  3+1-dimensional Weyl fermions in interacting systems are described by
effective quasi-relativistic Green's functions parametrized by a 16 element
matrix $e^\mu_\alpha$ in an expansion around the Weyl point. The matrix
$e^{\mu}_{\alpha}$ can be naturally identified as an effective tetrad field for
the fermions. The correspondence between the tetrad field and an effective
quasi-relativistic metric $g_{\mu\nu}$ governing the Weyl fermions allows for
the possibility to simulate different classes of metric fields emerging in
general relativity in interacting Weyl semimetals. According to this
correspondence, there can be four types of Weyl fermions, depending on the
signs of the components $g^{00}$ and $g_{00}$ of the effective metric. In
addition to the conventional type-I fermions with a tilted Weyl cone and
type-II fermions with an overtilted Weyl cone for $g^{00}>0$ and respectively
$g_{00}>0$ or $g_{00}<0$, we find additional "type-III" and "type-IV" Weyl
fermions with instabilities (complex frequencies) for $g^{00}<0$ and $g_{00}>0$
or $g_{00}<0$, respectively. While the type-I and type-II Weyl points allow us
to simulate the black hole event horizon at an interface where $g^{00}$ changes
sign, the type-III Weyl point leads to effective spacetimes with closed
timelike curves.
