What the heck is a manifold.

After the failure in describing geodesics from Schwarzschild solution I've decided to come back to the origins. Starting from the basics. Manifolds. Yeah. Manifolds rock!

General Relativity -uppercase means it's important- is defined in a 4 dimensional pseudo Riemannian manifold or more specifically, a Lorentzian manifold.

So, before continuing, what is a manifold?

Some people could say mathematicians are weird. I share this appreciation with a deep and sincerely respect (sometimes a little bit of fear) for mathematicians. Level of mental abstraction in Math is something out of my grasp, so I can only understand it by drama.

That means, in 4 steps:

1. I've got a problem I need to solve but nothing I understand so far helps, so...
2. I find someone's work who already has solved the issue and I try to grasp the Math behind his/her solution...
3. I believe I have understood the Math, so I try to solve the problem again.
4. I come back to the first point if the problem is not solved yet. More than once, 'cause I can be a real donkey sometimes...

Quite algorithmic. You have to remember I'm an engineer...

So instead of explaining what is a manifold, I'm going to explain what I understand a manifold is...

Coming back soon!

So... 2013... Nice!

Hey, don't look at me that way. It's been a couple of hard months. More than a couple, I get it, but I needed to focus in real life problems, like ending a master and looking for a job. Which I already have... For the moment. The job, I mean. I'm still working in the master.

It's hard, OK? Come to Madrid, they said. You'll have fun, they said... :)

Anyway, thanks for the waiting. I'm returning to the subject as soon as I start to re-read all I have posted before. Last thing I tried was obtaining a parametric form for an easy geodesic, and I failed so hard it's been 7 months since the last entry. I have pending a couple of book reviews and lot of extra work, but I'll do my best.

See you in a couple of months!

Edit: It's been so long I almost forget how to post. Sorry for the mistakes!

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: