Methinks it works like how a 2D space (a flat sheet of paper) bends in 3D space and leaves behind 3D space. — TheMadFool
I don't think physicists actually think that any bending of space of
n dimensions needs to involve a flat spacetime of
n+1 dimensions surrounding it. Think about it like this. Suppose you are writing a computer program in which you define a series of variables that are sort of connected in a chain, where some sort of information can be moved from one to another through a "link". Information is not allowed to skip over links. So, for example, you might define one like the following. A is connected to B and B is connected to C and C is connected to D and D is connected to A. Information cannot move directly from A to C. It must first go through B or through D. You could visualize it as a loop, like this:
But to visualize it this way is a bit misleading, as we are "bending" it in two dimensions. And the way we have defined it, we haven't defined any space at all. We have only defined how the elements are connected, what is linked to what. You could go on to define much more complex networks like this with any imaginable space-like topology. You can imagine easily creating one that is like the surface of a cylinder. Just define something like a grid and then connect all the nodes along one edge to all the nodes along the opposite edge. But notice that space language like "grid" is still misleading, as we wouldn't be drawing a grid or a cylinder. We would just define "adjacencies". A1 is connected to B1 and to A2. B1 is connected to A1 and C1 and also to B2. B2 is connected to B1, B3, A2, and C2. Get the idea? No space. Just connections.
But consider if there were an incredibly large network like this, with a astronomical number of nodes. You could do things like Conway's Game of Life in this network. But you could have any imaginable topology. And the topology could change according to certain dynamical rules. Connections could be formed and broken. Nodes could be created or destroyed. The effective topology could have any number of "dimensions" and any imaginable "curvature".
Imagine that, like Conway's game of life, changes propagate through the network at a max speed of one link per clock cycle. Clearly, you can't skip links. This establishes a speed limit. And interestingly, this speed limit is one link per clock cycle. It seems conspicuously like the speed of light in a way, which is 1 Planck length per Planck time. In other words, it is equivalent to the smallest possible step in the smallest possible duration.
Perhaps the space we live in is like this. What is "closer" is simply what involves fewer links.
There are actually some new ideas in physics that try to marry quantum mechanics and general relativity that treat spacetime as a network. Such an approach is showing some promise. One is loop quantum gravity. Another is EPR=ER. The latter is especially interesting to me. An interesting article on it:
https://www.nature.com/news/the-quantum-source-of-space-time-1.18797
There is also this one I just came across (from:
link
Another approach that aims to reconcile the apparent passage of time with the block universe goes by the name of causal set theory. First developed in the 1980s as an approach to quantum gravity by the physicist Rafael Sorkin — who was also at the conference — the theory is based on the idea that space-time is discrete rather than continuous. In this view, although the universe appears continuous at the macroscopic level, if we could peer down to the so-called Planck scale (distances of about 10–35 meters) we’d discover that the universe is made up of elementary units or “atoms” of space-time. The atoms form what mathematicians call a “partially ordered set” — an array in which each element is linked to an adjacent element in a particular sequence. The number of these atoms (estimated to be a whopping 10240 in the visible universe) gives rise to the volume of space-time, while their sequence gives rise to time. According to the theory, new space-time atoms are continuously coming into existence. Fay Dowker, a physicist at Imperial College London, referred to this at the conference as “accretive time.” She invited everyone to think of space-time as accreting new space-time atoms in way roughly analogous to a seabed depositing new layers of sediment over time. General relativity yields only a block, but causal sets seem to allow a “becoming,” she said. “The block universe is a static thing — a static picture of the world — whereas this process of becoming is dynamical.” In this view, the passage of time is a fundamental rather than an emergent feature of the cosmos. (Causal set theory has made at least one successful prediction about the universe, Dowker pointed out, having been used to estimate the value of the cosmological constant based only on the space-time volume of the universe.)
By the way, the PacMan gameworld has a cylindrical topology. Go off the right side of the screen and you'll come in on the left. But this space is not bent through 3D space to do that. In fact, even the 2D space you see when playing is only there because it is mapped onto a screen for you to see what's happening. The way the information is being processed doesn't involve any space. It is more like the node network I've been describing.
Regardless of whether our space is discrete and network-like or not, what I am saying here about networks might help make more intuitive how a space could have geometry other than the familiar flat Euclidean while not being "bent" inside some higher space. Rather, it might just have to do with the causal structure of the universe.