Ernst equation

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The Ernst's equation governing the complex scalar function ${\displaystyle Z}$ is given by^[4]

${\displaystyle \Re (Z)\nabla ^{2}Z=\nabla Z\cdot \nabla Z}$

where ${\displaystyle \nabla }$ is the two-dimensional gradient operator with axisymmetry; for instance, if ${\displaystyle Z=Z(\rho ,z)}$, then

$${\displaystyle \Re (Z)\left[{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial Z}{\partial \rho }}\right)+{\frac {\partial ^{2}Z}{\partial ^{2}z}}\right]=\left({\frac {\partial Z}{\partial \rho }}\right)^{2}+\left({\frac {\partial Z}{\partial z}}\right)^{2}.}$$

and if ${\displaystyle Z=Z(t,\rho )}$ (with ${\displaystyle c=1}$), then^[5]

$${\displaystyle \Re (Z)\left[{\frac {\partial ^{2}Z}{\partial ^{2}t}}-{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial Z}{\partial \rho }}\right)\right]=\left({\frac {\partial Z}{\partial t}}\right)^{2}+\left({\frac {\partial Z}{\partial \rho }}\right)^{2}.}$$

where ${\textstyle \Re (Z)}$ is the real part of ${\textstyle Z}$. If ${\displaystyle Z}$ is a solution of the Ernst's equation, then ${\displaystyle Z/(1+icZ)}$ (so is ${\displaystyle Z^{-1}}$) is also a solution where ${\displaystyle c}$ is an arbitrary real constant. The transformation ${\displaystyle Z\to Z/(1+icZ)}$ belongs to the so-called Ehlers transformation.

Often, one introduces

${\displaystyle Z=-{\frac {1+E}{1-E}}}$

so that we have

${\displaystyle (1-|E|^{2})\nabla ^{2}E=-2E^{*}\nabla E\cdot \nabla E.}$

The Ernst equation is derivable from the Lagrangian density

${\displaystyle {\mathcal {L}}={\frac {\nabla Z\cdot \nabla Z}{\Re (Z)^{2}}}={\frac {\nabla E\cdot \nabla E^{*}}{(1-|E|)^{2}}}.}$

For its Lax pair and other features see e.g.^[6][7] and references therein.

The Ernst equation is employed in order to produce exact solutions of the Einstein's equations in the general theory of relativity.