Living in a Hyperbolic Universe

 A few years ago, there was a question on the Worldbuilding StackExchange site about living on a hyperbolic plane. This turns out to be a pretty fascinating setting, but it runs into all the same sorts of problems as infinite flat Euclidean worlds do--how does the sun work? If there's gravity, why don't mountains destabilize the whole thing and fragment it into ball-worlds? Etc.

So, I started thinking; what if you just had a planet in a hyperbolic space?

It turns out, the surfaces of spheres in hyperbolic spaces are not themselves hyperbolic manifolds; just like Euclidean spheres, they end up having positive curvature all over. So, you don't get all the interesting features of living on a hyperbolic plane, like the entire surface area of the Earth being within walking distance and the navigational problems that that implies--but there are other interesting features of such worlds!

First, let's talk about exactly how we're going to construct a hyperbolic universe. It turns out, you can do this in General Relativity, but a relativistic universe with hyperbolic space turns out to also be an exponentially expanding universe as well. The reason for this is that parallel lines in a hyperbolic space diverge, and in a relativistic universe you can't distinguish objects in motion from objects at rest--which means that parallel world lines of some collection of objects that are at relative rest to each other must diverge, no matter how that collection is moving relative to anything else. Since geodesics diverge in space, they must diverge in time as well, and the result is an expanding universe. If the curvature is strong enough to be interesting on human scales, that means the universe blows up before anything interesting can happen, so we won't be putting out planet in a relativistic universe after all!

Instead, we'll use a 3-dimensional hyperbolic space with Newtonian absolute time. Now, in a Euclidean Newtonian world, we still have Galilean relativity; any object moving inertially can be declared to be at rest, and physics is unchanged. But in this hyperbolic universe, that is no longer true; motion implies geodesic divergence, which means moving objects will feel strain from pseudoforces trying to blow them apart perpendicular to their direction of motion, so it's easy to determine a universal absolute state of rest, and absolute measurement of velocity. Travel speeds won't be limited by the speed of light, but they will be limited by the tensile strength of materials--move too fast, and you will blow up!

This also applies to a planet orbiting a sun. At different parts of its orbit, a planet will have different absolute velocities, producing seasonally-varying strains trying to distort it, and the orientation of those strains relative to the surface features will vary as the planet rotates throughout its day--meaning that there will be tides with no need for a moon! Incidentally, this also puts limits on where planets can exist and how big they can be, lest the divergence forces of orbital motion tear them apart! Fortunately, gravitational force drops much more rapidly with distance in hyperbolic space than it does in Euclidean space (exponentially, in fact), so orbits can be rather slow, which will keep our planet from blowing up even if it orbits very close to a star--which it will need to do to get enough exponentially-decaying light!

Now, the characteristics of the planet itself: we want the hyperbolic curvature to be strong enough to be interesting and noticeable, but not so strong that it makes human life impossible. Suppose we choose a hyperbolic scale of 10km. On that scale, a planet with about 55 thousand square kilometers less surface area than Earth would have a radius of a mere 71.4998km. Why am I using so many decimal places? Because if you increase the radius by a mere 10cm, to 71.4999km, the surface area jumps to 9,695 square kilometers bigger than the Earth! Fortunately, the circumference of the world changes by only about 0.02% over the 2-meter height of a tall human, so you can in fact walk on such a world without your feet leaving your head behind!

So, humans can certainly walk around on that kind of world; but what if we make the curvature more extreme? At a hyperbolic scale of 1km, a planet with a radius of a mere 9.45257km exceeds Earth in surface area, and the variation in arc lengths over a 2-meter height is still only about 0.2%. Meanwhile, if you dig down 50 meters (a typical subway tunnel depth), all distances contract by 5%. And if you travel upwards by even 150m (not even quite meeting FAA requirements for minimum flight altitudes), distances expand by 16%. So, unlike living on a hyperbolic plane, the entire surface of the world is not within walking distance over that surface, and navigation on the surface isn't too different from navigation on Earth... but flying is significantly disincentivized, and everything is within 20 km if you are willing to take the most direct route tunneling through the interior of the planet! So, subways rather than airlines turn out to be the way to go for fast travel!

And don't forget, the core of the planet is only 9 kilometers down. The deepest mines on Earth are not-quite-4km deep--a significant fraction of that distance. Which means this hyperbolic planet is much easier to mine than Earth--almost all of it is extremely close to the surface. Precious (and mundane) metals won't all be inaccessible in a deep core.

And you know what that means: hyperbolic space is pretty much made for dwarfs!