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–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:= tr − 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 \(g_{tt} g_{rr} = -1\)?]; 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−3AC 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.

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