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.,
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 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
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. 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
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; 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