Schwarzschild metric (V).

Did you really think we were done with Schwarzschild metric? We have finally found that...
\begin{equation}
ds^2= \left ( 1- \frac{2GM}{c^2r}\right )c^2dt^2- \frac{dr^2}{\left( 1- \frac{2GM}{c^2r}\right)}-r^2(d\theta^2+sin^2\theta d\varphi^2)
\end{equation}
... so... Good for us.

But what does that mean? Why is it important? Why is it useful?

Schwarzschild metric (IV)

This Schwarzchild Schwarzschild topic is getting out of hand!

Just two more posts. In this one we are going to find the final form of the metric. Next one is about interpretations, conclusions and such, which turns out is a lot more faster than deriving the metric itself...

So, here we go again... Math ahead!

Schwarzschild metric (III): the search for simplification

Stubborn as I am, I'm not going to continue with the demonstration demostration until I completely understand why $C=D=1$. I'm not saying it's wrong. Just saying I don't understand why it's right... Yet.
Spoiler: I have found a way of understanding it. Not sure is correct, but it's good for me at the moment. Yes. I do believe too that I have overthought this too much.

Schwarzschild metric (II)

Are you ready?
We were supposed to end up with the demonstration, but it actually is pretty damn long, so in this post I'm going to just obtain the equations as functions of A, B, C and D. And I'll let the final steps of the demonstration for the next post. So two points already pendant:

1. Why metric tensor has only diagonal components? That is, why there are no cross-terms?
2. Applying $g_{\mu\nu}$ into the $R_{\mu\nu}$ equation.

Warning! Math ahead!

Schwarzschild metric (I)

Maybe I should start explaining what a metric is. Or curvature. Or the energy-stress tensor...

Nahhh... :)

Let's run before walk with an example. It's funnier this way.

Legend says Karl Schwarzschild was a german artillery officer during WWI, but the truth is he was a physicist and astronomer before that. He died several months after finding the first analytical solution to EFE's, of an illness called Pemphigus (click at your own risk). You have read right: Schwarzchild did find an analytical solution to Einstein Field Equations. Exactly. Analytically. Manly.

Truth be said, he did it for the "simplified" case of a spherical static object, but it's a good aproximation, specially for people like me who need to learn. Are you ready? I prevent you I have (finally) learned how to use latex in blogger...