Deriving Ernst's Equation for Complex Metrics

In summary, there is an equation called the Ernst equation that is equivalent to the Einstein equation in case of a metric with two commuting killing vectors. It can be derived and there is a simple procedure that allows for the construction of many solutions using only one equation. This is covered in Chapter 13 of the book Exact Space-Times in Einstein's General Relativity by Jerry B. Griffiths and Jiri Podolsky.
  • #1
paweld
255
0
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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  • #2
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.
 

FAQ: Deriving Ernst's Equation for Complex Metrics

What is Ernst's Equation for Complex Metrics?

Ernst's Equation for Complex Metrics is a mathematical formula used in the field of complex metrics, which is a subfield of mathematics that deals with complex-valued functions of a complex variable.

How is Ernst's Equation derived?

Ernst's Equation is derived from the Cauchy-Riemann equations, which are a set of necessary conditions for a complex function to be differentiable at a point.

What are the applications of Ernst's Equation?

Ernst's Equation has various applications in the study of complex metrics, including its use in solving boundary value problems, conformal mapping, and complex analysis.

Are there any limitations to Ernst's Equation?

Yes, Ernst's Equation is only applicable to complex metrics and cannot be used for real-valued functions. It also assumes that the functions being studied are differentiable.

Can Ernst's Equation be extended to higher dimensions?

Yes, Ernst's Equation can be extended to higher dimensions by using the concept of the Wirtinger derivatives, which are generalizations of the Cauchy-Riemann equations for complex functions of several complex variables.

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