How does GR handle metric transition for a spherical mass shell?

PAllen

Then it is not a straw men. Your last sentence is a mathematical falsehood, with no possible valid interpretation. That is, anything you can detect restricted to a 2-surface (or thin 3-analog of it) will tell you nothing about presence of absence of spatial curvature.

PAllen

No, this is false. As long as you restrict to a 2-surface, and use physical procedures (which naturally pick out the flattest possible interpretation of local reality), you will detect no deviation. To make this plausible, go right back to my argument that you accused of being strawman - 2-sphere embedded in Euclidean 3-space. The 2-surface is curved, the 3-space is flat. As a result, anything you find restricted to a 2-surface cannot be distinguished from an embedding artifact; further, in seeking straightness, you will pick out a flat 2-surface and find no deviation.

The discussion I had with Peter went further than this. That if spacetime curvature is simple enough, then even fully general measures of 3-space curvature (spanning large regions) can show exact flatness. In each case Peter mentioned it is the 'most natural observer' - the inertial one, or the one who sees CMB radiation as isotropic - that detects no spatial at curvature at all, by any means. His K, in SC geometry, only applies to stationary observers, who are non-inertial. One can detect similar things for non-inertial observers in flat spacetime, so even K is related more to non-inertial motion than intrinsic spatial curvature.

PeterDonis

What is the triangle made of? The "triangle" is not a physical object; it's an abstraction. To "move" it, you have to realize the abstraction somehow. How?

How are the sides going to be kept geodesically straight? As an abstraction, yes, a "triangle" in a curved spacetime, with geodesically straight sides, will have its angles add to something different from the Euclidean sum--more if the spacetime is positively curved, less if it's negatively curved. That much is simple geometry. But to move from that to the actual physical behavior of a physical object, you have to bring in, well, physics.

Btw, in the examples we've been discussing, both in 2-D and 3-D, the "hoops" and 2-spheres are *not* necessarily composed of geodesics. In the 2-D case, the "hoops" are lines of latitude, which are not geodesics except for the equator. In the 3-D case, the 2-spheres themselves have geodesic tangent vectors, but the static objects sitting between them do not if you include the time dimension; static objects hovering above a gravitating body are accelerated.

How will you constrain the triangle to have straight sides (where I assume "straight" means "geodesic in the curved surface"--again, precise terminology really helps in these cases)? If you work it out, you will see that, if you imagine taking a triangle of actual, physical material and moving it from flat to curved space this way, constraining the sides as you suggest will necessarily cause stresses in the material. As soon as that happens, you can no longer use the material as a standard of distance to compare things with flat space, because in flat space the material was unstressed.

In the sense that you can't "wrap" a flat surface onto a curved surface without distorting it, yes--so if you don't distort the flat surface, it won't fit properly onto the curved surface. That's a quick way of summarizing what you've been saying. But the distortion is a *physical change* in the flat object, so once that object is distorted you can't use it to compare the flat with the curved surface. That's what you appear to be missing.

I won't bother commenting on the 3D case because it works out exactly the same.

None of what you say is incorrect. It just doesn't mean what you appear to think it means.

Added in edit, after seeing PAllen's post: Here's another way to say what I just said. I say none of what you have said is incorrect; but at the same time, PAllen is right when he says the comment he quoted is false. That's because I am saying you are correct as a matter of abstract geometry only, while he is saying you are incorrect in how you are trying to relate the geometry to the physics. *Both* of our comments are valid. Can you see why?

