*[This is a transcript of the video embedded below.]*

If the universe expands, what does it expand into? That’s one of the most frequent questions I get, followed by “Do we expand with the universe?” And “Could it be that the universe doesn’t expand but we shrink?” At the end of this video, you’ll know the answers.

I haven’t made a video about this so far, because there are already lots of videos about it. But then I was thinking, if you keep asking, those other videos probably didn’t answer the question. And why is that? I am guessing it may be because one can’t really understand the answer without knowing at least a little bit about how Einstein’s theory of general relativity works. Hi Albert. Today is all about you.

So here’s that little bit you need to know about General Relativity. First of all, Einstein used from special relativity that time is a dimension, so we really live in a four dimensional space-time with one dimension of time and three dimensions of space.

Without general relativity, space-time is flat, like a sheet of paper. With general relativity, it can curve. But what is curvature? That’s the key to understanding space-time. To see what it means for space-time to curve, let us start with the simplest example, a two-dimensional sphere, no time, just space.

That image of a sphere is familiar to you, but really what you see isn’t just the sphere. You see a sphere in a three dimensional space. That three dimensional space is called the “embedding space”. The embedding space itself is flat, it doesn’t have curvature. If you embed the sphere, you immediately see that it’s curved. But that’s NOT how it works in general relativity.

In general relativity we are asking how we can find out what the curvature of space-time is, while living inside it. There’s no outside. There’s no embedding space. So, for the sphere that’d mean, we’d have to ask how’d we find out it’s curved if we were living on the surface, maybe ants crawling around on it.

One way to do it is to remember that in flat space the inner angles of triangles always sum to 180 degrees. In a curved space, that’s no longer the case. An extreme example is to take a triangle that has a right angle at one of the poles of the sphere, goes down to the equator, and closes along the equator. This triangle has three right angles. They sum to 270 degrees. That just isn’t possible in flat space. So if the ant measures those angles, it can tell it’s crawling around on a sphere.

There is another way that ant can figure out it’s in a curved space. In flat space, the circumference of a circle is related to the radius by 2 Pi R, where R is the radius of the circle. But that relation too doesn’t hold in a curved space. If our ant crawls a distance R from the pole of the sphere and you then goes around in a circle, the radius of the circle will be less than 2πR. This means, measuring the circumference is another way to find out the surface is curved without knowing anything about the embedding space.

By the way, if you try these two methods for a cylinder instead of a sphere you’ll get the same result as in flat space. And that’s entirely correct. A cylinder has no intrinsic curvature. It’s periodic in one direction, but it’s internally flat.

General Relativity now uses a higher dimensional generalization of this intrinsic curvature. So, the curvature of space-time is defined entirely in terms which are internal to the space-time. You don’t need to know anything about the embedding pace. The space-time curvature shows up in Einstein’s field equations in these quantities called R.

Roughly speaking, to calculate those, you take all the angles of all possible triangles in all orientations at all points. From that you can construct an object called the curvature tensor that tells you exactly how space-time curves where, how strong, and into which direction. The things in Einstein’s field equations are sums over that curvature tensor.

That’s the one important thing you need to know about General Relativity, the curvature of space-time can be defined and measured entirely inside of space-time. The other important thing is the word “relativity” in General Relativity. That means you are free to choose a coordinate system, and the choice of a coordinate system doesn’t make any difference for the prediction of measurable quantities.

It’s one of these things that sounds rather obvious in hindsight. Certainly if you make a prediction for a measurement and that prediction depends on an arbitrary choice you made in the calculation, like choosing a coordinate system, then that’s no good. However, it took Albert Einstein to convert that “obvious” insight into a scientific theory, first special relativity and then, general relativity.

So with that background knowledge, let us then look at the first question. What does the universe expand into? It doesn’t expand into anything, it just expands. The statement that the universe expands is, as any other statement that we make in general relativity, about the internal properties of space-time. It says, loosely speaking, that the space between galaxies stretches. Think back of the sphere and imagine its radius increases. As we discussed, you can figure that out by making measurements on the surface of the sphere. You don’t need to say anything about the embedding space surrounding the sphere.

Now you may ask, but can we embed our 4 dimensional space-time in a higher dimensional flat space? The answer is yes. You can do that. It takes in general 10 dimensions. But you could indeed say the universe is expanding into that higher dimensional embedding space. However, the embedding space is by construction entirely unobservable, which is why we have no rationale to say it’s real. The scientifically sound statement is therefore that the universe doesn’t expand into anything.

Do we expand with the universe? No, we don’t. Indeed, it’s not only that we don’t expand, but galaxies don’t expand either. It’s because they are held together by their own gravitational pull. They are “gravitationally bound”, as physicists say. The pull that comes from the expansion is just too weak. The same goes for solar systems and planet. And atoms are held together by much stronger forces, so atoms in intergalactic space also don’t expand. It’s only the space between them that expands.

How do we know that the universe expands and it’s not that we shrink? Well, to some extent that’s a matter of convention. Remember that Einstein says you are free to choose whatever coordinate system you like. So you can use a coordinate system that has yardsticks which expand at exactly the same rate as the universe. If you use those, you’d conclude the universe doesn’t expand in those coordinates.

You can indeed do that. However, those coordinates have no good physical interpretation. That’s because they will mix space with time. So in those coordinates, you can’t stand still. Whenever you move forward in time, you also move sideward in space. That’s weird and it’s why we don’t use those coordinates.

The statement that the universe expands is really a statement about certain types of observations, notably the redshift of light from distant galaxies, but also a number of other measurements. And those statements are entirely independent on just what coordinates you chose to describe them. However, explaining them by saying the universe expands in this particular coordinate system is an intuitive interpretation.

So, the two most important things you need to know to make sense of General Relativity is first that the curvature of space-time can be defined and measured entirely within space-time. An embedding space is unnecessary. And second, you are free to choose whatever coordinate system you like. It doesn’t change the physics.

In summary: General Relativity tells us that the universe doesn’t expand into anything, we don’t expand with it, and while you could say that the universe doesn’t expand but we shrink that interpretation doesn’t make a lot of physical sense.