn't been unified with quantum mechanics, so we can't "build it up" from the gravitational interactions of atoms. Hence my comment.

This got me thinking about what kind of correspondence principle GR really has. If you accept that you are going to have to build it up from classically behaving elements, you can find the gravity of a star made of protons and electrons because you can still define statistical concepts like mass density and pressure for such quantum particles (what publius meant by an equation of state). But is there ever a point where GR actually needs more information for an exact treatment than you can get from classical dynamics? In other words, does the dynamics ever require a "top down" holistic treatment, rather than a "bottom up" reductionist treatment?

Put differently, will you need to track some "global behavior" to even have enough information to do the reductionist dynamics, which you may then make consistent with the global behavior? That's still not a correspondence principle because you could never get it without adding something global to all the local behaviors. In fact, it is more like an inverse correspondence principle, like the way thermodynamics takes on its own emergent characteristics (irreversibility, ergodicity) without contradicting the statistical mechanics of its constituents, but not describing them either. It's a strange state of affairs. So let's see what more concrete questions can emerge by consulting an expert:

This one I think we are good on-- GR gives Newtonian in the weak-field slow-speed non-quantum corner of regime space. But does Newtonian gravity work in a quantum mechanical Hamiltonian? I doubt we'll ever know.
This sounds like it would admit a superposition principle, so has a built-in correspondence principle like Newtonian gravity and allows the introduction of causality. So that seems fine.
That's penetrating more deeply into the correspondence, built up through nonlinear superposition. But it's still static, so we still can't say that "the whole is the sum of its parts" if the parts are dynamical.
Not sure on this one, the doing of physics at the level of "manipulating symmetries" has always been at the outside edge of my arsenal! But at least we would need to go outside Weyl vacuums for the reason above.
This sounds like it is getting right to the issue. It seems to suggest a dynamical incompleteness in GR, stemming from its lack of unification with the other forces. In other words, it is consistent with any number of unphysical possibilities, so it requires some external constraints to reign it into our universe. The correspondence of GR by itself may not be specifiable for that reason-- it may depend on the constraints we apply. We would likely use constraints that themselves demonstrate a correspondence principle, but would that correspondence translate into the application of GR, or could non-correspondent "emergent" properties appear in the GR solutions? This is as close as I can come to what I am asking here-- because if GR added to other theories that have a correspondence principle can result in emergent non-corresponding phenomena, then I have my answer to "don't you imagine the atoms of the Earth interact gravitationally with the atoms in me": no.
I don't know if coordinatizations enter directly into these issues, as one would expect correspondences to be coordinate independent. If that's not true, then it would seem to introduce preferred systems.

__________________
If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

That's the spirit!

Good. Did I mention that my Ph.D. is in math? I learned everything I know about gtr from reading books/papers and making computations/cogitations, so it -is- possible to learn gtr entirely outside the classroom. (But I don't think I could have done it had I lacked the benefit of a fine formal training in mathematics.) Funnily enough, my diss was on generalized Penrose tilings; I sometimes wonder if Penrose himself might not be the only other entity which knows about both Penrose tilings and NP formalism.

But note well: I've never been able to muster the intestinal fortitude to seriously study quantum mechanics, much less QFT, even though I possess the mathematical prerequisites like operator theory and Hilbert spaces, because, you know, it Just Doesn't Make Sense. As a non-physicist I figure I am permitted such luxuries!--- I obviously couldn't recommend this option to physics students.

Yes, indeed. Even in the context of classical gravitation, consider a binary star system. It produces gravitational radiation, and using the weak-field approximation we can compute the energy, momentum, polarization &c. of this radiation, but if we ask linearized theory to tell us where exactly the radiation is -created-, as it were, we ask in vain!

(I know hundreds of exact solutions describing propagating gravitational waves and beams of gravitational radiation, some of which I have found myself, but I know of no convincingly physically realistic exact solutions describing production of same. Using Bondi formalism one can write down good approximations to isolated sources of radiation, but again I don't see how to use this to answer such questions.)

when you throw in the slow motion condition: sources of the field are varying slowly, test particles are moving slowly wrt sources. Weak field also covers stuff like linearized gravitational radiation, first order approximations to isolated massive objects, and so on.

