Conformal Structures and Transformations  

Conformal Structures > s.a. Compactification; geodesics; spacetime and models [axiomatic]; types of spacetimes [conformally flat]; Weyl Manifold.
* Idea: A conformal structure on a manifold M is an equivalence class of metrics under conformal transformations, i.e., where [g]:= {g' | g' = Ω2 g, Ω: M → \(\mathbb R\)}; It can also be seen as a way of defining angles between line elements in a differentiable manifold (including the notion of orthogonality) and, in the Lorentzian case, infinitesimal light cones; To recover the full information on the metric one needs to add the volume element or determinant, \(|g|^{1/2}\).
@ General references: Herranz & Santander JPA(02) [of some important spaces]; Nurowski JGP(05)m.DG/04 [and differential equations].
@ Related topics: Dray et al JMP(89) [and duality operation]; Matveev & Scholz a2001 [compatibility with projective structure].

And Spacetime Structure / Gravity > s.a. formulations and modified versions of general relativity; types of dark matter.
@ General references: Jadczyk IJTP(79); Barut et al FP(94) [conformal spacetimes]; Schmidt PRD(95)gq/01; Faraoni et al FCP(99)gq/98 [conformal frames in alternative theories]; Forte & Laciana CQG(99) [as an isolated degree of freedom]; Dąbrowski et al AdP(09)-a0806 [rev]; Valiente Kroon 16 [conformal methods].
@ Conformal factor in cosmology: in Mukhanov et al PRP(92) [perturbations and degrees of freedom]; Barbashov et al ht/04-conf [as time]; Tsamparlis et al GRG(13)-a1307 [conformally related metrics, Lagrangians and cosmology]; > s.a. scalar-tensor theories.
@ Quantization: in Narlikar & Padmanabhan PRP(83); Padmanabhan PRD(83); Hu PLA(89); Forte & Laciana CQG(99)-a1109; > s.a. quantum gravity, approaches, and phenomenology.
> Specific metrics: see solutions with matter; types of metrics and spacetimes [conformally flat].

Conformal Transformation > s.a. analytic transformation (in 2D).
* Idea: A transformation of the metric preserving angles, or the conformal structure.
* And dimension: By Liouville's theorem, in 3 or more dimensions conformal transformations form a finite-dimensional group, but not in the 2-dimensional case.
$ Def: A transformation of the metric of the form g \(\mapsto\)g' = Ω2 g, for some (non-vanishing) function Ω on M.
* Restricted: Restricted conformal transformations have been defined as those such that ∇2Ω = 0; In 4D, all curvature scalars such as R 2, Rab R ab and Rabcd R abcd are invariant under these transformations [Edery & Nakayama PRD(14)-a1406].
* Other geometrical quantities: In an n-dimensional manifold, if we define Ccab by ∇'a kb = ∇a kbCcab kc, we find that

Ccab = Ω−1 gcd (gbdaΩ + gadbΩ − gabdΩ) = 2 δc(ab) ln Ω − gab gcddln Ω ,

C'abcd = Cabcd

R'ab = Rab − (n−2) Ω−1ab Ω − Ω−1 gab2Ω + 2 (n−2) Ω−2 (∇a Ω) (∇b Ω) − (n−3) Ω−2 gab gmn (∇m Ω) (∇n Ω)

R' = Ω−2 R − 2 (n−1) Ω−32Ω − (n−1) (n−4) Ω−4 gmn (∇m Ω) (∇n Ω)
= Ω−2 [R − 2 (n−1) ∇2 ln Ω − (n−2) (n−1) gmn (∇m ln Ω) (∇n ln Ω)]

G'ab = Gab − (n−2) Ω−1ab Ω + (n−2) Ω−1 gab2Ω
+ 2 (n−2) Ω−2 (∇a Ω) (∇b Ω) − \(1\over2\)(n−2)(n−5) Ω−2 gab gmn (∇m Ω) (∇n Ω)

' 2φ' = Ωs−12φ + (2s+n−2) Ωs−1 gmn (∇m Ω) (∇n φ) + s Ωs−3 (∇2Ω) φ
+ s (s−3+n) Ωs−4 gmn (∇m Ω) (∇n Ω) φ,   if   φ' = Ωs φ.

