# Conformal Killing vector field

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Vector field in conformal geometry

In [conformal geometry](/source/Conformal_geometry), a **conformal Killing vector field** on a [manifold](/source/Manifold) of [dimension](/source/Dimension) *n* with [(pseudo) Riemannian metric](/source/Metric_tensor) g {\displaystyle g} (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field X {\displaystyle X} whose (locally defined) [flow](/source/Flow_(mathematics)) defines [conformal transformations](/source/Conformal_transformation), that is, preserve g {\displaystyle g} up to scale and preserve the conformal structure. Several equivalent formulations, called the **conformal Killing equation**, exist in terms of the [Lie derivative](/source/Lie_derivative) of the flow e.g. L X g = λ g {\displaystyle {\mathcal {L}}_{X}g=\lambda g} for some function λ {\displaystyle \lambda } on the manifold. For n ≠ 2 {\displaystyle n\neq 2} there are a finite number of solutions, specifying the [conformal symmetry](/source/Conformal_symmetry) of that space, but in two dimensions, there is an [infinity of solutions](/source/Conformal_field_theory#Two_dimensions). The name Killing refers to [Wilhelm Killing](/source/Wilhelm_Killing), who first investigated [Killing vector fields](/source/Killing_vector_field).

## Densitized metric tensor and Conformal Killing vectors

A [vector field](/source/Vector_field) X {\displaystyle X} is a [Killing vector field](/source/Killing_vector_field) if and only if its flow preserves the metric tensor g {\displaystyle g} (strictly speaking for each [compact](/source/Compact_(topology)) subsets of the manifold, the flow need only be defined for finite time). Formulated mathematically, X {\displaystyle X} is Killing if and only if it satisfies

- L X g = 0. {\displaystyle {\mathcal {L}}_{X}g=0.}

where L X {\displaystyle {\mathcal {L}}_{X}} is the Lie derivative.

More generally, define a *w*-Killing vector field X {\displaystyle X} as a vector field whose (local) flow preserves the densitized metric g μ g w {\displaystyle g\mu _{g}^{w}} , where μ g {\displaystyle \mu _{g}} is the volume density defined by g {\displaystyle g} (i.e. locally μ g = | det ( g ) | d x 1 ⋯ d x n {\displaystyle \mu _{g}={\sqrt {|\det(g)|}}\,dx^{1}\cdots dx^{n}} ) and w ∈ R {\displaystyle w\in \mathbf {R} } is its weight. Note that a Killing vector field preserves μ g {\displaystyle \mu _{g}} and so automatically also satisfies this more general equation. Also note that w = − 2 / n {\displaystyle w=-2/n} is the unique weight that makes the combination g μ g w {\displaystyle g\mu _{g}^{w}} invariant under scaling of the metric. Therefore, in this case, the condition depends only on the [conformal structure](/source/Conformal_structure). Now X {\displaystyle X} is a *w*-Killing vector field if and only if

- L X ( g μ g w ) = ( L X g ) μ g w + w g μ g w − 1 L X μ g = 0. {\displaystyle {\mathcal {L}}_{X}\left(g\mu _{g}^{w}\right)=({\mathcal {L}}_{X}g)\mu _{g}^{w}+wg\mu _{g}^{w-1}{\mathcal {L}}_{X}\mu _{g}=0.}

Since L X μ g = div ⁡ ( X ) μ g {\displaystyle {\mathcal {L}}_{X}\mu _{g}=\operatorname {div} (X)\mu _{g}} this is equivalent to

- L X g = − w div ⁡ ( X ) g . {\displaystyle {\mathcal {L}}_{X}g=-w\operatorname {div} (X)g.}

Taking traces of both sides, we conclude 2 d i v ⁡ ( X ) = − w n div ⁡ ( X ) {\displaystyle 2\mathop {\mathrm {div} } (X)=-wn\operatorname {div} (X)} . Hence for w ≠ − 2 / n {\displaystyle w\neq -2/n} , necessarily div ⁡ ( X ) = 0 {\displaystyle \operatorname {div} (X)=0} and a *w*-Killing vector field is just a normal Killing vector field whose flow preserves the metric. However, for w = − 2 / n {\displaystyle w=-2/n} , the flow of X {\displaystyle X} has to only preserve the conformal structure and is, by definition, a *conformal Killing vector field*.

