Deriving Ernst's Equation for Complex Metrics

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I look for derivation of Ernst equation - an equation for complex function which is
equivalent to Einstein equation in case of metric with two commuting killing vectors.
I know this equation but I wonder how it may be derived. I also heard that teher is
a simple procedure which allow to construct many solution of this equation using only
one. Does anyone know how it works?
 
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paweld said:
I look for derivation of Ernst equation - an equation for complex function which is
equivalent to Einstein equation in case of metric with two commuting killing vectors.
I know this equation but I wonder how it may be derived. I also heard that teher is
a simple procedure which allow to construct many solution of this equation using only
one. Does anyone know how it works?

It may be a little too brief, but this stuff is covered in Chapter 13, Stationary axially symmetric space-times, from the recent book Exact Space-Times in Einstein's General Relativity by Jerry B. Griffiths and Jiri Podolsky.
 

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