Conformal Killing vector field

A conformal Killing vector field generates a local flow that preserves the metric up to a position-dependent scale factor.

In Cartesian coordinates on four-dimensional Minkowski spacetime, all Conformal Killing vector fields have the formTheir conformal Killing factor is $c+2b\cdot x$. There are four translation generators, six Lorentz transformation generators, one dilation generator, and four special conformal transformation generators. These form a fifteen-dimensional Lie algebra under the Lie bracket of vector fields.

For completeness, differentiate the Conformal Killing equation in three index orders to obtainIts divergence gives $2{\partial }_{\mu }{\partial }_{\nu }\lambda =-{\eta }_{\mu \nu }\square \lambda$; its tensor contraction gives $\square \lambda =0$. Hence $\lambda$ is an affine function. Subtracting the displayed quadratic and dilation terms leaves a Killing vector field whose second partial derivatives vanish, proving completeness of the list. The vector fields are globally smooth, but some special conformal transformations have incomplete flows on affine Minkowski spacetime; the dilation flow $x\mapsto e^{s}x$ is global.

Let $X$ be a Conformal Killing vector field with conformal Killing factor $\lambda$, and let $U$ be the tangent vector of a geodesic with affine parameter $s$. The product rule, metric compatibility and geodesic equation giveEvery null geodesic therefore has the conserved quantity $g(X,U)$. On a timelike geodesic with proper time as parameter, the derivative is $-\lambda$ in metric signature $(-,+,+,+)$. With a nonaffine parameter satisfying ${\nabla }_{U}U=fU$, an extra term $fg(X,U)$ appears; the conserved expression for a null geodesic is tied to its affine parameter normalization.

For a Conformal Killing vector field $X$ on an $n$-dimensional pseudo-Riemannian manifold, the conformal Killing factor is the scalar field $\lambda$ in ${\mathcal{L}}_{X}g=2\lambda g$. For the Levi-Civita connection, taking the tensor contraction of the Conformal Killing equation gives $\lambda ={\nabla }_{\mu }{X}^{\mu }/n$. A Killing vector field has $\lambda =0$, while the dilation generator $x^{\mu }{\partial }_{\mu }$ on Minkowski spacetime has $\lambda =1$.

In $d$ dimensions, the conformal Killing equation is

For a flat Conformal Killing vector field, let $\rho ={\partial }_{\mu }{\omega }^{\mu }$. Differentiate the Conformal Killing equation in three index orders, add two equations and subtract the third to obtain the displayed identity. Tracing gives $\Delta {\omega }_{\alpha }=(2-d){\partial }_{\alpha }\rho /d$, whose divergence implies $\Delta \rho =0$ for $d>1$. Taking the divergence of the original identity gives $(d-2){\partial }_{\beta }{\partial }_{\gamma }\rho =-{\delta }_{\beta \gamma }\Delta \rho$. Thus $\rho$ is affine and $\omega$ is at most quadratic when $d>2$.