  A particular family of time- and space-dependent discrete-time quantum walks
(QWs) is considered in one dimensional physical space. The continuous limit of
these walks is defined through a new procedure and computed in full detail. In
this limit, the walks coincide with the propagation of a massless Dirac fermion
in an arbitrary gravitational field. A QW mimicking the radial propagation of a
fermion outside and inside the event horizon of a Schwarzschild black hole is
explicitly constructed and simulated numerically. Finally, the limiting
procedure and the main result itself are carefully discussed.
