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ernst equation wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 description 2 usage 3 references toggle the table of contents ernst equation 6 languages العربية español 日本語 한국어 türkçe 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia equation used in general relativity not to be confused with nernst equation in general relativity the ernst equation 1 is an integrable non linear partial differential equation named after the american physicist frederick j ernst sl who published it in 1968 2 3 description edit the ernst s equation governing the complex scalar function z displaystyle z is given by 4 ℜ z 2 z z z displaystyle re z nabla 2 z nabla z cdot nabla z where displaystyle nabla is the two dimensional gradient operator with axisymmetry for instance if z z ρ z displaystyle z z rho z then ℜ z 1 ρ ρ ρ z ρ 2 z 2 z z ρ 2 z z 2 displaystyle re z left frac 1 rho frac partial partial rho left rho frac partial z partial rho right frac partial 2 z partial 2 z right left frac partial z partial rho right 2 left frac partial z partial z right 2 and if z z t ρ displaystyle z z t rho with c 1 displaystyle c 1 then 5 ℜ z 2 z 2 t 1 ρ ρ ρ z ρ z t 2 z ρ 2 displaystyle re z left frac partial 2 z partial 2 t frac 1 rho frac partial partial rho left rho frac partial z partial rho right right left frac partial z partial t right 2 left frac partial z partial rho right 2 where ℜ z textstyle re z is the real part of z textstyle z if z displaystyle z is a solution of the ernst s equation then z 1 i c z displaystyle z 1 icz so is z 1 displaystyle z 1 is also a solution where c displaystyle c is an arbitrary real constant the transformation z z 1 i c z displaystyle z to z 1 icz belongs to the so called ehlers transformation often one introduces z 1 e 1 e displaystyle z frac 1 e 1 e so that we have 1 e 2 2 e 2 e e e displaystyle 1 e 2 nabla 2 e 2e nabla e cdot nabla e the ernst equation is derivable from the lagrangian density l z z ℜ z 2 e e 1 e 2 displaystyle mathcal l frac nabla z cdot nabla z re z 2 frac nabla e cdot nabla e 1 e 2 for its lax pair and other features see e g 6 7 and references therein usage edit the ernst equation is employed in order to produce exact solutions of the einstein s equations in the general theory of relativity references edit weisstein eric w ernst equation mathworld a wolfram web ernst f j 1968 new formulation of the axially symmetric gravitational field problem physical review 167 5 1175 biography of frederick j ernst archived from the original on 2018 01 04 retrieved 2017 05 09 chandrasekhar s 1998 the mathematical theory of black holes vol 69 oxford university press chandrasekhar s 1986 cylindrical waves in general relativity proceedings of the royal society of london a mathematical and physical sciences 408 1835 209 232 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 v t e 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 textbooks phenomena time dilation mass energy equivalence e mc 2 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 melvin others pp wave ozsváth schücking alcubierre ellis in computational physics numerical relativity scientists poincaré lorentz einstein hilbert schwarzschild de sitter weyl eddington friedmann lemaître milne robertson chandrasekhar zwicky wheeler choquet bruhat kerr zel dovich novikov ehlers geroch penrose hawking taylor hulse bondi misner yau thorne weiss others category retrieved from https en wikipedia org w index php title ernst_equation oldid 1367716784 categories nonlinear partial differential equations mathematics of general relativity integrable systems hidden categories articles with short description short description is different from wikidata this page was last edited on 4 august 2026 at 19 26 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents ernst equation 6 languages add 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