# Tag Archives: cosmology

## Flat Space of the Torus

Flat Earth!

To describe how space could be flat, finite, and yet unbounded, science writers sometimes use an analogy involving the surface of a torus (the mathematical abstraction of the doughnut shape). Such a surface has no boundary — no edge.  And despite being embedded in three-dimensional space, the torus surface, if seen in terms of compensating surface metric, is indeed flat.

Yet a natural issue people have is that the three-dimensional embedding is clearly curved, not flat. It’s easy to see how wrapping a flat 2D sheet into a cylinder doesn’t distort it, but hard to see why wrapping a cylinder around a torus doesn’t stretch the outside and compress the inside.

In fact it does, but there are ways to eat our cake (doughnut).

## Hard Problems

Among those who study the human mind and consciousness, there is what is termed “The Hard Problem.” It is in contrast to, and qualitatively different from, problems that are merely hard. (Simply put, The Hard Problem is the question of how subjective experience arises from the physical mechanism of the brain.)

This post isn’t about that at all. It’s not even about the human mind (or about politics). This post is about good old fundamental physics. That is to say, basic reality. Some time ago, a friend asked me what was missing from our picture of physics. This is, in part, my answer.

There is quite a bit, as it turns out, and it’s something I like to remind myself of from time to time, so I made a list.

## Entropy and Cosmology

Last time I started talking about entropy and a puzzle it presents in cosmology. To understand the puzzle we have to understand entropy, which is a crucial part of our view of physics. In fact, we consider entropy to be a (statistical) law about the behavior of reality. That law says: Entropy always increases.

There are some nuances to this, though. For example we can decrease entropy in a system by expending energy. But expending that energy increases the entropy in some other system. Overall, entropy does always increase.

This time we’ll see how Roger Penrose, in his 2010 book Cycles of Time, addresses the puzzle entropy creates in cosmology.

## The Puzzle of Entropy

I’ve been chiseling away at Cycles of Time (2010), by Roger Penrose. I say “chiseling away,” because Penrose’s books are dense and not for the fainthearted. It took me three years to fully absorb his The Emperor’s New Mind (1986). Penrose isn’t afraid to throw tensors or Weyl curvatures at readers.

This is a library book, so I’m a little time constrained. I won’t get into Penrose’s main thesis, something he calls conformal cyclic cosmology (CCC). As the name suggests, it’s a theory about a repeating universe.

What caught my attention was his exploration of entropy and the perception our universe must have started with extremely low entropy.