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.