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

[PeterDonis]

Orienting the tube axis in the tangent plane will give some reading for height of fluid in the capillary (think of old style mercury thermometer). Orient tube along radial direction, at same mean radial position r, and the level in capillary will drop - differential rate of 'volume expansion/contraction' along r direction is such that 'expanded volume' in tube portion nearest source of gravity wins over opposite effect in portion furtherest from source. this is just a reinterpretation of physical implications of K factor imo.

No, the K factor does not imply this. Remember that a spherical object (more precisely, an object that in flat spacetime, under zero stress, is spherical) will still be spherical if placed at radial coordinate r; the K factor does not cause any distortion in the object. There is no "distortion" in the effect on the capillary tube either, for the same reason.

Further, one could take a fluid filled spherical container (again with a capillary tube sticking out of it), and find that for inwardly directed radially displacement, fluid level in capillary will drop.

No, it won't. See above.

This might be interpreted as a weird volumetric expansion of containment vessel - one without explanation in terms of any mechanical stress/strain.

This is not possible; if the physical volume of the container were expanded, the containment vessel would *have* to show strain. That's part of what "physical volume" means.

Think again about what the K factor means. It does not mean that "the physical volume of a particular piece of space is expanded". That's impossible. It means that there are *more* "pieces of space", more physical volume, per unit radial coordinate than Euclidean geometry would lead one to expect. But as I said in a previous post, to view this as somehow a "distortion of space" implies that the Euclidean state is the "natural" state, so any variation from it is a "distortion" and requires some physical manifestation. That's wrong. There is nothing privileged about Euclidean geometry in physics, and the fact that the geometry of space is non-Euclidean along the radial dimension in the spacetime surrounding a gravitating object is just that: a fact about the geometry of that spacetime. Just as the fact that, in my "house at the North Pole" scenario, there is "more distance" along a given unit of the radial coordinate I defined than Euclidean geometry would lead one to expect is simply that: a fact about the geometry of the surface of the Earth. None of these facts change the behavior of physical objects locally; they only change the global structure of the geometry.

 

[PAllen]

I'll add one more bit to what Peter said. Your stated goal of having tidal effects ignorable guarantees you can't detect Euclidean deviations. Tidal effects are the first order influence of curvature, thus they define the minimum scale needed to detect curvature. However, if you are willing to span a relatively large distance, and have near mathematically ideal measuring devices, you can detect Euclidean deviation as follows:

You pick a configuration of 5 points in space (e.g. the vertices of the figure made by joining two tetrahedra). You set up distances and angles between them per Euclidean predictions (e.g. using round trip laser time to define distance, and laser path the define straight lines). Then, at the very end, with all angles and all but one edge length set up, the last edge will be the wrong length.

J.L. Synge, in his 1960 book, develops this 5 point curvature detector. He shows that 5 points is the minimum needed to make this work (because, for example, flat Euclidean planes can be embedded in general 4-manifolds).

[EDIT: as for scale, if you use a 10 meter device near earth, your final deviation would be 10^-20 centimers or so. Less than a millionth the radius of a proton. ]

 

[Q-reeus]

Originally Posted by Q-reeus:
"Orienting the tube axis in the tangent plane will give some reading for height of fluid in the capillary (think of old style mercury thermometer). Orient tube along radial direction, at same mean radial position r, and the level in capillary will drop - differential rate of 'volume expansion/contraction' along r direction is such that 'expanded volume' in tube portion nearest source of gravity wins over opposite effect in portion furtherest from source. this is just a reinterpretation of physical implications of K factor imo."

No, the K factor does not imply this. Remember that a spherical object (more precisely, an object that in flat spacetime, under zero stress, is spherical) will still be spherical if placed at radial coordinate r; the K factor does not cause any distortion in the object. There is no "distortion" in the effect on the capillary tube either, for the same reason.

