© 2005 Pearson Prentice Hall, Inc

The graph on the left shows the trajectory of an object that is launched upward and comes back down. The graph on the right shows an object that escapes. Notice that the curve becomes a straight line. The object just keeps going, never going back to the axis.

As these diagrams depict, a flat universe would increase in size, and then stop increasing. A closed universe would increase in size and then decrease. An open universe would keep increasing in size.

The fate of the universe involves a concept similar to escape speed, and it also depends on the initial velocity and the density of the early universe. If the initial speed of the matter of the universe was low enough, and mass of the universe was big enough (at a small size) and the the matter in the universe would continue to move apart for awhile, than fall back inward, ending in a big crunch. We call that a closed universe.

If, say, the initial speed was the same but the density of matter was much higher, the particles would all get away. We call that an open universe. A flat universe would happen right at critical density. The matter would be moving at escape speed.

© 2005 Pearson Prentice Hall, Inc

This graph shows the curves for a bound universe (closed), a marginally bound universe (flat) and an unbound universe (open). Notice that our time is shown sometime after the "launch." At the present time, all we can do is observe what we see now. The amount of time that has passed since the big bang, in our graph, depends on whether the universe is bound, marginally bound or unbound.

These graph show another way of visualizing the fate of the universe. For a long time, the debate focused on whether or not the early universe was dense enough for the universe to be closed. No one thought that the last model was really even a possibility, that the expansion of the universe would accelerate. That would be like throwing a ball upward, only to see that not only did it not fall back down, it started moving faster and faster.

To everyone's surprise, the last model turned out to be true, at least to the best understanding of the data that we have now. The universe is accelerating its expansion. Ordinary matter and dark matter would not make the universe behave this way. This brings us to assume that there must be something else making up part of the universe, causing the expansion to increase. We don't know what that would be, but we can give it a name. We call it dark energy.

- Spacetime is curved
- The degree of curvature depends on density

The main idea of general relativity is that the density of matter curves spacetime. The denser the matter, the stronger the curvature. Also, the curvature of spacetime tells particles how to move. The stronger the curvature of spacetime, the more curved the paths of the particles.

The mass of the particles also adds to the density of matter. The curvature of spacetime tells matter how to to move. The density of matter tells spacetime how to curve. The particles affect the curvature, which affects the particles, which affects the curvature, which affects the particles, which affects the curvature,...

In classical Newtonian physics, light is not affected by gravity. In general relativity, photons follow curved paths in curved spacetime, even though they have no mass.

Einstein showed that there is an equivalence between matter and energy. Since E = mc2, we need to consider the contribution of energy to the curvature of spacetime. To be more accurate, we need to say that the curvature of spacetime tells matter/light how to to move. The density of matter/energy tells spacetime how to curve.

The geometry of the universe is really just a way of discussing the curvature of spacetime in the universe. The geometry of the universe depends on the density of matter/energy. The critical density is defined as the density of the early universe that would produce just enough gravity for the universe to collapse back together in a big crunch.

© 2005 Pearson Prentice Hall, Inc

In flat space, the sum of the angles inside a triangle equal 180 degrees. In space that has positive curvature, like on the surface of a sphere, the angles of a triangle add up to more than 180 degrees. On a negatively curved space, like this saddle-shape, the angles of a triangle add up to less than 180 degrees.

Notice that it is not possible for the edges of the negatively curved space to close together into a single surface. It would be possible, though, for a closed surface to have negative over a small region, as long as the space had more positive curvature than negative curvature. In other words, space could be locally negative, but still have overall positive curvature. Also, space could average out to be flat overall, with local regions of positive and negative curvature.

If we turn back the clock, the receding galaxies would rush back together, joining in the beginning of the universe. What happens if we fast-forward the clock? How does it all end? Or does it?

Let's consider the escape speed of the universe.

The escape speed of a massive object from a planet depends on the mass and radius of the planet. The more massive the planet, the faster the object has to go to get away. The smaller the radius, the faster the object has to go to get away.

On earth, if you throw a ball upward with an initial speed of, say 10 m/s, it will go about 5 meters high before falling back down (neglecting air resistance). If you throw it upward at 20 m/s, it will go about 20 meters up, and then fall back down. At an initial speed of 100 m/s, it will stop at about 500 meters high before falling back down. If you launched it upward with an initial speed of 11,700 m/s, the ball would stop at infinity and fall back down. If it was moving faster when it launched, it would not fall back down. It would get away. 11,700 m/s is escape speed for the surface of the earth.

For a planet with the same size as Earth, but a bigger mass, (a higher density) the escape speed would be higher.