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

[PAllen]

I haven't seen Synge's derivation (I think I tried to find it once, but I know I never managed to get a hold of it), but I'm confident you can detect intrinsic curvature of a plane with 4 points, so I find it logical to believe you can detect space curvature with 5.

Whether you use my method for detecting the intrinsic curvature of a plane (which involves comparing the ratio of the diagonals of a square to the side of a square, the square by definition having four equal sides and two equal diagonals), or the marble packing method suggested by Q, you can detect the intrinsic curvature of a plane embedded in a 3d space.

What you can't do is tell if said intrinsic curvature is due to the way the plane is embedded. Thus information on the intrinsic curvature of a single plane embedded in a higher 3d space-time doesn't directly tell you anything about the intrinsic curvature of the space it's embedded in, the intrinsic curvature of the plane could result from the way it's embedded.

A trivial example: The Earth's surface is curved, and not flat. In fact, thinking about ways to detect the curvature of the Earth's surface (while staying on the surface) is a good way to get comfortable with the concepts and properties of curvature.

But the fact that the Earth's surface is curved (has an intrinsic curvature) doesn't tell you anything about whether or not space or space-time the Earth is in is curved.

I've heard that you can decompose the Rieman into "sectional curvatures" of planes, but I'm a bit hazy about the details. Clearly, though, you need information on the intrinsic curvature of a lot of planar slices of your space-time, not just one.

It would have to be some pattern in the planar slices. Obviously, you can embed concentric 2-spheres in flat Euclidean 3-space. That does not imply anything about the geometry of the space the in which the spheres are embedded.

 

[PeterDonis]

I thought Q-reeus was completely filling the interior of a hoop with marbles -- in other words he is comparing the area of a circular disk with its circumference.

You may be right as regards that particular example; I may well have misread him.

I prefer to compare the space enclosed between two adjacent concentric circles with circumference C and C + dC, because it allows one to use just one value of the K factor, the one that applies at the particular "r" coordinate corresponding to the circumference C. (Or in the 3-D spatial slice of spacetime case, to evaluate the volume enclosed between two adjacent concentric spheres of area A and A + dA, we only need to use the value of r corresponding to A.) To fully compare the area enclosed by a circle around the "North Pole" with its circumference, on the Earth, we would need to integrate the K factor over a range of r values. (In the spacetime case, it's even worse because the K factor's dependence on r is different in the vacuum exterior region and in the interior of the gravitating body at the center; in the case of a black hole, there isn't even a spacelike slice at in exterior Schwarzschild coordinates that reaches to r = 0.)

Any deviation from the expected 2D Euclidean value would be apparent only for relatively large (i.e. "non-local") hoops. Hoops that are small enough to be regarded as "local" would be too small for the deviation to be measurable -- that's pretty much what we mean by "local" as used in the Equivalence Principle.

I'm not sure about this way of stating it; I think we need to clarify the meaning of "local" and "non-local" in this connection.

Consider a pair of 2-spheres concentric on the Earth, with areas A and A + dA, where dA << A, and A is just a bit larger than the Earth's surface area--just large enough so that the spheres are unarguably in the "exterior vacuum region" of the Earth's Schwarzschild spacetime geometry. The volume enclosed between these 2-spheres will be greater than Euclidean geometry would predict based on the difference in their areas, by about the ratio of Earth's Schwarzschild radius to its actual radius, or about 1 part in a billion. We can't measure that today, but I see no difficulty in principle in doing so, and we may well have enough accuracy to do it in practice in the foreseeable future. That difference is what I am calling the K factor; for the case I just described, K is about 1 + 10^-9.

The measurement I have just described is indeed "non-local", in the sense that it can't be done without enclosing the entire Earth with a pair of 2-spheres, which is not a local measurement. However, it is "local" in the sense that I can make dA very, very small compared to A, and the K factor will still be the same; it will just be harder to detect. The K factor depends on the Schwarzschild r coordinate, hence it depends on A, but it does *not* depend on dA once A is fixed. So I would be hesitant to say that the measurement is not "local", period; because of the spherical symmetry, it is not unreasonable to describe as "local", for some purposes, a measurement like this that is "local in r", so to speak, but not local in space as a whole.

 

[Q-reeus]

Originally Posted by DrGreg:
"I thought Q-reeus was completely filling the interior of a hoop with marbles -- in other words he is comparing the area of a circular disk with its circumference."

You may be right as regards that particular example; I may well have misread him.

Right in both cases. Looks like words getting in the way again. Had used the marbles thing as a direct 2D carry-over from your 3D description of K in terms of 'little counting spheres' between concentric shells earlier. Seemed self-evident that hoop is not torus, that hoop as defined perimeter, sits on a surface, and one proceeds to fill the enclosed area with marbles. And that the count will be a function of surface curvature. Just seemed a natural way to continue that analogy. While your concentric circles around north pole analogy in #99 talked in terms of perimeter-to-radius ratio, one could equally talk in terms of an enclosed surface area-to-perimeter ratio of a dished annulus (numerically different, but having in common dependence on surface curvature). Once you see it the latter way, the hoop thing springs out as a more evident manifestation that local phenomena will exist, which is why I used it.

Deary me - should have just gone straight to DrGreg's use of area vs circumference. But one still needed something like 'marble count' to get it that the ratio was changing with changed surface curvature - and most importantly - it will physically manifest (gaps opening up between marbles). And ergo - go 3D and fluid level in a container responds to changed 3-curvature. Also, as DrGreg mentioned in #117, the larger the hoop, the larger the relative effect, in the same way that the ratio of surface area of a chorded section cut from a spherical surface to that of a circular plate of the same diameter is negligibly different from unity for small chord size, but grows non-linearly with chord diameter. Ties in with comments in #113 about triangles. And with comments in #103 etc that counting spheres are a standard precisely because of this non-linear scale dependence.

I think while angles not adding to 180 degrees for triangles is common fare, extension to 3D volume effects appears not to be. However PAllen's comments in #118 and pervect's in #119 to e.g. Synge's 5-point method of detecting curvature show that 'practical' methods for detection have been devised that can operate over a 'local' region of space. But as argued in #103 and #105, there is directionality involved - functional dependence is on r, not on tangent directions. And despite what I've heard, seems natural to interpret K as the contraction ratio ∂'r'/∂r, 'r' being the radius in coordinate measure. In fact, to answer your first query in #116: "What does "look smaller" mean?" - inferred with reference to coordinate measure, the only handle on all this that makes sense to me.

Originally Posted by DrGreg:
"Any deviation from the expected 2D Euclidean value would be apparent only for relatively large (i.e. "non-local") hoops. Hoops that are small enough to be regarded as "local" would be too small for the deviation to be measurable -- that's pretty much what we mean by "local" as used in the Equivalence Principle."

I'm not sure about this way of stating it; I think we need to clarify the meaning of "local" and "non-local" in this connection.