Agreed. Certainly if we assume that Nature adores the quantum, then gtr cannot be telling us "how gravitation really works" for the simple reason that it is not a quantum theory. But I doubt we can really say that QED allows us to understand "how EM really works", however.

Did you see the recent arXiv eprint by Luminet on the supposed death of science? He argues that we are near, not the death of science, but the birth of science, and furthermore, we should expect to always be near the birth of science. My sentiments exactly.

I guess everyone here knows that a paradigm seems to be emerging in which viable gravitation theories must be constrained by thermodynamical considerations, and furthermore, many notions which were originally discovered in the context of gtr and thus thought of as "gravitational phenomena" may one day be seen as thermodynamical phenomena.

BTW, my training was in ergodic theory so I have the mathematical prerequisites for that discussion also, although I can hardly claim to be a true expert in the huge discipline of ergodic theory.

Dunno, but I suppose I could ask John Baez if he has any comment. Maybe we should try to refine the question first...

[EDIT: oh, I think I see, you meant: can we take some simple example of quantum mechanical Lagrangian (Legendre-Young dual of the Hamiltonian) and add a Newtonian gravitational potential?]

BTW, did I misunderstand what kind of correspondence you are looking for? Are you looking for a quantum-classical gravitation dictionary?

Sorry, I'm lost. Can you try to reexpress that?

Symmetry: these days I am thinking about "geometric figures" in finite Kleinian geometries. A rambling general comment, possibly irrelevant here: Cartanian geometry, which has recently undergone a modest revival in physics, has been described as the common generalization of Kleinian geometry and Riemannian geometry. Over at the n-category cafe, Urs Schreiber, John Baez, and other luminaries are discussing things like groupoidification and n-categorification of mathematics and thus, of physics. Exciting times--- and maybe a bit brain-baffling for those who don't yet appreciate the mental labor saving virtues of the categorical approach to mathematics. One aspect of this area which I am particularly interested in right now is how the ideas of Galois, Klein, Joyal, Fraisse, Cameron, and others intersect in first order logic, enumerative combinatorics via "structors" (or "combinatorial species"), graph theory (esp. "random graphs"), group actions, topology, information theory, and discrete dynamical systems. For some hints of a the idea behind structors, see the eprint by Baez and Dolan, "From Finite Sets to Feynman Diagrams", http://arxiv.org/abs/math/0004133, which also has hints of a quantum version. Kleinian geometry is also part of the background for the lovely theory of reflection groups which does much to explain the special role played in mathematical physics by "finite simple groups of Lie type", and John has posted some lovely material relating a computation of the cohomology of the Lie group associated to a root lattice to computations using q-deformed binomial numbers. Which brings us right to finite projective spaces and closes the circle with finite Kleinian geometries.

Evidently I am somewhat reluctant to start thinking about gtr right now, huh?!

(In another thread, someone took umbrage to my remark that I've spent some of this week "slumming" at BAUT while waiting upon computations to terminate. If nothing else, I just gave a bit more indication of what those computations involve.)

Don't close that book, you might eventually find either that it can make more sense to you, or that it shouldn't need to. I think we mistake familiarity for understanding-- we are all familiar with the concept of a trajectory and much fuzzier about the concept of a wave, but if you think about it, trajectories are a much more bizarre concept than a wave. A wave constantly recreates itself, as everywhere it passes becomes a source for the next stage in its propagation and a sink that cancels where the wave has been-- out with the old, in with the new, makes perfect sense. A trajectory, on the other hand, asks us to believe that somehow a single particle can "know" its speed and direction, so it can manifest an inertia. How exactly does it do that anyway, and how did such a bizarre notion ever get to be thought of as more sensible than wave propagation? Was Galileo any less a genius of his day to understand this strange concept of inertia than was Schroedinger for understanding that the time evolution of a wave function was the key concept?

That's interesting, it speaks very much to the way solutions are emergent of theories, rather than explained by theories.

