“When the universe expands, into what is it expanding? In what medium is it expanding? Is the universe like a bubble in a higher dimension something? [Anonymous], Indiana, USA”

Black hole embedding diagram.

Only the surface itself has physical meaning.

The surrounding space is for visual purposes.

[Image source: Quora

This is a very good question and one, I should add, I get frequently. It is, I believe, to no small part caused by the common illustrations of a curved universe: it’s a rubber-sheet with a bowling-ball on it, it’s an inflating balloon, or – in the rarer case that someone tries to illustrate negative curvature, it’s a potato chip (because really I have no idea what a saddle looks like).But in each of these cases what the illustration actually shows is a two-dimensional surface embedded in a non-curved (“flat”) three-dimensional space. That’s good because you can draw it, but it’s bad because it raises the impression that to speak of curvature you need to put the surface into a larger space. That, however, isn’t so: Curvature is a property of the surface itself.To get an idea of how this works, consider the simplest example of a curved surface, a ball. On the ball’s surface the angles of triangles will not add up to 180 degrees. You can calculate the curvature from measuring all the angles in all triangles that you could draw onto the ball. This is a measurement which can be done entirely on the surface itself. Or by ants crawling on the surface, if you wish, to use another common analogy.Curvature, hence, is an intrinsic property of the surface – you do not need the embedding space to define it and to measure it. Also note that the curvature is a local property; it can change from one place to the next, just that a ball has constant curvature.General relativity uses the same notion of local, intrinsic curvature, just that in this case we aren’t dealing with two dimensions of space and ants crawling on it, but with three dimensions of space, one dimension of time, and humans crawling around in it. So the math is more complicated and all the properties of space-time are collected in something called the curvature-tensor, but that is still an entirely internal construct. We can measure it by tracking the motion of particles, and it’s this curvature that creates the effect we usually refer to as gravity.Now, what cosmologists mean when they speak of the expansion of the universe is a trend of certain measurement results that, using Einstein’s equations, can be interpreted as being due to an increasing distance between galaxies. Again, this expansion is an entirely internal notion. It is defined and measured in our universe. You do not have to embed this four dimensional space-time into anything else to quantify it. You do not need a medium and you do not need a larger space. Einstein’s theory is entirely self-contained with a four-dimensional, internally curved space-time.While you do notembed space-time in a higher-dimensional flat space, you can. Indeed it can be mathematically proved that you can embed any curved four dimensional space-time into a ten dimensional flat space-time. The reason physicists don’t normally do this is that these additional dimensions are superfluous and they don’t aid the math either.We do, however, on occasion use what is called an “embedding diagram”, whichcan be useful to visualize the extrinsic curvature of certain slices of space-time. This is, for example, what gives rise to the idea that when matter collapses to a black hole, space develops a long throat with a bubble that eventually pinches off. But please keep in mind that these are merely visual aids. They have their uses as such, but one has to be very careful in interpreting them because they depend on the chosen embedding.Now you ask what does the universe expand into? It doesn’t expand into anything, it just expands. That the universe expands is a statement about what happens inside the universe, supported by measurements inside the universe. It’s an entirely internal notion that does not require us to speak of an outside of the universe or a medium into which it is embedded.Thanks for an interesting question!