Astronomers sometimes say that an open universe has the geometry of a sphere, and a closed universe has the geometry of a saddle. What do they mean by that?
The sphere and the saddle are models to show that plane geometry (with the circumference of a circle exactly equal to the diameter of the circle times the number π (pi), and parallel lines never crossing) is not the only geometry that a universe might have, and that you can figure out information about the "shape" of the universe in a higher dimension than you can observe.
For our Universe, the sphere and the saddle don't relate to the three-dimensional shape of the Universe but to the four-dimensional shape.
Imagine creatures that live in only two dimensions, with length and width but no height. They can see their two-dimensional universe, but nothing outside of that. If the universe in which those creatures live is flat if you could see it in three dimensions (like a sheet of paper), then the circumference of a circle in their universe will always be equal to the diameter of that circle times the number π, and the sum of the three angles in a triangle will always be equal to 180 degrees. The geometry that applies in that case is plane geometry.
If their universe weren't flat but has the shape of a sphere when seen in three dimensions, then straight lines that are parallel somewhere can yet cross each other, and the circumference of a circle in that universe will always be less than π times the diameter, and the sum of the three angles in a triangle will always be more than 180 degrees.
Things work the same way on the two-dimensional surface of the Earth. The equator of the Earth is a circle that is everywhere equally far from the North Pole, so as seen in the two-dimensional surface of the Earth the North Pole is the center of the equator, and the equator is a circle around the North Pole.
To go from the equator along a straight line via the North Pole to the equator on the other side, you must travel half of the circumference of the Earth, so the diameter of the equator, measured on the two-dimensional surface of the Earth, is equal to half of the circumference. The length of the equator is equal to the circumference, so the circumference of the equator is equal to two times the diameter, which is less than π times the diameter (because π is approximately 3.14).
The ratio is closer to π for circles that are closer aroudn the North Pole than the equator is, and the ratio is even smaller than 2 for circles that are further away from the North Pole than the equator is.
If you walk from the North Pole to the equator along a meridian, then turn 90 degrees to the right, walk some distance along the equator, turn 90 degrees to the right again, and walk along a meridian again, then you end up at the North Pole again and have traveled along a triangle on the spherical surface of the Earth. To get back to the meridian that you started on at the beginning of your journey, you need to rotate through an angle that is related to how far you walked along the equator. The sum of the three angles in the triangle is then equal to 90 degrees + 90 degrees + the size of the stretch along the equator, so that is always more than 180 degrees. For smaller triangles, the sum of the angles will be closer to 180 degrees.
The geometry that goes with this is the spherical geometry. That there is less circumference (for a given diameter) in spherical geometry than in plane geometry is also the reason for what you see when you squash something that is spherical (such as the part of an egg carton that holds one egg): then the squashed thing gets tears that get wider the further you go from the center. Those tears occur because circles in that thing need less circumference than circles on a table do.
If their universe as seen from three dimensions does not have the shape of a sphere but of a saddle, then the circumference of a circle is always more than π times the diameter, and the sum of the three angles in a triangle is always less than 180 degrees. If you draw a circle around the center of such a saddle then that circle goes up a bit over here and down a bit over there, so it takes a longer path than a flat circle would, so it has a greater circumference than a flat circle has. If you squash such a saddle then it doesn't get torn but rather gets folds because there is more material than is needed to cover a flat area. The geometry that goes with this is the hyperbolic geometry.
If their two-dimensional universe as seen from three dimensions has the shape of a sheet of paper, then the circumference of a circle in that universe is equal to π times the diameter, and the rules of plane geometry hold. You can lay such a sheet flat without getting tears or folds, but you can also bend such a sheet until it looks like a letter "U" when seen from the side. That makes no difference for the kind of geometry that holds inside the sheet: that remains flat. That also means that two-dimensional creatures in a two-dimensional universe cannot figure out from measurements or observations inside that universe whether their universe as seen in three dimensions is flat or bent in the shape of a letter "U".
The important point is that from measurements and observations only inside your own universe you can figure out what kind of geometry holds in that universe: spherical, plane, or hyperbolic. Two-dimensional creatures can do that on their sphere or saddle, and us three-dimensional creatures can do that in our universe. We can (in principle) measure whether our universe is flat, but not if it is bent in higher dimensions like a two-dimensional sheet of paper can be bent in three dimensions.
That the geometry near a particular location looks like that of a sphere, a saddle, or a sheet of paper does not mean that the geometry has to be like that everywhere. It might even be that the geometry of our universe looks like that of a sphere in some places, and looks like that of a saddle in other places.
And, for all kinds of geometries, if the circle is sufficiently small (compared to the typical scale of the universe), then the circumference of a circle will be so close to π times the diameter (as for plane geometry) that you cannot detect the difference, and then the sum of the angles in a triangle will be so close to 180 degrees that you cannot detect the difference. At sufficiently small scales, you can use the rules of plane geometry.
How much the geometry deviates from plane geometry is indicated by the curvature or radius of curvature of the space. If the curvature of the two-dimensional space is like that of a sphere, then the radius of the sphere that fits the best at that location is the radius of curvature of space at that location. (That radius can be different in different directions, but let's forget about that for now.) The curvature is one divided by the radius of curvature, because the smaller the sphere, the greater the curvature.
For plane geometry, the curvature is equal to zero and the radius of curvature is infinitely great. How much the circumference of a circle differs from π times the diameter, and how much the sum of the three angles in a triangle differs from 180 degrees, depends on how large that circle is compared to the radius of curvature of space. Turning this argument around, you can figure out the curvature of space from the diameter and circumference of circles, and from the size and the sum of angles of a triangle. For plane geometry the curvature is equal to zero, for spherical geometry (like on a sphere) it is positive, and for hyperbolic curvature (like on a saddle) it is negative. You can figure out the curvature in three or more dimensions in the same way.
Why is an open universe like a saddle and a closed universe like a sphere? With the preceding explanation, this means that in an open universe the curvature of space is negative and the geometry is that of a saddle (hyperbolic), and in a closed universe the curvature of space is positive and the geometry is that of a sphere (spherical). You can take that for a definition: a universe is called open if it has hyperbolic geometry, and closed if it has spherical geometry. I guess that a universe with the geometry of a sphere is called closed because a sphere is closed, and a universe with the geometry of a saddle is called open because a saddle surface is not closed.
One might invent other definitions of "open" and "closed", for example that a universe with a finite volume is closed, and a universe with an infinite volume is open. I don't know if a universe that is open according to one of these definitions of "open" must also be open according to the other one, and likewise for "closed".
In our universe the curvature of space is very small, and the curvature radius therefore very large ― greater than the size of the visible universe itself. The curvature is so close to zero (and the geometry so close to plane geometry) that we cannot yet be certain if the curvature is positive or negative.
More Universe Wallpapers in FullHD
Niciun comentariu:
Trimiteți un comentariu