[math-fun] Curved space paradox
The following intuitive line of reasoning leads to a contradictory conclusion, and I don't know what's wrong with it. Obviously one of my assumptions or beliefs about spaces with constant Gaussian curvature is wrong, but I can't figure out which. Suppose we have a 3-dimensional space of constant, negative Gaussian curvature. For some large enough R, construct the locus of points at distance R from a fixed center C. This forms a 2-dimensional surface which is topologically an ordinary sphere. Select a point P on this sphere, and begin drawing circles of increasing radius r on the sphere centered at P. I mean r to be measured along the surface of the sphere, not straight through the hyperbolic 3-space in which the sphere is embedded. Now, this sphere must itself have constant Gaussian curvature; I don't have a fast proof but surely there is a symmetry argument that can establish it. Watch how the circumference of a circle, centered at P, changes with its radius r. For small r, it looks like 2 pi r. But as r gets larger, the negative curvature of the ambient hyperbolic space begins to make itself felt, and the circumference goes above 2 pi r, by some factor related to sinh. (If it doesn't, you didn't make R big enough -- that is exactly what I meant by "large enough R" above.) But later, as r becomes commensurate with R, the circumference rises to a maximum and begins to shrink again, because the sphere itself has finite circumference. On a surface of constant curvature, the circumference should either fall below 2 pi r, or it should exceed it forever. It should not do both. Surely I am not the first person to think these confusing thoughts. What is wrong with my reasoning?
Isn't a sphere in hyperbolic 3-space isometric to a sphere in Euclidean 3-space? So the sphere has *positive* constant Gaussian curvature, and the circles have circumference 2 pi k sin(r/k)? [Certainly, if the sphere has constant Gaussian curvature -- as you deduced from your symmetry argument -- then it must be positive by global Gauss- Bonnet.] -- APG.
Sent: Monday, February 19, 2018 at 7:30 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Curved space paradox
The following intuitive line of reasoning leads to a contradictory conclusion, and I don't know what's wrong with it. Obviously one of my assumptions or beliefs about spaces with constant Gaussian curvature is wrong, but I can't figure out which.
Suppose we have a 3-dimensional space of constant, negative Gaussian curvature.
For some large enough R, construct the locus of points at distance R from a fixed center C. This forms a 2-dimensional surface which is topologically an ordinary sphere. Select a point P on this sphere, and begin drawing circles of increasing radius r on the sphere centered at P. I mean r to be measured along the surface of the sphere, not straight through the hyperbolic 3-space in which the sphere is embedded.
Now, this sphere must itself have constant Gaussian curvature; I don't have a fast proof but surely there is a symmetry argument that can establish it.
Watch how the circumference of a circle, centered at P, changes with its radius r. For small r, it looks like 2 pi r. But as r gets larger, the negative curvature of the ambient hyperbolic space begins to make itself felt, and the circumference goes above 2 pi r, by some factor related to sinh. (If it doesn't, you didn't make R big enough -- that is exactly what I meant by "large enough R" above.)
But later, as r becomes commensurate with R, the circumference rises to a maximum and begins to shrink again, because the sphere itself has finite circumference.
On a surface of constant curvature, the circumference should either fall below 2 pi r, or it should exceed it forever. It should not do both.
Surely I am not the first person to think these confusing thoughts. What is wrong with my reasoning? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 19/02/2018 19:30, Allan Wechsler wrote:
The following intuitive line of reasoning leads to a contradictory conclusion, and I don't know what's wrong with it. Obviously one of my assumptions or beliefs about spaces with constant Gaussian curvature is wrong, but I can't figure out which.
The bit where I immediately felt "uh-oh, handwaving going on here, not at all convinced" was this:
But as r gets larger, the negative curvature of the ambient hyperbolic space begins to make itself felt, and the circumference goes above 2 pi r, by some factor related to sinh.
I don't see why this should be true. -- g
On Mon, Feb 19, 2018 at 2:30 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Watch how the circumference of a circle, centered at P, changes with its radius r. For small r, it looks like 2 pi r. But as r gets larger, the negative curvature of the ambient hyperbolic space begins to make itself felt, and the circumference goes above 2 pi r, by some factor related to sinh. (If it doesn't, you didn't make R big enough -- that is exactly what I meant by "large enough R" above.)
I'm not convinced of this. Why do you think the positive curvature of the sphere won't have a greater effect than the negative curvature of the ambient space? Both effects depend on R, so I don't see why "make R large" guarantees that one is a greater effect than the other. I may be missing something, but I think at least you need to fill i some detail here to sharpen the paradox. Andy
On Monday, February 19, 2018, 1:01:00 PM PST, Andy Latto <andy.latto@pobox.com> wrote: On Mon, Feb 19, 2018 at 2:30 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Watch how the circumference of a circle, centered at P, changes with its radius r. For small r, it looks like 2 pi r. But as r gets larger, the negative curvature of the ambient hyperbolic space begins to make itself felt, and the circumference goes above 2 pi r, by some factor related to sinh. (If it doesn't, you didn't make R big enough -- that is exactly what I meant by "large enough R" above.)