Didn't really expect this to go down quietly. Still having great difficulty reconciling that bit (and the remainder of your comments) with just this excised bit of mine from #103:
"If fractional excess volumetric particle count between two concentric shells is a function of radius r, this 'must' be true for subdivided portions - conic sections through the shells say. So I'm under the strong impression it really boils down to a kind of spatial divergence - the small counting spheres are only capable of being a reference if their relative volumetric expansion is negligible compared to much larger container volume..."

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?!

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'?

Originally Posted by Q-reeus:
Orienting the tube axis in the tangent plane will give some reading for height of fluid in the capillary (think of old style mercury thermometer). Orient tube along radial direction, at same mean radial position r, and the level in capillary will drop - differential rate of 'volume expansion/contraction' along r direction is such that 'expanded volume' in tube portion nearest source of gravity wins over opposite effect in portion furtherest from source. this is just a reinterpretation of physical implications of K factor imo.

No, the K factor does not imply this. Remember that a spherical object (more precisely, an object that in flat spacetime, under zero stress, is spherical) will still be spherical if placed at radial coordinate r; the K factor does not cause any distortion in the object. There is no "distortion" in the effect on the capillary tube either, for the same reason.

I see what you are saying but re my previous argument, something, albeit exceedingly tiny, is physically happening here.

Originally Posted by Q-reeus:
"This might be interpreted as a weird volumetric expansion of containment vessel - one without explanation in terms of any mechanical stress/strain."

This is not possible; if the physical volume of the container were expanded, the containment vessel would *have* to show strain. That's part of what "physical volume" means.

Not if one accepts a physical gradient of length scale operates - gradient non-euclidean is necessary if any non-euclidean at all, yes? I have given the 2-D example above - hoop+marbles on egg re seemingly impossible effects.

Think again about what the K factor means. It does not mean that "the physical volume of a particular piece of space is expanded". That's impossible. It means that there are *more* "pieces of space", more physical volume, per unit radial coordinate than Euclidean geometry would lead one to expect. But as I said in a previous post, to view this as somehow a "distortion of space" implies that the Euclidean state is the "natural" state, so any variation from it is a "distortion" and requires some physical manifestation. That's wrong...

Hope this doesn't bog down into arguing over meaning of things. Would you say that an apple falling to the ground represents physics, or 'just' an expression of non-euclidean geometry? For me, excess counts owing to non-euclidean geometry manifest as physical phenomena. Taking your example of North-pole (or anywhere on a spherical surface), surface curvature means more marbles between concentric circles than on flat ground. Yes I can call that just geometry, but since it is mass that causes the 3-space curvature in gravitational situation, that's physics to me. Stubborn me.

 

[Q-reeus]

I'll add one more bit to what Peter said. Your stated goal of having tidal effects ignorable guarantees you can't detect Euclidean deviations. Tidal effects are the first order influence of curvature, thus they define the minimum scale needed to detect curvature.

My stated aim was simply to say 'we know tidal effects will matter in practice - but let's fully account for them and just look at what remains'.

...However, if you are willing to span a relatively large distance, and have near mathematically ideal measuring devices, you can detect Euclidean deviation as follows:...[EDIT: as for scale, if you use a 10 meter device near earth, your final deviation would be 10^-20 centimers or so...]

Interesting and ingenious but I'm struggling here over principle - goes without saying nothing practical to patent. (caught your post after preparing response to Peter, so some things are repetitive - sorry)

 

[PeterDonis]

Q-reeus, first of all, did you read my post in the other thread about a similar issue?

https://www.physicsforums.com/showpost.php?p=3580102&postcount=6

I think it might be relevant here.

I'm going to comment on this particular thing you say first because it may be the key to the issue:

Not if one accepts a physical gradient of length scale operates - gradient non-euclidean is necessary if any non-euclidean at all, yes?

The K factor is not a "gradient of length scale". One meter is one meter, physically, regardless of what radial r coordinate you are at. (Or one nanometer, or one width of an atomic nucleus.) The only thing the non-Euclideanness of space affects is *how many meters* are between two concentric spheres; if K > 1, there are more meters between the spheres than the Euclidean formula predicts. That's all. It doesn't change what a meter is at all.

It may be worth thinking about this for a bit before reading the rest of what I have to say below. I'll be drawing on it.