Consider a pair of 2-spheres concentric on the Earth, with areas A and A + dA, where dA << A, and A is just a bit larger than the Earth's surface area--just large enough so that the spheres are unarguably in the "exterior vacuum region" of the Earth's Schwarzschild spacetime geometry. The volume enclosed between these 2-spheres will be greater than Euclidean geometry would predict based on the difference in their areas, by about the ratio of Earth's Schwarzschild radius to its actual radius, or about 1 part in a billion. We can't measure that today, but I see no difficulty in principle in doing so, and we may well have enough accuracy to do it in practice in the foreseeable future. That difference is what I am calling the K factor; for the case I just described, K is about 1 + 10^-9.

The measurement I have just described is indeed "non-local", in the sense that it can't be done without enclosing the entire Earth with a pair of 2-spheres, which is not a local measurement. However, it is "local" in the sense that I can make dA very, very small compared to A, and the K factor will still be the same; it will just be harder to detect. The K factor depends on the Schwarzschild r coordinate, hence it depends on A, but it does *not* depend on dA once A is fixed. So I would be hesitant to say that the measurement is not "local", period; because of the spherical symmetry, it is not unreasonable to describe as "local", for some purposes, a measurement like this that is "local in r", so to speak, but not local in space as a whole.

Aha. Just what I thought. Go back and read the strike-through part I wrote in #103. That was my thinking about what you could possibly imply by saying there is K > 1 only for concentric shells, not for a volume segment 'cut out' from those shells. I left it there as evidence of my thinking and as a chance to comment on it, but none came. But it doesn't make sense - to repeat what was written in #108:
"Thought I had it conceptually pinned down there. Do we agree that if K factor applies to excess volume between complete concentric shells, it must apply to partitioned portions. Apply a soccer-ball style tesselation over shell surface and cut through radially at the boundaries.That defines intimately joined volume segments. An observer in each segment does a count. How could the excess count by each observer not add to give just that for the whole shells? Ergo - there is an non-euclidean effect observable in a 'container'. No?!"

I can conceive no way around that. How can there be?

 

[PAllen]

I think while angles not adding to 180 degrees for triangles is common fare, extension to 3D volume effects appears not to be. However PAllen's comments in #118 and pervect's in #119 to e.g. Synge's 5-point method of detecting curvature show that 'practical' methods for detection have been devised that can operate over a 'local' region of space.

These methods are not local. You must span a region big enough that (to your level of precisions) tidal effects are detectable. Put it this way: propose a level measurement precision; then there is a minimum size region in which you can detect tidal gravity effects, thus encompassing curvature significant to the precision. Now assume your length and time measurements are comparable in geometric units (this typically means much more sensitive, in practice, or much larger region required). Then, by various global measurements of this scale, you can detect Euclidean deviation. You cannot localize it to any plane, let alone a linear direction. You can always get a finite size 2-surface, in any orientation, that is (mathematically) exactly flat (perhaps unless the 4-manifold is very pathological).

 

[Q-reeus]

Originally Posted by Q-reeus:
"I think while angles not adding to 180 degrees for triangles is common fare, extension to 3D volume effects appears not to be. However PAllen's comments in #118 and pervect's in #119 to e.g. Synge's 5-point method of detecting curvature show that 'practical' methods for detection have been devised that can operate over a 'local' region of space."

These methods are not local. You must span a region big enough that (to your level of precisions) tidal effects are detectable. Put it this way: propose a level measurement precision; then there is a minimum size region in which you can detect tidal gravity effects, thus encompassing curvature significant to the precision. Now assume your length and time measurements are comparable in geometric units (this typically means much more sensitive, in practice, or much larger region required). Then, by various global measurements of this scale, you can detect Euclidean deviation.

Agree entirely with that, which is why I put 'practical' and 'local' in single quote marks. The question is whether it can in principle be done at all, without having to have the measuring region enclose the gravitational source. In crude analogy, is 3-curvature to be considered a kind of 'volume charge density' having a local non-zero divergence which is measurable locally, or divergence free except for the source region itself? Remember - this is just rough analogy.

You cannot localize it to any plane, let alone a linear direction. You can always get a finite size 2-surface, in any orientation, that is (mathematically) exactly flat (perhaps unless the 4-manifold is very pathological).

Not quite up with how to interpret that - does it invalidate the principle behind what I was saying in #105 for instance?

 

[PeterDonis]

Seemed self-evident that hoop is not torus, that hoop as defined perimeter, sits on a surface, and one proceeds to fill the enclosed area with marbles. And that the count will be a function of surface curvature.

Yes, but as I pointed out, to calculate the count, you need to know the K factor for a whole range of "r" values, from r = 0 out to the "r" of the hoop, which is its circumference divided by 2 pi. This brings in additional complications which are not present if you consider the area between two nearby circles of circumference C and C + dC.

Deary me - should have just gone straight to DrGreg's use of area vs circumference. But one still needed something like 'marble count' to get it that the ratio was changing with changed surface curvature - and most importantly - it will physically manifest (gaps opening up between marbles).

Huh? Where has anyone said anything about gaps between marbles? I thought it was understood through all of this that we are packing whatever area (or volume) we're concerned with as tightly as possible with the marbles (or whatever small identical objects we are using). DrGreg even said so explicitly. If you don't do that, how can you possibly get reliable measurements?

And ergo - go 3D and fluid level in a container responds to changed 3-curvature.

Still an issue here--see below.

And despite what I've heard, seems natural to interpret K as the contraction ratio ∂'r'/∂r, 'r' being the radius in coordinate measure.

You have the ratio upside down. If we use s for "physical" distance measure and r for coordinate measure, then K is ds/dr. Or, if we use the definition of the r coordinate we've been using, K is ds/dsqrt(A), where A is the area of the 2-sphere at coordinate r. So K > 1 means an increase in how much actual distance s corresponds to a unit of coordinate r.

"Thought I had it conceptually pinned down there. Do we agree that if K factor applies to excess volume between complete concentric shells, it must apply to partitioned portions. Apply a soccer-ball style tesselation over shell surface and cut through radially at the boundaries.That defines intimately joined volume segments. An observer in each segment does a count. How could the excess count by each observer not add to give just that for the whole shells?

It does, as long as we're packing marbles correctly, or the equivalent with tesselations. But that does *not* imply the following:

Ergo - there is an non-euclidean effect observable in a 'container'. No?!"

No.

I can conceive no way around that. How can there be?

Because you are not correctly analyzing the physics of the container. Let's consider that example in more detail.

Suppose I have a cubical container with side length s; that is, when I measure its sides in some region of spacetime far away from all gravitating bodies, I measure each side to be identical in length, and the side length to be s. Now I take this container and lower it to some radial coordinate r above a gravitating body, where r is such that the K factor is measurably greater than 1. What will the container look like when I measure its sides again?

