Axisymmetric / Axially Symmetric Spacetimes  

In General > s.a. types of metrics; types of spacetimes / Circularity; general relativity solutions with symmetries; yang-mills gauge theory.
$ Def: A spacetime is axisymmetric (axially symmetric) if it has an isometry group whose orbits are spacelike and closed.
* Line element:
@ References: Mars & Senovilla CQG(93)gq/02 [with conformal symmetry]; Dain JPCS(11)-a1106 [numerical and analytical perspectives]; Brink et al PRD(13)-a1303 [rev]; Beetle & Wilder a1401 [characterization and general properties].

As Solutions of Einstein's Equation > s.a. generating solutions; gravitational energy; models in numerical relativity; multipoles.
* Status: 1983, There are many axially symmetric vacuum solutions, but we don't know of any asymptotically flat one with matter of compact support.
* Gamma metric: A two-parameter family of axially symmetric, static solutions of Einstein's equation found by Bach; Contains the Schwarzschild solution for a particular value of one of the parameters, that rules a deviation from spherical symmetry.
Other examples: Cylindrically symmetric (e.g., simple cosmic string solutions); Stationary and static; The Kerr, and Kerr-Newman metrics.
@ General references: Synge 60; Tomimatsu & Sato PTP(73); MacCallum ed-85; Van den Bergh & Wils CQG(85); Waylen PRS(87) [vacuum, time-dependent]; Lubo ht/01 [U(1) gauge theory]; Antoci et al CQG(01)gq, AN(03)gq/01 [Bach's gamma metric]; Dain JDG(08), CQG(12) [black holes, geometric inequalities]; Witten a1607 [vacuum, new formulation].
@ Non-stationary: Wagh & Muktibodh gq/99; Hollands CQG(12)-a1110 [n-dimensional, horizon area-angular momentum inequality].
@ Static: Waylen PRS(82) [vacuum, general solution]; Gutsunaev et al G&C(04) [electrovac]; Hernández-Pastora & Ospino a1010/GRG [vacuum]; González & López-Suspes a1104 [stability of equatorial circular geodesics]; Hernández-Pastora et al a1607 [procedure].
@ Stationary, vacuum: Dain CQG(06)gq/05 [as critical points of the total mass]; Harmark PRD(04)ht, Harmark & Olesen PRD(05)ht [D ≥ 4, sources].
@ Stationary, fluid: Mars & Senovilla CQG(94)gq/02, CQG(96)gq/02; Kyriakopoulos MPLA(99) [fluid Petrov I].
@ Stationary, other matter: Ernst PR(68), PR(68); Belinskii & Zakharov JETP(79); Van den Bergh & Wils CQG(85) [axis]; Meinel & Neugebauer PLA(96)gq/03; Schaudt & Pfister PRL(96) [boundary-value problem solvable]; Turakulov & Dadhich MPLA(01)gq [magnetic dual of Kerr]; Bonnor CQG(02)gq [2 massless spinning particles]; García & Campuzano PRD(02) [conformally flat], gq/03 [classification]; Doran & Lasenby CQG(03); Gutsunaev & Hassan G&C(03) [vacuum]; Harmark PRD(04)ht [D ≥ 4]; Assafari et al a1606 [constant Ricci scalar].
@ Electrovac: Gopala Rao JPA(74) [from vacuum Weyl solutions]; Dadhich & Turakulov CQG(02) [with separable equations of motion]; Goyal & Gupta PS(12).
@ Cylindrically symmetric: Sharif JKPS(00)gq/07 [static, pfluid]; Sharif & Aziz IJMPA(05)gq, IJMPD(05)gq; > s.a. Conformal Gravity.
@ Properties: Chandrasekhar & Friedman ApJ(72) [stability]; Carot CQG(00) [rev]; Radinschi gq/02 [Mřller energy].
@ Related topics: von der Gönna & Pravdová JMP(00)gq [asymptotically flat, null dust]; Barnes CQG(01)gq [symmetry groups].
@ Higher-dimensional: Tan BSc(03)-a0912 [D–2 orthogonal commuting Killing vectors in D dimensions]; Charmousis & Gregory CQG(04)gq/03 [arbitrary]; Godazgar & Reall CQG(09)-a0904 [algebraically special]; Delice et al GRG(13)-a1205 [cylindrically symmetric or Kasner-type, electrovac].
> Other metrics: see black holes; cosmic strings; cosmological models in general relativity [Einstein-Straus]; kerr-newman spacetime; Manko-Novikov Solutions; Papapetrou Solution; solutions of general relativity [Einstein-Yang-Mills].

Ernst Equation > s.a. black holes; general relativity solutions; lanczos tensor.
* Idea: A method for generating axisymmetric solutions in general relativity with electromagnetic charge from axisymmetric vacuum spacetimes.
$ Def: The equation for the Ernst potential ε = f + i ψ given by

Re ε ∇(ρε) = ρε · ∇ε .

* Applications: It arises in the stationary axisymmetric reduction of real general relativity, or of self-dual Yang-Mills theory.
@ General references: Ernst PR(68), PR(68) [vacuum and electrovac]; Korotkin & Nicolai PRL(95)ht/94 [Hamiltonian form]; Klein & Richter JGP(97), JGP(99)gq/98 [Riemann-Hilbert form]; Barbosa-Cendejas et al a1103-conf [matrix generalization]; Astorino JHEP(12)-a1205 [with cosmological constant].
@ Geroch conjecture: Hauser & Ernst GRG(01)gq/00 [hyperbolic, proof].
@ Solutions: Meinel & Neugebauer CQG(95)gq/03 [asymptotically flat, with reflection symmetry]; Klein & Richter PRL(97), PRD(98)gq [realistic]; Masuda et al JPA(98) [Neugebauer-Kramer]; Alekseev gq/99-conf [monodromy transform solution]; Gariel et al CQG(02)gq/01 [new, vacuum]; Ansorg et al PRD(02)gq/01 [Bäcklund-type]; Bergamini & Viaggiu gq/03, CQG(04)gq/03; Sotiriou & Pappas JPCS(05)gq; Ernst et al CQG(06)gq/07, CQG(07)gq [equatorial symmetry/antisymmetry]; Chruściel & Szybka APPB(08)-a0708 [smoothness at ergosurface].
@ Related topics: Papachristou & Harrison PLA(94) and PLA(94) [Lax pair]; Schief JPA(01) [dual as Loewner system].

Ernst Spacetime > s.a. cosmic censorship.
* Idea: A solution of the Einstein-Maxwell equations describing two charged black holes accelerating apart in a uniform electric (or magnetic) field; As the field approaches a critical value, the black hole horizon appears to touch the acceleration horizon.

In Other Theories > see brans-dicke theory; higher-order theories; teleparallel gravity.

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