Kerr
Solutions| Kerr
Solutions |
In General > s.a. [black holes;
general relativity solutions with symmetries]; horizons; numerical
models [collapse].
* 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 
2(r,
):=
r2 + a2 cos2 and
(r):= r2 – 2GMr +
a2,
ds2 = 2
(dr2/ +
d2)
+ (r2+a2)
sin2 d
2– dt2 +
(2GMr/2)
(a sin2 d –
dt)2 = 
ab dxa
dxb – 
la lb
dxa dxb ,
where = 2GMr3/(r 4+a2z2),
and la = (1, (rx+ay)/(a2+y2),
(ry–ax)/(a2+y2),
z/r) is null wrt ab.
* Parameters: M represents the mass and Ma the angular momentum
measured
at infinity.
* 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–a2cos2]1/2
.
* Killing tensor: The
tensor Kab
= l(a l'b) +
r2gab [@
Ludvigsen]; > s.a. killing fields [Killing-Yano
tensor].
@ General references: Kerr PRL(63);
Kerr & Schild in(65), re GRG(09);
Carter in(73)
[nice derivation, based on separability of the wave equation]; O'Neill 95;
Deser & Franklin AJP(07)mar-gq/06 [and
time-independence, pedagogical]; Visser a0706-in
[introduction]; Kerr a0706-in,
Dautcourt GRG(09)-a0807 [historical];
Ferrando & Sáez CQG(09)-a0812 [intrinsic
characterization]; Wiltshire et al ed-09.
@ Variations: Wang et al PRD(98)
[extreme case, geometry]; > s.a. generalized kerr
metrics; quantum
black holes.
Related Topics > s.a. energy; lanczos
potential; matter [interior]; perturbations [including
instability];
schwarzschild space [quantum corrections].
* 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: Boyer & Lindquist JMP(67);
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].
@ Papapetrou gauge: Bergamini & Viaggiu CQG(04); Moreno & Núñez GRG(05).
@ Properties: Cohen JMP(68)
[angular momentum]; Pretorius & Israel CQG(98)
[light cones]; Jerie et
al CQG(99),
comment Hall & Keane CQG(00)
[symmetries]; Beyer
CMP(01)ap/00 [stability];
Lake GRG(03)gq,
GRG(04)gq/03 [invariants];
Bai et al PRD(07)gq [light-cone
structure near null infinity]; Marsh gq/07 [infinite-redshift
surfaces]; Jacobson & Soong CQG(09)-a0809 [ergosurface]; > s.a. uniqueness
and
hair.
@ Other topics: Ge & Leng PLA(94)
[approximations]; de Felice & Preti
CQG(99)
[separation constants]; Mars CQG(99)gq [characterization];
Doran PRD(00)gq/99 [coordinates];
Loinger gq/99/NCB
[?]; > s.a. black-hole thermodynamics [phase
transitions]; Ergosphere, Hypersurfaces; petrov
classification; Smarr
Formula; tests of general relativity with
light.
Particles and Fields > s.a. gravitomagnetism;
lensing; spinning
particles.
* Aschenbach effect: For Kerr black holes with a spin parameter a > 0.9953,
the velocity has a positive radial gradient for geodesic, stable, circular
orbits in a small radial range close to the black hole horizon.
@ Geodesics: Walker & Penrose CMP(70),
Woodhouse CMP(75)
[integrability]; Hughes PRD(01)gq [horizon-skimming];
Schmidt CQG(02)gq;
Khanna PRD(04)gq/03 [elliptic/inclined
orbits]; Chicone & Mashhoon A&A(05)ap/04 [ultrarelativistic],
CQG(06)gq [tidal
dynamics]; Barausse et al PRD(07)
[nearly horizon skimming]; Rosquist et al IJMPD(09)-a0710 [Carter's
constant]; Levin & Pérez-Giz PRD(08)-a0802 [classification];
Bozza PRD(08)
[optical caustics]; Levin & Pérez-Giz PRD(09)-a0811,
Pérez-Giz & Levin PRD(09)-0811 [homoclinic
orbits]; Komorowski CQG(09)
[elliptical last stable orbits]; Fujita & Hikida CQG(09)-a0906 [bound
timelike geodesics, analytical].
@ Scattering:
Barrabès & Hogan PRD(04)gq [particles];
Barrabès et al CQG(05)gq
[high-speed black hole, particles + waves].
@ Astrophysical aspects: Gammie ApJ(04)ap
[magnetorotational instability].
@ Aschenbach effect: Aschenbach A&A(04); Stuchlik et al PRD(05)gq/04.
@ Other particle effects: Hartle PRD(71)
[no long-range neutrino forces]; Ottewill & Winstanley PLA(00)
[quantum thermal state]; Bozza et
al PLA(01)gq [maximal
acceleration]; Sibgatullin AL(01)gq [precession];
de Felice et al CQG(04)
[magnetized particles]; Bañados et al PRL(09),
comment Berti et al PRL-a0911 [as
high-energy particle accelerators].
@ Fields: Finster et al BAMS(09)-a0801 [scalar
+ Dirac, review]; > s.a. electromagnetism; dirac
fields; klein-gordon
fields [including tails].
@ Kerr-cft correspondence:
Guica et al a0809;
Chow
et al a0812; Azeyanagi
et al PRD(09)-a0812 [and
string theory]; Peng & Wu PLB(09)-a0901 [extremal,
5D Gödel spacetime]; Matsuo et al NPB(10)-a0907.
> Quantum fields: see quantum
field theory
in curved spacetime.
>
Related topics: see light
deflection; monopoles; orbits
of gravitating bodies; self-force [radiation
reaction];
sources of gravitational waves.
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 13
nov 2009