The answer is clear from what I've already said: the container will still be cubical, and its side lengths will still be s. The K factor has no observable effect on the size of the container, because K does not cause any stress on objects.

However, now consider the following experiment: I take my container, in a region of spacetime far from all gravitating bodies, and I sandwich it between two concentric 2-spheres, with the areas A and A + dA of the spheres chosen such that two opposite faces of the container are just tangent to the two spheres. I ask, what is the relationship between the side length s of the container and the area A of the inner sphere? The answer is, it is the relationship which Euclidean geometry predicts. In other words, the K factor here is 1.

Now I lower the container to a radial coordinate r above a gravitating body, such that the area A corresponding to r (A = 4 pi r^2) is *exactly* the same as the area A of the inner sphere I used above. In other words, the "bottom" surface of the container is now tangent to the sphere with radius A, exactly as it was when everything was far away from all gravitating bodies. I now ask: if I consider a second sphere of area A + dA in this situation, where dA is exactly the same dA I used above, will the "top" surface of the container be tangent to that sphere? The answer is *no*: the container's top surface will not quite reach the second sphere, because the side length s of the container, which is unchanged, is now not quite as long as the distance between the two spheres, because the K factor is now greater than 1. That is what I mean by saying that there is "more distance" between the spheres than there would be if the space geometry were Euclidean, but the size of a given unit of distance, such as the container side length s, is unchanged.

Note, please, that this is *not* saying that K is not physically observable. The failure of the top surface of the container to reach the second sphere is a physical observable--it's direct physical evidence of the K factor being greater than 1. It may not be the evidence your intuition was expecting, but it's certainly evidence.

 

[PAllen]

I wonder if the following would be true (it seem intuitively plausible based on Synge's results combined with Peter Donis's findings; I wouldn't rely on this without calculating it, though):

Take a ruler with marks r1 and r2 on it. In a region of curvature, use it to lay out two concentric spherical surfaces of a chosen solid angle. The relation of surface areas to r1, r2, and solid angle will be strictly Euclidean. But what about the volume between them measured with Peter's little marble idea? My guess is that it will not match the Euclidean prediction.

 

[Q-reeus]

Originally Posted by Q-reeus:
"Seemed self-evident that hoop is not torus, that hoop as defined perimeter, sits on a surface, and one proceeds to fill the enclosed area with marbles. And that the count will be a function of surface curvature."

Yes, but as I pointed out, to calculate the count, you need to know the K factor for a whole range of "r" values, from r = 0 out to the "r" of the hoop, which is its circumference divided by 2 pi. This brings in additional complications which are not present if you consider the area between two nearby circles of circumference C and C + dC.

If I read that right, you are perhaps inadvertently agreeing with my notion of 'spatial gradients'. That length measured with a microscopic ruler will not correspond exactly with length measured with a large ruler, even 'locally', because of local gradients. I maintain it's this that allows the marbles etc to act as a standard length. Blow them up to near hoop size, and evidence of curvature is lost. As said earlier, it means the larger the hoop, the proportionately larger the area excess becomes - the more sensitive a gauge of curvature one has. This is just repeating what's been said many times before. We are trying to get a handle on the rate of change with potential, of microscopic measure (marbles) to macroscopic measure (hoop).

Originally Posted by Q-reeus:
"Deary me - should have just gone straight to DrGreg's use of area vs circumference. But one still needed something like 'marble count' to get it that the ratio was changing with changed surface curvature - and most importantly - it will physically manifest (gaps opening up between marbles)."
Huh? Where has anyone said anything about gaps between marbles? I thought it was understood through all of this that we are packing whatever area (or volume) we're concerned with as tightly as possible with the marbles (or whatever small identical objects we are using). DrGreg even said so explicitly. If you don't do that, how can you possibly get reliable measurements?

Well gaps must open before one can pop in a new marble, yes? But I could have made that statement a bit clearer, sure.

Originally Posted by Q-reeus:
"And despite what I've heard, seems natural to interpret K as the contraction ratio ∂'r'/∂r, 'r' being the radius in coordinate measure."

You have the ratio upside down. If we use s for "physical" distance measure and r for coordinate measure, then K is ds/dr. Or, if we use the definition of the r coordinate we've been using, K is ds/dsqrt(A), where A is the area of the 2-sphere at coordinate r. So K > 1 means an increase in how much actual distance s corresponds to a unit of coordinate r.

Got me! EXpressed it wrong - meant K^-1 but it slipped me - a bit like that single typo 'loop' instead of 'hoop' you so quickly picked me up on earlier. Anyway we seem to agree there's more than one way to express the meaning of K - not just as differential volume-to-area ratio. That a coordinate related meaning is justified, not just as 'quasi-local' measure.

Suppose I have a cubical container with side length s; that is, when I measure its sides in some region of spacetime far away from all gravitating bodies, I measure each side to be identical in length, and the side length to be s. Now I take this container and lower it to some radial coordinate r above a gravitating body, where r is such that the K factor is measurably greater than 1. What will the container look like when I measure its sides again?
The answer is clear from what I've already said: the container will still be cubical, and its side lengths will still be s.

Stop right there. You torpedoed my hoop thing on the basis of K being a function of hoop radius, implying I suppose that perimeter was to a certain extent undefined because K is non-constant throughout a locally defined radial displacement. Precisely my point! It has negligible effect on the counting spheres owing to their being so small - they 'sample' K gradient only slightly. And likewise that cube still measuring s by a macroscopic ruler does not imply there has been no differential change between that and the microscopic ruler measure of those counting spheres. Run the micro rulers along the macro ruler - they disagree. Move to a location in lower potential - they disagree more. And the larger the value of s, the larger the counting anomaly will become for a given potential. Upon that I claim container *can* be used. As you argued re hoop - there are gradient effects here.

The K factor has no observable effect on the size of the container, because K does not cause any stress on objects.

I would have general doubts here. For a solid sphere, seems to me contraction factor varying with r implies the interior pulls on the exterior regions - not as a simple minded application of K factor, but a more subtle function. That would accord with what I wrote in e.g. #113. So tangential compression going to tangential tension on descent to the center. For a fluid sphere, obviously not. For empty container - negligible probably.

However, now consider the following experiment:
...The answer is *no*: the container's top surface will not quite reach the second sphere, because the side length s of the container, which is unchanged, is now not quite as long as the distance between the two spheres, because the K factor is now greater than 1. That is what I mean by saying that there is "more distance" between the spheres than there would be if the space geometry were Euclidean, but the size of a given unit of distance, such as the container side length s, is unchanged...

