WTF, Universe? The Earth is round, but you are flat

You could have had hyperbolic geometry, but no. Boring.

“What is the geometry of the Universe?”

It turns out that this is a completely reasonable question to ask, it is possible to answer it, and we have answered it. Mind-altering drugs are not even required.

In this 5th post about our weird-as-fuck Universe, we will explore its global curvature. In the last post, I talked about the small curvatures in space-time caused by everything that exists and has mass. (Yep, you too! You are definitely and imperceptibly curving space-time, right now!) Perhaps the weirdest thing about this is that the merger of very dense objects, like black holes or neutron stars, causes the fabric of space-time to ripple like a wave! Earlier posts explored how the Universe is super fucking large, expanding, and a time machine.

I fully acknowledge that none of this makes much sense, but so far it is all definitely, measurably true. In future posts, we will start exploring weird things that we’re not quite sure about, but not yet.

It’s. All about. Space. And time.

To enhance your experience, please look up and play “Space and Time (Phil Asher Mix)” by Soul Drummers on your favorite music service before continuing. Thank you and you’re welcome.

It is perhaps not too hard to think about objects in space having a geometry. Of course, they are all usually spheres… the Sun, the Moon, the Earth, all spheres.

The surface of the Earth is curved, even though it seems flat to us tiny humans who live there. (Credit: NASA)

But of course, the Universe is not an object in space. It is all of space and time and everything in it. It is up and down, forward and backward, left and right, past and future. We can picture the curvature of a 2-dimensional surface in 3-dimensional space, but trying to picture the curvature of 3-dimensional space, much less 4-dimensional space-time, is just super fucking weird. Especially because there is no larger-dimensional space in which the Universe is embedded. The Universe is all there is.

The reason it’s possible for the Universe to have curvature has something to do with Einstein’s general theory of relativity, which I completely and exhaustively explained in the last episode (just kidding! that shit is confusing!) To sum up: space and time don’t make sense separately, so we combine them into space-time. Matter makes this 4-dimensional space-time curve and ripple and, maybe, pokes it with holes.

Well, just as tiny things like galaxies and black holes distort space-time, so too can the combination of lots of stuff give the Universe a global curvature. And we can measure this global curvature by drawing some triangles on the Universe! In school, you might have been taught that the 3 angles of a triangle add up to 180 degrees. While this is not exactly a bold-faced lie, it relies on the assumption that you are drawing your triangles on a flat surface. Draw a triangle on a curved surface instead, and the angles could add up to more or less than 180, depending on the curvature.

Unfortunately, we have to resort to imagining that 4-dimensional space-time is just a 2-dimensional surface and picturing how that surface might curve, because humans are shit at visualizing 4-dimensional curvature. (Don’t feel bad, it’s not just you).

Some metaphors for the geometry of the Universe. Not very good ones, really, but they will have to do. (Credit: NASA WMAP)

If the Universe is positively curved, it would be like a sphere. You could travel in one direction for a VERY LONG TIME and end up back where you started! If it is negatively curved, it would look like a saddle or a Pringle chip. In a universe with this type of hyperbolic geometry, straight lines would diverge away from each other. (Now the next time you are eating Pringles you can mention how they have hyperbolic geometry, for maximum nerd points.) If, on the other hand, the triangles add up to 180 degrees, you are in boring-old flat geometry.

So, where should we draw our triangles on the Universe?

It turns out, our old friend the Cosmic Microwave Background radiation saves the day again. (To recap: the CMB is radiation (i.e. light) that was emitted everywhere in the Universe about 300,000 years after the Big Bang.) You see, some physics happened in the early Universe that left a characteristic scale on the CMB pattern. We know how big this scale is, and we can measure it, too — and the measurement is based on an angle on a triangle almost as big as the entire fucking observable Universe.

The angle we measure from the CMB tells us whether the Universe is cool and hyperbolic, giving a smaller angle (in purple), or dumb and flat, giving a boring angle (in red). (Image Credit: Me and the ESA Planck Mission)

As you may have noticed, I gave away the answer already. The Universe is flat. Not spherical, nor hyperbolic. Boringly flat. Disturbingly flat. So flat, in fact, that we think some weird shit went down right after the Big Bang in order to make the Universe almost unnaturally flat, which is the subject of the next post.

One cool thing we can say about our flat Universe, though, is that it is probably infinite. A positively curved, spherical geometry would have a finite size, which we could determine if we could measure the value of the curvature. But both hyperbolic (negatively curved) and flat (zero curvature) geometries lead to infinite universes. Now, we don’t know for sure that the Universe is infinite because we can’t see past our observable horizon. We would have to have a pretty good reason for space-time to just stop existing, but maybe there’s one out there, just waiting for someone to think of it.