At 3:06 a.m., Stan is home reading Mathematics and the Imagination when his cell phone announces a call from Herb by playing the beginning of the third movement of Beethoven’s Pathétique piano sonata. Stan pushes the call button, puts the phone to his ear and says with mock indignation, “So I can’t call you in the middle of the night, but you can call me?”
Herb laughs a little nervously and says, “Well, you usually stay up all night. Was I wrong about that this time?”
“No, no,” says Stan, clearing his throat and looking down at the book in his lap. “I’m up reading that math and imagination book you loaned me.”
“Good. I thought you’d be awake, anyway”
“And I thought you’d be asleep.”
“I couldn’t sleep after I started working on the math for that thought experiment of yours. I worked out something really neat, and I figured you’d like to hear about it. Did you work on it some more?”
“I did, but I ran into a problem right away. I don’t know what X is. I’ve got the odometer reading—the distance measurement from the streetlight, but I don’t have a measurement of X. Were you figuring I’d be able to find do the classic algebra trick and find X ? ”
“Yes and no—I only realized after we talked that it’d be a problem. It’s like an imaginary mark in your rest frame showing where the streetlight was when it flashed. A standard thing to do in deriving length contraction and time dilation. I thought I’d let you struggle with it, with the algebra, to try to figure it out. But then the distance measurement thing kept bugging me, and I found a way to write the equations for the time of the flash just in terms of the later time—the time the light arrives at the rearview mirror—and the speed of the car and the speed of light. That is, just t, v and c.”
“TV and see,” Stan repeats.
“t, v, and c, for each case—well, the beauty of it of course is v and c are the same for both observers. Let me give you what I’ve got and then I’ll call you back later.”
Stan lets out a barely audible sigh, then says, “Hold on while a get a pen and paper…. All right—I’m ready.”
“Well, a couple of things I need to mention first. When the car passes the streetlight, t is set to zero for both the street rest frame and the car rest frame—”
“But—”
“Yeh, I thought you’d have a question about that. But the car passing the streetlight is just a single event, and the people in both frames, at the streetlight location, can use the event to set their clocks to zero. It’s only later that the times get out of sync when they’re compared between different frames of rest. Let me put it a different way. Two people in relative motion can use the event of their passing each other to set their clocks and distance measurements to zero.”
“But I thought we were going to call the time of the flash ‘time zero’.”
“Oh—that’s what’s bugging you!” Herb pauses, remembering their previous conversation. “You’re right, we did do that. Well let’s just call it ‘time of flash’ and label it t sub f. Okay?”
“Okay…” Stan now pauses to recall his early understanding of the situation. “So in my rest frame I’m just sitting there in my car, I see the darkened streetlight approaching, and I set my clock to zero just as it passes?”
“That’s it. And somebody, let’s say Juanita—”
“Let’s not say Juanita.”
“All right, if you think she doesn’t love you any more.”
“I don’t have to think. She told me she doesn’t.”
“Well, this is the ideal situation to forget about her. Here’s what I was going to say. Juanita’s house is at the streetlight and she sets her clock to zero when she sees you pass. Ideally, that’s the last you ever see of her.”
“She’d like that—let’s say it that way. I’m better off without her.”
“I doubt it. You think she’s better off without you? I mean it’s kind of reciprocal, y’know, at least if there was some kind of equality in the relationship. At least you could still be friends…”
There is silence until Stan realizes the meaning of reciprocal. Then he says, “Let ex equal one over ex, eh?”
Herb laughs one of his roaring laughs and says “Hey that’s pretty good for somebody who hates math! You mean e-x don’t you? I see why you stay up late—your mind is in high gear in the wee hours of morning. Now the question is, can you solve that equation?”
“Just a second,” Stan says. He writes down x = 1/x. “Sure. Just multiply both sides by x and you get x2 = 1. So, x equals the square root of one…”
“Okay, you’re getting there, but just break the rules for a minute and try to think about anything else that could possibly equal one over itself.”
“I am thinking about that,” says Stan, testily. “That’s the first thing I thought about.”
“Okay, okay,” Herb says quickly, “The thing is, one reason I laughed, x = 1/x is a beautiful little strange equation. It’s just not something any math person writes down. But x2 = 1 is totally standard, totally uninteresting.”
“Not to me,” Stan says, with a hint of anger in his voice. “I came up with it on my own, so I happen to feel quite attached to x2 = 1.”
Herb laughs out loud again, knowing Stan wasn’t intending to be funny, but also knowing Stan well enough to think he won’t be offended.
“I love x2 = 1!” Stan almost shouts, then starts laughing himself, his rising anger dissipating immediately.
“I know what you love, brother, and it ain’t no equation,” Herb responds, smiling as he says it. “But since it’s your baby, what’s the solution of x2 = 1?”
“x equals one or negative one.”
“I wondered if you’d recall the negative one solution.”
“It came to me while we were discussing the beauty of x = 1/x,” Stan replied. “And I also remembered quadratic equations always have two solutions.”
“I bet a lot of people who didn’t flunk algebra don’t even remember that,” Herb says. “And, thank you, that also makes me wonder something about the equations I found for the time of the flash. But for now I’ll just let you write them down. Ready?”
“Ready.”
“Your frame, the car rest frame, has: time of flash equals your clock time when the flash reaches you, divided by the quantity one plus v over c.”
“Okay. Let me read that back to you, in equation form. t sub f equals t divided by, open parenthesis, one plus v divided by c, close parenthesis. Right?”
“Right. And the t sub f and t of the car frame need to have primes on them—they need different labels, and an apostrophe on the variable is the standard way of handling it in relativity. The t sub f and t of the street rest frame don’t have primes on them. We just have to be careful and not assume that time passes equally in each frame—really we know this from relativity, but we are re-deriving it, or at least I hope we do when we get to that point.”
“I guess that’s why you like this little project.”
“One reason, yeh. All right, here’s the other time-of-flash equation for the street frame: t sub f equals t times, open parenthesis, one minus v divided by c, close parenthesis. Isn’t that cool, the symmetry between the plus and minus signs, and the multiplication and division?”
“Weehhl, I can’t say I can see much symmetry.”
“I guarantee you’d be impressed if you’d done many relativity calculations. Anyway, I’ll just let you work with those equations and the speed you were traveling and see if you become sufficiently impressed later.”
“I’ll take a look at it, but I have to go find the speed reading from the computer, since I never used it in what I was working on before. I’ll give you a call when I become sufficiently impressed.”
“Sounds good, but make it after 8 in the morning. I’m going to bed now.”
“Oh, it’ll be after eight all right. I really don’t know if it’ll even be in the next 24 hours. I’m somehow starting to lose interest in this little project.”
“Sorry, Stan. I guess I’ve kind of co-opted it away from you. But I hope you won’t give up on it. Anyway, goodnight.”
“Goodnight,” says Stan brightly, in order to keep Herb from thinking he’s unhappy.
After he hangs up, Stan looks at the two equations he’s written down,
t’f = t’/(1+v/c)
tf = t(1- v/c).
He shakes his head, then slowly gets up and stretches and starts to go out to his car to check the value of v stored on his car computer from the previous morning. For once he doesn’t feel like whistling. He doesn’t like Herb’s taking over the experiment, even though he called Herb to get his help. In his kitchen, on his way to the carport, he stops and hits his fist softly on the countertop near the back door and says, “This time, I’m not going to do it Herb’s way.”
