Conformal Killing vector field: Difference between revisions
Lithopsian (talk | contribs) bold the title in the lead sentence, add the abbreviation CKV, apparently used for this topic (see CKV) |
m style |
||
Line 1: | Line 1: | ||
{{noref|date=February 2019}} |
{{noref|date=February 2019}} |
||
In [[conformal geometry]], a '''conformal Killing vector field''' on a [[manifold]] of [[dimension]] ''n'' with [[Metric_tensor|(pseudo) Riemannian metric]] <math>g</math> (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field <math>X</math> whose (locally defined) [[flow (mathematics)|flow]] defines [[conformal transformation]]s, |
In [[conformal geometry]], a '''conformal Killing vector field''' on a [[manifold]] of [[dimension]] ''n'' with [[Metric_tensor|(pseudo) Riemannian metric]] <math>g</math> (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field <math>X</math> whose (locally defined) [[flow (mathematics)|flow]] defines [[conformal transformation]]s, that is, preserve <math>g</math> up to scale and preserve the conformal structure. Several equivalent formulations, called the '''conformal Killing equation''', exist in terms of the [[Lie derivative]] of the flow e.g. <math>\mathcal{L}_{X}g = \lambda g</math> |
||
for some function <math>\lambda</math> on the manifold. For <math>n \ne 2</math> there are a finite number of solutions, specifying the [[conformal symmetry]] of that space, but in two dimensions, there is an [[Conformal_field_theory#Two_dimensions|infinity of solutions]]. The name Killing refers to [[Wilhelm Killing]], who first investigated [[Killing vector field]]s |
for some function <math>\lambda</math> on the manifold. For <math>n \ne 2</math> there are a finite number of solutions, specifying the [[conformal symmetry]] of that space, but in two dimensions, there is an [[Conformal_field_theory#Two_dimensions|infinity of solutions]]. The name Killing refers to [[Wilhelm Killing]], who first investigated [[Killing vector field]]s. |
||
==Densitized metric tensor and Conformal Killing vectors== |
==Densitized metric tensor and Conformal Killing vectors== |
||
A vector field <math>X</math> is a [[Killing vector field]] |
A vector field <math>X</math> is a [[Killing vector field]] if and only if its flow preserves the metric tensor <math>g</math> (strictly speaking for each compact subsets of the manifold, the flow need only be defined for finite time). Formulated mathematically, <math>X</math> is Killing if and only if it satisfies |
||
: |
:<math>\mathcal{L}_X g = 0.</math> |
||
where <math>\mathcal{L}_X</math> is the Lie derivative. |
where <math>\mathcal{L}_X</math> is the Lie derivative. |
||
More generally, define a ''w''-Killing vector field <math>X</math> as a vector field whose (local) flow preserves the |
More generally, define a ''w''-Killing vector field <math>X</math> as a vector field whose (local) flow preserves the densitized metric <math>g\mu_g^w</math>, where <math>\mu_g</math> is the volume density defined by <math>g</math> (i.e. locally <math>\mu_g = \sqrt{|\det(g)|} \, dx^1\cdots dx^n </math>) and <math>w \in \mathbf{R}</math> is its weight. Note that a Killing vector field preserves <math>\mu_g</math> and so automatically also satisfies this more general equation. Also note that <math>w = -2/n</math> is the unique weight that makes the combination <math>g \mu_g^w</math> invariant under scaling of the metric. Therefore, in this case, the condition depends only on the [[conformal structure]]. |
||
Now <math>X</Math> is a ''w''-Killing vector field |
Now <math>X</Math> is a ''w''-Killing vector field if and only if |
||
:<math>\mathcal{L}_X \left(g\mu_g^{w}\right) = (\mathcal{L}_X g) \mu_g^{w} + w g \mu_g^{w -1} \mathcal{L}_X \mu_g = 0. |
:<math>\mathcal{L}_X \left(g\mu_g^{w}\right) = (\mathcal{L}_X g) \mu_g^{w} + w g \mu_g^{w -1} \mathcal{L}_X \mu_g = 0.</math> |
||
Since <math>\mathcal{L}_X \mu_g = \operatorname{div}(X) \mu_g</math> this is equivalent to |
Since <math>\mathcal{L}_X \mu_g = \operatorname{div}(X) \mu_g</math> this is equivalent to |
||
:<math> \mathcal{L}_X g = - w\operatorname{div}(X) g</math> |
:<math> \mathcal{L}_X g = - w\operatorname{div}(X) g.</math> |
||
