
#109
Oct2511, 05:26 PM

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#110
Oct2511, 09:16 PM

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Qreeus, 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: 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. Also, once again, there will be a "nonEuclideanness" 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. 



#111
Oct2511, 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 + (11/h) / r^2 (6.2.4) 8 pi P = (r f h)^1 df/dr (11/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 (11/h) ] If we envision a thick shell where rho=0, this immediately implies that r(11/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 rewrite 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/(12M/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] 



#112
Oct2611, 10:48 AM

P: 1,115

"If we envision a thick shell where rho=0, this immediately implies that r(11/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. 



#113
Oct2611, 10:53 AM

P: 1,115

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 (minihoops), 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 nonuniform 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 handwavy, but can one not see an analogue with the well known example of triangles in curved space. Angles add to more than 180^{0} in positively curved space, but the effect is a nonlinear function of triangle size  virtually nonexistent 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 nonuniformity is miniscule. In my opinion that is. 



#114
Oct2611, 11:23 AM

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#115
Oct2611, 02:42 PM

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#116
Oct2611, 03:19 PM

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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. 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. 



#117
Oct2611, 04:57 PM

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Any deviation from the expected 2D Euclidean value would be apparent only for relatively large (i.e. "nonlocal") 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. 



#118
Oct2611, 05:23 PM

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Unless Synge is wrong, even over a large region, you cannot detect Euclidean deviation on a plane. You need something 3d, 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.




#119
Oct2611, 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 spacetime 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 spacetime 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 spacetime, not just one. 



#120
Oct2611, 06:21 PM

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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 KS coordinates). MTW makes note of his clear priority as the first ever to work it all out.




#121
Oct2611, 06:35 PM

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#122
Oct2611, 07:53 PM

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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 3D 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.) Consider a pair of 2spheres 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 areajust 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 2spheres 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 "nonlocal", in the sense that it can't be done without enclosing the entire Earth with a pair of 2spheres, 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. 



#123
Oct2711, 09:49 AM

P: 1,115

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 3curvature. 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 nonlinearly 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 nonlinear 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 5point 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. "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 soccerball 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 noneuclidean effect observable in a 'container'. No?!" I can conceive no way around that. How can there be? 



#124
Oct2711, 10:36 AM

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#125
Oct2711, 10:55 AM

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#126
Oct2711, 10:58 AM

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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 2spheres, 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 observableit'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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