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?!

Not if you specify the size of the container in meters. (Or in some unit of distance, anyway.) If you specify it in coordinate units, that's different; for example, if you specify it in units of radial coordinate r. But in your original container scenario, you didn't; you specified the size of the container in physical distance units. I suspect you didn't because you realized, subconsciously, that the "physical" definition of size is in meters, not coordinate units. If K > 1, there are more meters in a given unit of radial coordinate r, but each meter itself is still the same.

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'?

Of course there will be an observable change in the K factor; in fact, if you go back and read my "North Pole" post carefully, you will see that K varies even in that scenario, because of the way I defined the r coordinate. Constant curvature does not necessarily imply constant K.

Also, once again, there will be a "non-Euclideanness" in the number of marbles that can fit between a pair of hoops, but each marble itself remains the same size. Marbles, in this scenario, are like meters; they are the physical measure of distance. They themselves don't change, but how many of them fit between a pair of hoops does.

Hope this doesn't bog down into arguing over meaning of things. Would you say that an apple falling to the ground represents physics, or 'just' an expression of non-euclidean geometry?

It represents physics, and the non-Euclidean geometry is one way of modeling the physics.

For me, excess counts owing to non-euclidean geometry manifest as physical phenomena. Taking your example of North-pole (or anywhere on a spherical surface), surface curvature means more marbles between concentric circles than on flat ground. Yes I can call that just geometry, but since it is mass that causes the 3-space curvature in gravitational situation, that's physics to me. Stubborn me.

I never said it was *just* geometry. I said explicitly, when I defined the K factor, that it was a physical observable. But you have to be careful when you think about *which* physical observable it is.

 

[pervect]

I'm not really following the philosophical end of this discussion, but I think I can write a bit how to enclose the photon gas metric I previously presented in a shell to join the two together properly, which should serve as an actual concrete example.

Going back to the very basics, we can use Wald's metric

-f(r)*dt^2 + h(r)*dr^2 + r^2 (d theta^2 + sin(theta) dphi^2)

and Wald's results 6.2.3, 6.2.4

(6.2.3) 8 pi rho = (r h^2)^-1 dh/dr + (1-1/h) / r^2
(6.2.4) 8 pi P = (r f h)^-1 df/dr -(1-1/h) / r^2

rho and P are not the 'coordinate' density and pressure, but the densities in the orthonormal basis given by Wald in 6.1.6, i.e. they represent the "physical" density and pressure seen by an observer in a local Minkowskii frame.

6.2.3 can be written as 6.2.6

8 pi rho = (1/r^2) d/dr [r (1-1/h) ]

If we envision a thick shell where rho=0, this immediately implies that r(1-1/h) is constant through the shell.

If we shrink the shell to zero width, (a thin shell) then we say simply that h is the same inside the shell and outside, h being the spatial coefficient of the metric. So h must match where we join together the vacuum Schwarzschild metric with our photon gas metric.

If we add together 6.2.3 and 6.2.4 we can write

8 pi (rho + P) = (dh/dr) / r h^2 + (df/dr) / rfh

which we can re-write as

d/dr (f h) / (r f h^2) = 8 pi (rho +P)

Because this is NONZERO, we can say definitely that the product of f and h is not constant. We know that h is constant. Therefore we know that f changes. With a thick shell, f changes as we progress through the shell. As we shrink the shell to zero thickness, in the limit, know that f must 'jump' suddenly, because f*h can't be constant.

This doesn't tell us "how much" the jump is, and it's rather inconvenient to use this approach to actually match the metrics, but it does tell us something important, it tells us to expect 'f' to jump suddenly.