To put it more shortly - this means a simple subdivision of shell volume into 'containers' implies a radial stretching/stressing must occur after gravity is 'switched on'. And that this 'stretching' is what allows the excess count that a 'free' container will not experience. Fair argument; hadn't thought about it that way before and will have to consider full implications. I think though it simply implies that if one 'lets go' a stretched sub-volume to become a free container, there is simply an increased density of excess count. My 'in the meantime' response is this. Now that we are all clear on what hoop meant, how do you understand changing marble count with curvature - and extending that to molecule count in a 3d container? Is your argument taking full account, as I wrote above, of micro-to-macro K gradients, and how that changes with r? Still a problem here.

[Hardening my stance on this. Comments in #123 apply: "While your concentric circles around north pole analogy in #99 talked in terms of perimeter-to-radius ratio, one could equally talk in terms of an enclosed surface area-to-perimeter ratio of a dished annulus (numerically different, but having in common dependence on surface curvature). Once you see it the latter way, the hoop thing springs out as a more evident manifestation that local phenomena will exist, which is why I used it." There is a marble count excess for dished annulus sitting on 2-sphere (2D analogue of spherical shells experiencing 3-curvature). Same general effect must apply to a hoop, and likewise 3D container]

 

[PeterDonis]

If I read that right, you are perhaps inadvertently agreeing with my notion of 'spatial gradients'. That length measured with a microscopic ruler will not correspond exactly with length measured with a large ruler, even 'locally', because of local gradients.

The only thing that would affect a large ruler, as opposed to a very small ruler, is tidal gravity. That's the only kind of "gradient" in the field that can cause actual physical stress on a ruler and thus change its physical length. The K factor does *not* do this. Remember I said the K factor is *not* the same as curvature; it's related but it's not the same. Spacetime curvature is tidal gravity, not the K factor.

Well gaps must open before one can pop in a new marble, yes? But I could have made that statement a bit clearer, sure.

What do you mean, "pop in a new marble"? We are not talking about that kind of experiment; we are talking about packing marbles into various pre-existing spaces. If you're envisioning a "gap" opening up, you're envisioning something like this: we take a circular "hoop" of circumference C and place it on a sphere, centered on the North Pole. Then we take a second circular "hoop" made of some elastic material, so it can stretch; we place it on the sphere starting with circumference C + dC(0), and then slowly move it away from the first circle, so dC gradually gets larger compared to dC(0). As we do that, yes, open space will appear that we now need to pack with more marbles. But also, as we do that, the K factor will vary between the two circles, so things become more complicated. It would be really good if you would stick to the purely "local" case first, to avoid confounding factors, like for example tidal gravity--see my comments elsewhere in this post.

Stop right there...As you argued re hoop - there are gradient effects here.

Not if we're ignoring tidal gravity. See above. The only "gradient" effect that will actually stretch a container and change its physical size is tidal gravity. The K factor will *not* do this.

This is why I keep saying we should stick to the local case first; it avoids introducing confounding factors like tidal gravity that are *not* the same as the K factor. If you keep muddling these things together, you will keep on being confused. Once again: the K factor does *not* cause any stress on objects. Therefore, the K factor *cannot* change the physical size of a container; that would require causing stress on the container's walls. This is a basic point of the physics involved, and if it's not clear, we need to stick to the local case until it is.

I would have general doubts here. For a solid sphere, seems to me contraction factor varying with r implies the interior pulls on the exterior regions - not as a simple minded application of K factor, but a more subtle function. That would accord with what I wrote in e.g. #113. So tangential compression going to tangential tension on descent to the center. For a fluid sphere, obviously not. For empty container - negligible probably.

Now you're bringing in yet *another* different case--a non-vacuum region, inside a solid object. It would *really* help to stick to the simplest case first! We are talking about the exterior *vacuum* region, with two 2-spheres very close together, with areas A and A + dA, and what effect the K factor has in *that* case alone. We really need to get that case straight first before bringing in complications.

(If you insist on something about the non-vacuum case, inside a solid object, the stress-energy tensor is non-zero, so yes, there are additional forces "pulling" on a small object. But those forces also are *not* the K factor; they are related to it, in the sense that they are also functions of the radial coordinate r, but they are *not* the same. We can try to work that out after we get the "local", vacuum case clear.)

To put it more shortly - this means a simple subdivision of shell volume into 'containers' implies a radial stretching/stressing must occur after gravity is 'switched on'.

This implies that you're subdividing the shell volume by *coordinate* r, *not* by physical size of a very small object. You are still insisting on "labeling" every small point in the volume by its *coordinate*, r, which can be thought of as its "Euclidean coordinate", because it's derived from the area of the associated 2-sphere via the Euclidean geometry formula. But that labeling is *not* physical--it is coordinate-dependent. Physically, as I keep saying, Euclidean geometry is *not* privileged, and the "Euclidean coordinate" of a particular point has *no* physical meaning.

Now that we are all clear on what hoop meant, how do you understand changing marble count with curvature - and extending that to molecule count in a 3d container? Is your argument taking full account, as I wrote above, of micro-to-macro K gradients, and how that changes with r? Still a problem here.

See my comments above. The 3d container continues to contain the same number of atoms (I like that word better than "molecule" for solids, since they might be metals which don't really have "molecules") regardless of where it is placed in the gravitational field. And since the K factor does not cause any internal stresses in the container, the atoms maintain the same physical distance between themselves as they did when the container was far away from all gravitating bodies. Therefore, the container maintains the same physical size.

 

[PeterDonis]

Take a ruler with marks r1 and r2 on it. In a region of curvature, use it to lay out two concentric spherical surfaces of a chosen solid angle. The relation of surface areas to r1, r2, and solid angle will be strictly Euclidean. But what about the volume between them measured with Peter's little marble idea? My guess is that it will not match the Euclidean prediction.

If there is a good way to determine "solid angle", i.e., to determine what fraction a given surface is of the full 2-sphere that it is part of, then I agree, the volume between them would increase over the Euclidean prediction by the K factor. My only reservation is that I'm not sure exactly how the solid angle would be measured. There are two possible cases: (1) there is a gravitating body like the Earth in the center; measuring solid angle would, it seems to me, require having unobstructed sight lines from the center of the Earth to the surface in question, so the angle at the center of the Earth subtended by the surface can be measured; or (2) there is a black hole at the center; in this case there is not even the "in principle" possibility of setting up sight lines in this way.

 

[Q-reeus]

Originally Posted by Q-reeus:
"If I read that right, you are perhaps inadvertently agreeing with my notion of 'spatial gradients'. That length measured with a microscopic ruler will not correspond exactly with length measured with a large ruler, even 'locally', because of local gradients."

The only thing that would affect a large ruler, as opposed to a very small ruler, is tidal gravity. That's the only kind of "gradient" in the field that can cause actual physical stress on a ruler and thus change its physical length. The K factor does *not* do this.

Remember I said the K factor is *not* the same as curvature; it's related but it's not the same. Spacetime curvature is tidal gravity, not the K factor.