I didn't see it, but it sounds like something I'd agree with heartily. I always laugh at the concept of a "theory of everything", because I'm reminded of Galileo's comments to the effect that any time you truly discover something, you are not rewarded with a sense of how well you understand the universe, but rather, you are awed by the realization of how much more there is to understand. The ever-present "yes, but why", I imagine.I don't know how well we know that, many of us know that the surface area of a black hole is to be associated with having entropy so you can get one to form despite the loss of entropy everywhere else. But connecting that to a ball rolling down an incline isn't terribly obvious! Still, I may see where you are going with that.
Yes, maybe can we do gravity in QM like the way the energy levels of hydrogen are calculated without QED, and maybe not. The problem would be ever testing the result. My point is merely that we might imagine we can do this by analogy with electric forces, but if very strong gravitation isn't unifiable with such forces, maybe very weak ones aren't either, and they only overlap "in the middle".
It's possible that I'm using the term "correspondence" in a nonstandard way, or a way that is not well known outside the physics community. I think of a "correspondence principle" as the statement that one theory is in effect a strict subset of another, superceded as it were, with no new insights not available in the more fundamental theory, but possibly some computational advantages based on approximations or idealizations suggested by the superceded theory. When quantum mechanics was first formed, it was a point of some note that it did exhibit a correspondence to classical mechanics, in that with the appropriate simplifications applied at the QM end of things, all results of CM were consistent with QM predictions. CM at that point became a superceded theory that was only used for computational convenience when quantum accuracy was not needed.

The reason such correspondence is relevant is that if you think the goal is not to find a set of axioms that works in some new situation, but rather a more fundamental theory that subsumes all the old ones with which it overlaps, then you have to have a correspondence principle. But it seems to me that a crucial violation of such a correspondence is what one might call "emergent properties" of some particular theory. Like the way irreversibility is emergent of thermodynamic treatments of a gas, whereas the statistical mechanical equations, or the quantum mechanical ones for that matter, are entirely reversible. The reversibility of the "more fundamental" theory gets supplanted by the irreversibility of the "more practical" simplified over-theory. The property of determinism that exists in CM is lost by any real application of its uncertainties, so it is kind of an illusory propery compared to reality, and the over-theory sees through the illusion and replaces it with something that works better. In that sense the over-theory does not exhibit correspondence to the fundamental theory that is supposed to give rise to it-- the whole is greater than the parts, and contains a new concept that is not present in the purportedly more fundamental theory.
What I'm suggesting is, since GR is compatible with any imaginary physical interactions that don't violate its axioms, whether they are possible or not, finding a self-consistent GR solution to any particular physical interactions you select would seem to involve interplay between GR and those interactions. So the EFE seem more like a prescription for unifying any cockamamie physics you like into a holistic entity that includes gravity, and as such, it is not built on the concept of correspondence, it simply borrows whatever correspondences exist in the cockamamie physics and then in a single flourish calculates the dynamical evolution of such a system. If so, it has no general concept of correspondence built into it, it is an entirely "emergent" phenomenon in general.

__________________
If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

Thanks to you, Chris, and all. She was my mother's last living sibling, and it just happened all of the sudden, no warning at all. Just came out of the blue.

I'll be leaving tonight, and I'm sure I'll have some very interesting reading in this thread when I get back.


-Richard

You raise many interesting points, Ken! Some quick feedback:

I agree that is a common pitfall, one you and I try not to fall into!

Not sure I understand (even after reading what follows), but FWIW one point I have often tried to emphasize (following Bonnor and others) is finding solutions (typically one determines local solutions and tries to find maximal extensions) is only the beginning; the hard part is often finding a plausible physical interpretation. Einstein himself had quite a bit to say about this (see for example "hole problem" in D'Inverno's textbook) but unfortunately the proper mathematical tools were not widely available in his time.

