When people write short stories or novels, they’re creating something a physicist would call a thought experiment. The results of the experiment, for the reader at least, are unknown, which of course is what keeps things interesting. Until the story is finished, the writer is likely not to know how it will turn out either. Thus, the following is a thought experiment on the subject of time.
Let’s start with the car-on-the-roadway example, which shows how the time of a distant event depends on which rest frame you’re in. Any event you see, by the way, is a distant event. Looking at yourself in a mirror is the observation of a series of distant events, namely the emission of photons from the atoms of the mirror. If you’re looking at your face in the bathroom mirror, you observe these events about a nanosecond after they happen, so you never see your face as it is, but only as it was about two nanoseconds prior to your observation.
Now you’re ready to think about light travel time and the case of the flashing streetlight.
The driver of the car is our only observer. He's a philosopher. He's driving on a long straight road, and he’s thinking about how it's possible to say he's not moving and the road and all the scenery are in motion instead. Philosophers think about things like this in their idle time. Of course, the gas gauge is showing a continuous use of fuel, but he realizes this is going to happen whether the car or the road is moving, because of the fact that air (stationary with respect to the road, let’s say) is pushing against the car and fuel is needed to just to keep the car stationary against the force of the air. But then you have to ask, “Stationary with respect to what?” So, in the ideal case, which philosophers dearly love, the air resistance and the friction of moving parts in the car are ignored.
In such a frictionless environment, no force is needed to keep either the car or the road moving at constant speed. That is why as far as motion itself is concerned, the philosopher in the car has a choice. He can say he's moving or everything else is moving. But this gets him nowhere as far as drawing conclusions about the time of a distant event. So let's let him try to determine the time of a flash of light—a basic physical measurement—and see if he still has the same choice of saying he's moving or everything else is moving.
Stan O'Stanley is the guy's name, by the way. He passes what appears to be a burned out streetlight and sets his trip odometer to zero, just because he's the suspicious type. "You can't trust these streetlights anymore," he says to himself. "They seem to go off and on as if some jokester is playing a game with you." Sure enough, a while later in his rearview mirror he sees the streetlight flash briefly.
In his philosophical mind’s eye, Stan creates two pictures, one with the streetlight moving away from the rear bumper of the car, and one with the car moving away from the streetlight. Both pictures exclude everything except the streetlight and car, giving an ideal picture of relative motion: two objects in empty space moving at a constant speed relative to each other. The question Stan wants to answer is simple. “I want to know when the light left the streetlight,” he says, out loud. “I know relativity says time is relative, but what that means exactly, I’m not sure. So I’ll do a little thought experiment.” He is about to use his cell phone to call a physics-trained friend of his so they can discuss the matter, but he realizes it’s too early in the morning to do that. So he instead checks some data his car’s computer has collected.
Besides being a philosopher, Stan is a gadget nut. He’s got an electronic photodetector built into his rearview mirror, and a computer that keeps track of the timing of all the gadgets in his car. He types a few commands on the keyboard with his right hand and retrieves the odometer reading and the time the light from the streetlight was first detected by the rearview mirror, both shown in a super-large font. The odometer reading is 0.400000001 miles and the time is 4:25:03.090909092 a.m. Stan has been out late and is headed home, which is not part of this story, so just observe that he has a very precise odometer and clock in his car, a necessity for doing this particular experiment. Leave it to Stan to have the right stuff.
One thing he knows is that in calculating the time of the beginning of the flash, relativity says he must assume the speed of light leaving the streetlight is unaffected by the motion of the streetlight or the motion of the car.
First, he thinks of the streetlight as stationary. He assumes—not having dealt with relativity calculations before—the distance light traveled to get to him is just the reading on his odometer, and using that distance and the speed of light, he can calculate the time of the beginning of the flash. He taps a few keys on his keyboard and, connecting the computer to his cell phone line, looks up the exact speed of light at the National Institute of Standards and Technology website (http://physics.nist.gov).
He feels happy and starts whistling “If I Only Had a Brain” from The Wizard of Oz, thinking of what he might eat when he gets home. Then the exact value for the speed of light shows up on the screen: 299,792,458 meters per second. He stops whistling and hits the steering wheel with his open palm. “Blast it!” he shouts.“ I have to convert the odometer reading to meters! Well, no big deal.” Keeping an eye on the road, which is practically deserted anyway, he types in a few keystrokes and “644.000002 meters” appears on the screen.
“Now if I divide that by the speed of light,” he says with satisfaction, “I’ll have the time light took to reach the car.” (There is or soon will be a math appendix at the end of this story, for the mathematically curious.)
Stan hits a few more keys and the flat screen display shows the number 0.0000002148, the time in seconds the light took to reach the rearview mirror. “One more little calculation,” he says, resuming his whistling with renewed intensity. He retrieves the time-of-light-arrival reading, 4:25:03.090909092 a.m., and subtracts 0.0000002148 from it to get 4:25:03.090906944 a.m. Light travels fast, so not much time has passed. Stan is proud of himself and his gadgets for being able to do the precise calculation. He loved doing simple math in grade school but hated algebra in high school and has never gotten over his math inferiority complex.