We agreed on first part earlier, no need to repeat. But the last bit 'spacetime curvature is tidal gravity' is surely too restrictive a definition. It would leave out redshift for one. I mean, you earlier agreed curvature is what allows K to exceed unity, but if curvature = tidal gravity (grad(grad(potential))), and tidal forces don't effect K, something is missing here. I use the term curvature a bit loosely maybe, but we had a mutual understanding I thought that K is a manifestation of that curvature, in a distinctly different way to tidal forces. And that gradient of K is not tidal forces at work. Unless restraints are imposed, no stresses from varying K - always understood ('solid sphere' argument excepted).

Where did this start? Oh yes - your argument in #126 that varying K, just over the radius of a hoop, throws out the ability to accurately determine local curvature via marble count:
"Yes, but as I pointed out, to calculate the count, you need to know the K factor for a whole range of "r" values, from r = 0 out to the "r" of the hoop, which is its circumference divided by 2 pi. This brings in additional complications which are not present if you consider the area between two nearby circles of circumference C and C + dC."

Are we yet again misunderstanding each other's words? I took the above to mean, since it was not specified any more clearly, that 'varying r' was from the center of the hoop to it's periphery. And you meant something different? What exactly? If not, you are saying gradient of K locally matters re count - just as I thought. If not, how should one take it to mean? By the time that bit was written, you were quite aware of what I meant by hoop.

Originally Posted by Q-reeus: "Well gaps must open before one can pop in a new marble, yes? But I could have made that statement a bit clearer, sure."

What do you mean, "pop in a new marble"? We are not talking about that kind of experiment; we are talking about packing marbles into various pre-existing spaces. If you're envisioning a "gap" opening up, you're envisioning something like this: we take a circular "hoop" of circumference C and place it on a sphere, centered on the North Pole. Then we take a second circular "hoop" made of some elastic material, so it can stretch; we place it on the sphere starting with circumference C + dC(0), and then slowly move it away from the first circle, so dC gradually gets larger compared to dC(0). As we do that, yes, open space will appear that we now need to pack with more marbles. But also, as we do that, the K factor will vary between the two circles, so things become more complicated. It would be really good if you would stick to the purely "local" case first, to avoid confounding factors, like for example tidal gravity--see my comments elsewhere in this post.

Completely wrong at the start - more misunderstanding. Once you got the right idea of 'hoop' as a circular perimeter, why go bringing in this business of adding another one? Go right back to #108 where the hoop thing was introduced:
"Let's take your analogy of north pole - or anywhere on a curved spherical surface. Instead of concentric circles, just take a hoop, fill it with tiny marbles. We know that non-euclidean surface curvature means being able to fit more marbles inside the hoop than would be the case on a flat surface. But the analogy is flawed - we can move the hoop anywhere over a spherical surface and marbles fit the same. The proper analogy is more like a surface in the shape of an egg - with pointy end corresponding to the source of gravity in 'real' case. We note now that our hoop, despite having a fixed locally measured perimeter, fits more and more marbles within upon approach to the pointy end. Do you still say there will be no observable 'delta K factor'?"

Notice - one hoop, sampling a varying surface curvature. As it does so, the marble packing density alters - gaps will open - and to maintain packing density, one every now and then 'pops an extra one in'. Hope this part at least is perfectly bedded down. Sheesh.

Originally Posted by Q-reeus: "Stop right there...As you argued re hoop - there are gradient effects here."

Not if we're ignoring tidal gravity. See above. The only "gradient" effect that will actually stretch a container and change its physical size is tidal gravity. The K factor will *not* do this.

This is why I keep saying we should stick to the local case first; it avoids introducing confounding factors like tidal gravity that are *not* the same as the K factor. If you keep muddling these things together, you will keep on being confused. Once again: the K factor does *not* cause any stress on objects. Therefore, the K factor *cannot* change the physical size of a container; that would require causing stress on the container's walls. This is a basic point of the physics involved, and if it's not clear, we need to stick to the local case until it is.

Unless you can prove that the marble filled hoop will *not* experience changed packing density (restraint = fixed marble count) in heading towards pointy end, you have to face the fact that locally observed effects are present. And one possible *interpretation* by a local flat-land observer, who can't discern curvature directly, is varying hoop size, or alternately, shrinking marbles. Stresses can't explain it, but effects normally put down to changing container size and/or marble size are there. All you have to do to end that argument, is what I asked above - can the hoop packing density/number be independent of surface curvature? Around and around it all goes. Where it ends, nobody knows!

Originally Posted by Q-reeus:
"To put it more shortly - this means a simple subdivision of shell volume into 'containers' implies a radial stretching/stressing must occur after gravity is 'switched on'."

This implies that you're subdividing the shell volume by *coordinate* r, *not* by physical size of a very small object. You are still insisting on "labeling" every small point in the volume by its *coordinate*, r, which can be thought of as its "Euclidean coordinate", because it's derived from the area of the associated 2-sphere via the Euclidean geometry formula. But that labeling is *not* physical--it is coordinate-dependent. Physically, as I keep saying, Euclidean geometry is *not* privileged, and the "Euclidean coordinate" of a particular point has *no* physical meaning.

In order to properly match the s sided cube to spherical shells, which fails to fit re your scenario used earlier, one must stretch it further. That's what I meant. That stretch factor - what one would need to do, was my way of understanding your point about the misfit. But note my square bracketed comments in #128. :zzz:

 

[PAllen]

But the last bit 'spacetime curvature is tidal gravity' is surely too restrictive a definition. It would leave out redshift for one.

Gravitational redshift can occur in flat spacetime (e.g. uniform acceleration in SR produces it, yet the spacetime has no curvature).

Peter is correct that if there is curvature of spacetime in a region, then there is tidal gravity. There may also be non-euclidean effects on space alone, these being much harder to detect.

 

[PeterDonis]

But the last bit 'spacetime curvature is tidal gravity' is surely too restrictive a definition.

No, it's the exact definition in GR. Tidal gravity is spacetime curvature. There's a geometric way to express the meaning of this in more detail, but I won't go into it unless you want me to, as it is rather a tangent relative to the topic of this thread. But it's worth seeing how some of the other things you cite relate to the definition.

It would leave out redshift for one.

Redshift can be observed between accelerating observers in flat spacetime; curvature is not required. It can also be caused by curvature, of course, as it is around a gravitating body. But it is not the same thing as curvature.

I mean, you earlier agreed curvature is what allows K to exceed unity

Not quite; I said (or meant to say; I don't think I've gone into any detail about this yet, so it may be that you are reading too much into something I said rather quickly) that if K is not unity, there must be curvature present. The converse is *not* true; it is possible for there to be curvature present but still have K = 1. For example, "Painleve observers" who are free-falling towards a black hole from rest "at infinity" see K = 1, even though the spacetime is curved.

but if curvature = tidal gravity (grad(grad(potential))), and tidal forces don't effect K, something is missing here.

