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It is not the case that A geodesic directing field (guiding field / projective structure) must be posited to provide a standard of zero 3-acceleration.
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Reasons For
2 perspectives
Reason for 1 of 2
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1.
The projective structure can be read off from the metric tensor, making it a derived feature rather than an independently posited field.
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2.
In general relativity, the metric already encodes geodesic structure, so positing a separate guiding field introduces unmotivated ontological duplication.
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3.
Weyl's own conformal-projective decomposition program presupposes metrical underdetermination that standard GR does not exhibit.
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Reason for 2 of 2
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1.
Dynamical approaches to spacetime structure, as defended by Brown and Pooley, ground inertial standards in the symmetries of matter laws rather than autonomous geometric fields.
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2.
If geodesic behavior is explained by the dynamical dispositions of matter, the projective structure is an unnecessary postulate that violates Occam's razor in its strongest form.
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Reasons Against
1 perspective
Reason against
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1.
There is no unique standard of zero 3-acceleration intrinsic to the differential topological structure of spacetime.
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2.
Physical theories require a determinate standard for zero acceleration.
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3.
The projective structure provides such a standard by supplying the missing geometric field extrinsic to bare topology.
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