What we can do instead is say that the mass function for our metric must equal the Schwarzschild mass paramter M

i.e.

m(r) = $\int$ 4 pi r^2 rho(r)

Furthermore, we know what M is, because we know that the h coefficents must match, and h has the value 7/4 in our photon gas metric This implies that M/r = 3/14, as

h = 1 / (1 - 2M/r)

So if we look at the photon gas metric

$$\frac{7}{4}\, dr^2 + r^2 \,d \theta^2 + r^2 sin^2 \theta \, d\phi^2 - \sqrt{\frac{7}{3}}\,r\,dt^2$$

We see that (1/(1-2M/r)) must be 7/4, since the value of h must match. This implies that (M/r) must be 3/14.But we know that 8 pi rho(r) = 3 /( 7 r^2) from Wald's 6.2.3, and
we just have to solve for r such that m(r) = (3/14) r, where m(r) is given by the integral of 4 pi r^2 rho(r) dr , which is just the intergal of (3/14) dr.

Unless I'm mistaken, this is satisfied for all values of r, so we pick any value of r we like, set M = (3/14) r, and use that for the exterior solution.

IF we chose r = 1

we have the original metric for r<1

$$\frac{7}{4}\, dr^2 + r^2 \,d \theta^2 + r^2 sin^2 \theta \, d\phi^2 - \sqrt{\frac{7}{3}}\,r\,dt^2$$

and for r>1

$$\frac{dr^2}{1-\frac{3}{7r}}+ r^2 \,d \theta^2 + r^2 sin^2 \theta \, d\phi^2 - \left( 1-\frac{3}{7\,r} \right) dt^2$$

 

[Q-reeus]

I'm not really following the philosophical end of this discussion, but I think I can write a bit how to enclose the photon gas metric I previously presented in a shell to join the two together properly, which should serve as an actual concrete example...

pervect: thanks for all your work there, but of what I can follow, this is throwing me:
"If we envision a thick shell where rho=0, this immediately implies that r(1-1/h) is constant through the shell." Which states that h is a function of r. But later: "We know that h is constant." I'm reading the latter to merely follow as a limit of imposing zero shell thickness - ie dr -> 0. No doubt that is missing it somehow, but can't see where. I can say nothing about the derivation of the orthonormal basis stuff - whether there are any subtle assumptions that 'chop off' higher order gradient effects for instance. You will have read my reply to Peter, so perhaps let me know where you think it all comes apart, because I maintain there must be physical effects as described earlier.

 

[Q-reeus]

Q-reeus, first of all, did you read my post in the other thread about a similar issue?

https://www.physicsforums.com/showpost.ph...02&postcount=6

I have now - and you can read my comments on that in turn!

The K factor is not a "gradient of length scale". One meter is one meter, physically, regardless of what radial r coordinate you are at. (Or one nanometer, or one width of an atomic nucleus.) The only thing the non-Euclideanness of space affects is *how many meters* are between two concentric spheres; if K > 1, there are more meters between the spheres than the Euclidean formula predicts. That's all. It doesn't change what a meter is at all.

Strictly locally, I agree that 1 meter = 1 meter. My sense is though, to explain real effects, there must be a sense to '1 meter here looks different to 1 meter there'. I call that a gradient effect. Without it, there is magic - meters just slip in somehow.

Originally Posted by Q-reeus:
"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'?"

Of course there will be an observable change in the K factor; in fact, if you go back and read my "North Pole" post carefully, you will see that K varies even in that scenario, because of the way I defined the r coordinate. Constant curvature does not necessarily imply constant K.

Also, once again, there will be a "non-Euclideanness" in the number of marbles that can fit between a pair of hoops, but each marble itself remains the same size. Marbles, in this scenario, are like meters; they are the physical measure of distance. They themselves don't change, but how many of them fit between a pair of hoops does.

You refer to a pair of hoops here, which seems to imply this will be a peculiarity of using polar coords - 'north pole effect'. But note carefully my example was using just one hoop - 2D counterpart of the 3D container earlier referenced to re fluid levels in capillary. As you seem to agree that marble count for that single enclosing hoop will be a function of surface curvature - *the coord system independent geometric object*, how can you then argue there will be no counterpart in 3D container, influenced by coord system independent spatial 3-curvature? What applies between two concentric hoops must apply within one container hoop. Likewise for concentric shells vs a 3D container.

