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.
@ 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]; > 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–22 .

* In null coordinates: The 4D line element is (dΩ2 = dθ2 + sin2θ dφ2)

ds2 = –(1–2GM/r) dv2 + 2 dv dr + r22 = x–2 [2 du dxx2(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 dvdrr22 ,

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) dr2r22 ;

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 gtt grr = –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–3AC δBD – δAD δBC) .

* Curvature invariants: The value of the 4D Kretschmann invariant is Rabcd Rabcd = 48 (GM)2 r–6.
* Singularities: While the θ = 0, π and r = 2GM singularities 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.


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