I am reading some lecture notes on General relativity (undergraduate level) and I do not understand a sequence of statements about the topics in the title. 

After stating that the for matter-free space the components of the Ricci tensor vanish, they go on to say that non-trivial solutions to these equations would represent the propagation of gravitational waves through otherwise empty space. Then, they proceed to using that equation to derive the [Schwarzschild metric](http://en.wikipedia.org/wiki/Schwarzschild_metric), assuming a stationary and radially symmetric situation. After getting the expression for the line element they say that even though the Ricci tensor is zero for $r>0$ this does not mean that it is zero identically and that the Schwarzschild metric is the **metric of a point mass at the origin of spacial coordinate**.

I **don't understand** why this is the case and I thought that we were under the assumption that we were in **empty space**?

Furthermore, they then state the [Birkhoff's theorem](http://en.wikipedia.org/wiki/Birkhoff%27s_theorem_%28relativity%29) and say that it means that a **radially pulsating star cannot emit gravitational waves**.

Then when investigating black hole geometry they assume that the **source of the Schwarzschild metric is some massive object within its Schwarzschild radius**.

This raises several questions for me. Firstly, I thought the Schwarzschild metric was for empty space so I don't understand how we can be talking about its source being a massive object. Secondly, I thought that the Schwarzschild metric was a non-trivial solution to the matter free equations and according to their initial statements this would represent gravitational waves. But this conflicts with Birkhoff's theorem. Could you help me identify what concept(s) I am misunderstanding?