When he arrives home, he makes a cheese, lettuce and tomato sandwich, opens a bag of corn chips and Vanilla Cream Soda, and sits down to do some more calculating. “Now how is it different if I consider the streetlight to be moving away from the car?” he wonders—out loud, of course.
Herbert J. Steingold Jr. is having an elaborate dream when the ringing of his bedside phone wakens him at what his clock radio shows to be 5:12 a.m. Not being a call-screener, Herbert eschews Caller ID, and in fact enjoys picking up the phone without knowing who’s there. But he doesn’t enjoy a five a.m. call that wakes him up.
“Who the hell is calling me at five in the morning?!” he grumbles loudly as he reaches for the receiver. Before he picks it up, he thinks of two answers: somebody he knows has died, or his friend Stan is calling him at an inappropriate hour, again. With due respect for the first possibility, he answers with a fairly quiet “Hello?”
“Sorry if I woke you up, I –”
“Stanley! I asked you not to call me in the middle of the night!”
“But –”
“Okay, Mr. Literal, it’s not the middle of the night, I know. Man, do I ever wish I hadn’t said to call me anytime you have a physics question! I guess you do have a question…”
“Yes.”
“All right, all right. Maybe you’re onto something and it’s worth talking about now, but I can’t think about physics with a full bladder, so let me call you right back.”
“Thanks, Herb, but I can just wait. This is free cell phone time for me.”
“I guess you’re aware that free cell phone time is destroying what’s left of personal privacy—but never mind, that’s not the issue, just hang on a minute.”
Stan hangs on and whistles the theme song from Green Acres until Herbert returns and says, “Last time we were talking about simultaneity. Is that what you want to ask me about?”
“Sort of—well, yes, that’s it. It’s the question of how to determine the time of a distant event, and how the time is different in different reference frames, I mean rest frames, since that’s what you said I should say instead of reference frames.”
“All right. And the question is…?”
“I’m in a car and I want to say the car’s stationary and the roadway in motion. A streetlight flashes on and off after I pass it and I know how far I am from the streetlight when the light reaches me. How can I find the time the streetlight flashed as measured in each ref—I mean rest frame?”
“Well, okay, yeh, that’s one of those simple questions usually covered under the time-dilation heading in the textbooks. . . Hohuuum!” Herbert lets out a loud yawn. “But, of course, Einstein is going to be found wrong eventually, and it’s probably going to be in some basic philosophical way, and you can’t get much more philosophical than talking about the meaning of time.”
“I’m trying to do the math, actually.”
“I should have guessed you wouldn’t be calling me about a mere philosophical question.”
“But math is philosophy, too--you said so yourself.”
“Yeh, but I'm not feelin' very philosophic right now. But anyway, what you need in your case of the flashing streetlight is to have an imaginary observer in the rest frame of the car who passes the streetlight just as it flashes. Or I should say, whom the streetlight passes just as it flashes. Try to draw a picture of that and label it, and that should help. We can meet later today to talk about it.”
“It’s really the same problem we talked about earlier, the age of the galaxies and how far away they are. That’s one reason I started thinking about it, you know.”
“I know—I know! You and your thinking. Keep it up.” Herbert smiles as he imagines Stan sitting at home, staring at a wall, lost in his thoughts.
“By the way, Herb, what were you dreaming about?”
Momentarily at a loss for words, since he’s chagrined at Stan for guessing he was dreaming and for interrupting the dream, Herbert says defensively, “What makes you think I was dreaming?”
“Just a guess.”
“Well, yes, I was, and it was a good dream and I’m pissed that you interrupted it, to tell you the truth.”
“Sorry about that. But now you can remember it, and you can write it down, if you hurry.”
“Well, I probably can’t go back to sleep, so I may do that.”
“I’ll let you go, and call you later.”
“Later.”
“Bye.”
“All right. How many times must a man write down the meaning of relative time, before he gets it right? The answer, my friend, is obviously N. So, for the Nth time, let’s do the thought experiment.
“First, there is of course the concept of cosmic time, which assigns an expanding coordinate system to the whole universe. The problem is whether one coordinate system can really be superimposed on the universe and everything synchronized by cosmic time and the fundamental observers, or if you take time dilation and length contraction as applying on a cosmic scale. Just pick up a ruler, and hold it horizontally in front of you. One end of the ruler is pointing over the horizon, into the sky, and toward a galaxy speeding away from our galaxy. Well both ends are, but let’s keep it as simple as possible. The farther away the galaxy is, the faster it’s moving. If you apply the standard idea of length contraction, then the concept of “far away” changes. There is an imaginary scale, like the ruler but a lot longer, attached to the galaxy, and in the standard interpretation of length contraction, that scale and the distance to the galaxy are length-contracted.”
