To see just what is paradoxical about the statement c = constant, imagine Alicia standing beside a road and Barbara driving by in a supercharged rocket car. Just as Barbara comes abreast of Alicia, a light wave going in the same direction as Alicia passes them. Later on Alicia meets Barbara and remarks, “Did you see that light wave come by us this afternoon? I happened to notice that it was traveling at exactly 300 meters per microsecond.” “You must be in error,” replies Barbara. “ It passed me at a relative speed of 300, and I was doing 100 myself. “No, no,” says Alicia, “I clocked both you and the light wave. Its true that you were doing 100, but the light wave passed you at a relative speed of only 200.” Our common-sense point of view is that either Alicia or Barbara or both must be in error. But according to Einsteins postulate, both are indeed correct. Yet something has to give. If we are prepared to admit that the light wave was moving at the same speed c relative to both Alicia and Barbara, we must admit the possibility that there is some intrinsic difference in the way they are defining speed. Since a measurement of speed involves measurements of both distance and time, perhaps they disagree about length measurements or about time measurements. In fact they must disagree about both.
The hypothetical conversation reported above is, of course, farfetched, not merely because Barbara is reported to have traveled at a quite phenomenal speed, but because in a world where such speeds are commonplace, the constancy of the speed of light and the relativity of time would be so well known in everyday experience that no such controversy would arise. But suppose that the rocket car is a recent invention and that people sometimes fall back into their old ways of thinking. “Sorry,” says Alicia. “I cant stay to argue. Its nearly midnight and I am quite tired.” “What!” replies Barbara. “Why its only 10:00 by my watch, and I feel quite fresh after cruising about all day.”
The conclusion that we are forced to is that the constant speed of light is paradoxical—even impossible—unless observers moving with respect to each other disagree about distance and/or time measurements. It is easy to construct innumerable thought experiments that demonstrate this fact. As a slight variant on the observations of Alicia and Barbara reported above, suppose that on some other occasion Barbara switches on her headlights just as she passes Alicia. “Well,” she reports later to Alicia, “I verified that light travels at 300 meters per microsecond with respect to my rocket car, for exactly 10 –8 seconds after switching on my lights, the light beam had reached a point just 3 meters ahead of me.” “Quite impossible,” replies Alicia wearily. “I took some measurements on that beam myself. After 10 –8 seconds, it had progressed exactly 3 meters from me and was only 2 meters ahead of you. But wait,” says Alicia as an afterthought. “Wasnt your watch running slowly the other day? Perhaps what you took to be 10 –8 seconds was really longer, giving the light beam some extra time to get three meters ahead of you. “
The discrepancies of observations of Alicia and Barbara reported so far could be explained either by a difference in their length scales, or, as Alicia suggests, by a difference in their time scales. Another simple thought experiment will show that their time scales at least must be different. Suppose that Barbara switches on the interior light in her rocket car as she speeds past Alicia. If the light is located just at the center of the car, Barbara will conclude that its illumination reaches the front and the back of her car simultaneously, for she consistently finds that light travels at a fixed speed with respect to her car. But Alicia as adamantly claims to see light always traveling at the same fixed speed with respect to her. From Alicias point of view the light will spread at equal speed in each direction and reach the back of the car, which is approaching the point where the light was turned on, before it reaches the front of the car, receding from that point. Two events—the arrival of light at the front and at the back of the car—are judged by Barbara to be simultaneous and by Alicia to occur at different times. Clearly there must be something different about time itself for the two observers. This particular thought experiment demonstrates wha is known as the relativity of simultaneity.