Quote by Passionflower
The metric distance between two points on the manifold is observer independent.
That is not true, the metric distance is not ambiguous and observer independent.
Then you clearly misapply the notion of 'locally Lorentzian'.

There's really no point in saying "locally lorentz," since the Lorentz Transformations apply to every event in spacetime, it is either globally Lorentz, or not Lorentz at all. Perhaps, MTW should use some other set of words to describe what they are talkng about.
For instance, I would recommend talking about how, within the gravitational pull of a planet, the rate of proper time is a function of the distance from the planet. My suggestion would be to say that in this region, we have a situation where somehow, the geometry seems to differ from Lorentz in some fashion, for it is in these local regions where we find spacetime to be curved. The particle, traveling on a straight path in its own coordinates, ends up traveling on a curved path in another body's coordinates. I think that the theory behind General Relativity is strong enough, that it does not need to rely on ambiguously defined terms and attacking the fundamentals of Special Relativity. It should stand constructively on its foundationsnot try to dismiss them as "only valid locally." On the global level, such slowing of proper time (implicit in general relativity) won't make a difference, because the end result, from a gloabl perspective, is just the local slowing of the speed of light. It's no more paradoxical than having glass, with its index of refraction slowing the speed of light.
But claiming that physics is somehow "locally lorentz" implies that physics is not "globally lorentz." This is something that MTW tries to do throughout "Gravitation" is dismiss Special Relativity as being somehow incompatible with General Relativity. However, I have not yet found any logic in any of their arguments. Only weird claims, like this "locally lorentz" one.