I didn't say tidal forces don't affect K. See above for how the two are related. With regard to the marbles and containers and so forth, I said that K does not cause stress in objects, whereas tidal gravity does.

I use the term curvature a bit loosely maybe, but we had a mutual understanding I thought

Apparently not. See above for clarification.

And that gradient of K is not tidal forces at work. Unless restraints are imposed, no stresses from varying K - always understood ('solid sphere' argument excepted).

This is true; K and variation in K does not cause stresses in objects, whereas tidal gravity does.

Where did this start? Oh yes - your argument in #126 that varying K, just over the radius of a hoop, throws out the ability to accurately determine local curvature via marble count:

That's not what I said. You even quoted what I said, but apparently failed to notice that I was talking about determining the marble count itself, *not* determining local curvature via marble count. Actually, you can't determine local curvature from the marble count, even if we restrict ourselves to a small enough range that K can be considered constant. K is not curvature, and all that the marble count allows us to measure is K. Measuring curvature is more complicated, as others' posts have illustrated.

Are we yet again misunderstanding each other's words? I took the above to mean, since it was not specified any more clearly, that 'varying r' was from the center of the hoop to it's periphery. And you meant something different? What exactly? If not, you are saying gradient of K locally matters re count - just as I thought. If not, how should one take it to mean? By the time that bit was written, you were quite aware of what I meant by hoop.

You've lost me here; I don't understand how this relates to the part of my post that you quoted. But again, all this about what happens when K varies is PREMATURE. Sorry for shouting, but I made a point of repeating this several times in my last post. You *need* to get the "local" case, with constant K, figured out *first*, before even *thinking* about gradients of any kind. Most of the rest of your post is the same thing. Let's get the constant K, local case agreed first. As far as I know, you still believe that even in that case, there is some effect that causes objects to "change size" somehow, when there isn't. We need to get that cleared up first. Until we do, I can't respond to any talk about what happens when there is a gradient in K, because we don't have a common base to start from.

 

[PeterDonis]

Peter is correct that if there is curvature of spacetime in a region, then there is tidal gravity.

And it's also true that if there is tidal gravity, then there is curvature of spacetime. So the two are equivalent, which is why I made the stronger claim I did in my post.

I should note that by "tidal gravity" I include *any* effect that causes initially parallel geodesics to converge or diverge. That's exactly what the Riemann curvature tensor captures.

 

[PAllen]

Unless you can prove that the marble filled hoop will *not* experience changed packing density (restraint = fixed marble count) in heading towards pointy end, you have to face the fact that locally observed effects are present. And one possible *interpretation* by a local flat-land observer, who can't discern curvature directly, is varying hoop size, or alternately, shrinking marbles. Stresses can't explain it, but effects normally put down to changing container size and/or marble size are there. All you have to do to end that argument, is what I asked above - can the hoop packing density/number be independent of surface curvature? Around and around it all goes. Where it ends, nobody knows!

Forget hoops, filled or otherwise. You cannot detect curvature in a 4-manifold with anything restricted to a 2-surface, in any orientation (anything you think you might detect this way will be a an embedding feature, similar to embedding a curved 2-sphere in flat 3-space). You need lots of measurements of a substantial spatial region, as in the examples Peter and I have been discussing (his volume examples, and Synge's many point, many measurement examples).

 

[PeterDonis]

In order to properly match the s sided cube to spherical shells, which fails to fit re your scenario used earlier, one must stretch it further. That's what I meant. That stretch factor - what one would need to do, was my way of understanding your point about the misfit.

I'm responding to this separately because it actually *is* about the local issue, when K is constant. Why would you want to stretch the cube? That invalidates the property that makes the cube useful in the first place: that it has sides of a known length. There is no force due to the K factor that *makes* the cube stretch, so if you think of it as being stretched, you're bringing in some external force, *not* due to the K factor, to stretch it. Why do that? It just complicates things without any benefit in understanding what we're trying to understand, which is the physical effects of the K factor.

However, the last part of that paragraph is *not* about the local issue:

But note my square bracketed comments in #128. :zzz:

Your square bracketed comments in #128 were:

[Hardening my stance on this. Comments in #123 apply: "While your concentric circles around north pole analogy in #99 talked in terms of perimeter-to-radius ratio, one could equally talk in terms of an enclosed surface area-to-perimeter ratio of a dished annulus (numerically different, but having in common dependence on surface curvature). Once you see it the latter way, the hoop thing springs out as a more evident manifestation that local phenomena will exist, which is why I used it." There is a marble count excess for dished annulus sitting on 2-sphere (2D analogue of spherical shells experiencing 3-curvature). Same general effect must apply to a hoop, and likewise 3D container]

The enclosed surface area to perimeter ratio is *not* local; it requires sampling K over a range of perimeters (circumferences in my terminology), over which K varies. The "annulus" scenario I used does not; that's why I used it. So this part is out of bounds: I won't discuss it until we've got the local, constant K part figured out.

 

[pervect]

Forget hoops, filled or otherwise. You cannot detect curvature in a 4-manifold with anything restricted to a 2-surface, in any orientation (anything you think you might detect this way will be a an embedding feature, similar to embedding a curved 2-sphere in flat 3-space). You need lots of measurements of a substantial spatial region, as in the examples Peter and I have been discussing (his volume examples, and Synge's many point, many measurement examples).

You could, I suppose, also replace the hoops with spheres, which seems more in spirit with the discussion.

 

[PAllen]

You could, I suppose, also replace the hoops with spheres, which seems more in spirit with the discussion.

Q-reeus introduced hoops (to simplify? to detect orientation? not quite sure). So, I want to emphasize it will never work.

The simplest thing you can do with spheres to detect curvature is up for discussion. I don't think anyone here so far claims to know what the simplest construction involving spheres or parts of spheres that would detect curvature is.

[The hard part of detecting curvature from purely spatial measurements is avoiding embedding artifacts. Anything dependent on a particular foliation doesn't cut it. So, if one hypothesized that volume to surface area ration differed from 1/3, you would have to show that there does not exist any foliation in which the ratio is 1/3.]

 

[pervect]

Taking a large, hollow sphere, and counting the number of smaller spheres you can pack into it, to measure it's volume, would (at least in principle) give you a measure of spatial curvature. But it wouldn't give a measure of space-time curvature, it would measure the spatial curvature of some particular spatial slice.

I think that's what was wanted, though I haven't been following in detail and the thread is too long to try and catch up.

Another minor issue is that the Riemann of a plane only has 1 component, but the Riemann of a three-space should have 3. So the circle-packing tells us as much as we can know about the curvature of a plane, but sphere-packing doesn't tell us everything about the curvature of some particular spatial slice.

 

[PeterDonis]

Taking a large, hollow sphere, and counting the number of smaller spheres you can pack into it, to measure it's volume, would (at least in principle) give you a measure of spatial curvature. But it wouldn't give a measure of space-time curvature, it would measure the spatial curvature of some particular spatial slice.

