The question as to the character of both time and light has been the most vexing problem for physicists since the nineteenth century. Shedding 'light' on this problem was the ultimate objective of both Lorentz and Einstein. Here their results are summarized and reevaluated.
The following argument, using only simple algebra, shows that the Lorentz factor, gamma, is a pure number, comparing the time of travel of light under two models of physical reality -and it is not necessary or correct to use either a square root or a division by two in calculating this factor.
The prevailing belief of physicists at the end of the 19th and beginning of the 20th century was that light is carried by a medium, called the 'aether'. Further, that the aether is stationary in the universe with the sun at the center. So that an experiment with two points on earth, and a mirror at the second point, so oriented as to let the light travel in the direction of the motion of the earth, and return in the opposite direction, will show the total time of travel to be
using the fact that time = distance/velocity.
This can then be written as
where L is the distance between the mirrors and v is the orbital velocity of the earth.
In the alternative model where there is no carrier the time is the same in both directions and is given by
The ratio of these is a pure number
which is the square of what Lorentz calls gamma.
But comparing the total time and the average time for a one-away path leads to the same ratio - a pure, dimensionless factor. We are comparing two models of reality, and the ratio is the same for a round trip as for an average one-way trip.
So the corrected gamma factor is neither the square-root, nor 1/2 of the ratio of the total times. THE SQUARE OF GAMMA is the correct comparison of these two models.
If we then use the largely negative results from various attempts at duplicating the results of the Michelson-Morley experiment, we come to the conclusion that the ratio is most likely one - that there is no carrier for light.
Einstein derived the Lorentz Transformation, not by assuming that light is carried and has the speed c only in the coordinate system in which the aether is at rest, but instead that it has the speed c in all coordinate system that are in relative motion. Einstein believes, apparently, that any observer will measure the speed of light to be the same constant, c, no matter weather the source was in motion relative to the coordinate system in which the observer received and measured the speed of the beam of light.
As a concrete example: We know that the velocity of a plane is measured with respect to the air stream through which it moves. If we fly from, say, San Francisco to New York and back and the air is still, it will take six hours each way, a total of 12 hours, at 500 miles per hour. If the air is moving at 100 miles an hour, from west to east, and the plane flies at 500 miles per hour, with respect to the air stream, the distance of about 3000 miles is covered in five hours (at 500+100 miles per hour) and the return trip takes 7.5 hours (at 500-100 miles per hour). So the total time is not 12 hours but 12.5 hours! The gain and loss, due to the movement of the air, don’t quite balance out. Analogously, that would be the situation if the velocity of light is measured with respect to the aether, and the earth moves through the aether.
Here c=500, v=100, and L=3000, so the ratio t(v)/t(0) = 1.04
It is easy to show that if the round trip distance is reduced by 240 miles, or the distance between SF and NY by 120 miles, that is, by one-half this amount, it takes about 4.7 hours going, 7.3 hours returning, and the total time will be 12 hours – the same as it would be if there were no jet stream.
We cannot use the square root i.e. the Lorentz Transformation, since it would reduce the distance by about one-half this amount – not enough to bring the time back to 12 hours. The point to notice, from this example, is that even if the sun were the center of the universe, and the aether exists and is stationary in that universe (as Lorentz believed), taking the square root is unjustified and leads to a false conclusion.
This hypothesis leads him to the same expression, gamma, but it has a different meaning! Einstein's view implies that time is relative - observers in relatively moving coordinate system will not measure time the same way. (It is hard to summarize his lengthy and tortuous discussion clearly. He wants the formulas to do that work for him.)
It probably does not need to be pointed out that no body, except possibly light, has a velocity in itself. The term 'velocity' always implies a reference to an inertial, or coordinate system, with respect to which the object is moving, or at rest, (and presumably the observer is at rest in that coordinate system in arriving at the 'speed' he is measuring.)
Lorentz did his work prior to Einstein's Special Relativity. He thought about light in the context of the Michelson-Morley experiment, which assumed that the velocity of the observer's frame of reference relative to the aether would show up in that experiment.
We can illustrate Einstein's thinking by means of a down to earth example: Suppose we have a train moving at 80 mph and on the road next to the tracks a car is moving at 40 mph. If we want the train to move at 80 mph with respect to the car we can increase the time so that one hour is twice as long in the car when viewed from the ground. So with respect to this 'relative' time the train moves at 80 mph with respect to the car. Of course Einstein only intended the speed of light to be the same, c, in all systems in relative motion, (not the speed of the train) - and the time dilation required for light turns out to be given by the same expression that Lorentz called gamma.
On a more serious note, we are now able to answer the question: A QUESTION OF TIME? The answer depends on understanding and pulling together the discussion of the Doppler effect, the evidence from deep space astronomy, and the results of the Michelson Morley experiment.
The Doppler red shift, in the case of light, can be the result of motion by the source away from the receiver. But the same result could also be produced if the receiver moves away from the source.
If the speed of light were independent of the motion of the source, any Doppler effect could only be due to the movement of the receiver away from the source, not the movement of the source itself. This means that the extreme red shifts observed from very distant supernovae would be due to the movement of the earth, or perhaps the solar system, relative to the path of the light reaching us from each of the ancient supernovae. But we receive such Doppler shifts from many directions (the cosmology project of 1998 identified over 40 type 1A high red shift supernovae). This fact rules out the conjecture that we could be moving away from each of these detected explosions. Therefore the Doppler effect must be attributed to the sources, and that implies that the speed of light relates to the movement of the source relative to earth, as well as an expansion of the universe, to explain these large Doppler red shifts. (At the very least there was expansion of the universe at the time these signals originated, billions of years ago!) In any case, the speed of light is not independent of the movement of the source as Einstein claimed. In other words Einstein's basic belief, the basis and substance of Special Relativity, is false. TIME IS NOT RELATIVE. The question is answered!