Re: Why does the no boundary proposal mean no singularity?

Believe it or not, the more important question is actually why there must be a singularity in the "normal" universe! It turns out that the no boundary proposal doesn't exclude the existence of singularities (in fact, they will still appear in black holes), and some recent work work has shown that cosmological singularities can appear in solutions to the no boundary problem. The crucial point is rather that, due to some theorems proved by Stephen Hawking and Roger Penrose in the late 1960s, without the no boundary hypothesis the universe must have begun with a singularity. Many physicists don't like that because general relativity cannot describe areas close to singularities, and any theory that predicts areas where its rules don't apply is not self-consistent.

Proving that the Big Bang theory must contain a singularity is not trivial, to say the least. The essential point (which I will attempt to describe in non-mathematical terms) is to note that a "singularity" is basically equivalent to "incomplete geodesics." A geodesic is simply the analog to a straight line in curved space. There are many equivalent definitions, but perhaps the simplest is that geodesics either minimize or maximize the length between two given points. For example, on a sphere, great circles are geodesics - that's why airplanes fly along great circles! In general relativity, particles and photons all move along geodesics. This is why general relativity is considered to be so beautiful - in some sense, particles all take the "simplest possible path." Note that in the case of general relativity, the geodesics go through space and time.

If a geodesic approaches a singularity, it tends to get trapped. This is easiest to see with the singularity associated with the center of a black hole. As particles approach the black hole, they get trapped and pulled into the center, never to escape. The geodesic "ends" at the singularity, rather than continuing forward forever; this is why it is called "incomplete."

This makes it fairly easy to see why the Big Bang must have been a singularity: time "began" then, so all geodesics must come to an end in time at that point. Therefore the Big Bang is a singularity. (I must warn you that this is not mathematically rigorous, it is just the most straighforward way to describe the essentials. The theorems of Hawking and Penrose that actually prove that the Big Bang was a singularity are extremely complicated and technical - if you want to take a glance at them in all of their glory, try The Large Scale Structure of Spacetime by Hawking and Ellis).

The no boundary proposal weasels out of this problem by simply saying that there is no boundary! Therefore the geodesics don't have to end at the Big Bang...they sort of pass through it and come back out the other side (whatever that means). Think of a great circle that passes through the South Pole - it needn't end on the South Pole, even though we sometimes think of that as the "edge" of the earth. You get rid of the boundary by "analytically continuing" time into the imaginary domain. Again, this is a highly mathematical technique, but it is not quite so weird as it sounds.

For more information, check out Hawking's books and try searching our archives for other information on the
no boundary proposal
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