Geodesics: a way to grasp curvature.

I'll go deeper with time in several concepts I'm starting to relate here, please be patient. I'm reading Misner, Thorne and Wheeler's "Gravitation" with the objective in mind of making a summary in the first "Books" post, but for now, the content of this post is an interpretation about I've already read and understood regarding geodesics.

Allow me to start with Einstein field equation -the "easy" one.

Sorry about using an image. I've tryed to use third parties LaTEX renderers, but no results. I'll improve my blogging skills in the future.

So, the equation states -for the moment-, there's a G thing, which corresponds to a T thing multiplied by a fancy constant. Well, these things are tensors, and they're pretty serious stuff because this simple equation gives us a partial diferential equation system, with 10 coupled equations. Too fast, too soon. You have to believe for the moment the G thing is were we are watching the curvature, while the T thing is a way to tell how energy is distributed. Energy distribution in space, by the way, is the responsible for that curvature.

But this post is about geodesics. Why I'm giving you all this crap?

Well I hate metaphores, so let's go to the point: a geodesic is the natural trajectory for a free fall object inside a curved spacetime. When I talk about free fall I always picture a BASE jumper. It's quite a strong image: a guy jumping from a cliff, or a bridge. Look at her trajectory -in case she's not using wings, neither other fancy stuff. She falls to a certain death in what it looks like a straight line to the center of Earth.

We all are falling to the center of our planet at every moment. Even ISS crew. In case of orbiting astronauts, it's their speed and therefore their inertial forces which prevents them from actually "go down". In our case, mere mortals, Earth surface reaction forces are responsibles for not allowing us to follow what spacetime tell our matter to do: follow the geodesics.

A geodesic is technically much more, but allow me for the moment retaining this concept.

Because, can you picture what whould happen if geodesics were pointing in another directions? We would fall to other directions, with no energy consumption at all. Well, we should use a massive quantity of energy for warping the geodesics -with special caution in not creating a singularity. But that's something we'll deal with later.

So, is it warping geodesics what I understand for warping spacetime? So far, as I know, that's the point. Roughly. Very, very, very roughly. So many details stay out, it seems easy and it is not. Maybe it is not even possible. What works at a conceptual level, have to face the mathematical truth.

I'm not forgeting about the equation, I'm just letting it for the end. That G thing is where geodesics are hidden into. Where? Well, you can follow it with more detail here but, we can see for now:

That little g is where something called metric is described.

And geodesics, those free fall trajectories I want to control, come from the metric.

The details will be given -as fast as I learn about them! - but for now, several questions are quite more important than how a metric is transformed into a set of geodesics:

• What are those cute little greek letters?
• What those uppercase letters stand for?
• Is that equation complete -I mean, dude, the cosmological constant should be there, right?
• When you said a massive quantity of energy, what was the number you were thinking about?

Well, I'll try to explain as fast as I'm sure of understanding things. By now, I keep the fact I find funny how not knowing things lead me to learn another concepts.

Concepts I surely would have find boring in case of starting the quest from them. :)