Q-reeus

From #151 "In flat spacetime we have a flat equilateral triangular surface formed from a fine tiling of much smaller uniform equilateral triangles."
Miss that bit - tiling, as in tiles? A composite, physical object.
Like gaps appearing for instance. I gave two restraint examples - gaps appear, or sides curve. Not physics? Assuming some kind of elastic glue joining the triangle tiles, naturally, forcing the outer triangle's to be straight requires imposing stresses - evidence of living in curved spacetime. Gaps or curved sides or stresses. Not there, or not needed in flat spacetime. Can't see any physics at work; no makings of an in principle 'detector'?
Fine, but in present scenario, whether triangles have perfectly 'straight' sides is just a convenience - we are only really interested in detecting change. Straight sides just makes the job easier. Point is, there is detectable change - choose a restraint - see the effect.
Gone over that above. But another angle on it - what assumptions are you making in saying one must force the outer triangle to have straight sides? That assumes, as I guessed in answering above, that the tiles are glued together somehow. That's one configuration. Another could be tiles just sitting together snuggly but loosely on a plane. Moved en masse to curved space, why would they not simply drift apart slightly owing to curvature? There are any number of possible constraints - everyone I have considered results in detectable change - my idea of physics at work.
Huh!? We *want* physical change - that's the detection of being in curved spacetime. Things change. Please, no endless circling here.

PeterDonis

Warning: long post!

Yes, things change. And the change means you can't use the things as standards of distance.

Here's another way of stating what I just said: the things you are saying change as a result of curvature, change because you are specifying that the objects have forces exerted on them in order to match the curvature. Those forces are external forces; they are not due to the curvature itself, at least not in the scenarios you are describing.

At the risk of piling on scenario after scenario, consider the following:

Take a piece of paper and try to wrap it around a globe. You can't; the paper will buckle. That's a physical realization of one scenario you were describing, the one where the sides of the triangles remain straight in the Euclidean sense; if we imagine the paper tesselated with tiny triangles, the paper behaves in such a way as to keep the triangles as flat Euclidean triangles. The paper may buckle along the edges between triangles, but the triangles themselves maintain their shape. So the paper can't possibly conform to any curved surface.

Now take a piece of flat rubber and wrap it around the same globe. You can do it, but only by exerting force on the rubber to deform it into the shape of the globe. This is a physical realization of another scenario you described, where we allow the triangles to adjust to the curvature of the surface; if we imagine the rubber tesselated by tiny triangles, the deformation of the rubber will deform the triangles too. But the deformation is not somehow magically caused by the globe's curvature; it's caused by us, exerting external force on the rubber to change its shape.

We can use the rubber to measure the K factor as follows: suppose that, when sitting on a flat Euclidean plane, the paper and the rubber have identical shape and area. We cut the paper into little tiny squares, and use them to tile the rubber. They fit exactly. Now we take the rubber and stretch it over the globe, and it deforms; we can see the deformation by watching the little squares and seeing that they no longer exactly tile the rubber; we might, as you say, see little gaps open up between the squares, assuming that the rubber is being deformed appropriately. But in order to know exactly how the squares will tile the rubber once it's wrapped around the globe, we have to specify *how*, exactly, the rubber is being wrapped.

Here's one way of specifying the wrapping. Take a circular disk of rubber instead of a square, and a circular piece of paper that, on a flat Euclidean plane, is exactly the same size. We cut the paper up into tiny shapes (triangles, squares, whatever you like) and verify that on a flat Euclidean plane, they tile the rubber exactly. Now take a sphere and draw a circle on it, centered on the North Pole, such that the circumference of the circle is exactly the same as the circumference of the rubber circle when the latter is sitting on the flat Euclidean plane. If we wrap the rubber circle onto the sphere in such a way that its circumference lies exactly on the circle we drew on the sphere, we will need to stretch the rubber; its area will now be larger than it was on the flat plane, and we can verify this by trying to tile it with the little pieces we cut out of the paper and seeing that there is still area left over. This is a manifestation of the K factor being greater than 1. But again, we had to physically stretch the rubber--exert external force on it--to get it to fit the designated area of the sphere.

(As I've noted before, the K factor will vary over the portion of the sphere being used for this experiment, but the average will be greater than 1; if we wanted to limit ourselves to a single value for K, we could cut a small annular ring of rubber on the flat plane, draw two circles on the sphere that matched its outer and inner circumferences, and stretch the rubber between them, and verify that the area increased by tiling with little pieces cut out of a paper annulus that exactly overlapped the rubber on a flat plane. We'll need this version for the 3-D case, as we'll see in a moment.)

