Conformal Killing vector field: Difference between revisions
→Densitized metric tensor and Conformal Killing vectors: avoid confusing between the metric tensor and its dee |
→Densitized metric tensor and Conformal Killing vectors: Finnish avoiding confusing between the metric tensor and its determinant, with a bit of polish |
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==Densitized metric tensor and Conformal Killing vectors== |
==Densitized metric tensor and Conformal Killing vectors== |
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A vector field <math>X</math> is a [[Killing vector field]] iff its flow ( |
A vector field <math>X</math> is a [[Killing vector field]] iff 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). This can be formulated infinitesimally (and more conveniently) as <math>X</math> is Killing iff it satisfies |
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: <math>\mathcal{L}_X g = 0.</math> |
: <math>\mathcal{L}_X g = 0.</math> |
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where <math><math>\mathcal{L}_X</math> is the Lie derivative. |
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⚫ | More generally, define a ''w''-Killing vector field <math>X</math> as a vector field whose flow preserves the |
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⚫ | More generally, define a ''w''-Killing vector field <math>X</math> as a vector field whose (local) flow preserves the densitised 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 only depending on the [[conformal structure]]. |
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Now <math>X</Math> is a ''w''-Killing vector field iff |
Now <math>X</Math> is a ''w''-Killing vector field iff |
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:<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> |
:<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> |
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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 |
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:<math> \mathcal{L}_X g = - w\operatorname{div}(X) g</math>. |
:<math> \mathcal{L}_X g = - w\operatorname{div}(X) g</math>. |
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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> merely has to preserve the conformal structure and is, by definition, a ''conformal Killing vector field''. |
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> merely has to only preserve the conformal structure and is, by definition, a ''conformal Killing vector field''. |
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==Equivalent formulations== |
==Equivalent formulations== |
Revision as of 07:51, 26 November 2020
In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric (also called a conformal Killing vector, or conformal colineation), is a vector field whose (locally defined) flow defines conformal transformations, i.e. 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 that preserve a Riemannian metric and satisfy the Killing equation .
Densitized metric tensor and Conformal Killing vectors
A vector field is a Killing vector field iff its flow preserves the metric tensor (strictly speaking for each compact subsets of the manifold, the flow need only be defined for finite time). This can be formulated infinitesimally (and more conveniently) as is Killing iff it satisfies
where is the Lie derivative.
More generally, define a w-Killing vector field as a vector field whose (local) flow preserves the densitised 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 only depending on the conformal structure. Now is a w-Killing vector field iff
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 merely 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