When I said "state of motion", I was talking about whether motion is inertial or non-inertial.
What does this mean in terms of the mathematics? My understanding (which you seem to confirm) is that "inertial motion" refers to geodesic paths. But those are precisely the paths which (the theory says) describe all motion! In other words, there's no such thing as "non-inertial motion". (Remember that GR is a theory of gravity alone -- as far as it's concerned, the other forces of nature don't exist, so everything is always "in freefall under gravity" at all times.)
your definition of Lorentzian manifold is incorrect
How so? Your link agrees with my definition.
Spacetime is represented by a Lorentzian manifold in special relativity as well
I thought that in special relativity, spacetime was represented by a specific Lorentzian manifold: Minkowski space itself. (Or, perhaps more precisely, an affine space over Minkowski space.) In other words, the manifold is required to be flat (have zero curvature). Whereas in general relativity, the curvature is determined by the equations of motion.
Special relativity is supposed to be what general relativity reduces to in the local limit: it's what goes on in the tangent space at a point. Right?
[Eliezer] says, for instance:
This meant you could never tell the difference between firing your rocket to accelerate through flat spacetime, and firing your rocket to stay in the same place in curved spacetime.
Coupled with his subsequent claim that epiphenomenal distinctions are, as a rule, illusory, he seems to be strongly suggesting that there is in fact no difference between these two states of affairs, which would imply that there is no objective fact of the matter about whether spacetime is flat or curved
It seems to me that you're mixing up the local and global structures of spacetime. There is no fact of the matter about whether spacetime is flat or curved locally, because many of the permissible coordinate changes turn straight lines into curves and vice-versa. However, there is a fact of the matter about the global curvature of the manifold.
Consider the twin paradox: the twin who leaves Earth has the right to say that he/she was at rest the whole time (thus traveling along a path that appeared locally "straight"), but must admit that the region of spacetime through which he/she traveled had nonzero global curvature. (Here, of course, we're assuming that the journey was caused by gravity rather than a rocket ship, in order for GR to be strictly applicable.)
My understanding (which you seem to confirm) is that "inertial motion" refers to geodesic paths. But those are precisely the paths which (the theory says) describe all motion! In other words, there's no such thing as "non-inertial motion". (Remember that GR is a theory of gravity alone -- as far as it's concerned, the other forces of nature don't exist, so everything is always "in freefall under gravity" at all times.)
Some worldlines satisfy the geodesic equation, others don't. The ones which do are geodesics. It's not true...
Today's post, Mach's Principle: Anti-Epiphenomenal Physics was originally published on 24 May 2008. A summary (taken from the LW wiki):
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was My Childhood Role Model, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
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