Deriving Schwarzschild Metric: How to Come Up with a Proper Ansatz?

[space-time]

In this source explaining how to derive the Schwarzschild metric, the person came up with an ansatz and worked from there.

Source: http://www.thescienceforum.com/physics/30059-solving-einstein-field-equations.html

I just want to know this: How do you come up with a proper ansatz? It looks like the guy in the source just chose any space-time interval of his desire (the one for spherical coordinates in this case) and then multiplied a few of the terms in the interval by some unknown functions of r. How would he know how many unknown functions of r he needed and which terms to multiply them by?

Also, recently, I just went about deriving the Christoffel symbol for 4D cylindrical coordinates. I also derived the metric tensor, the inverse metric tensor and the space-time interval before deriving the Christoffel symbol.

How would I know what terms within my cylindrical coordinates space time interval should be multiplied by some unknown functions when I create my ansatz? Also, how many terms should be multiplied by some unknown function?

If possible, can someone link me to any sources that teach you how to propose a proper ansatz for the Einstein field equations?

 

[Nugatory]

I just want to know this: How do you come up with a proper ansatz? It looks like the guy in the source just chose any space-time interval of his desire (the one for spherical coordinates in this case) and then multiplied a few of the terms in the interval by some unknown functions of r. How would he know how many unknown functions of r he needed and which terms to multiply them by

Choosing an ansatz is more of art than a science, and it takes some practice to develop the necessary intuition. In this case, it's most likely that author of that science forum piece already knew the final answer, which makes it a lot easier to choose a good starting point. However, Schwarzschild didn't have that advantage, and he was able to solve the problem; his thought process was probably something along the lines of:
- Because the physical problem has spherical symmetry, the solution is likely to be most easily expressed in spherical coordinates.
- Because the physical problem has spherical symmetry, the surfaces of constant $r$ and $t$ are likely to have the same topology as the surface of a sphere. That determines the metric coefficients for $d\theta$ and $d\phi$, and leaves only $dr$ and $dt$ to attach unknown functions to.

How would I know what terms within my cylindrical coordinates space time interval should be multiplied by some unknown functions when I create my ansatz? Also, how many terms should be multiplied by some unknown function?

it depends on what physical problem you're trying to solve. For flat spacetime, you're done - you have all the information needed to describe flat space-time using cylindrical coordinates. For anything else, you'd have to further specify the problem.