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Vectors are generally involved when dealing with forces and moving entities. If a rocket is launched from a moving platform, its overall velocity (when taking a simplified view of rocket dynamics) can be determined by adding the velocity imparted by the rocket propulsion system (v) and the velocity imparted from the moving platform to the rocket (u) as vectors. This would result in an overall rocket velocity of relative to the ground. However, problems arise when trying to apply this same reasoning to light velocity. This section discusses some of the apparent reasons for this and develops a somewhat different set of rules for vector addition when dealing with light. It might be noted that this discussion applies to the motion of photons relative to the time gradient through which they are passing, whereas the TAV discussed elsewhere is generally applied to the velocity of photons relative to an arbitrary observer.
In the previous section Examples: The Room, the photons emitted from a source were assumed to have the velocity of the source imparted to them. These photons would initially travel within the time gradient (TG) associated with the source, which can be considered to be traveling with the source. It is perhaps more accurate to say that the source continuously generates the TG so that the proximal areas of the TG appear to be traveling with the source. Thus the velocity of the source (and the TG) is essentially imparted to the photons, and the photons will travel within the origin’s TG in their original direction at a speed determined by the clock rate of that TG.
If the source were stationary relative to its immediate external environment, then the TG of the source and the TG of the environment would have the same ‘relative velocity’ as a result of any motion common to both. Thus when the photons travel from the source TG to the environment TG they would retain their original direction and would adjust their speed to conform to the clock rate of the different TG in which they find themselves. They would also (apparently) retain the velocity imparted by the source.
The situation becomes more interesting if the source is moving relative to its immediate environment. In this case, because of the velocity imparted to the photons by the source, the photons and the TG of the environment would not have the same ‘motion induced velocity’. This means that while the photons adjust their speed in the original direction of travel to conform to the new clock rate, the source imparted velocity would act to alter the direction of travel through the new TG and cause an increase or decrease in the overall speed of the photons relative to the TG according to normal velocity addition. What apparently happens is that the two velocities do alter the original direction of the photons, but the speed of the photons in the new direction relative to the TG is determined by the local clock rate. When this occurs, it effectively cancels the velocity imparted by the source to the photons, or at least that component relative to the immediate environment.
Figure 1 illustrates the difference between the normal addition of velocities and the addition of velocities when one of them is light and the other affects the velocity of the light through the local time gradient.
Figure 1
Now suppose that light is traveling through some environment and enters the TG of a body that has some relative motion to the environment (such as starlight entering the TG of the earth). Any velocity that was imparted to the light by the source was lost when the light left the TG of the source (and any including environments). Therefore the photons entering the TG of the body will maintain the same direction of travel. However, it will appear to the photons that they are traveling in a direction relative to the TG as determined by the vector addition of the forward velocity of the photons to the reverse velocity of the TG relative to their original direction. Thus, according to this modified form of vector addition, their speed will be equal to C in that direction, which is the speed and direction the photons would appear to be going to an observer on the body. This means that the photons would actually be going slower than C in their original (and current) direction while passing through the TG of the body. Figure 2 illustrates this situation. After the velocity of the photon has adjusted to the local time gradient field, then any component vectors related to the photon path would be determined by normal vector manipulation. Thus the velocity component of the photon in its original direction would be less than its velocity relative to the TG.
Figure 2
The alternative to this latter situation is that the photons continue on their original path with their velocity in that direction determined by the local clock rate. In that case, an observer on the body would measure the velocity of the light as being greater than C. This is illustrated by an example using a moving train and a couple of balls.
Consider a train with an open top boxcar traveling at some velocity v relative to the tracks and bank, or v’ as determined by an observer on the train. A person, O, is situated on some structure by the track which the train must pass. O holds a ball, B, in a position that is over where the boxcar will pass and at a height that is just at the top of the boxcar side. Another person, O’, is riding with the boxcar and is holding a ball, B’, at a height of the boxcar side. When O’ just reaches the position of O, both balls are dropped. Since both balls are falling with respect to the same ‘gravitational field’, they will both reach the boxcar floor at the same time, t’ with respect to the train. However, O’ sees B falling at an angle that causes it to reach the floor at a distance -v’t’ from the position that was under it when it was dropped, while B’ falls to the position that was directly under it when it was dropped. As far as O’ is concerned, B traveled a greater distance than B’ in the same time, and was therefore going faster than B’.
Figure 3 illustrates the vector magnitudes discussed
above.
Figure 3
As an interesting aside, using the Pythagorean theorem we can write
.
Multiplying the right side by 1 in the form of u’/u’, after some rearrangement we get
.
Thus the square root factor can be used to convert between the average velocity due to gravity and the perceived average velocity of B’ as measured by O’. The similarity of this to the factor is clear. The reader may wish to convert this to the form of the equation as it would appear to O using the values of a, v, and u that would be measured by O.
The situation as shown in Figure 2 appears to present a better representation of the behavior of light in that situation. If the velocity of light behaved as in the above example, then light entering the earth’s TG from different angles would appear to be traveling at different speeds (faster than C for forward angles and less than C for rearward angles) relative to earthbound observers.
