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'* = 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].

@ __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'* =
Ω^{2} *g*, for some
(non-vanishing) function Ω on *M*.

* __Restricted__: 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];
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* : *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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