Apologies for the (hopefully now somewhat less) clickbait-y title. Now, of course, I know that the Big Bang did not happen at any point connected to a single point in our current $3$-dimensional observable universe by a one-dimensional causal curve. I also know that at any point in the universe, all other points seem to be moving away from that point. However, according to our current understanding of physics, the universe is (at least) $4$-dimensional. Just like how in the classical "balloon" analogy for an expanding universe, the points do in fact all move away from a common point on the interior of the balloon, all spacetime points do move away from the Big Bang, or at least some kind of cosmological horizon which surrounds it – this is how I understand going forward in time, at least. Does it make sense to think of this as a sort of "center" for the full, $4$-dimensional spacetime? Or are there further subtleties to this situation?
If we consider the universe to be four-dimensional, does the Big Bang lie in its center
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Best Answer
There are two important points in here. The first was raised in a previous answer (and I'll comment on it also), but there is another nuance.
The Big Bang is not a point in spacetime
Yeah, that's pretty much it. The Big Bang is a singularity. The precise definition of a singularity is essentially that it is a "hole" in spacetime (you can see the rigorous definitions on the books by Wald or Hawking & Ellis). In the particular case of the FLRW model, the Big Bang singularity also involves curvature scalars blowing up. Now, any point in the manifold must have well-defined metric and curvatures. It is not possible for the curvatures to blow up at any point: that would mean the manifold itself stops making sense at that "point", and hence it does not belong to the manifold.
This is very different from electromagnetism, for example, in which the electromagnetic field can blow up at a point. That is because the field is defined on top of spacetime and only the field is blowing up. In general relativity, a curvature singularity means spacetime itself is blowing up, and there is no physically or mathematically meaningful way of considering that point as a point in spacetime.
The universe is not embedded in anything
This point was raised in another answer, and I'll try to address it with different words in here. I recently entered a discussion on Twitter about a SciComm article mentioning the universe might have the shape of a torus (a doughnut, in layman's terms), and a lot of confusion arose due to the question of "what lies in the empty space in the middle of the torus?". I can't help but notice the similarity of your balloon question with this question. I will discuss the torus example, because it has a wonderfully simple analogy that becomes a bit more intricate in the balloon case.
Both the balloon and our day-to-day experience of a torus are what we call embedded manifolds. I don't like the term embedding: in Portuguese we say "diving" (literal translation), which I think is more pictorial. You "dive" the space you are considering inside a larger space, just like a swimmer dives inside a pool. My point being that your pictorial vision of the balloon or of the doughnut is not of the object in and of itself, but rather of how it is embedded/inserted/depicted inside three-dimensional space. You only need two dimensions to describe the surface of the balloon or of the doughnut, but you add an extra one for the sake of visualization.
This is fine, and sometimes useful and interesting, but it has a problem: it introduces features that are due to the embedding, and not due to the space itself. When you look at the balloon, you are not only thinking about the balloon, but also about the way you inserted it in three-dimensional space. It is hard to separate which properties are due to the balloon and which are due to the embedding.
Now here's the reason I wanted to bring up the doughnut/torus: there is a beautiful visualization of a torus that does not depend on any particular embedding, and which is familiar to many people. Namely, the Pacman world. Pacman lives in a torus because when you go up the screen, you reappear at the bottom, and similar with the sides. Mathematically, these properties are (roughly speaking) what actually defines a torus. When we discuss spacetime, we are interested in these particular properties, not on how spacetime is embedded in a larger-dimensional space.
Now, with the Pacman example in mind, I ask you: what is in the "middle" of the torus? The answer is nothing. In fact, one could even question whether the question makes sense at all: there is no "middle". The "middle" only exists in your embedding, it is not a property of the torus.
What you found is a problem with the balloon visualization. It makes it seem as if there was some kind of center, but there isn't. That is a feature of the embedding, not of spacetime. If you think of spacetime in and of itself, the question doesn't even makes sense.
This is an important nuance in differential geometry and relativity. We always take an intrinsic point of view on spacetime, because it doesn't really make physical sense to imagine in what spacetime is embedded: we can't access this abstract region, so for all physical purposes it isn't real (and it is not unique mathematically, so there is no mathematical reason to expect it exists either). Without this embedding, the question disappears.