I find it difficult to get the big picture of GR, i.e. to connect the various components described by Reichenbach and Norton. These components are the principle of equivalence, the intrinsic curvature of spacetime that arises in the vicinity of masses, the spacetime and space-space sheets, the role of free fall etc. I'm also not sure I understand the notions Reichenbach discusses introducing the underlying concepts, like the conventionality of motion, the "transforming away of gravitation" and the distinction of local and astronomical inertial systems.
So the general context of GR is Einstein's aim to extend his theory of special relativity to gravitation, since it was incompatible with Newton's theory of gravitation. This new gravitational theory then postulates gravitation not as a force, but as an intrinsic curvature of spacetime around masses. It is no longer a force acting between bodies but a geometrical property of spacetime. This is often depicted using this image of masses in curved space like the surface of a blanket around a heavy object on it, which is inaccurate because the curvature is not merely a feature of space but also of spacetime, different sheets over time.
The basic idea of the principle of equivalence is that the effects of gravitation and inertial effects in an accelerated system (which in Newtonian physics seem to correspond coincidentally) are really the same thing, which arises from the fact that acceleration creates a gravitational field.
So is gravitation both inertial motion in curved spacetime and the effect of accelerated systems? I don't get this connection. Does acceleration create curvature of spacetime or is it equivalent to inertial motion in curved spacetime? Is it in this sense that free fall is a special kind of motion, in that it is both inertial motion in curved spacetime and accelerated motion creating such a field? And does GR abandon the idea of forces altogether, or does it merely postulate the equivalence of descriptions of forces in terms of curved spacetime?
The example of balls in free fall inside the earth in Norton's book shows how free fall is now described in terms of curved spacetime, where the trajectory of the balls traces geodics as the straightest lines in non-Euclidean geometry.
Lastly, I wonder what the role of the conventionality of motion for the theory is and what this concept of local inertial systems means?
Reichenbach spends a lot of time talking about what you might call philosophical origins or underpinning or inspiration for relativity, but I wonder how deep we should consider this to be---just because something inspired Einstein it doesn't mean that it is a correct philosophical moral to draw from GR.
ReplyDeleteFor instance Mach's solution to Newton's bucket thought experiment: that inertial effects are caused by motion relative to nearby matter. It's not clear to me how much this carries through in GR since Minkowski Spacetime solves the GR field equations for no matter, and yet we can still have inertial effects in rotating systems in SR.