I think that's what was wanted, though I haven't been following in detail and the thread is too long to try and catch up.

Kinda sorta. You bring up a good point, we've been using the word "curvature" without always being clear about what kind.

When I said that tidal gravity is the same as curvature, I meant specifically *spacetime* curvature. (I said so explicitly at least once.)

There is also, as you say, the curvature of a spatial slice. That, of course, depends on how you cut the slice, so to speak, out of spacetime. Also, as you note, the sphere packing, which measures what I've been calling the K factor, is not a complete measure even of the spatial curvature. (Also, as I've noted, the measurement you describe samples the K factor over a range of radial coordinates, or sphere areas, so it's more complicated than just measuring the K factor between two spheres that are very close together. I'm trying to stick to the "local" case, where K is effectively constant, until we get that sorted out, before bringing in variation in K.)

(Another minor point is that what you described is the *intrinsic* curvature of the spatial slice; there is also the extrinsic curvature of the slice, which is something else again.)

 

[PAllen]

Taking a large, hollow sphere, and counting the number of smaller spheres you can pack into it, to measure it's volume, would (at least in principle) give you a measure of spatial curvature. But it wouldn't give a measure of space-time curvature, it would measure the spatial curvature of some particular spatial slice.

I think that's what was wanted, though I haven't been following in detail and the thread is too long to try and catch up.

Another minor issue is that the Riemann of a plane only has 1 component, but the Riemann of a three-space should have 3. So the circle-packing tells us as much as we can know about the curvature of a plane, but sphere-packing doesn't tell us everything about the curvature of some particular spatial slice.

Well I was interested in something intrinsic. I see no reason you can't construct a non-euclidean spacelike 3-surface in Minkowski flat spacetime. What would be the physical significance of that? Whereas, with spacetime curvature present, while you can generally find a flat 2-surface, you cannot find a flat 3-surface (a while back I opened a thread on embedding like this, and determined this based on number of coordinate conditions that can be imposed on a metric). So, to have real meaning, the condition to look for isn't ability to find a curved spatial slice; instead, it is inability to find a flat one.

Separately, I don't know if every 3-surface with non-vanishing Riemann tensor must deviate from the Euclidean sphere area/volume ratio. Have you determined that this is true?

 

[pervect]

Kinda sorta. You bring up a good point, we've been using the word "curvature" without always being clear about what kind.

When I said that tidal gravity is the same as curvature, I meant specifically *spacetime* curvature. (I said so explicitly at least once.)

There is also, as you say, the curvature of a spatial slice. That, of course, depends on how you cut the slice, so to speak, out of spacetime. Also, as you note, the sphere packing, which measures what I've been calling the K factor, is not a complete measure even of the spatial curvature. (Also, as I've noted, the measurement you describe samples the K factor over a range of radial coordinates, or sphere areas, so it's more complicated than just measuring the K factor between two spheres that are very close together. I'm trying to stick to the "local" case, where K is effectively constant, until we get that sorted out, before bringing in variation in K.)

(Another minor point is that what you described is the *intrinsic* curvature of the spatial slice; there is also the extrinsic curvature of the slice, which is something else again.)

I agree with everything you wrote, though I'm still not sure which one of the various aspects of curvature is of interest. I suspect that the idea is just to overall describe curvature.

I don't regard extrinsic curvature as being physically very interesting, because we'd have to stand outside of space-time to do define it. So I'm mostly interested in intrinsic curvature. I suppose that the extrinsic curvature might be of some use if you're doing ADM stuff, but it's outside the scope of my current interests.

As regards intrinsic space-time curvature, I'd go with the perhaps overly mathematical approach that says that it's completely defined by the Riemann tensor, and that tidal forces are described by one part of the Riemann tensor, the part that's sometimes called the electro-gravitic part in the Bel decomposition.

There are two other parts of the Bel decomposition in the 4d spacetime of GR. One of them is the topo-gravitic part. This describes the purely spatial part of the curvature.

The remaining part is the magneto-gravitic part, that describes frame dragging effects.

So my take is that tidal gravity is part of the mathematical entity (the Riemann) that completely describes all the aspects of space-time, curvature, but it's not the complete story.

Though I think that if you have the tidal forces for observers in all state of motion (rather than just the tidal forces for one observer), you can recover the Riemann, just as you can do it from a set of multiple sectional curvatures of planar slices, though I couldn't write down exactly how to perform either operation.

If we focus on the deviation between a reference geodesic and nearby geodesics (via the geodesic deviation equation), we can neatly categorize the various parts of the Bel decompositon as follows:

The geodesic deviation (the relative acceleration between nearby geodesics) will depend on both the spatial separation (and be proportional to it), and will also depend on the relative velocity (said velocity being measured in the fermi-normal frame of the reference geodesic).

The deviation turns out to be quadratic with respect to the velocity. The terms independent of velocity, presnet at zero velocity, will give rise to the electro-gravitic part of the tensor, and are described by the tidal forces.

The parts that are proportional to velocity describe the magnetic part.

The parts that are proportional to velocity squared are due to the spatial curvature (i.e. the topogravitic part of the tensor). They're rather analogous to the v^2/R type forces that an object moving in a circular path of radius R with velocity v experiences.

 

[PeterDonis]

Whereas, with spacetime curvature present, while you can generally find a flat 2-surface, you cannot find a flat 3-surface (a while back I opened a thread on embedding like this, and determined this based on number of coordinate conditions that can be imposed on a metric).

You can in certain special cases. One has been mentioned in this thread: Painleve coordinates in Schwarzschild spacetime; the slices of constant Painleve time are flat. Another is FRW spacetime with k = 0; spacetime as a whole is curved but the spatial slices of constant "comoving" time are flat. I don't know of any other such cases off the top of my head.

So, to have real meaning, the condition to look for isn't ability to find a curved spatial slice; instead, it is inability to find a flat one.

Provided it isn't one of the special cases.

 

[PeterDonis]

I don't regard extrinsic curvature as being physically very interesting, because we'd have to stand outside of space-time to do define it.

Not the extrinsic 3-curvature of a spatial slice taken out of a 4-d spacetime. But I agree extrinsic curvature isn't very interesting compared to intrinsic; certainly not for the topic of this thread.

As regards intrinsic space-time curvature, I'd go with the perhaps overly mathematical approach that says that it's completely defined by the Riemann tensor,

Agreed.

and that tidal forces are described by one part of the Riemann tensor, the part that's sometimes called the electro-gravitic part in the Bel decomposition.

There are two other parts of the Bel decomposition in the 4d spacetime of GR. One of them is the topo-gravitic part. This describes the purely spatial part of the curvature.

The remaining part is the magneto-gravitic part, that describes frame dragging effects.

So my take is that tidal gravity is part of the mathematical entity (the Riemann) that completely describes all the aspects of space-time, curvature, but it's not the complete story.

