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: Can be parametrized in different ways, e.g.,
ds^{2} = –f(r, t) dt^{2} + h(r, t) dr^{2} + k(r, t) dΩ^{2} or – e^{ν(r, t)} dt^{2} + e^{λ(r, t)} dr^{2} + r^{2} dΩ^{2} .
* Most general static form: Without loss of generality, can be written as
ds^{2} = –f(r) dt^{2} + h(r) dr^{2}
+ r^{2} dΩ^{2}.
* Painlevé-Gullstrand form:
A form in which the spatial metric is flat, with line element ds^{2} =
–[c^{2}(r, t)
– f^{ 2}(r, t)]
dt^{2} – 2 f(r, t)
dr dt + dr^{2} + r^{2} dΩ^{2}.
@ 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 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]; > 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
Γ^{0}_{00} = \(1\over2\)ν^{·} = \(1\over2\)f^{ ·}/f Γ^{0}_{10} = \(1\over2\)ν' = \(1\over2\)f '/f Γ^{0}_{11} = \(1\over2\)λ^{·} exp{λ–ν} = \(1\over2\)h^{·}/f Γ^{1}_{00} = \(1\over2\)ν' exp{ν–λ} = \(1\over2\)f '/h |
Γ^{1}_{01} = \(1\over2\)λ^{·} = \(1\over2\)h^{·}/h Γ^{1}_{11} = \(1\over2\)λ' = \(1\over2\)h'/h Γ^{1}_{22} = –exp{–λ}r = –r/h Γ^{1}_{33} = –exp{–λ}r sin^{2}θ |
Γ^{2}_{12} = r^{–1} Γ^{2}_{33} = –sinθ cosθ Γ^{3}_{13} = r^{–1} Γ^{3}_{23} = (tanθ)^{–1} . |
* Riemann tensor: The non-equivalent, non-vanishing components are
R^{0}_{101} = \(1\over2\)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' | |
R^{0}_{202} = –\(1\over2\)ν'
exp{–λ}r R^{0}_{303} = –\(1\over2\)ν' exp{–λ}r^{2} sin^{2}θ R^{0}_{212} = –\(1\over2\)λ^{·} exp{–ν}r R^{0}_{313} = –\(1\over2\)λ^{·} exp{–ν}r sin^{2}θ |
R^{1}_{212} = \(1\over2\)λ'
exp{λ}r R^{1}_{313} = \(1\over2\)λ' exp{λ}r sin^{2}θ R^{2}_{323} = (1–exp{–λ}) sin^{2}θ . |
Generalizations > s.a. 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 ds^{2} = –s^{2} [(1–x^{2})^{–1} dx^{2} – (1–x^{2})
dt^{2}].
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.
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feb
2017