Axisymmetric / Axially Symmetric Spacetimes| Axisymmetric / Axially Symmetric Spacetimes |
In General
> s.a. types of metrics; types of spacetimes
/ Circularity; models in canonical quantum gravity.
$ 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. general relativity solutions with symmetries;
generating solutions; gravitational energy.
* 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];
Gudapati GRG(18) [with cosmological constant];
Gambini et al CQG(19)-a1812 [in terms of real Ashtekar-Barbero variables].
@ 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 CQG(16)-a1607 [procedure];
Turimov et al PRD(18)-a1810 [scalar field].
@ 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];
Makino a1908.
@ 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 [Møller 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; models in numerical relativity; multipoles;
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 ∇(ρ∇ε) = ρ ∇ε · ∇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; yang-mills gauge theory.
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