  The large $j$ asymptotic behavior of $4$-dimensional spin foam amplitude is
investigated for the extended spin foam model (Conrady-Hnybida extension) on a
simplicial complex. We study the most general situation in which timelike
tetrahedra with timelike triangles are taken into account. The large $j$
asymptotic behavior is determined by critical configurations of the amplitude.
We identify the critical configurations that correspond to the Lorentzian
simplicial geometries with timelike tetrahedra and triangles. Their
contributions to the amplitude are phases asymptotically, whose exponents equal
to Regge action of gravity. The amplitude may also contains critical
configurations corresponding to non-degenerate split signature $4$-simplices
and degenerate vector geometries. But for vertex amplitudes containing at least
one timelike tetrahedron and one spacelike tetrahedron, critical configurations
only give Lorentzian $4$-simplices, while the split signature and degenerate
$4$-simplices do not appear.