You say 'meter hasn't changed, there are just more meters there than expected by Euclidean measure'. But how did those extra meters slip in exactly? Just because we are using SC's? Surely it's got to be the geometric object at work, totally independent of any coords used. Take that single bounding hoop example again. To keep it consistently 2D, rather than marbles, fill it with identically shaped tiny circular rings (mini-hoops), and demand that the ring count, for fixed packing density, remain constant irrespective of surface curvature. Only means to gaurantee that is one of two ways. Stress the containing hoop in compression, or stress the rings in tension, as surface curvature increases. Notice the manifestation of curvature now is stresses - and corresponding strains - rather than perceived perimeter expansion of containing hoop (or alternately, shrinking diameters of rings), from the pov of local observer, who just notices 'weirdness'.

One may wish to argue the interpretation as to what's behind it all ('more meters' vs non-uniform meter'), but for sure, there are physical effects - as I maintain there must be. And for me, that 'constant meter' idea is the problem here. Can't see other than a length scale gradient effect at work - not observable 'at a point'. This is hand-wavy, but can one not see an analogue with the well known example of triangles in curved space. Angles add to more than 180⁰ in positively curved space, but the effect is a non-linear function of triangle size - virtually non-existent in the small. And by analogy, those marbles/rings/water molecules act as 'standard meters' for the same basic reason - their distortion by spatial metric non-uniformity is miniscule. In my opinion that is.

 

[PeterDonis]

I have now - and you can read my comments on that in turn!

I did, and I responded to them there.

Strictly locally, I agree that 1 meter = 1 meter. My sense is though, to explain real effects, there must be a sense to '1 meter here looks different to 1 meter there'.

What is wrong with putting it the way I did in my last post? That is: 1 meter here = 1 meter there, but how many meters fit between two spheres with area A and A + dA is different here than it is there.

I call that a gradient effect. Without it, there is magic - meters just slip in somehow.

If you think extra meters "slipping in" is "magic", why is that? The only reason I can see, is that you think the Euclidean relationship between areas of concentric spheres and the volume enclosed between them is somehow privileged. It isn't. So no "magic" is required for extra meters to be present. I don't see why this is so hard to grasp.

You refer to a pair of hoops here, which seems to imply this will be a peculiarity of using polar coords - 'north pole effect'.

No, it isn't. I have said all along that the K factor is an invariant physical observable. You could adopt Cartesian coordinates in my north pole scenario and it would still be there; it would just be a lot harder to express in those coordinates.

But note carefully my example was using just one hoop - 2D counterpart of the 3D container earlier referenced to re fluid levels in capillary. As you seem to agree that marble count for that single enclosing hoop will be a function of surface curvature - *the coord system independent geometric object*, how can you then argue there will be no counterpart in 3D container, influenced by coord system independent spatial 3-curvature? What applies between two concentric hoops must apply within one container hoop. Likewise for concentric shells vs a 3D container.

No, you didn't just use one hoop. You used hoops placed at different points on an egg-shaped surface, instead of on a spherical surface. (At least, that's the example I think you're referring to; if it's another, please re-state it or give a direct reference.) Did you read my response? I said constant curvature doesn't necessarily imply constant K. That means the K factor is *not* the same as curvature. It may be *related* to curvature, but it is not the same thing.

You say 'meter hasn't changed, there are just more meters there than expected by Euclidean measure'. But how did those extra meters slip in exactly?

See above. You are assuming the Euclidean expectation is privileged. It isn't.

 

[Q-reeus]

Originally Posted by Q-reeus:
"Strictly locally, I agree that 1 meter = 1 meter. My sense is though, to explain real effects, there must be a sense to '1 meter here looks different to 1 meter there'."

What is wrong with putting it the way I did in my last post? That is: 1 meter here = 1 meter there, but how many meters fit between two spheres with area A and A + dA is different here than it is there.

Maybe it is semantics getting in the way on that one. I think your '1 meter here = 1 meter there' statements are continually trying to clear up a non-existent conception on my part - that one would/could *locally* observe a meter changing just by moving around in a gravitational potential. No, have tried to make it abundantly clear I have never believed in such an absurdity. Rather, that move the meter rod over there into a lower grav potential, and in general it will look smaller than if done in flat spacetime. Yes, no doubt just calculating distance moved is a complication, but one that can be taken into account. So maybe we really are on the same (redshifted?) wavelength.

