Conformal Structures and Transformations| 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 kb
− Ccab
kc, we find that
Ccab = Ω−1 gcd (gbd ∇aΩ + gad ∇bΩ − gab∇dΩ) = 2 δc(a ∇b) ln Ω − gab gcd ∇dln Ω ,
C'abcd = Cabcd
R'ab = Rab − (n−2) Ω−1 ∇a∇b Ω − Ω−1 gab ∇2Ω + 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−3
∇2Ω − (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−1
∇a∇b
Ω + (n−2) Ω−1
gab
∇2Ω
+ 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−1 ∇2φ
+ (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 \(\mapsto\)
Ω−dψ 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 : 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 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 Pb − ηab Pc; [Ka, Jbc] = ηac Kb − ηab Kc ; [Pa, Kb] = Jab + ηab D ;
[Jab, Jcd] = ηac Jbd − ηad Jbc + ηbd Jac − ηbc 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 gab
− Fab,
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 ∇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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