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 > s.a. Extremal Surface; metric matching; Pseudosphere; rotations; Trapped Surface.
* Most general form: Can be parametrized in different ways, e.g.,

ds² = –f(r, t) dt² + h(r, t) dr² + k(r, t) d²   or   – exp{(r, t)} dt² + exp{(r, t)} dr² + r² d² .

* Most general static form: Without loss of generality, can be written as ds² = –f(r) dt² + h(r) dr² + r² d².
@ References: Husain et al PRD(02)gq/01 [flat foliations]; Gundlach & Martín-García gq/03 [discretely self-similar].
> Special forms: see bianchi metrics; kantowski-sachs; FRW; schwarzschild.

Connection and Curvature > s.a. sphere.
* Connection coefficients: The non-equivalent, non-vanishing ones are

000 = · = f ·/f 010 = ' = f '/f 011 = · exp{–} = h·/f 100 = ' exp{–} = f '/h

101 = · = h·/h 111 = ' = h'/h 122 = –exp{–}r = –r/h 133 = –exp{–}r sin2

212 = r–1 233 = –sin cos 313 = r–1 323 = (tan)–1 .

* Riemann tensor: The non-equivalent, non-vanishing components are

R0101 = ·· exp{–} – '' + (· )2 exp{–} – '2 – ·· exp{–} + ''
R0202 = – ' exp{–}r R0303 = – ' exp{–}r2 sin2 R0212 = – · exp{–}r R0313 = – · exp{–}r sin2	R1212 = ' exp{}r R1313 = ' exp{}r sin2 R2323 = (1–exp{–}) sin2 .

Spherical Symmetry in Classical Field Theory > s.a. distributions; spherical solutions in general relativity; solutions in gauge theory; theories of gravity.
@ Spherically symmetric perturbations: Seifert & Wald PRD(07)gq/06 [diffeomorphism-covariant theories, variational principle].

Spherical Symmetry in Quantum Field Theory > see loop quantum gravity.

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 20 jun 2008