Originally Posted by Q-reeus:
"But note carefully my example was using just one hoop - 2D counterpart of the 3D container earlier referenced to re fluid levels in capillary. As you seem to agree that marble count for that single enclosing hoop will be a function of surface curvature - *the coord system independent geometric object*, how can you then argue there will be no counterpart in 3D container, influenced by coord system independent spatial 3-curvature? What applies between two concentric hoops must apply within one container hoop. Likewise for concentric shells vs a 3D container."

No, you didn't just use one hoop. You used hoops placed at different points on an egg-shaped surface, instead of on a spherical surface. (At least, that's the example I think you're referring to; if it's another, please re-state it or give a direct reference.)

Yes I did - where do you find me using more than one? Check my 3rd passage in #108. But suppose I used different hoops, provided they were standardized with reference to some particular starting location, how would that effect the argument? Crux of the matter is, within a single enclosing perimeter of locally measured invariant shape and size (the hoop), marble stacking density varies with surface curvature. Again, how can this not carry over more or less directly to the case of fluid-filled (water molecules = nano size marbles) container re 3-curvature effect? A rising or lowering capillary level is measurable physics. You say it won't, but where is this breakdown of analogy occurring? That is deeply baffling.

Did you read my response? I said constant curvature doesn't necessarily imply constant K. That means the K factor is *not* the same as curvature. It may be *related* to curvature, but it is not the same thing.

Accept that there is no direct relation. Having gone and re-read your #99, I see what you are driving at, but to me that situation, where 'the marbles' just sit there static on the ground, is hiding potential physics. In order for K > 1 there must be curvature - so curvature is the key operator that non-euclidean K factor manifests. I'm going to keep coming back to this same point - if there is no potential physics going on, explain or refute the matter of a locally invariant loop experiencing a varying marble area density, just by moving said hoop+marbles to a region of higher curvature. Then go back to the fluid filled container analogue, and tell me why level in the capillary would not vary with change of radial location r for container. And note, it has nothing to do with 'distortion' of the capillary tube, which is merely an indicator of change. :zzz:

 

[PeterDonis]

Maybe it is semantics getting in the way on that one. I think your '1 meter here = 1 meter there' statements are continually trying to clear up a non-existent conception on my part - that one would/could *locally* observe a meter changing just by moving around in a gravitational potential. No, have tried to make it abundantly clear I have never believed in such an absurdity. Rather, that move the meter rod over there into a lower grav potential, and in general it will look smaller than if done in flat spacetime.

What does "look smaller" mean? Part of the issue may be the continual temptation to use ordinary English words that have imprecise or ambiguous meanings. It's very important to resist that temptation, and to phrase things carefully in terms of actual observables. (For one thing, "look smaller" involves light and how light paths are changed by gravity in the intervening space, and you've ruled all that out of bounds--we're supposed to assume that all that has been corrected for.)

Check my 3rd passage in #108.

Do you mean the following?

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

If so, I must have missed it previously, because I should have objected, or at least clarified. I see how it refers to one hoop, but I also think it's false, unless I'm misunderstanding how you're defining a "hoop". I was assuming "hoop" meant a *single* line of marbles going around the circumference of a circle centered on the "North Pole"; combined with the assumption that the marbles themselves are so small that they can be used as little identical objects to measure distances to any accuracy we need for the problem, then the number of marbles that fit inside a "hoop" is determined by the hoop's circumference and nothing else. Since the circumference is tangential, it is unaffected by any "spatial distortion", regardless of anything else; that's the fixed point of departure that we both agree on. So the quoted sentences above are false if "hoop" means what I think it means.

If, on the other hand, by "hoop" you mean "two circles of slightly different circumferences, C and C + dC, plus the space between them", then we've been using "hoop" to mean different things. In the following quote, it looks like you're using "hoop" in this other sense, but if you were doing that in post #108 I didn't understand that. I was using the word "circle" to avoid such ambiguity. However, I'm not entirely sure, because in the following quote you still seem to equivocate about how "local" a hoop is. See below.

