In her excellent blog,
Backreaction, phenomenologist Sabine Hossenfelder laments the difficulty of
finding a precise answer to the question, “What
is a Singularity?” She talks about several kinds of mathematical and
cosmological singularities and attempts to clear up the confusion about them.
Singularities are of concern
in our present discussion, namely two kinds of cosmological singularity. Both
arise in Einstein’s theory of general relativity. One type is the singularity
that is thought to exist at the center of a black hole, where a star has
collapsed to a single point of infinite mass density because its gravity has
overpowered its internal pressure. Sabine is of the opinion that these points
don’t exist, and expresses the hope that the eagerly sought theory of quantum
gravity will show that. We’ll eventually see that that’s correct. When you have
the correct spacetime model, which is our discrete model rather than general
relativity’s continuous one, it becomes obvious that it’s impossible for a
bunch of matter particles to collapse to a single point. But before we get
there we have to do a lot more spacetime physics
The second cosmological
singularity is the big bang, which is
thought to have occurred when a point of infinite mass density exploded to
begin our expanding universe. We already know something about this one from my last
post, although I didn’t call it the big bang there. I showed how a single self-thinking thought,
atemporal existence, logically implies an infinity of thoughts that occurs in
steps, each step taking N thoughts to 2N – 1. We get an expanding
spacetime simply by thinking of these thoughts as points and the steps as time
ticks. When you consider that the most likely number for the amount of time
represented by each time tick is the Planck time, 10‑43 second, you
realize how unbelievably fast this expansion is. It’s truly an explosion, but
not of an infinitely dense particle. There aren’t any particles yet.
So now we have the big bang.
When do we get particles? At the end of inflation. What’s that? It’s this
unbelievably fast expansion in the number of points in the universe. In the
standard cosmology, inflation is driven by a scalar field that comes out of the
big bang with potential energy higher than the minimum possible. This results
in a state called a false vacuum, in which the vacuum, or empty space, appears
to have an energy density. This exerts a gravitational pressure that is
negative, or repulsive, causing space to expand exponentially. As the scalar field drives the expansion it
loses potential energy, eventually arriving at a minimum, where it oscillates,
leaving space oscillating in size. The energy in these oscillations decays, and
voila!, particles.
That’s the standard picture.
Its discovery by Alan Guth in 1981 answered many of the then-existing puzzles
about the early universe, but it has also led to some bizarre conclusions, the
most bizarre being what’s called the multiverse. For various reasons, you can’t
just have a single point where inflation starts; you have to have lots of such
points, so the universe consists of lots of inflating bubble universes, each
having different physical parameters. Our universe is just one of those bubbles
that happens to support life. Of course, there’s no way to know if all those
bubble universes are really there, nor is there any way to explain why our
universe has the physical parameters it does.
Now let’s follow our
inflating universe of spacetime points and see if we can do better. We’ve got
two fields of points, fermionic and bosonic, and they’re coupled. The bosonic
points seek the same quantum state, and because they’re coupled to the
fermionic points, the fermionic points also seek the same quantum state. That’s
fine for a while, but since no two fermionic points can ever be in the same
quantum state (this is called the Pauli
exclusion principle), it’s got to end eventually. Let’s see how.
In the beginning, there are
only a few points. They have random position quantum numbers, so they’re very
likely quite far apart. But they’re herded together by the law of spin and
statistics, and there are more of them all the time, so the universe quickly
becomes dense with points. The universe is actually contracting at this point. Soon, the fermionic points begin to feel uncomfortable
because they can’t have the same position quantum number as any other fermionic
point. What they feel is called degeneracy
pressure. It becomes so intense that the fermionic points refuse to get any
closer together, while the bosonic points are still pushing them towards each
other. There’s only one way out—inflation has to end. The exclusion principle
suppresses the probability that a new point is a fermion because there isn’t
any more room. But the probability doesn’t go to zero, since the fluctuations
in the points’ positions will always leave a hole here and there that’s big
enough for a new fermionic point. So the expansion continues, but at a snail’s
pace compared to the original frantic pace. So in our spacetime we have
inflation and we have an ever expanding universe, but we don’t have a
multiverse. Inflation actually begins with the universe contracting, having started with just a few points possibly
infinitely far apart. At the end of inflation there are a zillion fermionic
points that are as close together as they can be, and they’re held in this
discrete lattice configuration by a bosonic point field that wants to continue
contracting but can’t because it’s coupled to the fermionic point field. The
positions of all points still fluctuate, but the fermionic points can’t move
very far without running into another point, which is forbidden.
The most likely average
separation between fermionic points at the end of inflation is the Planck
length, 10-33 cm. This is a huge difference from the standard
picture. Our universe starts out really big while the standard universe starts
out really small, even smaller than the Planck length. That’s a problem for the standard picture because
physics is thought to break down below the Planck length, so nobody can explain
how structure comes out of something that has none. In our picture the Planck
length is established only at the end
of inflation. Notice, too, that there are no bubbles here, no multiverse.
Both pictures of inflation,
ours and the standard one, end up at the same place. The universe has some
finite size from which it continues to expand slowly. Its size is oscillating
as the degeneracy pressure dukes it out with the bosonic attraction. Because
this is a quantum process, there are small variations in density that
eventually grow into large-scale structure under the influence of gravity. And
if you’re thinking that the bosonic attraction looks like gravity, you’re
absolutely right; that’s exactly what it is. The force of gravity is the
statistical effect of the law of spin and statistics as the bosonic points seek
the same position quantum number. Later we’ll see that this same attraction
applied to the other quantum numbers, time and spin, gives us the
electromagnetic and weak forces. That’s why I consider the law of spin and
statistics to be the fundamental “force” in the universe.
