  We define the entanglement entropy of free fermion quantum states in an
arbitrary spacetime slice of a discrete set of points, and particularly
investigate timelike (causal) slices. For 1D lattice free fermions with an
energy bandwidth $E_0$, we calculate the time-direction entanglement entropy
$S_A$ in a time-direction slice of a set of times $t_n=n\tau$ ($1\le n\le K$)
spanning a time length $t$ on the same site. For zero temperature ground
states, we find that $S_A$ shows volume law when $\tau\gg\tau_0=2\pi/E_0$; in
contrast, $S_A\sim \frac{1}{3}\ln t$ when $\tau=\tau_0$, and
$S_A\sim\frac{1}{6}\ln t$ when $\tau<\tau_0$, resembling the Calabrese-Cardy
formula for one flavor of nonchiral and chiral fermion, respectively. For
finite temperature thermal states, the mutual information also saturates when
$\tau<\tau_0$. For non-eigenstates, volume law in $t$ and signatures of the
Lieb-Robinson bound velocity can be observed in $S_A$. For generic spacetime
slices with one point per site, the zero temperature entanglement entropy shows
a clear transition from area law to volume law when the slice varies from
spacelike to timelike.