All of the above is consistent with what you've been saying about how curvature of a surface manifests itself, and nobody has been disputing it. Now carry the analogy forward to the 3-D case. Here, since we're going to have a gravitating body in the center, we need to use the "annular" version of the scenario, as I just noted. We go to a region far away from all gravitating bodies, so spacetime in the vicinity is flat, and we cut ourselves a hollow sphere of rubber, with inner surface area A and outer surface area A + dA. We also cut a hollow sphere of metal (we use this instead of paper as our standard for material that will maintain its shape instead of deforming) with the same inner and outer surface area. Then we cut the metal up into little tiny pieces and verify that the pieces exactly fill the volume of the rubber. (This is a thought experiment, so assume that we can do this "tiling" in the 3-D case as we did in the 2-D case.)

Now take the hollow rubber sphere and place it around a gravitating body, in such a way that its inner and outer surface area are exactly the same as they were in the flat spacetime region. (Again, this is a thought experiment, so assume we can actually do this as we did in the 2-D case.) We will find that we need to physically stretch the rubber in order for the inner and outer surface areas to match up; there is "more volume" between the two surfaces than there was in the flat spacetime region. Again, we can verify this by trying to tile the volume of the rubber with the little metal pieces, and finding that we run short; there is volume left over when all the pieces have been used up. This excess volume is a physical measure of the K factor. (If dA is small enough compared to A, K is basically constant in this scenario, as it was in the "annular" 2-D case.)

The key fact, physically, is that we have to apply an external force to the rubber, stretch it, to make it fit when it's wrapped around a gravitating body, just as we had to apply an external force to the 2-D rubber circle to make it fit in the specified way around the sphere. So we *cannot* say that the rubber was stretched "by the K factor"--the K factor wasn't what applied the force. We did. Remember that we are assuming that tidal forces, and any other "forces", are negligible; the *only* effect we are considering is the K factor. (Again, this is easier to do because we are now dealing with a "local" scenario, where K is constant; if we covered a wider area, so K varied, we would also find it impossible, or at least a lot harder, to ignore or factor out the effects of tidal gravity. In the "local" case it's easy to set up the scenario so that tidal gravity is negligible, while the K factor itself is not; we just choose the mass M of the gravitating body and the radial coordinate r at which we evaluate K appropriately.)

You may ask, well, what about if we *don't* exert any force on the rubber? What then? In the 2-D case, the answer is that if we exert no force on the rubber, we can't make it conform to the curved surface at all. We specified one way of fitting the rubber to the sphere--so that its circumference remained the same (we checked that by drawing in advance a circle on the sphere centered on the North Pole, to use as a guide). We could specify another way--keep the total area of the rubber constant. But if we do that, then the rubber's circumference will be smaller, so again the rubber has to deform. There is *no* way to make the flat piece of rubber exactly fit to the curved surface without deforming it somehow.

For the spacetime case, it's a little more complicated, because if we take the hollow sphere of rubber we made in a flat spacetime region, and move it around a gravitating body, *something* will happen to it if we exert no force on it (other than the force needed to move it into place). But the general point still applies: there is no way to "fit" the object into a curved spacetime region without deforming it somehow. Since we can't actually wrap a full hollow sphere around a gravitating body without breaking it, consider a portion of the sphere covering a given solid angle, as PAllen proposed in a previous post. Suppose we have done all the work described earlier in a flat spacetime region: we have the inner surface area A and outer surface area A + dA measured (now not covering the total solid angle 4 pi of the sphere, but something less), and we have a piece of metal of the exact same shape and size in flat spacetime cut up into little pieces, which tile the volume of the rubber exactly in flat spacetime.

Now we slowly lower the rubber into place "hovering" over a gravitating body. We have to specify *some* constraint for where it ends up. Suppose we specify that the inner surface lies in a portion of a 2-sphere such that the solid angle it covers is exactly the same as what it covered in flat spacetime--i.e., the ratio of the inner surface area A of the rubber piece to the total area of the 2-sphere is the same as it was for the 2-sphere that the inner surface was cut out of in flat spacetime. That tells us that the inner surface of the 2-sphere is not deformed; but now we have to decide what to do with the outer surface. We have the same problem as we did in the 2-D case: if we specify that the outer surface area remains the same, relative to the inner surface area (i.e., the outer surface "fits" into a corresponding 2-sphere the same way the inner surface does--the outer surface is not deformed), then the rubber will have to stretch to match up, and there will be extra volume in the rubber, according to the K factor (which we can verify by trying to tile with the little pieces of metal and seeing that there is volume left over). Or, if we specify that we want the volume of the rubber to remain the same (i.e., we want to be able to tile it exactly with the little pieces of metal), then the outer surface area will "fit" to a smaller 2-sphere, so the shape of the outer surface will deform. And, of course, if we relax the constraint on the inner surface, there are even more ways we could deform the rubber; but there is *no* way to fit it without *some* kind of deformation.

