Ernst equation

Mathematical equation used to find exact solutions of equations in the general theory of relativity

In mathematics, the Ernst equation^[1] is an integrable non-linear partial differential equation, named after the American physicist Frederick J. Ernst [Wikidata].^[2]

The Ernst equation

The equation reads:

\Re (u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}.

For its Lax pair and other features see e.g. ^[3]^[4] and references therein.

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

^ Weisstein, Eric W, Ernst equation, MathWorld--A Wolfram Web.
^ "Biography of Frederick J. Ernst". Archived from the original on 2018-01-04. Retrieved 2017-05-09.
^ Harrison, B. Kent (30 October 1978). "Bäcklund Transformation for the Ernst Equation of General Relativity". Physical Review Letters. 41 (18). American Physical Society (APS): 1197–1200. Bibcode:1978PhRvL..41.1197H. doi:10.1103/physrevlett.41.1197. ISSN 0031-9007.
^ Marvan, M. (2004). Recursion operators for vacuum Einstein equations with symmetries. Proceedings of the Conference on Symmetry in nonlinear mathematical physics. Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine. Vol. 50. Kyiv, Ukraine. pp. 179–183. arXiv:nlin/0401014.

Relativity

Special
relativity

Background	Principle of relativity (Galilean relativity Galilean transformation) Special relativity Doubly special relativity
Fundamental concepts	Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration Relativistic mass
Formulation	Lorentz transformation
Phenomena	Time dilation Mass–energy equivalence (E=mc²) Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Ladder paradox Twin paradox Terrell rotation
Spacetime	Light cone World line Minkowski diagram Biquaternions Minkowski space

General
relativity

Background	Introduction Mathematical formulation
Fundamental concepts	Equivalence principle Riemannian geometry Penrose diagram Geodesics Mach's principle
Formulation	ADM formalism BSSN formalism Einstein field equations Linearized gravity Post-Newtonian formalism Raychaudhuri equation Hamilton–Jacobi–Einstein equation Ernst equation
Phenomena	Black hole Event horizon Singularity Two-body problem Gravitational waves: astronomy detectors (LIGO and collaboration Virgo LISA Pathfinder GEO) Hulse–Taylor binary Other tests: precession of Mercury lensing (together with Einstein cross and Einstein rings) redshift Shapiro delay frame-dragging / geodetic effect (Lense–Thirring precession) pulsar timing arrays
Advanced theories	Brans–Dicke theory Kaluza–Klein Quantum gravity
Solutions	Cosmological: Friedmann–Lemaître–Robertson–Walker (Friedmann equations) Lemaître–Tolman Kasner BKL singularity Gödel Milne Spherical: Schwarzschild (interior Tolman–Oppenheimer–Volkoff equation) Reissner–Nordström Axisymmetric: Kerr (Kerr–Newman) Weyl−Lewis−Papapetrou Taub–NUT van Stockum dust discs Others: pp-wave Ozsváth–Schücking Alcubierre In computational physics: Numerical relativity