If physical coincidences and causal structure jointly fix the conformal factor, then zero 4-acceleration is determined up to the projective structure, not left wholly undetermined.
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causal structure(Paired with inertial structure as jointly sufficient to determine metrical structure.)
The structure of the world defined by causal relations, operationalized here via light signals.
projective structure(Contrasted with the infinite-parameter microsymmetry group present when only a differentiable structure exists.)
A post-differentiable topological geometric field on spacetime whose preservation reduces the microsymmetry group at any event to a 20-parameter Lie group.