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
is an equivalence class of metrics under conformal transformation, i.e., where
[*g*]:= {*g'* | *g'* = Ω^{2} *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].

@ __And duality operation__: Dray et al JMP(89).

**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 preservingangles, or the conformal structure.

$ __Def__: A transformation of the
metric of the form *g* \(\mapsto\)*g'* =
Ω^{2}
*g*, for some (non-vanishing) function Ω on *M*.

* __Types__: Restricted conformal transformations have been defined as those such that ∇^{2}Ω = 0; In 4D, all curvature scalars such as *R*^{ 2}, *R*_{ab}*R*^{ ab} and *R*_{abcd}*R*^{ abcd} are invariant under these transformations [Edery & Nakayama PRD(14)-a1406].

* __Other geometrical quantities__:
In an *n*-dimensional manifold, if we define *C*^{c}_{ab}
by ∇*'*_{a} *k*_{b}
= ∇_{a}* k*_{b} – *C*^{c}_{ab}* k*_{c}, we find that

*C*^{c}_{ab} = Ω^{–1} *g*^{cd} (*g*_{bd} ∇_{a}Ω +
*g*_{ad} ∇_{b}Ω – *g*_{ab}∇_{d}Ω)
= 2 δ^{c}_{(a}∇_{b)}
ln Ω –
*g*_{ab} *g*^{cd} ∇_{d} ln Ω ,

*C'*_{abc}^{d} = *C*_{abc}^{d}

*R'*_{ab} = *R*_{ab} – (*n*–2) Ω^{–1} ∇_{a}∇_{b} Ω – Ω^{–1}* g*_{ab}∇^{2}Ω +
2 (*n*–2) Ω^{–2} (∇_{a} Ω)
(∇_{b} Ω) – (*n*–3) Ω^{–2}* g*_{ab}
*g*^{mn} (∇_{m} Ω)(∇_{n} Ω)

*R'* = Ω^{–2}* R* – 2
(*n*–1) Ω^{–3} ∇^{2}Ω – (*n*–1)
(*n*–4) Ω^{–4} *g*^{mn} (∇_{m} Ω)
(∇_{n} Ω)

= Ω^{–2} [*R* – 2
(*n*–1) ∇^{2} ln Ω – (*n*–2)
(*n*–1) *g*^{mn} (∇_{m} ln Ω)
(∇_{n} ln Ω)]

*G'*_{ab} = *G*_{ab} – (*n*–2) Ω^{–1} ∇_{a}∇_{b} Ω +
(*n*–2) Ω^{–1} *g*_{ab }∇^{2}Ω

+
2 (*n*–2) Ω^{–2} (∇_{a} Ω)
(∇_{b} Ω) – \(1\over2\)(*n*–2)(*n*–5) Ω^{–2}* g*_{ab}
*g*^{mn} (∇_{m} Ω)
(∇_{n} Ω)

∇*'*^{ 2}*φ'* = Ω^{s–1} ∇^{2}*φ*
+ (2*s*+*n*–2) Ω^{s}^{–1}* g*^{mn} (∇_{m} Ω)
(∇_{n} *φ*)
+ *s* Ω^{s–3} (∇^{2}Ω)
*φ*

+ *s* (*s*–3+*n*) Ω^{s}^{–4}* g*^{mn} (∇_{m} Ω)
(∇_{n} Ω)
*φ*, if *φ'* = Ω^{s} *φ*.

* __Conformal weight__:
A.k.a. scaling dimension of a spinor or tensor field *ψ*;
The number *d* such that *ψ* \(\mapsto\) Ω^{–d}*ψ* when
the metric *g* \(\mapsto\) Ω^{2}*g* 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].

@ __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* : *M* → *M* 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 *P*_{a} and
3 rotations *J*_{ab} of the Poincaré
group + 3 special conformal transformations *K*_{a} +
dilation *D*, with (semisimple) Lie algebra

[*P*_{a}, *P*_{b}]
= [*J*_{ab}, *D*]
= [*K*_{a}, *K*_{b}]
= 0; [*P*_{a},
*D*] = *P*_{a}; [*K*_{a},*D*]
= –*K*_{a} ;

[*P*_{a}, *J*_{bc}]
= *η*_{ac} *P*_{b} – *η*_{ab} *P*_{c}; [*K*_{a}, *J*_{bc}]
= *η*_{ac}* K*_{b} – *η*_{ab} *K*_{c} ; [*P*_{a}, *K*_{b}]
= *J*_{ab} + *η*_{ab} *D *;

[*J*_{ab}, *J*_{cd}]
= *η*_{ac}* J*_{bd} – *η*_{ad}* J*_{bc} + *η*_{bd}* J*_{ac} – *η*_{bc}* J*_{ad} .

* __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} *k*_{b} =
*φ* *g*_{ab} – *F*_{ab},
with *F*_{ab} = *F*_{[ab]} the
conformal bivector, and *φ* some non-singular function; This is equivalent to
\(\cal L\)_{k} *g*_{ab} =
2*φ* *g*_{ab}.

* __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 ∇_{a}∇_{b}
*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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