Taking traces of both sides, we conclude <math>2\mathop{\mathrm{div}}(X) = -w n \operatorname{div}(X)</math>. Hence for <math>w \ne -2/n</math>, necessarily <math>\operatorname{div}(X) = 0 </math> and a ''w''-Killing vector field is just a normal Killing vector field whose flow preserves the metric. However, for <math>w = -2/n</math>, the flow of <math>X</math> |
Taking traces of both sides, we conclude <math>2\mathop{\mathrm{div}}(X) = -w n \operatorname{div}(X)</math>. Hence for <math>w \ne -2/n</math>, necessarily <math>\operatorname{div}(X) = 0 </math> and a ''w''-Killing vector field is just a normal Killing vector field whose flow preserves the metric. However, for <math>w = -2/n</math>, the flow of <math>X</math> has to only preserve the conformal structure and is, by definition, a ''conformal Killing vector field''. |
||
==Equivalent formulations== |
==Equivalent formulations== |
||
Line 30: | Line 30: | ||
Another index notation to write the conformal Killing equations is |
Another index notation to write the conformal Killing equations is |
||
:<math> X_{ |
:<math> X_{a;b}+X_{b;a} = \frac{2}{n}g_{ab} X^c{}_{;c}.</math> |
||
==See also== |
==See also== |
||
Line 37: | Line 37: | ||
* [[Einstein manifold]] |
* [[Einstein manifold]] |
||
* [[Homothetic vector field]] |
* [[Homothetic vector field]] |
||
* [[ |
* [[Invariant differential operator]] |
||
* [[Killing vector field]] |
* [[Killing vector field]] |
||
* [[Matter collineation]] |
* [[Matter collineation]] |
Revision as of 11:04, 5 June 2022
In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field whose (locally defined) flow defines conformal transformations, that is, preserve up to scale and preserve the conformal structure. Several equivalent formulations, called the conformal Killing equation, exist in terms of the Lie derivative of the flow e.g. for some function on the manifold. For there are a finite number of solutions, specifying the conformal symmetry of that space, but in two dimensions, there is an infinity of solutions. The name Killing refers to Wilhelm Killing, who first investigated Killing vector fields.
Densitized metric tensor and Conformal Killing vectors
A vector field is a Killing vector field if and only if its flow preserves the metric tensor (strictly speaking for each compact subsets of the manifold, the flow need only be defined for finite time). Formulated mathematically, is Killing if and only if it satisfies
where is the Lie derivative.
More generally, define a w-Killing vector field as a vector field whose (local) flow preserves the densitized metric , where is the volume density defined by (i.e. locally ) and is its weight. Note that a Killing vector field preserves and so automatically also satisfies this more general equation. Also note that is the unique weight that makes the combination invariant under scaling of the metric. Therefore, in this case, the condition depends only on the conformal structure. Now is a w-Killing vector field if and only if
Since this is equivalent to
Taking traces of both sides, we conclude . Hence for , necessarily and a w-Killing vector field is just a normal Killing vector field whose flow preserves the metric. However, for , the flow of has to only preserve the conformal structure and is, by definition, a conformal Killing vector field.
Equivalent formulations
The following are equivalent
- is a conformal Killing vector field,
- The (locally defined) flow of preserves the conformal structure,
- for some function
The discussion above proves the equivalence of all but the seemingly more general last form. However, the last two forms are also equivalent: taking traces shows that necessarily .
The conformal Killing equation in (abstract) index notation
Using that where is the Levi Civita derivative of (aka covariant derivative), and is the dual 1 form of (aka associated covariant vector aka vector with lowered indices), and is projection on the symmetric part, one can write the conformal Killing equation in abstract index notation as
Another index notation to write the conformal Killing equations is
See also
- Affine vector field
- Curvature collineation
- Einstein manifold
- Homothetic vector field
- Invariant differential operator
- Killing vector field
- Matter collineation
- Spacetime symmetries