You remember from all of our discussion of black holes, that space is curved, locally by the presence of mass, and that gravity can be thought of simply as 'bent space'. But it's also true that the Universe

- Positive curvature
- Zero curvature (flat)
- Negative curvature

- Positive Curvature In 2-D, this is like the surface of a balloon. If you are confined to the surface of the balloon, so that you are a balloon creature, living in 2-D, you can go up-down, or right-left, but in-out of the balloon makes NO SENSE to you. You can't even imagine it.

If the balloon is sufficiently large, and you are sufficiently small, it looks flat. (This is what happened to all those 'flat-Earth' people in Columbus' time. They thought the Earth was flat, because it looked that way. A person is small compared to the size of the Earth.)

This positively curved surface has several properties. It is finite, that is, not infinite. It has no boundary (you can go around and around on a ball with out ever running into a wall), and has no center. (Think about it this way---if there is no boundary, there CAN'T be a center---you define the center of a room by looking at where the walls are.

In a positively curved Universe, the rules of geometry are different than the ones you learned in grade school and high school. First, there are no straight lines. Any two parallel lines will converge at a point (like lines of longitude on the Earth, which are parallel at the equator, but come to a point at the North Pole). The sum of the angles of a triangle in curved space is**greater than 180 degrees**, and the circumference of a circle is**less than 2R**. By measuring these properties, a 2-D creature might be able to figure out that they lived in a positively curved, 2-D space.

So a good description of a 3-D positively curved Universe would be that it's finite, unbounded, has no center, and is expanding (from the Hubble Law). - Zero Curvature In 2-D, you want to imagine an infinite sheet of paper, extending off in all directions. A flat, 2-D space is infinite, unbounded and has no center. The geometry in a flat space is the kind of geomtry that you are used to. Parallel lines stay parallel, the angles of a triangle add up to 180 degrees, and the circumference of a circle is equal to 2R.
- Negative Curvature You can't even draw or imagine a 2-D negatively curved space, but on small scales, it looks like a 'Pringle'. A negatively curved space is infinite, has no boundary, and no center. Parallel lines diverge, the angles of a triangle add up to
**less than 180 degrees**, and the circumference of a circle is**less than 2R**.

If

If

If

If

This is the way the world ends;

This is the way the world ends;

Not with a bang, but with a whimper" Age of Universe The age of the universe is 1/H ONLY if the Universe has been expanding at the same rate the whole time. If the Universe is slowing down (which it