If you want to learn more about exact solutions of the EFE, I recommend that as your first stop you look for a two part paper by Bonnor (the second part has coauthors), which appeared years ago in General Relativity and Gravitation (I can provide the exact citations if neccessary), in which he emphasizes the importance of finding a physically plausible interpretation. A good on-line survey of exact solutions which also stresses this point is http://arxiv.org/abs/gr-qc/0004016 (two more recent papers by Bicak are much less extensive). The standard source for exact solutions is the monograph by Kramer et al, Exact Solutions of Einstein's Field Equations, which is a veritable gold mine of information but which is unfortunately hard to read, particularly concerning some of the most important solutions. A prerequisite for reading this would be some graduate level gtr textbooks plus the monograph by Hawking and Ellis, The Large Scale Structure of Spacetime. An excellent specialized monograph is Griffiths, Colliding Plane Waves, which the author has made available on-line at http://www-staff.lboro.ac.uk/~majbg/jbg/book.html
(There is a huge huge literature so these are just a few of hundreds of particularly valuable resources.)

Just so.

I was actually referring to a number of distinct lines of thought which I lack time to even try to truly explain. One idea which should rock your universe is a (not entirely rigorous) argument by Ted Jacobson that the EFE can be derived by demanding that dS = dQ/T hold for local "Rindler horizons"; see
http://arxiv.org/abs/gr-qc/9504004
http://arxiv.org/abs/gr-qc/0602001
Another is the suggestion that the gtr can be viewed as a general framework for studying how energy gravitates, in the same sense that thermodynamics can be viewed as a general framework for studying how energy can (cannot) be used to do useful work. Both emerge, it is argued, from fundamental "theories of matter" (and/or fields/interactions), but conversely, both rule out putative fundamental "theories of matter" which fail to be "thermodynamically self-consistent".

You should be interested in numerous long-ago postings to venues like sci.physics.research in which I tried to explain similar insights. I'm sure I've put this much better in the past, but my points included:

1. Physics seems to demand choosing a manifold category (C^p spacetimes? C^infty spacetimes? real analytic spacetimes?) for convenience of the moment, which raises troubling issues related to "Will the real gtr please stand up?" See for example arXiv:gr-qc/9507019 and the monograph by Griffiths for some related discussion.

2. The EFE places no limits on what can stand on the RHS. Einstein himself understood this pragmatically as "any energy-momentum tensor which is physically reasonable according to well-established physical theory", an attitude which would be hard to maintain today with so many wild and wooly proposal being touted in the absence of convincing experimental tests. It is widely appreciated that the various "energy conditions" (see for example Carroll's textbook) are inadequate attempts to propose a crude "filter", since there seems to be at least one experimentally well-established physical scenario which violates each of these conditions (consider for example Casimir force). This "vagueness" perhaps be can be understood in terms of Jacobson's ideas.

3. By its very nature, the EFE makes it hard to set up allowed initial conditions, particularly if you are looking for model in which two things actually interact in a physically interesting way. (For example, currently an exact solution modeling a binary black hole system seems entirely out of reach, although in principle such things clearly must exist in a suitable space of vacuum solutions.)

4. As everyone here probably is well aware, much effort has gone into developing formalisms for treating splitting the EFE into constraint equations (criteria for "legal initial conditions", including geometry of an "initial hyperslice" and initial values of physical fields on that initial hyperslice) plus evolution criteria (evolve the geometry of a family of hyperslices plus values of physical fields on those hyperslices). In particular, theorists seek formalisms in which the evolution equations have desirable properties in terms of the theory of nonlinear PDEs. This program must confront numerous obstacles: one of the most fundamental is the development of Cauchy horizons. Particularly striking is the fact that it is possible to write down simple exact solutions in which such horizons are also "weak null nonscalar curvature singularities"; Ellis and Schmidt have given (not entirely rigorous) arguments that small objects might survive an encounter with such singularities, but by definition gtr proclaims its failure to make unique predictions about what might happen after that!

That's pretty much what I thought you meant. I usually put it something like this: "in the weak-field slow motion limit [mathematical details ommitted] we recover the non-relativistic classical field theory formulation of Newtonian gravitation from gtr". Similarly for classical limits of QFTs.

I avoid saying that classical mechanics has been -replaced- by quantum mechanics, since it is -appropriate- to use the former in many circumstances not merely for computation convenience but for (valid) theoretical insight. Old bad theories (like N-rays) are well and truly dead, but old good theories never die, they simply are further developed as an important "sector" of science, even if they are in some sense "subsumed" by "more general" theories.

