I am not happy with your definition that v
= dr/dτ. I can't escape the idea that speed should be measured here, on my ds = rdτ worldline where I belong, not there where r=1. Is it a misconception of my part that velocity is a vector, whereas speed is a scalar? Has it anything to do, with our measurements in a 2-dimensional flat space-time ( cylindrical, conical, tilted planar ) embedded in 3-dimensions?
As for vectors, I think there exist manifolds where the dot product is not defined. ( Finslerian manifolds )
I believe that everything should follow from the metric, and from the metric only, since our clocks and rulers are measuring ds's and not arbitrary co-ordinate plantations.
Maybe, my question belongs to another thread... Anyway, thanks for the input.