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 + r22 .

* Most general static form: Without loss of generality, can be written as ds2 = −f(r) dt2 + h(r) dr2 + r22.
* 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 + r22.
@ 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{λν} = \(1\over2\)h·/f
Γ100 = \(1\over2\)ν' exp{νλ} = \(1\over2\)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{λν} −\(1\over2\)ν'' + \(1\over4\)(λ· )2 exp{λν} −\(1\over4\)ν'2 −\(1\over4\)λ·ν· exp{λν} + \(1\over4\)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.

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