{{Short description|Vector field in conformal geometry}} 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.
==Densitized metric tensor and Conformal Killing vectors== 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 (topology)|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.
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 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> 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> 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== The following are equivalent # <math>X</math> is a conformal Killing vector field, # The (locally defined) flow of <math>X</math> preserves the conformal structure, # <math>\mathcal{L}_X (g\mu_g^{-2/n}) = 0,</math> # <math> \mathcal{L}_X g = \frac{2}{n} \operatorname{div}(X) g,</math> # <math> \mathcal{L}_X g = \lambda g </math> for some function <math>\lambda.</math> 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 <math>\lambda = (2/n) \operatorname{div}(X)</math>.
The last form makes it clear that any Killing vector is also a conformal Killing vector, with <math>\lambda \cong 0.</math>
==The conformal Killing equation== Using that <math>\mathcal{L}_X g = 2 \left(\nabla X^\flat \right)^{\mathrm{symm}}</math> where <math>\nabla</math> is the Levi Civita derivative of <math>g</math> (aka covariant derivative), and <math>X^{\flat}=g(X,\cdot)</math> is the dual 1 form of <math>X</math> (aka associated covariant vector aka vector with lowered indices), and <math>{}^{\mathrm{symm}}</math> is projection on the symmetric part, one can write the conformal Killing equation in abstract index notation as :<math>\nabla_a X_b + \nabla_b X_a = \frac{2}{n}g_{ab}\nabla_{c}X^c.</math>
Another index notation to write the conformal Killing equations is :<math> X_{a;b}+X_{b;a} = \frac{2}{n}g_{ab} X^c{}_{;c}.</math>
==Examples== ===Flat space=== In <math>n</math>-dimensional flat space, that is [[Euclidean space]] or [[pseudo-Euclidean space]], there exist globally flat coordinates in which we have a constant metric <math>g_{\mu\nu} = \eta_{\mu\nu}</math> where in space with signature <math>(p,q)</math>, we have components <math>(\eta_{\mu\nu}) = \text{diag}(+1,\cdots,+1,-1,\cdots,-1)</math>. In these coordinates, the connection components vanish, so the covariant derivative is the coordinate derivative. The conformal Killing equation in flat space is <math display = block>\partial_\mu X_\nu + \partial_\nu X_\mu = \frac{2}{n}\eta_{\mu\nu} \partial_\rho X^\rho.</math> 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]] of isometries of flat space. Considering the ansatz <math>X^\mu = M^{\mu\nu}x_\nu,</math>, we remove the antisymmetric part of <math>M^{\mu\nu}</math> as this corresponds to known solutions, and we're looking for new solutions. Then <math>M^{\mu\nu}</math> is symmetric. It follows that this is a [[homothety|dilatation]], with <math>M^\mu_\nu = \lambda\delta^\mu_\nu</math> for real <math>\lambda</math>, and corresponding Killing vector <math>X^\mu = \lambda x^\mu</math>.
From the general solution there are <math>n</math> more generators, known as [[special conformal transformation|special conformal transformations]], given by :<math>X_\mu = c_{\mu\nu\rho}x^\nu x^\rho,</math> where the traceless part of <math>c_{\mu\nu\rho}</math> over <math>\mu,\nu</math> vanishes, hence can be parametrised by <math>c^\mu{}_{\mu\nu} = b_\nu</math>.
{| class="wikitable collapsible collapsed" ! General solution to the conformal Killing equation (in more than two dimensions)<ref name="BYB">P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', 1997, {{ISBN|0-387-94785-X}}</ref> |- | For convenience we rewrite the conformal Killing equation as <math display = block> \partial_\mu X_\nu + \partial_\nu X_\mu = f(\mathbf{x})g_{\mu\nu}.</math> (By taking traces we can recover <math>f(\mathbf{x}) = \frac{2}{n}\partial_\rho X^\rho.</math>)
Applying an extra derivative, relabelling indices and taking a linear combination of the resulting equations gives <math display=block>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.</math> Contracting on <math>\mu,\nu</math> gives <math display=block> 2 \partial^2 X_\rho = (2 - n)\partial_\rho f.</math> A combination of derivatives of this and the original conformal Killing equation gives <math display=block>(2-n)\partial_\mu\partial_\nu f = \eta_{\mu\nu} \partial^2 f,</math> and contracting gives <math display=block> (n-1)\partial^2 f = 0.</math> Now focussing on the case <math>n \geq 3</math>, the two previous equations together show <math>\partial_\mu \partial_\nu f = 0</math>, so <math>f</math> is at most linear in the coordinates. Substituting into an earlier equation gives that <math>\partial_\mu \partial_\nu X_\rho</math> is constant, so <math>X_\mu</math> is at most quadratic in coordinates, with general form <math display=block>X_\mu = a_\mu + b_{\mu\nu}x^\nu + c_{\mu\nu\rho}x^\nu x^\rho.</math> |}
Together, the <math>n</math> translations, <math>n(n-1)/2</math> Lorentz transformations, <math>1</math> dilatation and <math>n</math> special conformal transformations comprise the conformal algebra, which generate the [[conformal group]] of pseudo-Euclidean space.
==See also== * [[Affine vector field]] * [[Conformal Killing tensor]] * [[Curvature collineation]] * [[Einstein manifold]] * [[Homothetic vector field]] * [[Invariant differential operator]] * [[Killing vector field]] * [[Matter collineation]] * [[Spacetime symmetries]]
==References== {{reflist}}
=== Further reading === * Wald, R. M. (1984). General Relativity. The University of Chicago Press. [[Category:Differential geometry]] [[Category:Mathematics of general relativity]]