Ernst equation
{{Short description|Mathematical equation used to find exact solutions of equations in the general theory of relativity}}
{{Distinguish|Nernst equation}}
In mathematics, the Ernst equation[http://mathworld.wolfram.com/ErnstEquation.html Weisstein, Eric W, Ernst equation, MathWorld--A Wolfram Web.] is an integrable non-linear partial differential equation, named after the American physicist {{ill|Frederick J. Ernst|qid=Q31187904}}.{{Cite web |url=http://mypages.iit.edu/~segre/iit_physics_bios/ernst_f.html |title=Biography of Frederick J. Ernst |access-date=2017-05-09 |archive-url=https://web.archive.org/web/20180104233406/http://mypages.iit.edu/~segre/iit_physics_bios/ernst_f.html |archive-date=2018-01-04 |url-status=dead }}
The Ernst equation
The equation reads:
where is the real part of . For its Lax pair and other features see e.g. {{cite journal | last=Harrison | first=B. Kent | title=Bäcklund Transformation for the Ernst Equation of General Relativity | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=41 | issue=18 | date=30 October 1978 | issn=0031-9007 | doi=10.1103/physrevlett.41.1197 | pages=1197–1200| bibcode=1978PhRvL..41.1197H }}{{cite conference|last=Marvan |first=M. |year=2004 |title=Recursion operators for vacuum Einstein equations with symmetries |conference=Proceedings of the Conference on Symmetry in nonlinear mathematical physics| location=Kyiv, Ukraine|arxiv=nlin/0401014|journal=Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine|volume=50|pages=179–183}} 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.