  We address the fundamental question how the spatial Kondo correlations are
building up in time assuming an initially decoupled impurity spin $\vec{S}_{\rm
imp}$. We investigate the time-dependent spin-correlation function
$\chi(\vec{r},t) = \langle \vec{S}_\mathrm{imp} \vec{s}(\vec{r}) \rangle (t)$
in the Kondo model with antiferromagnetic and ferromagnetic couplings where $
\vec{s}(\vec{r})$ denotes the spin density of the conduction electrons after
switching on the Kondo coupling at time $t=0$. We present data obtained from a
time-dependent numerical renormalisation group (TD-NRG) calculation. We gauge
the accuracy of our two-band NRG by the spatial sum-rules of the equilibrium
correlation functions and the reproduction of the analytically exactly known
spin-correlation function of the decoupled Fermi sea. We find a remarkable
building up of Kondo-correlation outside of the light cone defined by the Fermi
velocity of the host metal. By employing a perturbative approach exact in
second-order of the Kondo coupling, we connect these surprising correlations to
the intrinsic spin-density entanglement of the Fermi sea. The thermal wave
length supplies a cutoff scale at finite temperatures beyond which correlations
are exponentially suppressed. We present data for the frequency dependent
retarded spin-spin susceptibility and use the results to calculate the
real-time response of a weak perturbation in linear response: within the
spatial resolution no response outside of the light cone is found.