As far as how the rubber would deform if we carefully exerted *no* force on it except to lower it into place, I think it would end up sitting with the inner surface not deformed and with the volume the same (meaning the outer surface would be deformed, as above). But even in this "constant volume" case, the shape of the rubber would still not be exactly the same. This is a manifestation of curvature, but I don't think you can attribute it to the K factor, because the K factor is measured by "volume excess" when I impose a particular constraint (having the inner and outer surfaces both the same shape as in flat spacetime, as above), and we're now talking about a *different* constraint, one where we keep the volume the same, since, as I said, that's the "natural" constraint that I think would hold if we carefully exerted no "extra" forces on the rubber.

Sorry for the long post, but this is not a simple question, and I wanted to try to get as much as possible out on the table for discussion. To summarize: the K factor is measured by "area excess" or "volume excess" under a particular physical constraint; with that constraint, K is a physical observable, but you have to be careful about interpreting what it means. More generally, curvature manifests itself as the inability to "fit" an object with a defined shape in a flat region, onto a curved region without changing its shape somehow. That change in shape requires external forces to be exerted on the object, which distort the object and make it unusable as a standard of "size" or "shape"; this is always true in the 2-D case we were discussing, and in the spacetime case, it is "almost always" true; there is *some* way to place the object without exerting any "extra" forces, and even in this case, the object's shape will not be the same as it was in flat spacetime, but its volume will be the same, as long as we can ignore tidal gravity.

Q-reeus

[QUOTE=PeterDonis;3585113]Warning: long post!
You are I deduce a speed typist - massive effort!
Understand and agree with that almost completely. Sure the curvature is not exerting any direct forces, so in that sense yes is not physics. Had thought though we could use the 'deformity correction' externally applied forces as a gauge of 'geometry'. More on that below.
This is where something, perhaps much different than straight K, should still allow measurement of a kind. Take an equalateral triangle composed of rigid tubes joined by free-hinging joints. In this configuration, one should expect vertex angles will exceed 60 degrees as discussed before. If small triangular gussetts were initially glued to the apexes, owing to their reduced susceptibility to angular change, ought to partially restrain angular expansion of the much larger triangle tube joints. In other words, there should be some internal stresses - miniscule but in principle measurable. I think.
Many thanks Peter for going to all that fuss - I do now appreciate more just why K factor, relating to a global geometry restraint, requires an implied stretching to make sense. So an isolated, stress-free container (or sheet in 2D case) cannot exhibit that unless forced into an equivalent geometry - like the solid angle segment mentioned. OK then agree that my earlier subdividing of concentric shells argument is not appropriate. And the hoop/marbles thing is presumably analogous. To explain a null result, we must assume there is no effective sampling of curvature, which is still hard for me to fathom re curved vs flat surface. Or is it a fundamental difference betwen 2D vs 3D , which may have been talked about elsewhere.
So how to interpret goings on in the case of say a very rigid tubular frame in the shape of a perfect tetrahedron in flat spacetime. It is the simplest polyhedron that is fully constrained re shape when all sides are connected via freely hinging ball joints. A 3D arrangement of four triangles. If we accept the vertex angles will be greater in +ve curved spacetime, there is a sense that the enclosed volume should be greater than in flat spacetime. Can we say just how that would not be true? S'pose it's down to fierce math proof - yes? Scratching head over that one.
Thanks again, Cheers

PeterDonis

Thanks to my Dad forcing me to learn to type one summer in high school, yes. I learned on the old Olympic typewriter that he had used to type his master's thesis, which was older than I was. So my fingers learned how to squeeze a decent word rate out of those old, half-rusted keys that took about ten pounds of effort each to strike, and now they just laugh at a typical computer keyboard, which takes practically no effort by comparison. (It probably helps that I play keyboards too.)

