2.2 Ocular velocity versus measured velocity

It is a common misconception that nothing can be seen to travel faster than the speed of light. What should instead be meant is that no object can appear to travel faster than 1c when it is measured from another frame of reference. Ocular velocity is an important aspect of the verification of true relativity, and it can certainly exceed 1c by any amount.

Let us say that two planets are located 10 lm (light-minutes) from each other, and let us send an object from planet A to planet B. The planets are aware of their spatial delay, and when they discuss the measured travel time of the object, they will agree that it took 20 minutes for the object to travel the 10 lm. The measured speed is 0.5c.

If the object were large enough to be optically seen at such a distance, planet B would see the object stand still for the first 10 minutes of the experiment, approach at a speed of 1c and then arrive at the planet after another 10 minutes. Planet A would see the object moving at 0.333c and arrive at planet B after 30 minutes because the ocular event of the landing also requires 10 minutes to reach planet A.

When calculating the ocular velocity, one must take the spatial delay into account by adding it to the travel time on the departure side and by subtracting it from the travel time on the arrival side.

 

Measured travel time	1000 min	100 min	20 min	15 min	12 min
Measured velocity	0.01c	0.1c	0.5c	≈0.667c	≈0.833c
Ocular travel time acc. to A	1010 min	110 min	30 min	25 min	22 min
Ocular velocity according to A	≈0.010c	≈0.091c	≈0.333c	0.4c	≈0.455c
Ocular travel time acc. to B	900 min	90 min	10 min	5 min	2 min
Ocular velocity according to B	≈0.011c	≈0.111c	1c	2c	5c

 

Apparently, there is no limit to the ocular velocity observed from planet B. The object can be seen to move at any speed, but the initiation of the movement of the object is always delayed by 10 minutes. Regardless of how much energy is conferred to the object, it will always require at least some time to travel in addition to the 10-minute spatial delay. The ocular velocity observed from planet B approaches infinity when the measured velocity approaches 1c.

(Remember that an object travelling rapidly toward the observer will appear to be blueshifted. At extremely high speeds, visible light from such objects will become gamma rays, and the term "see" must include "being able to detect electromagnetic waves of any frequency".)

The equation used to transform the ocular approaching velocity (β_oa) into the measured velocity (β_m) can also be used to transform the measured velocity (β_m) into the ocular departing velocity (β_od).

The above equation implies that β_a is less than β_b. To transform the velocities back, the reverse equation is used:

The following table summarizes the results of such transformations for various velocities:

 

βoa (ocular approaching)	βm (measured)	βod (ocular departing)
0.001	0.000999	0.000998
0.01	0.0099	0.0098
0.1	0.091	0.083
0.2	0.167	0.143
0.25	0.2	0.167
0.3333	0.25	0.2
0.5	0.3333	0.25
1	0.5	0.3333
2	0.667	0.4
5	0.833	0.45
100	0.99	0.4975
1000	0.999	0.499975

 

It should be noted that with increasing velocity,

• β_oa approaches infinity,
• β_m approaches 1 (the spatial delay) and
• β_od approaches 0.5 (half the spatial delay).