In general relativity, the Ernst equation is an integrable non-linear partial differential equation, named after the American physicist who published it in 1968.
The Ernst's equation governing the complex scalar function is given by
where is the two-dimensional gradient operator with axisymmetry; for instance, if , then
and if (with ), then
where is the real part of . If is a solution of the Ernst's equation, then (so is ) is also a solution where is an arbitrary real constant. The transformation belongs to the so-called Ehler's transformation.
Often, one introduces
so that we have
The Ernst equation is derivable from the Lagrangian density
For its Lax pair and other features see e.g. 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.