I keep changing the subject, but interestingly enough, a very interesting arena in which one can study in great detail how properties emerge is in Kleinian geometry, including finite Kleinian geometries. As a very simple example, consider how the projective plane over GF(q) relates to the affine plane over GF(q). It is best to start by considering these as point sets equipped with actions by certain groups, the symmetry groups of the geometries. The symmetry group of the former is PGL(3,q) and the symmetry group of the latter is the subgroup AGL(2,q), namely the subgroup which leaves invariant some choice of one line in the projective plane as "the ideal line" (aka "circle at infinity"). As a general rule, when you restrict to the action by some subgroup, some of the orbits under the original action will split up, showing that the "more rigid" geometry (affine geometry, in this comparision), makes finer distinctions and has more invariants (all other things being equal). For example in the case GF(3), the projective plane has 13 points and 13 lines, but restricting from PGL(3,3) to AGL(2,3), the orbit of points under PGL(3,3) splits into two orbits under AGL(2,3) (9 ordinary points plus 4 ideal points), and the orbit of lines under PGL(3,3) also splits into two orbits under AGL(2,3) (12 ordinary lines plus 1 ideal line).

But that's only the tip of the iceberg: according to Galois we should study the stabilizer-fixset lattices, and then (particularly when we add actions on lines to the mix) we can see that elements of these lattice enumerate the "geometric figures" which are possible in each geometry and how they are related. For example, we can see from these lattices that in the projective plane over GF(3), each point lies in 4 lines and each line contains 4 points, while in the affine plane, each ordinary line contains 3 ordinary points and 1 ideal point (the "point at infinity"), while each ordinary point lies in 4 ordinary lines. Following Boltzmann, Klein, and Planck, certain homogeneous spaces (complexions) then represent the possible "motions" (in sense of projective or affine geometry) of each kind of geometric figure (for example, the four lines concurrent on a point, in case of a projective plane), and numerical invariants of these are entropies satisfying the axioms laid down by Shannon. For example, in projective geometry, if we fix two points we still have some freedom to move the other points on the line through these two points, but in affine geometry, the entire line is pointwise fixed; these distinctions are reflected in the fact that the complexion of two points in affine geometry is smaller the complexion of two points in projective geometry (which again illustrates the notion that affine geometry is "more rigid than" projective geometry).

Even more striking observations occur when we ask how much information we gain about the motion of a line when we learn the motion of a disjoint point (for example). We can say that "Shannonian" entropies (conditional entropies), numerical quantities which obey certain axioms formulated independently of any probabilistic context, arise as numerical invariants of underlying algebraic objects, the complexions (conditional complexions), so in a sense Klein unified Galois theory and information theory (an achievement obscured by the historical oddity than on Earth, information theory appeared -later- than Kleinian geometry!).

Next, we can consider finite analogues of Euclidean geometry, in which new concepts "emerge", for example the euclidean notion of "distance". (Here, concept is also being used as a technical term; see Wille's theory of concepts in the undergraduate textbook by Davies and Priestly, Introduction to Lattices and Order. See also "closure operator" and note that "fixset of" is a closure operator.) Generalizing in a different direction, we obtain geometries in which another new concept make sense, "area". Or going off in yet another direction, we can obtain symplectic geometry with all the "emergent concepts" characteristic of that notion of geometry.

Before anyone asks, yes, these ideas should provide a useful tool in investigating Wheeler's old suggestion that geometry itself should emerge from more primitive notions, even from logic. I mentioned Fraisse theory, a topic in model theory in which one studies how properties emerge in first order logic from (essentially) Klein's point of view. These are very general ideas which apply to any group action, in particular any smooth action by a Lie group such as any of the "classical groups" studied by Weyl, but they are interesting only for nonabelian groups. Someone ought to write a book explaining all this good stuff.

Wow, that was quite a tour de force. My mathematics knowledge is woefully inadequate to follow it in detail, but at least I can form a general sense of the richness of the mathematical landscape that sits underneath each of these naive physical concepts. Above all it conveys the sense that even a fairly complete physics education is still nought but a tip of the iceberg as far as our mind's ability to model the phenomena we encounter-- and the "reality" must be orders of magnitude richer still. This seemed a particularly interesting avenue: