  The tunneling dynamics in relativistic strong-field ionization is
investigated with the aim to develop an intuitive picture for the relativistic
tunneling regime. We demonstrate that the tunneling picture applies also in the
relativistic regime by introducing position dependent energy levels. The
quantum dynamics in the classically forbidden region features two time scales,
the typical time that characterizes the probability density's decay of the
ionizing electron under the barrier (Keldysh time) and the time interval which
the electron spends inside the barrier (Eisenbud-Wigner-Smith tunneling time).
In the relativistic regime, an electron momentum shift as well as a spatial
shift along the laser propagation direction arise during the under-the-barrier
motion which are caused by the laser magnetic field induced Lorentz force. The
momentum shift is proportional to the Keldysh time, while the wave-packet's
spatial drift is proportional to the Eisenbud-Wigner-Smith time. The signature
of the momentum shift is shown to be present in the ionization spectrum at the
detector and, therefore, observable experimentally. In contrast, the signature
of the Eisenbud-Wigner-Smith time delay disappears at far distances for pure
quasistatic tunneling dynamics.