I think this is right, provided that the tetrahedron is large enough that spatial curvature is measurable over its size, and that we deform the tetrahedron to conform to the curvature. Consider the analogous case on a 2-D surface, where we make a triangle out of rigid tubes joined by free-hinging joints, and then measure the joint angles after carefully wrapping the triangle onto a sphere so it conforms to the sphere's curvature. This case is exactly like that of the circular rubber disk or annulus; to see any change in shape, we need to exert external force on the rubber to make it conform to the shape of the sphere, and we have to decide how it is going to conform. We also have to make the shape we are trying to wrap around the sphere large enough that it "sees" the curvature; if it's too small we will be unable to use it to detect any difference from a flat plane (like the tiny pieces of paper we cut out of the paper disk or annulus in my previous example).

Given that these conditions are met, yes, we can in principle use the triangle as described (or the tetrahedron in the 3-D case) to measure differences between the space we are interested in and flat space; but there is still the question of how we decide to constrain the deformation. In the case quoted above, you are essentially constraining the side lengths of the tetrahedron (or triangle) to be constant, and letting the angles vary. Consider the 2-D case first (the triangle), and note that to physically realize this, we not only need to let the joint angles expand; we also need to bend the sides of the triangle since they are no longer Euclidean straight lines, but geodesics of the sphere, i.e., segments of great circles. (At least, I assume this was your intention in specifying side lengths constant and letting angles vary.) For thought experiment purposes, we can stipulate that this can be done while keeping the length of the sides constant, and similarly, in 3-D, the sides of the tetrahedron will have to bend slightly, but we can stipulate that their lengths are still held constant. In both cases, however, the sides will clearly undergo deformation, and we will have to exert external force on the tetrahedron to effect this deformation.

With the constraint as described above, yes; I would expect the volume inside the tetrahedron (or the area of the triangle, in the analogous 2-D case) to be larger if the side lengths are stipulated to be held constant. But as I just noted, to realize this case we have to exert external force to deform the tetrahedron (or triangle), and this external force is what causes the volume to expand.

OTOH, we could stipulate a different constraint, that the tetrahedron should be "unstressed"--or more precisely, that we should not *impose* any stress on it by exerting external force. In this case, I'm not sure what would happen, other than that I would expect the tetrahedron's volume to remain constant--always assuming that we can neglect tidal forces (as I noted previously, we can always choose the mass M of the central body and the radius r appropriately to make K measurably different from 1 while still having tidal gravity negligible). I *think* that the angles would still expand, but the sides would shorten, keeping the resulting volume constant but changing the shape.

JDoolin

Hi, I apologize if this seems off topic. You guys are talking about a stationary shell, but another interesting idea to consider is an expanding shell.



The particles making up the shell need not be attached, but are simply Lorentz-contracted and time-dilated to an extremely high density.

This is a question I've been pondering for some time; precisely what would the effect of gravity be on the internal particles? It's quite possible that you would have large regions where the gravitational potential would be extremely great, but fairly constant, so there is no net force in any direction.

Since everything would be so uniform, would there be any observable effect at all?

Attached Thumbnails

 

PeterDonis

Lorentz-contrated and time-dilated relative to what?

It would depend on what the shell was made of and how you specified the initial conditions of the expansion.

For a shell made of "normal" matter, i.e., with pressure no greater than 1/3 its energy density, the gravity of the shell would cause the expansion to decelerate, similar to a matter-dominated expanding FRW model. Depending on the initial conditions (velocity of expansion vs. shell energy density and pressure), the shell might stop expanding altogether and re-contract, or it might go on expanding forever, continually slowing down but never quite stopping.

If the shell's density is uniform (i.e., uniform throughout the shell at any particular "time" in the shell's comoving frame--the density could still change with time, as long as it remained uniform spatially within the shell), then I think you are right to guess that there would be no observable effect from the "potential" within the shell itself. There might still be an effect relative to the potential in the spacetime exterior to the shell. (The potential interior to the shell would be the same as the potential on the shell's inner surface, just as for a static shell.)

