Cosmic Horizons

You'd think as a teacher of relativity, I would understand time a little better, but I seem to have little clue to where it all goes every week (luckily Sean Carroll over at Cosmic Variance points out that time is a more slippery customer than you may expect). It's been a little while, so I thought I would catch up on my 2D Universe . Those who have been following closely will have seen that we have derived our equations of motion over the surface of a sphere, and now all we need at the acceleration terms. This is where it starts to get a little sticky. The first part is the easy point. If you remember, we want a gravitational-like force, and this depends on the distance between the two objects. Now, again, there is more than one way to skin a rabbit (is there?) but I am going to take the computationally simple approach. Any point on a sphere is denoted by our two coordinates, (θ,φ); remember, it's a 2D surface, so no radius to worry about. But let's pretend it...

So, I have now generalized the 2-d Universe a little more, and here are two particles interacting with each other within the surface of a sphere. Cool, isn't it? So, how does one calculate such a pair of paths? As I mentioned previously, it's all standard non-Euclidean geometry and vectors and the like. So, let's go through the basics (and maths-types, please remember I am an astrophysicist and don't get grumpy about the words I use - it works :)). Starting point is that we are on a sphere, and so it makes sense to use spherical polar coordinates. Now, one painful thing is that different people define which angle is ϑ and which is φ, so I will be following the convention shown in the top figure on  Wikipedia . Remember, however, we are working in the surface of the sphere, and so we have no radial (r) coordinate. If we have two infinitesimal displacements in our coordinate (and assuming a unit sphere for convenience), then the separation between the displaced coo...

One last post for today, as I have conquered movie making in matlab and handbrake (for small values of conquered) to make a movie of particle motion in my 2-d universe. I have softened the interaction so that it is now 1/r (which seems more sensible as we are one dimension down here). Here's the path (again, with the blue dot being the primary mass). And for your viewing pleasure, here's a movie of the orbit

What this? If you said "That's a small mass orbiting a large mass in a 2-d spherical universe", then you're correct. I am going to be teaching general relativity in a couple of weeks time, and every time I do I start thinking about geodesics, not just through 4-d space-time, but also standard 3-d geometry. One of the problems (IMHO) with current physics degrees is that we don't really touch on curvilinear coordinates and tensors until their final year, and (especially when it comes to general relativity), this all comes as a bit of a shock. However, we can cast much of (all of?) physics in generalized coordinates, and we should be doing this from the start, showing how things like classical mechanics can be done in Euclidean or polar coordinates (or whatever), and the key thing being that the physical predictions come out to be the same. I also think this will help students understand things like conserved quantities a bit more, and realize that there is...