Weak hyperbolicity of the system (4.20, 4.21) with given shift and densitized lapse has also been pointed out in [254
]
based on a reduction, which is first order in time and space. However, there are several inequivalent such reductions, and so it
is not sufficient to show that a particular one is weakly hyperbolic in order to infer that the second-order-in-space
system (4.20, 4.21) is weakly hyperbolic.