Q-reeus

Having accepted your explanation of why K cannot apply to a simple container that does not enclose the source of curvature, there remains a point of disagreement here. My understanding of triangles adding to more than 180 degrees in +ve curved 3-space is that the weirdness here is precisely due to that, as measured by say laser theodolite, the tubular sides of said triangle will be *exactly* straight and thus entirely unstressed (assuming 'gussetts' are absent). It is understood here measurements are taken in the curved environment - not some distant coordinate reference. Hence the specification of free-hinging pinned joints, and rigid tubes that are not 'floppy'. Otherwise, what is implied is surely an intrinsic, locally measurable curvature of just one straight rod, in going from flat spacetime to curved. But in 3-curvature, how will the 'straight' rod/tube 'know' which way to bend?

I think the proper analogy here in going from flat to curved is not trying to wrap a flat object onto a curved surface. Rather, think of drawing an equalateral triangle on the surface of a large balloon (low surface curvature). 2D flat-landers living on the balloon surface cannot directly detect the surface curvature, but with their 2D confined 'laser theodolites' will confirm the triangle sides are straight, and the vertex angles are 'near enough' to 60 degrees. Now deflate the balloon to a much smaller radius. Flat-landers now attempt to construct another equalateral triangle of the same side lengths as before (meaning triangle occupies a much larger portion of the balloon surface than before). Their theodolites continue to say the sides are perfectly straight, but are puzzled to find the vertex angles now significantly exceed 60 degrees. That's how I got what curvature does here - there's a faint whiff of sanity to Dr Who's 'Tardis' if you like.
Yes, and is it not just this size related differential that allows flat-landers to detect curvature induced angular changes by means of a small, 'standard' protractor that minimally 'feels' curvature. Thoughts?

PAllen

One part you don't get is the issue of embedding. Please think about how a curved spherical surface is embedded in flat 3-space without telling you anything about the curvature of the 3-space. Similarly, in curved spacetime you can embed flat planes, and any procedure looking for straight lines will pick out this embedded flat plane. Thus no procedure limited to a plane can detect curvature of a 4-manifold.

I have explained this multiple times and you have continued to ignore it.

Q-reeus

Ignore is not perhaps the right word, as all I could pick up were assertions of how it is - may well be true, but to me they were just assertions.
Allright, given that is so, what sense does one make of the 'popular' statement that the internal angles of a triangle do not generally add to 180 degrees in curved spacetime? Having not studied the subject, I took my que from such as:
"In other words, in non-Euclidean geometry, the relation between the sides of a triangle must necessarily take a non-Pythagorean form." at http://en.wikipedia.org/wiki/Pythago...idean_geometry
[Another grab, from: http://en.wikipedia.org/wiki/Curved_...flat.2C_closed
"Triangles which lie on the surface of an open space will have a sum of angles which is less than 180°. Triangles which lie on the surface of a closed space will have a sum of angles which is greater than 180°. The volume, however, is not (4 / 3)πr³" In context this appears to me to be a generalized statement applicable to higher than 2D curvature. As I say , haven't studied this subject at all.]

PAllen

If you look at that link, the context is geometry on a 2-surface. Popular statements don't get into the issue of embedding - they are over-simplifed. I am not just making assertions, I am asking you to think, as I'll do again. Apply your idea to 3-space "if a triangle is non-pythagorean, the *space* is non-euclidean' to flat 3-space. On a 2-sphere in 3-space, you conclude, correctly, that the 2-sphere is curved (triangle is non-pythagorean). What does that tell you about the 3-space: nothing. The 3-space is still flat.

One specific argument that flat planes are embeddable in a 4-manifold is simply to note that for an arbitrary symmetric metric with 10 components, you can apply 4 coordinate conditions. This is sufficient to make e.g. the x,y components of the metric [[1,0],[0,1]], that is a Euclidean plane. As a result, any non-Euclidean behavior of a plane is just a function of coordinate choice, and is not telling you anything intrinsic about the manifold.

