One-way speed of light and clock sync

[pervect]

I see what you are doing here, and it makes sense. But the dot product above is frame invariant, which is not what the OP wants and is not what is usually meant by "velocity". Using this formulation the speed of light is c no matter what coordinates or synchronization convention you use. At that point, I would just go ahead and use four-vectors.

Ah - well, you're right in that I assumed standard, isotropic clock synchronizations in my analysis and wasn't considering non-standard ones.

I would have to agree that with non-Einsteinian clock synchronizations, one marks out some straight course of known distance, synchronizes the two clocks at start and end of the course, and divide the distance by the change in clock readings (which represents the time interval according to the synchronization convention used) to get the velocity.

This will be the average velocity, one needs the additional step of taking the limit as the distance goes to zero in general - taking the limit gets rid of the effects of acceleration or curvature that I was concerned about.

 

[nikolakis]

No, DaleSpam has insisted and continues to insist that v = dr/dt. c is a defined constant. There may be an alternative standard definition for velocity, but I am not aware of it.

What pervect is discussing is (to the best of both his knowledge and mine) not the standard definition of velocity. As he explicitly mentioned:

For four-vectors, the magnitude of the four-velocity of a light pulse is always 0, regardless of the coordinates.

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.