Review:
The author presents two most popular methods to determine the shape of our universe. Prior to explaining the principles of those methods, there is an extensive introduction to topology of space, and the possible theoretical shapes of the universe.
View the whole Science Section -->
If the Universe is finite and small enough, we should be able to see "all around" it because the photons might have crossed it once or more times. In such a case, any observer might recognize multiple images of the same light source, although distributed in different directions of the sky and at various redshifts. The main limitation of cosmic crystallography is that the presently available catalogues of observed sources at high redshift are not complete enough to perform convincing tests for topology. But the large and deep surveys (up to redshift z=6), such as the LSST (Large Synoptic Survey Telescope) planned in the next decade, should make such methods applicable.2. The Circles in the Sky method uses the 2D cosmic microwave background (CMB) maps. The last scattering surface from which the CMB is released represents the most distant source of photons in the Universe, and hence the largest scales with which we can probe the topology of the universe. One of the methods is the circles-in-the-sky test. It uses pairs of circles with the same temperature fluctuation pattern. So far, no positive results have been found.
An example of three-dimensional manifold is three-dimensional torus (also called three-torus) is a cube (room), where ceiling is glued to the floor, and front wall to the back wall, whilst the left wall is glued to the right one. In this three-torus if we walk towards the back wall and step into it, we will re-enter our room by the front wall, and so on.
Topology vs. Geometry: the aspect of a surface’s nature that is unaffected by deformation is called the topology of the surface. For example, an eggshell and a ping-pong ball have the same topology. A surface’s geometry consists of those properties that do change when the surface is deformed. An example here is curvature of surface.
Intrinsic vs. Extrinsic Properties: two surfaces have the same intrinsic topology, if the creatures living on the surface cannot (topologically) tell one from the other. Two surfaces have the same extrinsic topology if one can be deformed within three-dimensional space to look like the other. For example, a rubber band and a twisted rubber band (Mobius strip) have the same intrinsic properties. Another example is a sheet of paper and its bent version to make up a half-cylinder – a creature living on the sheet of pare could not detect whether the paper was bent or not
Local vs. Global Properties: local properties are those observables within a small region of the manifold whereas global properties require consideration of the manifold as a whole.
Homogeneous manifold is one whose local geometry is the same at all points. A sphere is a homogenous surface; a doughnut is non-homogenous. A flat torus (a piece of paper that enables to get from the top end to the bottom end, and from left to right, etc.) is homogenous.
A Klein bottle is made up of a square, where bottom left is joined to top right corner, bottom left side with the top right side, etc.
Manifolds that bring traveller back mirror-reversed are called nonoreintable manifolds. The sphere and torus are orientable surfaces. A Klein bottle is a nonorientable surface. A nonorientable three-manifold is different from its orientable version by gluing the front wall to the back wall with side-to-side flip – similar to the Klein bottle.
Spherical triangle – is made up of three great circles. Each great circle divides the sphere in two equal hemispheres. The spherical geometry belongs to elliptic geometry and has positive curvature.
There is an example of how to calculate the area of spherical triangle. Before we do that, we need to introduce spherical lune, which is an area on a sphere bounded by two half great circles which meet at antipodal points.
The whole area of sphere is equal to 4πr². In case of two lunes being the same, the highest angle (in radians) would be π and each of two half-spheres 2πr². If the angle is, say α=π/3, we expect the area of each lune to be 2πr²/3 = 2αr², where a is expressed in radians.
In order to determine an area of spherical triangle, with angles α, β and γ, we construct two lunes for each angle. This will result in the spherical triangle being shaded three times, one for each pair of lunes. All the other lunes will be shaded once only.
We can then add the individual areas, as follows:
1. Area of first double lune with angle α is 4αr²
2. Area of second double lune with angle β is 4βr²
3. Area of third double lune with angle γ is 4γr²
The total of those three above is equal to:
1. Total of all triangles shaded once only, which equals to area of sphere 4πr²
2. 2A – second marking of the triangle when marking lunes for angle β
3. 2A – third marking of the triangle when marking lunes for angle γ
Therefore, if we combine two above operations we can say that:
4αr²+4βr²+4γr²=4πr²+2A+2A, which leads to
A=(α+β+γ-π)r², where each angle is expressed in radians.
