Geometry
of Schwarzschild Spacetime| Geometry
of Schwarzschild Spacetime |
Slicings and Coordinates > s.a. foliations; schwarzschild [extensions].
@ Flat spacelike slices:
in Hawking & Hunter; Qadir & Siddiqui IJMPD(06)
[and Reissner-Nordström].
@ Maximal slicing: Beig & Ó Murchadha PRD(98);
Beig gq/00/AdP.
@ General references: Fukuyama & Kamimura MPLA(91)
[connection variables]; in
Kuchar PRD(94)gq;
Gergely JMP(98)
[harmonic coordinates]; Hernández-Pastora et al gq/01 [Lichnerowicz];
Malec & Ó Murchadha PRD(03)
[constant mean curvature]; Rosquist GRG(04)gq/03 [various];
Francis & Kosowsky AJP(04)gq/03 [general
form]; Pareja & Frauendiener PRD(06)gq [constant R];
Kol gq/06 [from
action]; Alvarez gq/07 [without
coordinates]; Bel a0709.
@ Painlevé-Gullstrand: Martel & Poisson AJP(01)gq/00;
Czerniawski CoP(06)gq/02.
@ Without coordinates: Álvarez gq/07 [using the bundle of orthonormal Lorentz
frames].
Line Element > s.a. Kruskal
Extension; spherical symmetry.
* Schwarzschild coordinates:
ds2 = –(1–2GM/r)
dt2
+ (1–2GM/r)–1 dr2
+ r2 d
2
.
* Null coordinates:
ds2 = –(1–2GM/r)
dv2
+ 2 dv dr + r2 d2 = x–2 [2
du dx – x2(1–2GMx)
du2
+ d2]
,
with u:= t – r – 2GM ln(r–2GM)
and x:= r–1
(
+).
* 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, 
, 
,
such that (d2 =
d2 +
sin2 d2)
ds2 = (1–2M/r)
dv2 – 2
dvdr – r2 d2 ,
with v:= t + r + 2 M ln(r–2M),
the advanced time parameter; Or t', r,
, ,
in terms of which
ds2 = (1–2M/r)
dt' 2 – (4M/r)
dt' dr – (1+2M/r) dr2 – r2 d2 ;
Their motivation is that they show that the metric is regular at r = 2M,
and can be used across the horizon.
* Isotropic coordinates:
ds2 = –[(1–GM/2r)/(1+GM/2r)]2 dt2 + (1+GM/2r)4 (dx2 + dy2 + dz2) .
@ References: Marolf GRG(99)gq/98 [embedding diagram]; Jacobson a0707-CQG [when is gtt grr = –1?].
Connection and Curvature
* Connection coefficients: The non-equivalent, non-vanishing ones
are
 010
= GM/[r(r–2GM)] |
133
= –(r–2GM) sin2 |
| 100 =
GM(r–2GM)/r3 |
212
= 313
= r–1 |
| 111
= –GM/[r(r–2GM)] |
233 = –sin cos |
| 122
= –(r–2GM) |
323
= (tan)–1 . |
* Curvature components: The
non-equivalent, non-vanishing ones are (
AB
is the standard metric on r = const)
R0A0B = R1A1B
= GMr–3AB
, RABCD
= 2GMr–3 (
AC BD – AD BC)
.
* Curvature invariants: The Kretschman invariant is Rabcd Rabcd = 48 (GM)2 r–6.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
16 jun 2008