Spherically Symmetric Geometries| Spherically Symmetric Geometries |
In General > s.a. trigonometry.
$ Idea: A spacetime is spherically symmetric
if it admits SO(3) as an isometry group, with 2D surfaces of transitivity.
Spacetime Metric
* Most general form:
It can be parametrized in different ways, e.g.,
ds2 = −f(r, t) dt2 + h(r, t) dr2 + k(r, t) dΩ2 or − eν(r, t) dt2 + eλ(r, t) dr2 + r2 dΩ2 .
* Most general static form: Without loss
of generality, can be written as ds2
= −f(r) dt2
+ h(r) dr2
+ r2 dΩ2.
* Painlevé-Gullstrand form: A form in which
the spatial metric is flat, with line element ds2
= −[c2(r, t)
− f 2(r, t)]
dt2 − 2 f(r, t)
dr dt + dr2
+ r2 dΩ2.
@ References: Husain et al PRD(02)gq/01 [flat foliations];
Gundlach & Martín-García PRD(03)gq [discretely self-similar];
Ferrando & Sáez CQG(10)-a1005,
CQG(17)-a1701 [intrinsic characterization, ideal labeling];
Parry AMP(14)-a1409 [survey];
Akbar PLB(17)-a1702 [spacelike geometries].
@ Painlevé-Gullstrand form: Martel & Poisson AJP(01)qg/00;
in Visser IJMPD(03)ht/01;
Fischer & Visser AP(03)cm/02 [effective geometry];
Natário GRG(09)-a0805 [for Kerr spacetime];
Lin & Soo PLB(09)-a0810 [generalized];
Finch a1401 [for the 5D Myers-Perry black hole];
MacLaurin a1911 [for Schwarzschild spacetime];
> s.a. Painlevé-Gullstrand Coordinates [other spacetimes];
schwarzschild spacetime;
Wikipedia page.
> Special types:
see bianchi metrics; kantowski-sachs solutions;
FLRW spacetime; schwarzschild spacetime.
> Related topics:
see Extremal Surface; Hypersurface;
killing fields; metric matching;
Pseudosphere; rotations;
Trapped Surface.
Connection and Curvature > s.a. sphere.
* Connection coefficients:
The non-equivalent, non-vanishing ones are
| Γ000
= \(1\over2\)ν·
= \(1\over2\)f ·/f Γ010 = \(1\over2\)ν' = \(1\over2\)f '/f Γ011 = \(1\over2\)λ· exp{λ−ν} = \(1\over2\)h·/f Γ100 = \(1\over2\)ν' exp{ν−λ} = \(1\over2\)f '/h |
Γ101
= \(1\over2\)λ·
= \(1\over2\)h·/h Γ111 = \(1\over2\)λ' = \(1\over2\)h'/h Γ122 = −exp{−λ}r = −r/h Γ133 = −exp{−λ}r sin2θ |
Γ212
= r−1 Γ233 = −sinθ cosθ Γ313 = r−1 Γ323 = (tanθ)−1 . |
* Riemann tensor: The non-equivalent, non-vanishing components are
| R0101 = \(1\over2\)λ·· exp{λ−ν} −\(1\over2\)ν'' + \(1\over4\)(λ· )2 exp{λ−ν} −\(1\over4\)ν'2 −\(1\over4\)λ·ν· exp{λ−ν} + \(1\over4\)λ'ν' | |
| R0202
= −\(1\over2\)ν' exp{−λ}r R0303 = −\(1\over2\)ν' exp{−λ}r2 sin2θ R0212 = −\(1\over2\)λ· exp{−ν}r R0313 = −\(1\over2\)λ· exp{−ν}r sin2θ |
R1212
= \(1\over2\)λ' exp{λ}r R1313 = \(1\over2\)λ' exp{λ}r sin2θ R2323 = (1−exp{−λ}) sin2θ . |
Generalizations
> s.a. spherical symmetry in general relativity [locally spherically symmetric spacetime].
$ Pseudospherically symmetric
spacetime: A spacetime, invariant under the action of the 3D Lorentz
group, whose surfaces of transitivity are timelike and 2D, with line element
ds2 = −s2
[(1−x2)−1
dx2 − (1−x2)
dt2].
Spherical Symmetry in Classical Field Theory
> s.a. distributions.
@ Spherically symmetric perturbations:
Seifert & Wald PRD(07)gq/06 [diffeomorphism-covariant theories, variational principle].
> In gravity theories: see
bimetric gravity; conformal gravity;
einstein-cartan theory; hořava gravity;
massive gravity; scalar-tensor theories;
spherical solutions in gravity theories [including higher-order theories and
theories with torsion].
> In other theories: see
klein-gordon theory; solutions in gauge theory.
Spherical Symmetry in Quantum Field Theory > see loop quantum gravity; singularities in quantum gravity.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 26 apr 2021