Well, the term "tidal gravity" may be a bit ambiguous. I posted earlier the definition I was using: *anything* that causes geodesic deviation. All parts of the Riemann tensor describe this in some form; you describe how later in your post.

I recognize now that my terminology may be non-standard. I first saw it in Kip Thorne's Black Holes and Time Warps, and it made sense to me. But my arguments in this thread are essentially unchanged if "tidal gravity" is taken to describe just the electrogravitic part of the full curvature tensor, as you say. It's still true that what I am calling the K factor is not the same as tidal gravity, and is not equivalent to it, nor is it equivalent to curvature in full.

Though I think that if you have the tidal forces for observers in all state of motion (rather than just the tidal forces for one observer), you can recover the Riemann, just as you can do it from a set of multiple sectional curvatures of planar slices, though I couldn't write down exactly how to perform either operation.

I think this is true, yes.

 

[PAllen]

You can in certain special cases. One has been mentioned in this thread: Painleve coordinates in Schwarzschild spacetime; the slices of constant Painleve time are flat. Another is FRW spacetime with k = 0; spacetime as a whole is curved but the spatial slices of constant "comoving" time are flat. I don't know of any other such cases off the top of my head.
Provided it isn't one of the special cases.

Ok, interesting. I only looked at the general case. I see you are right. So I would interpret that as saying that your K can be considered as an artifact of the 'natural' way stationary (not inertial) observers set up simultaneity.

Also interesting, is that G-P coordinates have the feature (just like SC) that (t,t) metric component becomes spacelike inside the horizon. It looks to me, then, that inside the horizon, where the metric is not static, you do not have flat spatial slices. I might guess that you can't, in this region.

 

[Passionflower]

You can in certain special cases. One has been mentioned in this thread: Painleve coordinates in Schwarzschild spacetime; the slices of constant Painleve time are flat.

Yes an observer traveling radially at escape velocity observes his space as flat in that the distance between two r-coordinate values is exactly the difference. Other observers, who are also on a radial path or stationary, observe this distance differently. One can obtain this distance by applying the Lorentz factor (local speed wrt a free falling at escape velocity observer) before integration between these two r-coordinates.

 

[PeterDonis]

Ok, interesting. I only looked at the general case. I see you are right. So I would interpret that as saying that your K can be considered as an artifact of the 'natural' way stationary (not inertial) observers set up simultaneity.

Yes, I agree.

Also interesting, is that G-P coordinates have the feature (just like SC) that (t,t) metric component becomes spacelike inside the horizon. It looks to me, then, that inside the horizon, where the metric is not static, you do not have flat spatial slices. I might guess that you can't, in this region.

Actually, you still can. G-P coordinates are very weird inside the horizon; *all four* coordinates are spacelike! That means that, even though the time coordinate T is spacelike, the surfaces of constant T are as well. I agree this is extremely odd, but it's true.

To be more precise in stating what this means: inside the horizon, the vector \frac{\partial}{\partial T} corresponding to Painleve time T *and* the vector \frac{\partial}{\partial r} corresponding to the radial coordinate r (which is defined the same for Painleve as for Schwarzschild coordinates) are spacelike. Since the angular vectors are also spacelike, you can form a spacelike 3-surface using the three "spatial" coordinates inside the horizon just like you can outside, and these 3-surfaces will be orthogonal to the worldline of an infalling "Painleve" observer. (Note that they are *not* orthogonal to the vector \frac{\partial}{\partial T}, obviously, because that vector is spacelike inside the horizon. But the 3-surfaces I've just defined aren't orthogonal to \frac{\partial}{\partial T} *outside* the horizon either, because of the dTdr term in the Painleve metric.)

 

[PAllen]

Yes, I agree.



Actually, you still can. G-P coordinates are very weird inside the horizon; *all four* coordinates are spacelike! That means that, even though the time coordinate T is spacelike, the surfaces of constant T are as well. I agree this is extremely odd, but it's true.

To be more precise in stating what this means: inside the horizon, the vector \frac{\partial}{\partial T} corresponding to Painleve time T *and* the vector \frac{\partial}{\partial r} corresponding to the radial coordinate r (which is defined the same for Painleve as for Schwarzschild coordinates) are spacelike. Since the angular vectors are also spacelike, you can form a spacelike 3-surface using the three "spatial" coordinates inside the horizon just like you can outside, and these 3-surfaces will be orthogonal to the worldline of an infalling "Painleve" observer. (Note that they are *not* orthogonal to the vector \frac{\partial}{\partial T}, obviously, because that vector is spacelike inside the horizon. But the 3-surfaces I've just defined aren't orthogonal to \frac{\partial}{\partial T} *outside* the horizon either, because of the dTdr term in the Painleve metric.)

Fascinating. After a little thought I agree. So I guess the SC geometry everywhere has too many symmetries to preclude construction of flat spatial slices.

 

[Q-reeus]

Originally Posted by Q-reeus:
"Unless you can prove that the marble filled hoop will *not* experience changed packing density (restraint = fixed marble count) in heading towards pointy end, you have to face the fact that locally observed effects are present. And one possible *interpretation* by a local flat-land observer, who can't discern curvature directly, is varying hoop size, or alternately, shrinking marbles. Stresses can't explain it, but effects normally put down to changing container size and/or marble size are there. All you have to do to end that argument, is what I asked above - can the hoop packing density/number be independent of surface curvature?"

Forget hoops, filled or otherwise. You cannot detect curvature in a 4-manifold with anything restricted to a 2-surface, in any orientation (anything you think you might detect this way will be a an embedding feature, similar to embedding a curved 2-sphere in flat 3-space). You need lots of measurements of a substantial spatial region, as in the examples Peter and I have been discussing (his volume examples, and Synge's many point, many measurement examples).

I'm putting it down to this: both you and Peter 'know' I probably have everything wrong, so what I actually argue tends to go in one ear and out the other, to be replaced with an image of what 'I probably meant', and then proceed to tear down that straw man. Point to anywhere, not just above but any previous entry, where I claimed what you think I did. I'm not so stupid as to imagine that running a hoop over an 'egg' allows one to detect 3-curvature at all. Have endlessly now referred to this as 2D analogue of 3D situation. Having said that, there *is* I think an in principle 2D manifestation of 3-curvature, which will be discussed in another posting.

 

[Q-reeus]

You could, I suppose, also replace the hoops with spheres, which seems more in spirit with the discussion.

And in #139:

"Taking a large, hollow sphere, and counting the number of smaller spheres you can pack into it, to measure it's volume, would (at least in principle) give you a measure of spatial curvature. But it wouldn't give a measure of space-time curvature, it would measure the spatial curvature of some particular spatial slice.
I think that's what was wanted, though I haven't been following in detail and the thread is too long to try and catch up."

Full marks from me for carefully reading what I have been arguing - yes got the drift admirably.