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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.

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 $\lambda =1$.

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.

In Cartesian coordinates on Minkowski spacetime, the Conformal Killing equation becomesDifferentiate the equations with index pairs $(\nu ,\rho )$, $(\mu ,\rho )$ and $(\mu ,\nu )$ by ${\partial }_{\mu }$, ${\partial }_{\nu }$ and ${\partial }_{\rho }$, respectively. Adding the first two and subtracting the third, Clairaut's theorem yields the flat conformal Killing integrability formulaApply ${\partial }^{\rho }$ and use ${\partial }^{\rho }{X}_{\rho }=4\lambda$. With the d'Alembert operator $\square ={\eta }^{\alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }$, this givesA tensor contraction with ${\eta }^{\mu \nu }$ gives $2\square \lambda =-4\square \lambda$, so $\square \lambda =0$. Returning to the preceding equation proves the stronger statementThus the conformal Killing factor is an affine function, not an arbitrary solution of the wave equation.

For a Conformal Killing vector field, the same metric compatibility calculation givesThus its Conformal Killing equation is ${\nabla }_{(\mu }{X}_{\nu )}=\lambda g_{\mu \nu }$. Taking the tensor contraction with the inverse metric, and using $g^{\mu \nu }{g}_{\mu \nu }=4$, determines the conformal Killing factor:Parentheses include the factor $1/2$ in tensor index symmetrization.

Write the affine function obtained in part (iii) as $\lambda =c+2b_{\mu }{x}^{\mu }$, with constant $c$ and $b_{\mu }$. A Conformal Killing vector field with precisely this conformal Killing factor isDirect partial differentiation verifies its Conformal Killing equation. Subtracting it from any other solution leaves a Killing vector field $Y$. The formula in part (iii), now with $\lambda =0$, says ${\partial }_{\mu }{\partial }_{\nu }{Y}_{\rho }=0$. Therefore $Y$ is affine, and the Killing equation makes its linear coefficient antisymmetric after lowering an index. Every solution is consequentlyThe four constant $a^{\mu }$ generate translations, the six independent ${\omega }_{\mu \nu }$ generate Lorentz transformations, $c$ generates a dilation, and the four $b^{\mu }$ generate special conformal transformations. Their constant, linear and quadratic terms are independent; within the linear terms, the antisymmetric coefficient cannot equal a nonzero multiple of the identity. Hence the conformal Killing algebra of Minkowski spacetime hasFor $\lambda =1$, the canonical choice is the dilation generator $D=x^{\mu }{\partial }_{\mu }$. Its flow of a vector field solves $d{x}^{\mu }/ds=x^{\mu }$, givingOne may add any Killing vector field without changing $\lambda =1$, so this choice is not unique. The displayed dilation gives a global one-parameter group of diffeomorphisms. The count of fifteen concerns independent Conformal Killing vector fields; some special conformal transformations have only local flows on all of affine Minkowski spacetime.

The product rule for the Lie derivative and the Killing equation giveConsequently $K$ is a Conformal Killing vector field for the rescaled metric tensor, with smooth conformal Killing factorIt remains an actual Killing vector field precisely when $K(\Lambda )=0$: the conformal factor must be constant along its integral curves of a vector field.