View the whole Science Section -->
Notes:
As of 2002, two research projects are underway to measure the shape of space:1. The method of Cosmic Crystallography looks for patterns in the arrangement of the galaxies.If the Universe is finite and small enough, we should be able to see "all around" it because the photons might have crossed it once or more times. In such a case, any observer might recognize multiple images of the same light source, although distributed in different directions of the sky and at various redshifts. The main limitation of cosmic crystallography is that the presently available catalogues of observed sources at high redshift are not complete enough to perform convincing tests for topology. But the large and deep surveys (up to redshift z=6), such as the LSST (Large Synoptic Survey Telescope) planned in the next decade, should make such methods applicable.2. The Circles in the Sky method uses the 2D cosmic microwave background (CMB) maps. The last scattering surface from which the CMB is released represents the most distant source of photons in the Universe, and hence the largest scales with which we can probe the topology of the universe. One of the methods is the circles-in-the-sky test. It uses pairs of circles with the same temperature fluctuation pattern. So far, no positive results have been found.
An example of three-dimensional manifold is three-dimensional torus (also called three-torus) is a cube (room), where ceiling is glued to the floor, and front wall to the back wall, whilst the left wall is glued to the right one. In this three-torus if we walk towards the back wall and step into it, we will re-enter our room by the front wall, and so on.
Topology vs. Geometry: the aspect of a surface’s nature that is unaffected by deformation is called the topology of the surface. For example, an eggshell and a ping-pong ball have the same topology. A surface’s geometry consists of those properties that do change when the surface is deformed. An example here is curvature of surface.
Intrinsic vs. Extrinsic Properties: two surfaces have the same intrinsic topology, if the creatures living on the surface cannot (topologically) tell one from the other. Two surfaces have the same extrinsic topology if one can be deformed within three-dimensional space to look like the other. For example, a rubber band and a twisted rubber band (Mobius strip) have the same intrinsic properties. Another example is a sheet of paper and its bent version to make up a half-cylinder – a creature living on the sheet of pare could not detect whether the paper was bent or not
Local vs. Global Properties: local properties are those observables within a small region of the manifold whereas global properties require consideration of the manifold as a whole.
Homogeneous manifold is one whose local geometry is the same at all points. A sphere is a homogenous surface; a doughnut is non-homogenous. A flat torus (a piece of paper that enables to get from the top end to the bottom end, and from left to right, etc.) is homogenous.
A Klein bottle is made up of a square, where bottom left is joined to top right corner, bottom left side with the top right side, etc.
Manifolds that bring traveller back mirror-reversed are called nonoreintable manifolds. The sphere and torus are orientable surfaces. A Klein bottle is a nonorientable surface. A nonorientable three-manifold is different from its orientable version by gluing the front wall to the back wall with side-to-side flip – similar to the Klein bottle.
Spherical triangle – is made up of three great circles. Each great circle divides the sphere in two equal hemispheres. The spherical geometry belongs to elliptic geometry and has positive curvature.
There is an example of how to calculate the area of spherical triangle. Before we do that, we need to introduce spherical lune, which is an area on a sphere bounded by two half great circles which meet at antipodal points.
The whole area of sphere is equal to 4πr². In case of two lunes being the same, the highest angle (in radians) would be π and each of two half-spheres 2πr². If the angle is, say α=π/3, we expect the area of each lune to be 2πr²/3 = 2αr², where a is expressed in radians.
In order to determine an area of spherical triangle, with angles α, β and γ, we construct two lunes for each angle. This will result in the spherical triangle being shaded three times, one for each pair of lunes. All the other lunes will be shaded once only.
We can then add the individual areas, as follows:
1. Area of first double lune with angle α is 4αr²
2. Area of second double lune with angle β is 4βr²
3. Area of third double lune with angle γ is 4γr²
The total of those three above is equal to:
1. Total of all triangles shaded once only, which equals to area of sphere 4πr²
2. 2A – second marking of the triangle when marking lunes for angle β
3. 2A – third marking of the triangle when marking lunes for angle γ
Therefore, if we combine two above operations we can say that:
4αr²+4βr²+4γr²=4πr²+2A+2A, which leads to
A=(α+β+γ-π)r², where each angle is expressed in radians.
The hyperbolic geometry has negative curvature. An example here is a hyperbolic plane.
No comments:
Post a Comment