The set P_f of freely falling particle paths cannot be us... — Carmelics

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The set P_f of freely falling particle paths cannot be used to establish the differential structure of spacetime without circularity

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Reasons For

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Reason for

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1.P_f is a subset of P that consists of paths of spherically symmetric, electrically neutral particles
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2.P_f can only be properly characterized after a coordinate system (differential structure) is available
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3.Using P_f to establish differential structure would require differential structure to already be in place
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Reasons Against

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Reason against 1 of 2

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1.Weyl's own constructive program shows P_f can be identified via purely topological and affine criteria prior to any smooth atlas.
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2.The characterization of free-fall paths requires only continuity and extremal properties, not a pre-given differential structure.
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3.Ehlers, Pirani, and Schild (1972) demonstrated that projective structure from particle paths logically precedes and grounds the differential manifold.
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Reason against 2 of 2

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