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


by Q-reeus
Tags: handle, mass, metric, shell, spherical, transition
Q-reeus
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Oct25-11, 05:26 PM
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Quote Quote by PAllen View Post
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
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Oct25-11, 09:16 PM
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Q-reeus, first of all, did you read my post in the other thread about a similar issue?

http://physicsforums.com/showpost.ph...02&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:

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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
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Oct25-11, 10:02 PM
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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) = [itex]\int[/itex] 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

[tex]
\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
[/tex]

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

[tex]
\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
[/tex]

and for r>1

[tex]
\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
[/tex]
Q-reeus
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Oct26-11, 10:48 AM
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Quote Quote by pervect View Post
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
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Oct26-11, 10:53 AM
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Quote Quote by PeterDonis View Post
Q-reeus, first of all, did you read my post in the other thread about a similar issue?

http://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
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Quote Quote by Q-reeus View Post
I have now - and you can read my comments on that in turn!
I did, and I responded to them there.

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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
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Oct26-11, 02:42 PM
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Quote Quote by PeterDonis View Post
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.
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Quote Quote by Q-reeus View Post
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.)

Quote Quote by Q-reeus View Post
Check my 3rd passage in #108.
Do you mean the following?

Quote Quote by Q-reeus
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.

Quote Quote by Q-reeus View Post
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".

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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.
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Quote Quote by Q-reeus View Post
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'?
Quote Quote by PeterDonis View Post
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.
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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.
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Oct26-11, 06:14 PM
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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.
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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
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Oct26-11, 06:35 PM
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Quote Quote by pervect View Post
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
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#122
Oct26-11, 07:53 PM
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Quote Quote by DrGreg View Post
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.)

Quote Quote by DrGreg View Post
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
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#123
Oct27-11, 09:49 AM
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Quote Quote by PeterDonis View Post
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
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#124
Oct27-11, 10:36 AM
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Quote Quote by Q-reeus View Post
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
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#125
Oct27-11, 10:55 AM
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Quote Quote by PAllen View Post
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
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#126
Oct27-11, 10:58 AM
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Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
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?

Quote Quote by Q-reeus View Post
And ergo - go 3D and fluid level in a container responds to changed 3-curvature.
Still an issue here--see below.

Quote Quote by Q-reeus View Post
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.

Quote Quote by Q-reeus View Post
"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:

Quote Quote by Q-reeus View Post
Ergo - there is an non-euclidean effect observable in a 'container'. No?!"
No.

Quote Quote by Q-reeus View Post
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.


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