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A contravariant vector has coordinate components obeying the displayed transformation law. A tangent vector to a parametrized curve provides the basic example: the chain rule transforms $d{x}^a/ds$ in precisely this way. Pairing it with a covector produces a coordinate-independent scalar. Conversely, if an array pairs invariantly with every contravariant vector, coefficient comparison forces the covector transformation law for that array.

Under a smooth coordinate change $x\mapsto x^{\prime }$, a contravariant vector obeysThe scalar-contraction hypothesis gives $U_a^{\prime }(\partial x^{\prime a}/\partial x^b)T^b=U_b{T}^b$ for every choice of $T^b$. Equality of all coefficients and multiplication by the inverse Jacobian matrix give $\boxed{U_a^{\prime }=(\partial x^b/\partial x^{\prime a})U_b}$, the covector law.

Let $V^a=d{x}^a/ds$ be a timelike geodesic tangent. Metric compatibility and the geodesic equation giveA Killing vector therefore gives a conserved contraction. Conversely, if that contraction is conserved along every timelike geodesic, then at any point its symmetric tensor $S_{ab}={\nabla }_{(a}{T}_{b)}$ satisfies $S_{ab}{V}^a{V}^b=0$ for every unit timelike initial tangent. Homogeneity extends this to the entire timelike cone, an open set of tangent vectors. A quadratic polynomial vanishing on an open set has every coefficient zero, so $\boxed{{\nabla }_{(a}{T}_{b)}=0}$. Considering a single selected geodesic would not suffice. This is the Converse geodesic characterization of a Killing vector.

For the given exterior transformation, differentiating at fixed $r$ givesThe transformation also gives $UV=(2m-r)e^{r/(2m)}$. This determines $r$ smoothly for $r>0$ throughout $UV<2m$. The flow of the displayed vector field rescales $U\mapsto e^{-\lambda /(4m)}U$, $V\mapsto e^{\lambda /(4m)}V$. It leaves $UV$, $r$, $dUdV$ and the angular metric unchanged. Thus it preserves the full metric everywhere in that region and extends as a global Killing field, not merely by a formal exterior coordinate substitution.

Its squared norm isIt is timelike for $UV<0$ ($r>2m$), spacelike for $0<UV<2m$ ($0<r<2m$), and a nonzero null vector on either horizon $UV=0$ away from $U=V=0$. At $U=V=0$ it is the zero vector, on the whole bifurcation surface of angular points. Zero must be distinguished from a nonzero null vector. This is the Schwarzschild time-translation Killing field in Kruskal coordinates.