Q-reeus

More inclusive quote from that same passage:
"The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Pythagorean theorem given above does not hold in a non-Euclidean geometry.[51] (The Pythagorean theorem has been shown, in fact, to be equivalent to Euclid's Parallel (Fifth) Postulate.[52][53]) In other words, in non-Euclidean geometry, the relation between the sides of a triangle must necessarily take a non-Pythagorean form. For example, in spherical geometry, all three sides of the right triangle (say a, b, and c) bounding an octant of the unit sphere have length equal to π/2, and all its angles are right angles, which violates the Pythagorean theorem because a2 + b2 ≠ c2."

I can only take that one way - in *any* non-Euclidean geometry. Just how that applies to triangle in 3-curvature is the question.
Not familiar with this to argue what you say here, other than to ask you to explain the full passage I quoted above. Put it simply please - are you saying that angles will add to 180 degrees in a generally 3-curved space, or not?

PAllen

The whole passage refers to geometry of a 2-surface. Euclid's Parrallel postulate is a postulate about plane geometry. Spherical geometry is the geometry of a 2-sphere - the *surface* of a sphere. And I ask you again to think about my simple embedding example. The surface of a globe is non-euclidean. The globe is sitting in a flat euclidean 3-space. Contradiction? No. The embedded space is curved, the space embedded in happens to be flat.

PAllen

I am saying if you try to make Euclidean triangles in a general 4-d semi-riemannian manifold, you will succeed. For 3-space, I'm not sure whether you can *always* do it. There are fewer degrees of freedom, and the simple counting argument I used for 4-manifold does not work. It is certainly true that not all 2-manifolds can be embedded in flat 3-space, so it is plausible that some 3-manifolds will not embed a section of flat 2-surface.

Also recall my discussion with Peter: the SC geometry allows embedding of completely flat 3-space regions. The K factor is actually a feature of a particular class of observers (static observers, which are non-inertial observers), not something intrinsic to the geometry. It is analogous to distortions seen in an accelerating rocket in flat spacetime - a feature of the observer, not the intrinsic spacetime geometry. GP observers in the same SC geometry, experience absolutely flat 3-space.

JDoolin

If you look carefully, there is one particle in the diagram that doesn't move. All motion in this diagram, then, is relative to that stationary particle.

This deserves further animations; I'd rather show you, if I can, rather than tell you. But suffice it to say, the animation above could be Lorentz transformed so as to place any particle in the center of the circle. However, because there are only a finite number of particles in the picture, there would be a gravitational asymmetry for every particle, except for one.


There are a couple different ways I might specify the initial conditions; (1) starting with an equipartition of rapidity (2) starting with an equipartition of momentum.

In this animation I assumed equal masses, and used equipartition of momentum.

Let

[tex]q\equiv \left \|\frac{\vec p}{m c} \right \| = \left \|\frac{(\vec v/c)}{\sqrt{1-v^2/c^2}} \right \|<3[/tex]


[tex]\frac{v}{c}=\sqrt{\frac{q^2}{q^2+1}}[/tex]

Once I calculated the velocities, I animated, finding position, by simply multiplying the velocities by t.



The animation renders about 79,000 dots of equal mass with random q between 0 and 3 in random directions. The particle at the center of this distribution should experience no net acceleration; however, closer to the edges, there is more and more gravitational force.

There's no limit to what q might be. If I set q=10^100, then all the particles that are in this animation would be so close to the center that they would experience essentially no net force.

I have a plan in mind to present the animation from the reference frame of a particle on the edge of the mass by Lorentz Transforming all of the velocities. Definitely, as long as the total mass of the distribution is finite, then there would be particles on the edge that experience extreme accelerations toward the center. If the mass were infinite (and thus no edge), you could invoke symmetry, and there would be no net force in any direction for any particle.

But then you'd have the problem, being inside a shell of infinite mass, that at any given point inside, you are at an infinite negative gravitational potential

On the other hand, with a finite mass, the particles at the edge could experience, at least for a time, acceleration equivalent to a black hole. I'm not at all sure what theoretical ramifications that would have.

Anyway, I'll work on the other animations, and hope that makes my meaning clearer.