[Robert]

There are three possible futures for our Universe. One is open, where the universe continues to expand forever. Another possibility is closed, where the universe will eventually stop expanding, turn around and collapse in on itself. The third possibility is really a special case of the first, the universe continues to expand but at ever slower speeds approaching zero velocity.

[AntonioLao]

the topology of the universe is that of a double torus with genus=2. It is closed in double one-dimensional spaces but open in all other dimensions 2D, 3D, 4D, etc.

These two 1D spaces cannot be transformed into each other. They are not topologically equivalent. Yet they are linked together in such a way that they are continuously feeding each other existence.

This topology allows each point in the unverse to "see" all the other infinite points. Yet each point will only move in the shortest paths and there are exactly six such shortest paths to choose from. Once a point chooses a path, it will keep moving along this path forever without ever reaching the end of eternity.

These closed paths are quantizable into 2 distinct topologies of linked closed paths.

[subversion]

the universe is open, but it will not expand forever

[Guille]

Who invented the idea of the universe being flat? that is compeltely impossible. Maybe it could be flat but 3d but not flat 2d bcause there are many things in the unvierse and all are 3d. I voted to closed because it would give more security, if it's open, what are the borders? no limit.

[AntonioLao]

Originally Posted by GUILLE Who invented the idea of the universe being flat?

the flatness problem wasn't invented. It's an experimental fact from astronomical observations. At the present epoch of the universe, the equilibrium between gravitational energy and expansion energy is undeniable and incontestable. The spacetime structure of the universe is very nearly Euclidean, and not Riemannian (closed - elliptic), nor Gaussian (open - hyperbolic).

Euclidean spacetime allows the metric components of nonzero values for equal values of index. g11, g22, g33, g44, g55, g66, g77, g88, g99,...And the 2D analoque is the Pythagorean theorem for determining distances.

Note: the flatness problem has been resolved by the inflationary model of the universe. There are many cosmological models in the hopper at the moment. All of them can be based on Einstein's general relativistic field equations with or without the cosmological constants.

[Guille]

Originally Posted by AntonioLao The spacetime structure of the universe is very nearly Euclidean, and not Riemannian (closed - elliptic), nor Gaussian (open - hyperbolic).

What is meant by hyperbolic? and how is math used to calculate it?

Originally Posted by AntonioLao Euclidean spacetime allows the metric components of nonzero values for equal values of index. g11, g22, g33, g44, g55, g66, g77, g88, g99,...And the 2D analoque is the Pythagorean theorem for determining distances.

How is the pythagorean therem used to detemrine distances?

Originally Posted by AntonioLao Note: the flatness problem has been resolved by the inflationary model of the universe.

How has it been solved?

[AntonioLao]

i'm going to try answering these questions one at a time starting with Q1.

hyperbolic geometry is based on the idea that infinite numbers of parallel lines can be drawn on the surface of a hyperboloid (a saddle shape surface). There exists one and only one minimum distance between two points. In 2D, the equation is that of a hyperbola given by



where d is the minimum distance.

For Q2, the Pythagorean theorem is stated as follow



this theorem is true iff a right triangle is defined and Euclid fifth postulate is also true. The distances are calculated by trigonometric circular functions, which are also based on the existence of right triangles and that the sum of interior angles is always exactly 180 degrees.

For Q3:
The inflationary model asserts that extreme curvature and high accelerated expansion happens only during the very early epoch of the universe (this event looks like an inverted delta function). After this accelerated expansion phase, the universe coagulated into stars and galaxies and the global spacetime structure appears homogeneous and isotropic.

[Guille]

Originally Posted by AntonioLao For Q2, the Pythagorean theorem is stated as follow this theorem is true iff a right triangle is defined and Euclid fifth postulate is also true.

1) Thanks for answering.

2) What is the euclid fifth postulate?

[AntonioLao]

Originally Posted by GUILLE 2) What is the euclid fifth postulate?

On the Euclidean plane, given a straight line and a point not on the line, one and only one straight line can be constructed passing throught the given point that is parallel to the given line.

[Guille]

Originally Posted by AntonioLao On the Euclidean plane, given a straight line and a point not on the line, one and only one straight line can be constructed passing throught the given point that is parallel to the given line.

o, true, and that is actually logical.