Geometry of Schwarzschild Spacetime |
Slicings and Coordinates > s.a. coordinates [Fermi coordinates];
schwarzschild spacetime [interior solutions].
@ Flat spacelike slices: in Hawking & Hunter CQG(96)gq;
Qadir & Siddiqui IJMPD(06) [and Reissner-Nordström].
@ Maximal slicing:
Beig & Ó Murchadha PRD(98);
Beig AdP?gq/00.
@ General references: Fukuyama & Kamimura MPLA(91) [connection variables];
in Kuchař PRD(94)gq;
Gergely JMP(98) [harmonic coordinates];
Hernández-Pastora et al gq/01 [Lichnerowicz];
Malec & Ó Murchadha PRD(03),
PRD(09) [constant mean curvature];
Rosquist GRG(04)gq/03 [various];
Francis & Kosowsky AJP(04)sep-gq/03 [general form];
Pareja & Frauendiener PRD(06)gq [constant R];
Kol gq/06 [from action];
Bel a0709;
Biswas a0809;
Cattani a1010,
Deser GRG(14)-a1307 [pedagogical derivations];
Unruh a1401 [various coordinate systems, pedagogical];
Fromholz et al AJP(14)apr [coordinates matter].
@ Painlevé-Gullstrand: Martel & Poisson AJP(01)apr-gq/00;
Czerniawski CoP(06)gq/02;
Lemos & Silva a2005 [maximal];
> s.a. spherical symmetry.
@ Without coordinates: Álvarez gq/07 [using the bundle of orthonormal Lorentz frames].
@ Related topics:
Kling & Newman PRD(99) [null cones];
Rama PLB(04) [in terms of branes and antibranes];
Ballik & Lake a1005 [invariant 4-volume];
Vakili AHEP(18)-a1806 [classical polymerization];
Röken a2009 [horizon-penetrating coordinates];
> s.a. foliations; Penrose Inequality.
Line Element and Related Geometrical Properties
> s.a. spherical symmetry; Tortoise Coordinate.
* In Schwarzschild coordinates:
The d-dimensional line element is
ds2 = −(1−2GM/r) dt2 + (1−2GM/r)−1 dr2 + rd–2 dΩ2 .
* In null coordinates: The 4D line element is (dΩ2 = dθ2 + sin2θ dφ2)
ds2 = −(1−2GM/r) dv2 + 2 dv dr + r2 dΩ2 = x−2 [2 du dx − x2(1−2GMx) du2 + dΩ2],
with u:= t − r − 2GM ln(r−2GM)
and x:= r−1
(\(\cal I\)+).
* Finkelstein extension:
An extension into the future, using v:= t
+ {r + 2M ln |r−2M|};
It is convenient (and sufficient) for studying the gravitational collapse of a star
[@ Finkelstein PR(58)].
* Eddington-Finkelstein coordinates:
The coordinates v, r, θ, φ,
such that the 4D line element can be written as
ds2 = (1−2M/r) dv2 − 2 dvdr − r2 dΩ2 ,
with v:= t + r + 2 M ln(r−2M), the advanced time parameter; Or t', r, θ, φ, in terms of which
ds2 = (1−2M/r) dt' 2 − (4M/r) dt' dr − (1+2M/r) dr2 − r2 dΩ2 ;
Their motivation is that they show that the metric is regular at r = 2M, and can be used
across the horizon; > s.a. black-hole solutions [evaporating].
* Isotropic
coordinates: The 4D line element is
ds2 = −[(1−GM/2r)/(1+GM/2r)]2 dt2 + (1+GM/2r)4 (dx2 + dy2 + dz2) .
@ General references: Marolf GRG(99)gq/98 [embedding diagram];
Jacobson CQG(07)-a0707 [when is \(g_{tt} g_{rr} = -1\)?];
Paston & Sheykin CQG(12) [classification of embeddings].
@ Causal properties: He & Rideout CQG(09)-a0811 [explicit causal structure].
@ Extensions:
Frønsdal PR(58);
Rosen AP(71);
Klösch & Strobl CQG(96);
Mitra ap/99 [??];
Abbassi PS(01)gq/99;
> s.a. Kruskal Extension.
Connection and Curvature
> s.a. geodesics [including Jacobi equation]
and particles in schwarzschild spacetime.
* Connection coefficients:
The non-equivalent, non-vanishing ones in 4D are
Γ010 = GM/[r(r−2GM)] | Γ133 = −(r−2GM) sin2θ |
Γ100 = GM(r−2GM)/r3 | Γ212 = Γ313 = r−1 |
Γ111 = −GM/[r(r−2GM)] | Γ233 = −sinθ cosθ |
Γ122 = −(r−2GM) | Γ323 = (tanθ)−1 . |
* Curvature components: The non-equivalent, non-vanishing ones in 4D are (σAB is the standard metric on r = const)
R0A0B = R1A1B = GMr−3σAB , RABCD = 2GMr−3 (δAC δBD − δAD δBC) .
* Curvature invariants:
The value of the 4D Kretschmann invariant is \(R_{abcd}\, R^{abcd} = 48\, (GM)^2 r^{-6}\).
* Singularities: While the singularities
at θ = 0, π and r = 2GM are removable, the one at
r = 0 is a true curvature singularity.
@ Singularity: Hellaby JMP(96);
Heinzle & Steinbauer JMP(02)gq/01 [distributional];
Qadir & Siddiqui IJMPD(09)-a1009;
> s.a. types of singularities.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 3 sep 2020