Spacetime geodesics (I).

Math, again, people!

A couple of posts ago I introduced myself to geodesics as free fall trajectories in curved spacetime. The problem with this description -one of them- is it's saying not enough. What I've learned since then about geodesics it's still not enough, but it's a step forward, so better than nothing.

This post was intended to solve geodesic equation for an easy case and show that spacetime nature depends on 4 coordinates -3 in space and 1 in time. The problem is the answer is not as straightforward as I expected. Let's see it anyway.

A geodesic is a curve. And like any curve, it can be described mathematically. General, index form is:

If you think previous equation is confusing you may be right. But you have to admit it's easier to write down than this one:

Which is easier to write down than the actual complete equation, in which every one of the Christoffel symbols must be expanded too, like -just in 0 coordinate case...

So, let's thank mathematicians love lazy notation! For those in pain after watching this formulae, the good news is in practice, lots of those Christoffel symbols are equal to cero. Bad news are this expansion should be applied three more times: one for every coordinate in our four dimensional manifold.

But, it's the same equation anyway. Same equation, same meaning. It's the expression for a curve in a 4^th dimensional pseudo-Riemannian manifold. It's complex in general form, but it's just a curve. Don't be scared!

 

So, the expresions above are still too general to be understood. In order to give this equation a more particular meaning, we have to choose a coordinate system and, this is very important too, solve the equation according to EFE. This is completely out of reach by now, so I'm going to believe a smarter guy than me who solved the EFE equations a long time ago and I'm going to choose a spherical coordinates reference system too...

 ...and show you what a simple geodesic looks like for a spherical distribution of mass-energy -quite similar to a star:

Thanks Spanish version of Schwarchild solution in wikipedia !

By the way, don't forget...

Job is not done yet people!!!

We have arrived at a weird system of differential equations, where geodesic nature is defined in a coordinate system -typically spherical coordinates, and time. But we don't have its form, yet. Look at that sexy coordinates. There are four of them. And that is the important point in this post: it's spacetime, not just space.

Those derivatives are not typical time derivatives, by the way, but proper time derivatives.

Summarizing a lot, a way to express any curve is using one only parameter. This parameter is proper time in geodesics, which is a consequence of Special Relativity. I'm not going to explain it yet, sorry. :)

Now, the fun doubles. I've tried to solve this system using Maple. Several problems arose:

• I need 8 initial constants -4 for the first derivatives and 4 for each of one of the functions.
• Even in case I fix one of the coordinates -not cheating, because I've got spherical simmetry-, the solution is a mess.

 ... And this is just a part of it...

 
Luckily, there are several analytical means in order to solve the system and finding the Schwarzchild geodesics. From these methods, two important consequences of General Relativity appear:

• Elliptical orbit precesssion.
• Light bending due to gravity.

But this post sadly ends here. I can't give for the moment a form in four coordinates for an easy metric. My objective was obtaining a fancy formula and representing it in graphs, for several cases. I must admit I'm a little overwhelmed. Even with several analytical solutions in front of me, I can't give a 4-dimensional formula, even knowing this formula must give the above phenomena and the typical Kepler formulation in no relativistic speeds...

I lost the battle, but I'm not surrending yet...
... I'll come back to this...