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Does the metric really represent "intrinsic" spacetime in plain English?
Does the metric really represent "intrinsic" spacetime in plain English? For a 2D curved surface, the metric, although intrinsic as a mathematical term, doesn't seem to me intrinsic in plain English - it describes the behaviour of the length of a string relative to the surface - and I would consider the string not to be part of the surface, and extrinsic in plain English. This makes more sense to me, since it follows that like the string, the metric is an additional structure; also, since a string curves in 2 directions, the metric should take 2 vectors to define a length. In this view, then there is also some physics in the metric, since we have to specify that we use a string, and not eg. chewing gum. Of course, if we think about it this way, there is also something intrinsic to the metric, since it contains information about the elasticity of the surface relative to the string. Nonetheless, the metric is not completely intrinsic since the string is needed. Furthermore, if the metric represents a string, then we are naturally led to ask - if 4D spacetime has a metric, what is the corresponding 4D string? - to which it seems plausible to say trajectories of particles or light - which prevents one from even thinking of a 'hole argument'.
Does the metric really represent "intrinsic" spacetime in plain English? For a 2D curved surface, the metric, although intrinsic as a mathematical term, doesn't seem to me intrinsic in plain English - it describes the behaviour of the length of a string relative to the surface - and I would consider the string not to be part of the surface, and extrinsic in plain English. This makes more sense to me, since it follows that like the string, the metric is an additional structure; also, since a string curves in 2 directions, the metric should take 2 vectors to define a length. In this view, then there is also some physics in the metric, since we have to specify that we use a string, and not eg. chewing gum. Of course, if we think about it this way, there is also something intrinsic to the metric, since it contains information about the elasticity of the surface relative to the string. Nonetheless, the metric is not completely intrinsic since the string is needed. Furthermore, if the metric represents a string, then we are naturally led to ask - if 4D spacetime has a metric, what is the corresponding 4D string? - to which it seems plausible to say trajectories of particles or light - which prevents one from even thinking of a 'hole argument'.