Completely agree, which I think is very interesting. In Newtonian mechanics, some scalar and vector quantities such as mass, internal energy, contact forces (stress), heat flux are frame-indifferent. Other, such as velocity and acceleration, are frame-dependent but we do have transformation rules for them. Some quantities – and quite important ones – such as momentum, are in a sort of limbo: they are frame-dependent, but there’s no clear transformation rule for them.

From the point of view of relativity theory, it’s interesting to note that for this particular case of coordinate transformation – note that it is not a Lorentz boost – we can actually calculate the spatial components of momentum in the new coordinate system, if the reported momentum is expressed as a covariant vector (p_µ). This is because its unknown temporal component (energy) gets multiplied by zero in this transformation. But the text is ambiguous on whether the reported components are covariant or contravariant.