We’ve now established a means for measuring the time of an event in a reference frame, so let’s begin to investigate the nature of time. The following “thought experiment” is very similar to one suggested by Einstein.

FIGURE 36.15 shows a long railroad car traveling to the right with a velocity v that may be an appreciable fraction of the speed of light. A firecracker is tied to each end of the car, just above the ground. Each firecracker is powerful enough so that, when it explodes, it will make a burn mark on the ground at the position of the explosion.

FIGURE 36.15 A railroad car traveling to the right with velocity $v$.

Ryan is standing on the ground, watching the railroad car go by. Priya is standing in the exact center of the car with a special box at her feet. This box has two light detectors, one facing each way, and a signal light on top. The box works as follows:

1. If a flash of light is received at the detector facing right, as seen by Ryan, before a flash is received at the left detector, then the light on top of the box will turn green.

2. If a flash of light is received at the left detector before a flash is received at the right detector, or if two flashes arrive simultaneously, the light on top will turn red.

The firecrackers explode as the railroad car passes Ryan, and he sees the two light flashes from the explosions simultaneously. He then measures the distances to the two burn marks and finds that he was standing exactly halfway between the marks. Because light travels equal distances in equal times, Ryan concludes that the two explosions were simultaneous in his reference frame, the reference frame of the ground. Further, because he was midway between the two ends of the car, he was directly opposite Priya when the explosions occurred.

FIGURE 36.16a shows the sequence of events in Ryan’s reference frame. Light travels at speed c in all inertial reference frames, so, although the firecrackers were moving, the light waves are spheres centered on the burn marks. Ryan determines that the light wave coming from the right reaches Priya and the box before the light wave coming from the left. Thus, according to Ryan, the signal light on top of the box turns green.

FIGURE 36.16 Exploding firecrackers seen in two different reference frames.

Full Alternative Text

How do things look in Priya’s reference frame, a reference frame moving to the right at velocity v relative to the ground? As FIGURE 36.16b shows, Priya sees Ryan moving to the left with speed v. Light travels at speed c in all inertial reference frames, so the light waves are spheres centered on the ends of the car. If the explosions are simultaneous, as Ryan has determined, the two light waves reach her and the box simultaneously. Thus, according to Priya, the signal light on top of the box turns red!

Now the light on top must be either green or red. It can’t be both! Later, after the railroad car has stopped, Ryan and Priya can place the box in front of them. Either it has a red light or a green light. Ryan can’t see one color while Priya sees the other. Hence we have a paradox. It’s impossible for Priya and Ryan both to be right. But who is wrong, and why?

What do we know with absolute certainty?

1. Ryan detected the flashes simultaneously.

2. Ryan was halfway between the firecrackers when they exploded.

3. The light from the two explosions traveled toward Ryan at equal speeds.

The conclusion that the explosions were simultaneous in Ryan’s reference frame is unassailable. The light is green.