It is very similar to Zeno's paradox.

I think it ultimately speaks to the need for calculus, or at least some method of determining whether the sum of an infinite series converges or diverges. You can add an infinite number of positive numbers together and get either infinity or a finite number, depending on how quickly the numbers in the list grow or shrink.

Another example that comes to mind is the idea of escape velocity. It's something most of us here probably understand quite well, but many people struggle with it at first. Knowing that the gravitational force has infinite range and is always attractive, it can be tempting to conclude that something hurled from a planet (and ignoring any other objects to pull on it), should always be slowing down and therefore eventually turn around and come back. (Also many people have heard the old saying "whatever goes up must come down.") But not so! The gravitational pull drops off as 1/r

^{2}, so there is some speed you can launch it so that even though it is always slowing down, it never turns around and never comes back.

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A-L-E-X, the balloon analogy is great for illustrating how metric expansion works. Choose any two points on the surface of an inflating balloon, and the distance, measured over the surface, increases at a rate which is proportional to the distance between them. From this we get a "Hubble Law" for the balloon, just like the Hubble Law for the universe.

However, like all analogies, this breaks down if taken too far. In many ways the balloon is not a good analogy for the expanding universe, and we must be careful to understand its limitations.

For example, an inflating balloon expands into the space around it. The 2D surface of it grows larger because it is "expanding into the third dimension". But this is not true for an expanding universe. There need not be any higher spatial dimension for the universe to expand into! This is extremely difficult to visualize or comprehend. How can we argue that it is true, then? Because the expansion of the universe, according to general relativity, only depends on what the universe contains. It is an

*intrinsic* property, not dependent on anything that lies outside of the universe. Unlike the balloon, it need not be embedded within a higher dimensional space at all.

While we cannot visualize how something can expand without having something outside of it to expand into, we can visualize something related to it. Let's use an example of

*curvature:*The surface of a sphere is curved. Obviously. You can tell just by looking at it. But how can you rigorously prove that it is curved? Well, you can draw a triangle on it and add up all the angles in the triangle. It will turn out to be more than 180°, which is a proof that the surface is curved. Alternatively, you can cut open the sphere and try to lay it flat. You'll discover that you

*can't*, not without distorting it. This is why all 2D maps of the Earth are distorted. These properties show that a sphere has "intrinsic" curvature: the curvature is a property of the surface itself, not how it is embedded in 3D space.

How about a cylinder? It is also obviously curved. Right? Well,

*no*, actually. Try drawing the triangle on it and measure the angles. Or even easier: cut it open and try laying it flat. It is perfectly flat! The only reason that a cylinder "looks curved" is because of how it was rolled up inside of three dimensions. But the surface itself was not curved at all. So the cylinder's curvature is an

*extrinsic* property rather than

*intrinsic.*Another problem with balloons: to continue to inflate a balloon, we must continue to add air to it. (Or decrease the pressure outside of it, I suppose). A universe does not work that way. A universe containing nothing (no matter, no radiation, no dark energy) will expand forever at a constant rate. No outside help required! In this sense, a better analogy for expansion might be inertia. It is more similar throwing a rock and tracing its trajectory than it is to inflating a balloon. The rock analogy can actually be taken pretty far and still be useful, even in mathematical rigor, to the true cosmic expansion, and I'll explore this again in more detail soon enough.

We are creatures stuck on a tiny planet and whose lifetimes are very short compared to the age of the universe. So the expansion of the universe is not something we understand intuitively from every day experience. There

*is* no perfect every-day analogy for it, though we can at least get some useful insight in a few ways. Otherwise, we require physics and mathematics to accurately describe it.