In his book "Dreams of a final theory" pg.144 Steven Weinberg says
"the circulation of the matter of the universe around the zenith seen by observers on a merry-go-round produces a field somewhat like the magnetic field produced by the circulation of electricity in the coils of an electromagnet. It is this Gravitomagnetic field that in the merry go round frame of reference produces the effects that in more conventional frames of reference are attributed to centrifugal force."
He goes on to say that, because of this, in general relativity we can not distinguish whether we are in a rotating laboratory or not.
This seems to be a bold, and wrong, acceptance of a very strong version of Mach's principle that
"all centrifugal effects are caused by the gravitational effects of the rotating stars when viewed from the rotating reference frame."
Weinberg's account seems contrary to modern views on Mach's principle. Yes, there is the Lense-Thirring effect and frame dragging, but that is a long way from saying that all inertial effects in a rotating reference frame are caused by the motion of the surrounding matter in the universe.
Can't we determine whether we are in a spinning spaceship or not (using centrifugal and inertial effect), without reference to the distant stars? Wouldn't Weinberg's claim mean that if we let the mass of the surrounding universe approach zero, the centrifugal effect in the rotating lab frame would go to zero as well?
In short, is Weinberg wrong about Mach's principle in general relativity here?