I'm not convinced of this. Why do you think the positive curvature of the sphere won't have a greater effect than the negative curvature of the ambient space? Both effects depend on R, so I don't see why "make R large" guarantees that one is a greater effect than the other. I may be missing something, but I think at least you need to fill i some detail here to sharpen the paradox. Andy The hyperbolic 3-space can be given the metric ds^2 = dr^2 + sinh^2 r (dθ^2 + sin^2 θ dφ^2). The 2-sphere of constant r is a plain ordinary 2-sphere of radius sinh r, and we may ignore that it happens to be embedded in a hyperbolic space. -- Gene
I have been stewing over this for several weeks now, and for some reason I am just not satisfied. The claim that has been made is that the negative curvature of the ambient space is overwhelmed by the positive curvature of the sphere constructed in it. But my difficulty comes from the fact that the negative curvature of the space is fixed at the outset, and we may grow the sphere as much as we want, making its local surface flatter and flatter. It is hard for me to see why this arbitrarily-flattenable surface never gets flat enough to be overwhelmed by the negative curvature of the space in which it is embedded. Let me say this another way. Suppose we embed a _plane_ in the hyperbolic space, defined as the locus of points equidistant from two given points. If we draw circles of increasing radius on that plane, and read off the circumferences, won't we be able to detect the hyperbolicity of the space as circumferences that rise above 2 pi r? And, are not local patches of a sphere better and better approximations to such a plane, as the radius of the sphere increases? Or is it the case that there is some kind of gap, in hyperbolic space, between spheres and planes? On Mon, Feb 19, 2018 at 4:26 PM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
On Monday, February 19, 2018, 1:01:00 PM PST, Andy Latto < andy.latto@pobox.com> wrote:
On Mon, Feb 19, 2018 at 2:30 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Watch how the circumference of a circle, centered at P, changes with its radius r. For small r, it looks like 2 pi r. But as r gets larger, the negative curvature of the ambient hyperbolic space begins to make itself felt, and the circumference goes above 2 pi r, by some factor related to sinh. (If it doesn't, you didn't make R big enough -- that is exactly what I meant by "large enough R" above.)
I'm not convinced of this. Why do you think the positive curvature of the sphere won't have a greater effect than the negative curvature of the ambient space? Both effects depend on R, so I don't see why "make R large" guarantees that one is a greater effect than the other. I may be missing something, but I think at least you need to fill i some detail here to sharpen the paradox.
Andy
The hyperbolic 3-space can be given the metric
ds^2 = dr^2 + sinh^2 r (dθ^2 + sin^2 θ dφ^2).
The 2-sphere of constant r is a plain ordinary 2-sphere of radius sinh r, and we may ignore that it happens to be embedded in a hyperbolic space.
-- Gene
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The 2-sphere has a surface of constant positive curvature even though it's embedded in a space of negative curvature. So your circle isn't on a surface of constant negative curvature. I don't think it's possible to choose a value of R so large that the 3-space negative curvature makes the circumference exceed 2pi*r for all r. The circle must shrink to a point on the opposite of the 2-sphere, just as it grew from a point where started with small values of r. Brent On 2/19/2018 11:30 AM, Allan Wechsler wrote:
The following intuitive line of reasoning leads to a contradictory conclusion, and I don't know what's wrong with it. Obviously one of my assumptions or beliefs about spaces with constant Gaussian curvature is wrong, but I can't figure out which.
Suppose we have a 3-dimensional space of constant, negative Gaussian curvature.
For some large enough R, construct the locus of points at distance R from a fixed center C. This forms a 2-dimensional surface which is topologically an ordinary sphere. Select a point P on this sphere, and begin drawing circles of increasing radius r on the sphere centered at P. I mean r to be measured along the surface of the sphere, not straight through the hyperbolic 3-space in which the sphere is embedded.
Now, this sphere must itself have constant Gaussian curvature; I don't have a fast proof but surely there is a symmetry argument that can establish it.
Watch how the circumference of a circle, centered at P, changes with its radius r. For small r, it looks like 2 pi r. But as r gets larger, the negative curvature of the ambient hyperbolic space begins to make itself felt, and the circumference goes above 2 pi r, by some factor related to sinh. (If it doesn't, you didn't make R big enough -- that is exactly what I meant by "large enough R" above.)
But later, as r becomes commensurate with R, the circumference rises to a maximum and begins to shrink again, because the sphere itself has finite circumference.
On a surface of constant curvature, the circumference should either fall below 2 pi r, or it should exceed it forever. It should not do both.
Surely I am not the first person to think these confusing thoughts. What is wrong with my reasoning? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (6)
-
Adam P. Goucher -
Allan Wechsler -
Andy Latto -
Brent Meeker -
Eugene Salamin -
Gareth McCaughan