* Conformal weight: A.k.a. scaling dimension of a spinor or tensor field ψ; The number d such that ψ \(\mapsto\) Ωdψ when the metric g \(\mapsto\) Ω2g for the field theory to be conformally invariant; If n is the spacetime dimension, d = (n−2)/2 for a scalar field, d = (n−1)/2 for a spinor field, and d = 0 for a vector field if n = 4.
@ General references: in Wald 84, app D; Krantz AS(99)sep [conformal mappings, I]; Nikolov & Valchev mp/04-conf [conformally invariant differential operators]; Carneiro et al G&C(04)gq [applications in general relativity]; Ho et al JPA(11) [finite conformal transformations].
@ Related topics: Minguzzi CQG(16)-a1606 [conformal transformation of the night sky]; Kapranov a2102 [enhancement of the conformal Lie algebra in n > 2].
@ And spacetime extensions: Aceña & Valiente Kroon a1103 [stationary spacetimes]; > s.a. asymptotic flatness, at spatial and null infinity; Penrose Diagram.
> Related topics: see conformal invariance; dualities; lorentzian geometry; singularities; solutions with matter.

Conformal Group > s.a. conformal invariance; killing fields [conformal killing spinor].
$ Def: The group of diffeomorphisms f : MM such that f*g = α g, for some α = α(x).
* In 2+1 Minkowski: It is isomorphic to SO(3, 2), with 10 generators, the 3 translations Pa and 3 rotations Jab of the Poincaré group + 3 special conformal transformations Ka + dilation D, with (semisimple) Lie algebra

[Pa, Pb] = [Jab, D] = [Ka, Kb] = 0;   [Pa, D] = Pa;   [Ka, D] = −Ka ;

[Pa, Jbc] = ηac Pbηab Pc;   [Ka, Jbc] = ηac Kbηab Kc ;   [Pa, Kb] = Jab + ηab D ;

[Jab, Jcd] = ηac Jbdηad Jbc + ηbd Jacηbc Jad .

* In 2+1 Euclidean space: It is isomorphic to SO(1, 4).
* In 3+1 Minkowski space: It is isomorphic to SU(2, 2).
@ References: Fulton et al RMP(62); Defrise-Carter CMP(75) [conformally equivalent isometry groups]; Fillmore IJTP(77); Wheeler ht/00 [extended, by grading]; Dolan JMP(06)ht/05 [higher-D, character formulae].

Conformal Killing Vector / Tensor > s.a. killing vectors / solutions of general relativity.
$ Def: A generator of the conformal group, i.e., a vector field k such that ∇a kb = φ gabFab, with Fab = F[ab] the conformal bivector, and φ some non-singular function; This is equivalent to \(\cal L\)k gab = 2φ gab.
* Examples: In Minkowski space, one conformal Killing vector field is the dilation vector field; The Edgar-Ludwig metric.
* Special conformal Killing vector field: A conformal Killing vector field with ∇ab φ = 0.
* Homothecy group, Killing vector: The case with α = constant, respectively φ = constant.
* Killing vector field: A conformal Killing vector field with φ = 0.
@ Conformal Killing vectors: Hall GRG(88) [special cases]; Hall JMP(90) [fixed points of conformal vector fields in 4D Lorentzian manfolds]; Carot GRG(90) [general relativity solutions]; Hall et al CQG(97) [conformal vector fields]; Saifullah MG11(08)-a0810 [and classification of spacetimes]; Khan et al a1510 [plane-symmetric spacetimes].
@ Conformal Killing tensors: Barnes et al gq/02-proc [Killing tensors from conformal Killing vectors]; Coll et al JMP(06)gq [spectral decomposition].
@ Homothecy transformations: Hall GRG(88) [with fixed points]; Hall & Steele GRG(90); Steele GRG(91); Shabbir & Iqbal a1110 [Kantowski-Sachs & Bianchi III].
@ Generalizations: García-Parrado JGP(06) [biconformal vector fields].

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