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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I asked a question here, probably over 15 years ago on entanglement and I appreciated the thoughtful answers I received back then. The intervening years haven't made me any more knowledgeable in physics, so forgive my naïveté ! If a have a piece of paper in an area of high gravity, lets say near a black hole, and I draw a triangle on this paper and 'measure' the angles of the triangle, will they add to 180 degrees? How about if I'm looking at this paper outside of the (reasonable)...
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Thread 'Is the variation of the metric ##\delta g_{\mu\nu}## a tensor?'

From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
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