Ernst equation

In general relativity, the Ernst equation is an integrable non-linear partial differential equation, named after the American physicist Template:Ill who published it in 1968.

Description

The Ernst's equation governing the complex scalar function $Z$ is given by

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

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

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

and if $Z=Z(t,\rho )$ (with $c=1$), then

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

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

Often, one introduces

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

so that we have

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

The Ernst equation is derivable from the Lagrangian density

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

For its Lax pair and other features see e.g. and references therein.

Usage

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