Kerr Solutions| Kerr Solutions |
In General
> s.a. black holes; solutions with
symmetries; particles and fields [including geodesics].
* Idea: A two-parameter family of
solutions to Einstein's equation, representing (the only) stationary vacuum black-hole solutions.
* Line element: In Boyer-Lindquist coordinates,
with ρ2(r,θ):=
r2
+ a2 cos2θ
and Δ(r):= r2 − 2GMr
+ a2,
ds2 = ρ2 (dr2/Δ + dθ2) + (r2+a2) sin2θ dφ2− dt2 + (2GMr/ρ2) (a sin2θ dφ − dt)2 = ηab dxa dxb − λ la lb dxa dxb ,
where λ = 2GMr3/(r
4+a2z2),
and la
= (1, (rx+ay)/(a2+y2),
(ry−ax)/(a2+y2),
z/r) is null with respect to ηab;
> s.a. Kerr-Schild Solutions.
* Parameters: M represents the mass
and Ma the angular momentum measured at infinity; Extremal solutions have a = GM.
* Inequalities: The three parameters
that characterize the Kerr black hole (M, a and horizon area) satisfy
several important inequalities, some of which remain valid also for dynamical black holes;
> s.a. black-hole geometry.
@ General references: Kerr PRL(63);
Kerr & Schild in(65),
re GRG(09);
O'Neill 95;
Deser & Franklin AJP(07)mar-gq/06 [and time-independence, pedagogical];
Visser in(09)-a0706 [introduction];
Kerr in(09)-a0706,
Dautcourt GRG(09)-a0807 [historical];
Wiltshire et al ed-09;
Teukolsky CQG(15)-a1410 [overview];
Heinicke & Hehl IJMPD(15)-a1503 [intro].
@ Derivations: Carter in(73)
[nice, based on wave equation separability];
Deser & Franklin GRG(10)-a1002 [pedagogical];
Dadhich GRG(13)-a1301.
> Generalizations: see
black-hole perturbations; generalized kerr metrics;
numerical models [collapse]; quantum black holes.
Coordinates and Geometry > s.a. Ergosphere;
horizons; Hypersurfaces;
petrov classification; Smarr Formula.
* Singularities and horizons:
They have a singularity at r = 0, horizons
at r = r±, and
an ergosurface at r = r0, where
r± = GM ± [(GM)2 − a2]1/2 , r0 = GM + [(GM)2 − a2 cos2θ]1/2 .
* Killing tensor:
The tensor Kab
= Δ l(a
l'b)
+ r2
gab [@ Ludvigsen];
> s.a. killing tensors [and Killing-Yano tensor].
* Boyer-Lindquist coordinates:
A coordinate system that allows to maximally extend the Kerr solution.
* Light-like limit: The
gravitational field relative to a distant observer moving at high speed
rectilinearly in an arbitrary direction is an impulsive plane gravitational
wave with a singular point on its wave front.
@ Coordinates and extensions:
Boyer & Lindquist JMP(67);
Doran PRD(00)gq/99;
Herberthson GRG(01) [extension at spi];
Fletcher & Lun CQG(03),
Bishop & Venter PRD(06) [generalized Bondi-Sachs];
Hayward PRL(04)gq [Kruskal-like, dual null];
Bini et al CQG(05)gq [static observers, Fermi coordinates];
Natário GRG(09)-a0805 [generalized Painlevé-Gullstrand];
Novello & Bittencourt G&C(11)-a1004 [Gaussian coordinate systems];
García-Compeán & Manko PTEP(15)-a1205 [physically inconsistency of maximal analytic extensions];
Dennison et al PRL(14)-a1409 [new family of analytical coordinate systems, trumpet slices];
Liberati et al CQG(18)-a1803 [progress towards a Gordon form];
Baines et al a2008 [unit-lapse form].
@ Papapetrou gauge:
Bergamini & Viaggiu CQG(04);
Moreno & Núñez GRG(05).
@ Light-cone structure:
Pretorius & Israel CQG(98);
Bai et al PRD(07)gq [near null infinity];
Riazuelo a2008 [visual, ray tracing].
@ Other geometric properties:
Jerie et al CQG(99),
comment Hall & Keane CQG(00) [symmetries];
Marsh gq/07 [infinite-redshift surfaces];
Jacobson & Soong CQG(09)-a0809 [ergosurface];
Castro et al PRD(10)-a1004 [hidden conformal symmetry];
Schinkel et al CQG(14)-a1310 [constant-mean-curvature slices];
Gibbons & Volkov PRD(17)-a1705,
comment Manko a1706 [zero-mass limit as a wormhole].
@ Invariants, intrinsic characterization: Lake GRG(03)gq,
GRG(04)gq/03;
Ferrando & Sáez CQG(09)-a0812;
Abdelqader & Lake PRD(15)-a1412 [horizon, M, a].
@ Extreme case, geometry:
Wang et al PRD(98);
Åman et al CQG(12)-a1206 [Killing-vector behavior].
Physical Properties and Related Topics
> s.a. black-hole thermodynamics [phase transitions]; energy.
@ General references: Cohen JMP(68) [angular momentum];
Berti et al PRL(16)-a1605 [testing the Kerr nature of a black hole with spectroscopy];
> s.a. black-hole uniqueness and hair.
@ Stability: Beyer CMP(01)ap/00;
Dotti et al CQG(08)-a0805;
Dotti et al CQG(12)-a1111,
IJMPE(11)-a1111-proc;
Lucietti & Reall PRD(12)-a1208 [extreme Kerr black hole];
Myung PRD(13)-a1309 [in f(R) gravity];
Gralla et al PRD(16)-a1608 [transient instability of near-extremal black holes];
Finster & Smoller a1609-in [outline of proof];
Finster a1811-ln;
Andersson et al a1903;
Giorgi et al a2002 [formalism for non-linear stability];
> s.a. matter near black holes.
@ Quantum corrections: Reuter & Tuiran PRD(11)-a1009;
> s.a. schwarzschild space.
@ Other topics:
Ge & Leng PLA(94) [approximations];
de Felice & Preti CQG(99) [separation constants];
Mars CQG(99)gq [characterization];
Loinger gq/99/NCB [?];
Camargo & Socolovsky a1405 [Rindler approximation];
Gralla et al PRD(16)-a1602 [magnetosphere];
Hernández-Pastora & Herrera PRD(17)-a1701,
Ravi & Banerjee NA(18)-a1705 [interior solution];
> s.a. initial-value formulation; lanczos potential.
> Phenomenology: see black-hole phenomenology
[including Kerr hypothesis]; particles and fields in kerr spacetimes.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 30 aug 2020