## Equivalent formulations

The following are equivalent

1. X {\displaystyle X} is a conformal Killing vector field,

1. The (locally defined) flow of X {\displaystyle X} preserves the conformal structure,

1. L X ( g μ g − 2 / n ) = 0 , {\displaystyle {\mathcal {L}}_{X}(g\mu _{g}^{-2/n})=0,}

1. L X g = 2 n div ⁡ ( X ) g , {\displaystyle {\mathcal {L}}_{X}g={\frac {2}{n}}\operatorname {div} (X)g,}

1. L X g = λ g {\displaystyle {\mathcal {L}}_{X}g=\lambda g} for some function λ . {\displaystyle \lambda .}

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 λ = ( 2 / n ) div ⁡ ( X ) {\displaystyle \lambda =(2/n)\operatorname {div} (X)} .

The last form makes it clear that any Killing vector is also a conformal Killing vector, with λ ≅ 0. {\displaystyle \lambda \cong 0.}

## The conformal Killing equation

Using that L X g = 2 ( ∇ X ♭ ) s y m m {\displaystyle {\mathcal {L}}_{X}g=2\left(\nabla X^{\flat }\right)^{\mathrm {symm} }} where ∇ {\displaystyle \nabla } is the Levi Civita derivative of g {\displaystyle g} (aka covariant derivative), and X ♭ = g ( X , ⋅ ) {\displaystyle X^{\flat }=g(X,\cdot )} is the dual 1 form of X {\displaystyle X} (aka associated covariant vector aka vector with lowered indices), and s y m m {\displaystyle {}^{\mathrm {symm} }} is projection on the symmetric part, one can write the conformal Killing equation in abstract index notation as

- ∇ a X b + ∇ b X a = 2 n g a b ∇ c X c . {\displaystyle \nabla _{a}X_{b}+\nabla _{b}X_{a}={\frac {2}{n}}g_{ab}\nabla _{c}X^{c}.}

Another index notation to write the conformal Killing equations is

- X a ; b + X b ; a = 2 n g a b X c ; c . {\displaystyle X_{a;b}+X_{b;a}={\frac {2}{n}}g_{ab}X^{c}{}_{;c}.}

## Examples

### Flat space

In n {\displaystyle n} -dimensional flat space, that is [Euclidean space](/source/Euclidean_space) or [pseudo-Euclidean space](/source/Pseudo-Euclidean_space), there exist globally flat coordinates in which we have a constant metric g μ ν = η μ ν {\displaystyle g_{\mu \nu }=\eta _{\mu \nu }} where in space with signature ( p , q ) {\displaystyle (p,q)} , we have components ( η μ ν ) = diag ( + 1 , ⋯ , + 1 , − 1 , ⋯ , − 1 ) {\displaystyle (\eta _{\mu \nu })={\text{diag}}(+1,\cdots ,+1,-1,\cdots ,-1)} . In these coordinates, the connection components vanish, so the covariant derivative is the coordinate derivative. The conformal Killing equation in flat space is ∂ μ X ν + ∂ ν X μ = 2 n η μ ν ∂ ρ X ρ . {\displaystyle \partial _{\mu }X_{\nu }+\partial _{\nu }X_{\mu }={\frac {2}{n}}\eta _{\mu \nu }\partial _{\rho }X^{\rho }.} The solutions to the flat space conformal Killing equation includes the solutions to the flat space Killing equation discussed in the article on Killing vector fields. These generate the [Poincaré group](/source/Poincar%C3%A9_group) of isometries of flat space. Considering the ansatz X μ = M μ ν x ν , {\displaystyle X^{\mu }=M^{\mu \nu }x_{\nu },} , we remove the antisymmetric part of M μ ν {\displaystyle M^{\mu \nu }} as this corresponds to known solutions, and we're looking for new solutions. Then M μ ν {\displaystyle M^{\mu \nu }} is symmetric. It follows that this is a [dilatation](/source/Homothety), with M ν μ = λ δ ν μ {\displaystyle M_{\nu }^{\mu }=\lambda \delta _{\nu }^{\mu }} for real λ {\displaystyle \lambda } , and corresponding Killing vector X μ = λ x μ {\displaystyle X^{\mu }=\lambda x^{\mu }} .