Crux of the matter is, within a single enclosing perimeter of locally measured invariant shape and size (the hoop), marble stacking density varies with surface curvature.

If the hoop has a "locally measured invariant shape and size", then it *must* be a "hoop" in the sense I was thinking--a *single* circle, with circumference C, and that's all. As I noted above, such a "hoop" must always contain the *same* number of marbles for a given "size" of hoop (i.e., circumference). If the number of marbles placed within a "hoop" of a given "size" can vary, then a "hoop" *cannot* be a single circle--it must be, as I noted above, two circles of slightly different circumferences, C and C + dC, plus the space between them. In this case, yes, the number of marbles placed within a hoop can vary, even if dC is held constant. But that just means the hoop does *not* have a "locally measured invariant shape and size".

In order for K > 1 there must be curvature - so curvature is the key operator that non-euclidean K factor manifests.

This is basically correct, with the proviso that K does not *equal* the curvature; it is *related* to it, but not the same.

if there is no potential physics going on, explain or refute the matter of a locally invariant loop experiencing a varying marble area density, just by moving said hoop+marbles to a region of higher curvature.

I did, by refuting your assumption that it is a "locally invariant loop". For a varying number of marbles to be seen in a "loop", the "loop" (you keep on changing words, and it doesn't help with clarity) cannot be "locally invariant"; that's obvious. More precisely, a region between two spheres of areas A and A + dA, or between two circles of circumference C and C + dC can vary in size as A or C change, even if dA or dC are held constant. That's the definition of the K factor, and I've said all along that it's a physical observable and represents "real physics" going on.

The only thing I am disagreeing with you about is that you are expecting this real physics to show up in a way that it does not, in fact, show up. The reason it does not show up the way you are expecting it to is that your expectation is based on giving a privileged status to the predictions of Euclidean geometry. In fact, there is no such privileged status. I've said that repeatedly, too, and you haven't picked up on it, or if you have, it hasn't shown in your posts. There's no point in continuing to wonder if this is about semantics, or if I think there's real physics going on. I've made all that clear multiple times. The thing to focus in on is why you believe Euclidean geometry has a privileged status, so that any departure from Euclidean geometry, meaning any K factor that is not equal to 1, requires some special manifestation over and above what I've already defined as the observable K factor.

 

[DrGreg]

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'?

I was assuming "hoop" meant a *single* line of marbles going around the circumference of a circle centered on the "North Pole"; combined with the assumption that the marbles themselves are so small that they can be used as little identical objects to measure distances to any accuracy we need for the problem, then the number of marbles that fit inside a "hoop" is determined by the hoop's circumference and nothing else. Since the circumference is tangential, it is unaffected by any "spatial distortion", regardless of anything else; that's the fixed point of departure that we both agree on. So the quoted sentences above are false if "hoop" means what I think it means.

If, on the other hand, by "hoop" you mean "two circles of slightly different circumferences, C and C + dC, plus the space between them", then we've been using "hoop" to mean different things. In the following quote, it looks like you're using "hoop" in this other sense, but if you were doing that in post #108 I didn't understand that. I was using the word "circle" to avoid such ambiguity. However, I'm not entirely sure, because in the following quote you still seem to equivocate about how "local" a hoop is.

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.

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.

 

[PAllen]

Unless Synge is wrong, even over a large region, you cannot detect Euclidean deviation on a plane. You need something 3-d, like Peter's concentric spheres (not circles). If Synge is right, the even over large regions, you can construct a Euclidean tetrahedron. You need one more vertex than a tetrahedron to detect Euclidean deviation.

 

[pervect]

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.

 

[PAllen]

A little side note on Synge is that well before Kruskal and Szekeres, Synge was the first to untangle the fully extended topology of SC geometry (though he didn't come up with K-S coordinates). MTW makes note of his clear priority as the first ever to work it all out.

 

[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.)