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

If the universe expands, what is it expanding into? That’s one of the most common questions I get, followed by “Are we expanding with the universe?”

I haven’t made a video about it yet, as there are already many videos about it. But then I thought if you kept asking, these other videos probably didn’t answer the question. And why is that I suspect it may be because you can’t really understand the answer without knowing at least a little bit about how Einstein’s general theory of relativity works. Hello Albert. Today it’s all about you.

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

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

You are familiar with this image of a sphere, but what you see is not just the sphere. You see a sphere in a three-dimensional space. This three-dimensional space is called “embedding space”. The embedding space itself is flat, it has no curvature. When you embed the sphere, you can immediately see that it is curved. But that’s NOT how it works in general relativity.

In general relativity, we wonder how we can find out the curvature of spacetime while living in it. There is no outside. There is no embedding space. So for the sphere that would mean we would have to wonder how we would find out that it is curved if we lived on the surface, maybe ants crawling around on it.

One way to do this is to remember that the interior angles of triangles in flat space are always 180 degrees. This is no longer the case in a curved space. An extreme example is a triangle that is at right angles to one of the poles of the sphere, goes down to the equator, and closes along the equator. This triangle has three right angles. They add up to 270 degrees. That is simply not possible in a flat space. So when the ant measures these angles, it can tell that it is crawling around on a ball.

There is another way the ant can find out that it is in a curved space. In flat space, the circumference of a circle is related to the radius of 2 Pi R, where R is the radius of the circle. But this relationship does not apply in a curved space either. If our ant crawls a distance R from the pole of the sphere and you then walk around in a circle, the radius of the circle will be less than 2πR. That said, measuring the perimeter is another way to find out if 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 will get the same result as in flat space. And that is absolutely correct. A cylinder has no intrinsic curvature. It is periodic in one direction, but internally it is flat.

General relativity now uses a higher dimensional generalization of this intrinsic curvature. So the curvature of spacetime is fully defined in terms that lie within spacetime. You don’t need to know anything about the embed speed. The space-time curvature is shown in Einstein’s field equations in these quantities called R.

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

That is the only important thing you need to know about general relativity, the curvature of spacetime can be fully defined and measured within spacetime. The other important thing is the word “relativity” in general relativity. This means that you can freely choose a coordinate system, and the choice of a coordinate system does not play a role in predicting measurable quantities.

It’s one of those things that sounds pretty obvious in hindsight. When you make a prediction for a measurement and that prediction depends on an arbitrary choice you made in the calculation, such as: B. Choosing a coordinate system, this is certainly not a good thing. However, it took Albert Einstein to turn this “obvious” insight into a scientific theory, first special relativity and then general relativity.

So with that background, let’s look at the first question. What is the universe expanding into? It doesn’t expand into anything, it just expands. The statement that the universe is expanding, like any other statement we make in general relativity, relates to the internal properties of spacetime. It says, roughly speaking, that the space between the galaxies is expanding. Think back to the sphere and imagine that its radius increases. As we discussed, you can find out by taking measurements on the surface of the sphere. You don’t need to say anything about the embedding space that surrounds the sphere.

You may now ask, but can we embed our 4-dimensional spacetime in a higher-dimensional flat space? The answer is yes. You can do that. It generally takes on 10 dimensions. But one could actually say that the universe is expanding into this higher-dimensional embedding space. However, due to the design, the embedding space is completely invisible, which is why we have no reason to call it real. The scientifically based statement is therefore that the universe does not expand into anything.

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

How do we know the universe is expanding and we are not shrinking? Well, to some extent, that’s a matter of convention. Remember Einstein said that you are free to choose whichever coordinate system you want. So you can use a coordinate system that has scales that expand at exactly the same speed as the universe. Using these you would conclude that the universe is not expanding in these coordinates.

Indeed you can. However, these coordinates do not have a good physical interpretation. That’s because they mix space with time. So you cannot stop at these coordinates. Whenever you move forward in time, you are also moving sideways in space. This is strange and that’s why we don’t use these coordinates.

The statement that the universe is expanding is actually a statement about certain types of observations, especially the redshift of light from distant galaxies, but also a number of other measurements. And these statements are completely independent of which coordinates you have chosen to describe them. However, explaining them by saying that the universe is expanding in that particular coordinate system is an intuitive interpretation.

The two most important things you need to know in order to understand general relativity are, first, that the curvature of spacetime can be fully defined and measured within spacetime. An embedding room is not required. Second, you can choose any coordinate system. It doesn’t change anything in physics.

In summary, general relativity tells us that the universe does not expand into anything, we do not expand with it, and while it could be said that the universe does not expand but we are shrinking, the interpretation doesn’t make much physical sense.