The Killing vector is $$ \xi_t~=~\sqrt{1~-~r_s/r}\partial_t $$ and reduces to you case in the asymptotic region, or on the rest frame of any observer. For $\xi_\mu U^\mu~=~\epsilon$ what this tell us is the metric may be expressed as $$ 1~=~\epsilon^2~-~\frac{1}{\sqrt{1~-~r_s/r}}(U^r)^2~-~r^2\left((U^\theta)^2~+~sin\theta(U^\phi)^2\right), $$ which defines a Hamiltonian.

The Killing vector is $$ \xi_t~=~\sqrt{1~-~r_s/r}\partial_t $$ and reduces to you case in the asymptotic region, or on the rest frame of any observer. For $\xi_\mu U^\mu~=~\epsilon$ what this tell us is the metric may be expressed as $$ 1~=~\epsilon^2~-~\frac{1}{{1~-~r_s/r}}(U^r)^2~-~r^2\left((U^\theta)^2~+~sin\theta(U^\phi)^2\right), $$ which defines a Hamiltonian.