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I should clarify this to forestall a question that might be coming from @pervect in view of the post of his that I just responded to. Since the ground observer's worldline is not a geodesic, the correct term for how the 4-momentum of emission of the light pulse is transported along that worldline is Fermi-Walker transport, not parallel transport. Or, more relevant to the current discussion, Lie transport (i.e., transport by the Lie derivative along the worldline), since that is what preserves the inner product of the 4-momentum with the Killing vector field.parallel transporting