What's the Geometry of the Universe?

To me, for the Big Bang to make sense, our universe must be closed. How, for example, can there be no such thing as spacetime at t=0 (no physical space, no time)and at t=1, we have an infinite open universe that expands? It makes more sense to me that our universe is a closed hyperdimensional structure (like a 4-sphere)that has no boundaries but finite space inside which expands inside hyperdimensional hyperspace. However, data from WMAP confirms the flatness of our universe (making it infinite). Could it be that we mesure our universe to be flat only because we aren't able to obtain data about our universe's geometry outside the visible horizon boundary? It is much like to determine wether or not the Earth is flat by analyzing its curvature on a football field. Maybe if we could see beyond the visible universe we would find a different geometry. Guess we will never find out...
- Anonymous

Bravo!  You have put your finger on several deep questions!

First, what is the geometry ("curvature") of the universe, and how do we know it?  Einstein's theory of gravity, General Relativity, allows for the three-dimensional space of the universe to have one of  three possible shapes or geometries:
(a) space can be "positively curved"--like the surface of a expanding balloon
(see http://background.uchicago.edu/~whu/beginners/expansion.html)
[but a 3-D upgrade of this idea!] and with an finite volume, or
(b) space can be "negatively curved"--like the seat of a saddle or a potato chip
[again upgraded to 3-D] and with an infinite volume, or
(c) space can be "flat"--lacking any curvature at all and infinite in volume. 

In each of these situations, geometry is different, meaning that the properties of circles and triangles are different. For example, the familiar Euclidean results about triangle interior angles summing to 180 degrees, or circle areas being pi*r^2, are only true in a "flat" universe and are false in the curved universes!  More discussion and illustration appears here:
 
Thus Einstein teaches that geometry is really a realm of physics, and it is an experimental task to measure which of these is the true geometry of the universe.  And in fact, this experiment has been performed, using observations of the cosmic microwave background, which are described here
http://map.gsfc.nasa.gov/universe/uni_shape.html and http://map.gsfc.nasa.gov/media/030639/index.html and ultimately amount to studying the geometry of huge triangles in the universe!

And the answer is: the universe shows no average curvature but rather has the "flat," Euclidean geometry on a large scale, within the error bars of the measurement.

Another question you ask is:  can this geometry be different elsewhere in the universe?
While in principle this is possible, all of the available evidence agrees with another prediction that goes back to Einstein: the universe is homogeneous, meaning that at
any time its large-scale properties (including geometry) are the same at
all points in space.  Thus if the observable universe (the amount anyone can
see since the big bang) is flat, the the entire universe is as well.

Finally, another question you raise is:   what properties does the universe have
at exactly t=0, the moment of the big bang?  This situation is known as a "singularity"
which is the technical way of saying that the known laws of physics
break down.  This means that nobody knows what happens then!  But
many ideas exist, in which people try to extend known physics to this extreme regime
by inventing theories of quantum gravity.  If you have a successful theory
of quantum gravity yourself, you should publish!

Brian F

(published on 07/04/2011)

When you speak of flat, do you mean flat like a solar system and a galaxy? If so, why the pattern?
- james (age 19)
usa

The meaning of "flat" here is very different from the meaning of "flat" in the case of collections of stars and planets. In the latter case, it just means that things are more or less arranged in a single plane within a 3-D space. For the universe, we're talking about something very different, the geometry of the space itself.

The key point here is Einstein's realization that our space does not have the same geometry as the Euclidean math we studied in school. The sum of the angles of a triangle isn't quite 180°. The circumference of a circle isn't quite pi times the diameter. All these geometrical relations are affected by the distribution of mass within the space. 

That leaves open the possibility that the very large-scale structure of the universe could be curved. One type of curvature would be the 3-D analog (really 4-D if you do the full picture including time) of a spherical shell, something that curves around to meet itself. Any two travelers on the sphere starting at the same spot and continuing "straight" at the same speed in any directions they initially choose will meet up again!
 That's one way that space itself can be curved. There's an opposite way, more like a saddle, in which two travelers who start off nearby and parallel end up getting farther and farther apart.

Is our 3-D space like either of those? There are experiments that can tell. Within the range of distances that can be detected, there's no measured average curvature. There are explanations of why that should be so, but they leave open the question of whether there is some tiny curvature that's too small to have been seen.

Mike W.

(published on 07/13/2011)