How Would Projectile Motion Be Described in Non-Euclidean Space?

Question

If I and a friend found ourselves in a world described by spherical geometry (as simulated in this linked video https://youtu.be/yY9GAyJtuJ0 ) how would the kinematics equations need to be augmented to describe the path and motion of a ball we tossed between us?

One thing in particular is that I'm not sure how to account for the fact that in this 3D spherical space the shortest distance between two points isn't necessarily a straight line.

I am trying to write code needed to describe these types of projectile motion in non-euclidean spaces but can't find any straightforward examples or direction elsewhere online.

(For context, I am a chemist by training and hobby programmer. I have a good amount of experience with physics and math but they haven't been the focus of my education nor work experiences thus far)

Answer 1

In the absence of any forces acting in your curved space (such as attraction of objects towards a particular point or toward each other), once you give an object an initial push, it will follow a trajectory called geodesic, a curved-space analogue of a straight line in the flat space. Same applies to the rays of light.

If you want to read some theory on this topic, you can pick a book on gravity of your choice. The standard references are the ones by Sean Caroll and Robert Wald.

If you want to go directly to simple equations, you can just google "geodesic equations sphere". This is probably the shortest explanation ever. I would suggest you, however, to become familiar with notions of metric and Christoffel symbols, as explained here (and in the references above). If you do so, switching from one curved space to another will be a simple matter of changing the metric in your initial equations.