In general relativity, the Ernst equationWeisstein, Eric W, Ernst equation, MathWorld--A Wolfram Web. is an integrable non-linear partial differential equation, named after the American physicist who published it in 1968.Ernst, F. J. (1968). New formulation of the axially symmetric gravitational field problem. Physical Review, 167(5), 1175.
Description
The Ernst's equation governing the complex scalar function is given byChandrasekhar, S. (1998). The mathematical theory of black holes (Vol. 69). Oxford university press.
where is the two-dimensional gradient operator with axisymmetry; for instance, if , then
and if (with ), thenChandrasekhar, S. (1986). Cylindrical waves in general relativity. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 408(1835), 209-232.
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 Ehlers 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.
Usage The Ernst equation is employed in order to produce exact solutions of the Einstein's equations in the general theory of relativity.
References