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Along an affinely parametrized geodesic, differentiating the contraction of a vector field $K$ with its tangent $V$ gives $d(K_a{V}^a)/ds=V^a{V}^b{\nabla }_{(a}{K}_{b)}$. If this is zero along every timelike geodesic, arbitrary timelike initial tangents at each point make the symmetric quadratic form vanish on the whole timelike cone. That cone is open, so its quadratic polynomial must vanish identically. Thus $K$ is a Killing vector. The converse follows directly from the same identity. One selected orbit cannot establish the converse; the quantifier over all initial timelike directions is essential.

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.

The timelike cone is the open set of nonzero tangent vectors with timelike squared norm. It has future and past components once a time orientation is chosen. Homogeneity extends an identity for unit timelike vectors to this whole open set. In particular, if a symmetric quadratic form vanishes on every timelike vector, its polynomial vanishes on an open set and therefore vanishes identically. This observation proves the converse from geodesic first integrals to the Killing equation.