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?


First, it means a lot of things. I'm going to summarize the few I understand. If I find anything new, i'll reopen the post.

• When $r \rightarrow \infty $ we recover the metric of a flat spacetime. This actually makes sense, because it can be expected that the influence of the spherical mass be actually zero at long distances. Something similar happens if $M \rightarrow 0$: flat space metric appears again. 
• When $\frac{2GM}{c^2r}=1$ we have a singularity. As you may have guessed, this happens for $r=\frac{2GM}{c^2}$. That's what serious people call Schwarzschild radius: a limit in which the metric stops working, not because it is wrong, but because model can not be applied there. Can be expressed as the radius in which escape velocity has the value of the speed of light. As a curiosity, Earth Schwarzchild radius is 0.8 cm. 
• There is another singularity when $r=0$, obviously. As I understand, this singularity is a little bit more serious and has to do (like the previous one) with black holes and stuff. No need to go there for a moment.

So far, so good. This is what everybody knows. Let's go a little bit deeper.

• No energy-stress tensor is defined (because vacumm solution, remember?). Well, it is defined, but is defined as zero. That's saying a lot. Like a pretty good deal. We have defined the metric almost by the sheer logic and mechanical calculations (which in the future I'm going to externalize to Maxima or other program like that). Summarizing a lot, it allows us to say, no matter what is the source of curvature: as long as we are outside it and there is no other close enough, coordinate system election and boundary conditions fix the problem. That has been shocking for me and didn't expect that at all, despite I had heard it in the past. As I said, I'm not a smart man. 
• Schwarzschild gives a good prediction for time dilation thanks to the time metric factor. Just look at the fact that if $d\theta=0, d\varphi=0,dr=0$, metric turns into 

\begin{equation}
d\tau^2= \left ( 1- \frac{2GM}{c^2r}\right )dt^2 \rightarrow \frac{d\tau}{dt} = \sqrt{1- \frac{2GM}{c^2r}}\\  \rightarrow \Delta \left(\frac{1}{\gamma} \right)=\frac{GM_{earth}}{c^2}\left (R_{earth}- R_{GPS}\right)
\end{equation}

Which translates into the correction formula for time dilation. This is actually the main source of error in GPS signal use and altogether with special relativity effects (due to satellites speed, which actually can reduce GR error), is responsible for the need of the change in clock frequency in GPS satellites with respect to Earth.

Is there anything else? Yes! A lot! Do you want to recover Kepler's third law? Got it! You just have to find out the Lagrangian of the metric and, there you go. Do you want to calculate how much a light ray deviates by a gravitational lens? Or maybe accounting for precession of orbits? Check the last sections of this beautiful explained lecture.

Final remarks

Deriving Schwarzschild metric has all the things I've never appreciated about physics. Non rigorous math, an incomplete model, the need for a non systematic approach in the resolution and a couple of U-turns regarding initial constants and simplifications. Yet, here it is. It explains things. It's real physics. You can make things with it. You can calculate time dilation, you can extract gravitation laws. That's cool.

Several things remain unsaid. For instance, I did not solve the metric "inside" the source. That's a more complicated problem (and much more relevant for my evil plans) and I'm not interested in it prepared for it at the moment. What happens if this stationary gravitational source rotates? What about if it changes? Questions for another post.

For now, I'm going to forget about this metric for a time.

Next step: installing and playing with Maxima :)

PS: Next post is going to take a while, probably. I have to make bottle rockets for the weekend and to revisite all the previous posts about Schwarzschild metric in order to correct mistakes.