From the general solution there are n {\displaystyle n} more generators, known as [special conformal transformations](/source/Special_conformal_transformation), given by

- X μ = c μ ν ρ x ν x ρ , {\displaystyle X_{\mu }=c_{\mu \nu \rho }x^{\nu }x^{\rho },}

where the traceless part of c μ ν ρ {\displaystyle c_{\mu \nu \rho }} over μ , ν {\displaystyle \mu ,\nu } vanishes, hence can be parametrised by c μ μ ν = b ν {\displaystyle c^{\mu }{}_{\mu \nu }=b_{\nu }} .

General solution to the conformal Killing equation (in more than two dimensions)[1] For convenience we rewrite the conformal Killing equation as ∂ μ X ν + ∂ ν X μ = f ( x ) g μ ν . {\displaystyle \partial _{\mu }X_{\nu }+\partial _{\nu }X_{\mu }=f(\mathbf {x} )g_{\mu \nu }.} (By taking traces we can recover f ( x ) = 2 n ∂ ρ X ρ . {\displaystyle f(\mathbf {x} )={\frac {2}{n}}\partial _{\rho }X^{\rho }.} ) Applying an extra derivative, relabelling indices and taking a linear combination of the resulting equations gives 2 ∂ μ ∂ ν X ρ = η μ ρ ∂ ν f + η ν ρ ∂ μ f − η μ ν ∂ ρ f . {\displaystyle 2\partial _{\mu }\partial _{\nu }X_{\rho }=\eta _{\mu \rho }\partial _{\nu }f+\eta _{\nu \rho }\partial _{\mu }f-\eta _{\mu \nu }\partial _{\rho }f.} Contracting on μ , ν {\displaystyle \mu ,\nu } gives 2 ∂ 2 X ρ = ( 2 − n ) ∂ ρ f . {\displaystyle 2\partial ^{2}X_{\rho }=(2-n)\partial _{\rho }f.} A combination of derivatives of this and the original conformal Killing equation gives ( 2 − n ) ∂ μ ∂ ν f = η μ ν ∂ 2 f , {\displaystyle (2-n)\partial _{\mu }\partial _{\nu }f=\eta _{\mu \nu }\partial ^{2}f,} and contracting gives ( n − 1 ) ∂ 2 f = 0. {\displaystyle (n-1)\partial ^{2}f=0.} Now focussing on the case n ≥ 3 {\displaystyle n\geq 3} , the two previous equations together show ∂ μ ∂ ν f = 0 {\displaystyle \partial _{\mu }\partial _{\nu }f=0} , so f {\displaystyle f} is at most linear in the coordinates. Substituting into an earlier equation gives that ∂ μ ∂ ν X ρ {\displaystyle \partial _{\mu }\partial _{\nu }X_{\rho }} is constant, so X μ {\displaystyle X_{\mu }} is at most quadratic in coordinates, with general form X μ = a μ + b μ ν x ν + c μ ν ρ x ν x ρ . {\displaystyle X_{\mu }=a_{\mu }+b_{\mu \nu }x^{\nu }+c_{\mu \nu \rho }x^{\nu }x^{\rho }.}

Together, the n {\displaystyle n} translations, n ( n − 1 ) / 2 {\displaystyle n(n-1)/2} Lorentz transformations, 1 {\displaystyle 1} dilatation and n {\displaystyle n} special conformal transformations comprise the conformal algebra, which generate the [conformal group](/source/Conformal_group) of pseudo-Euclidean space.

## See also

- [Affine vector field](/source/Affine_vector_field)

- [Conformal Killing tensor](/source/Conformal_Killing_tensor)

- [Curvature collineation](/source/Curvature_collineation)

- [Einstein manifold](/source/Einstein_manifold)

- [Homothetic vector field](/source/Homothetic_vector_field)

- [Invariant differential operator](/source/Invariant_differential_operator)

- [Killing vector field](/source/Killing_vector_field)

- [Matter collineation](/source/Matter_collineation)

- [Spacetime symmetries](/source/Spacetime_symmetries)

## References

1. **[^](#cite_ref-BYB_1-0)** P. Di Francesco, P. Mathieu, and D. Sénéchal, *Conformal Field Theory*, 1997, [ISBN](/source/ISBN_(identifier)) [0-387-94785-X](https://en.wikipedia.org/wiki/Special:BookSources/0-387-94785-X)

### Further reading

- Wald, R. M. (1984). General Relativity. The University of Chicago Press.

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Adapted from the Wikipedia article [Conformal Killing vector field](https://en.wikipedia.org/wiki/Conformal_Killing_vector_field) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Conformal_Killing_vector_field?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.