  We study the effects of spin-orbit coupling in the $d$-density wave (DDW)
phase. In the low-temperature orthorhombic phase of La$_{2-x}$Ba$_x$CuO$_4$, we
find that spin-orbit coupling induces a ferromagnetic moment in the DDW phase,
which is polarized along the [110] direction with a considerable magnitude.
This effect does not exist in the superconducting phase. On the other hand, if
the $d$-density wave order does not exist at zero field, a magnetic field along
the [110] direction always induces such a staggered orbital current. We discuss
experimental constraints on the DDW states in light of our theoretical
predictions.
