x-1 = 1/x

Herb and Stan met again at the coffee shop at 4:30 the next afternoon. The shop had several tables sitting outside under a couple of small city-beautification trees planted on a widened, brick-covered part of the sidewalk. The coffee shop side of the street was in the shade, and the air was cool enough, barely, for them to sit at the only unoccupied outside table.

Before they sat down, Stan and Herb both briefly noticed a good-looking couple sitting at one of the tables drinking iced coffee drinks. The young woman was reading a book and the young man was reading the Austin Chronicle—or, rather, flipping through it and not reading it. The two had a movie-star look about them, but neither Stan nor Herb was willing to look at them long enough to make a positive celebrity ID.

Across the street sitting in the sun by himself on a green bus-stop-style bench, a middle-aged homeless man in a heavy, dirty, black overcoat, dirty black pants and a big black ski cap smoked a cigar stub, and looked straight ahead, occasionally glancing at the folded newspaper in his lap. Stan and Herb saw the man every time they came downtown. Usually he was sitting on the same bench, but sometimes he was searching through one of the Congress Avenue garbage cans.

“I don’t see how that guy stands the heat of the sun in those heavy clothes,” Stan said as he and Herb sat down.

“I don’t see how that guy stands his own body odor,” Herb replied, just before taking a bite of cranberry bread.

“But have you noticed the cigar smell? He’s always smoking good cigars-maybe that helps negate the body odor,” said Stan, after his first swallow of hot tea.

“Maybe the cigar store owner down the street gives him free samples,” Herb said after swallowing his second sip of coffee and putting the large white china cup back in its saucer.

“Yeh-probably pre-smoked free samples,” said Stan, smiling one of his tight little smiles and squinting his eyes.

“Probably,” Herb agreed, grinning a little and emitting a slight chuckle. Taking a bite of cranberry bread and then a quick sip of coffee, he thought for a moment and then said, “You know, I was reminded of something I couldn’t quite put my finger on when we were talking about your affection for x =1/x last night. I just realized what it is.”

“You mean your affection for x =1/x. I was partial toward x squared equals one, myself. Something related to my unrequited affection for you-know-who, and how it should be obvious there’s only one solution-okay two solutions, negative and positive, but only one number. A song by Three Dog Night comes to mind.” Stan glanced toward the homeless man and took a bite of his walnut-apricot scone then a sip of tea.

“Well,” Herb replied testily, “they’re the same equation.” He paused for a moment and thought about changing the subject and depriving Stan of the thought he’d wanted to share with him. He decided to share it. “I suppose you’ve heard of the golden ratio.”

“Sure, I’ve heard a lot about that,” Stan said, putting his teacup in its saucer and sitting back in his chair. “In a visual context, it’s an eye-pleasing proportionality used by the Greeks in their architecture and also found in nature.”

Pausing and leaning back in his chair like Stan, Herb said, “Yeah, that’s part of the story.” Stan thought Herb was beginning to smirk a little, and indeed Herb looked away from Stan (to hide his smirk, Stan thought) as he continued, “But what is the equation for the golden ratio?” Herb looked back at Stan, waiting for a response.

In no hurry to provide a response, Stan took another bite of his scone, then a sip of his tea. He found the mixture to be very flavorful. “I know there’s a quadratic equation,” he replied after swallowing his tea and scone mixture, “that has a solution that’s a never-repeating decimal expression like 1.618... or something like that.”

“Right!” Herb said quickly, in a sort of enthusiastic whisper. In his more analytical, normal tone of voice he continued, “I’m impressed with your memory of the first digits of that decimal number. But the quadratic equation is very much like your beloved x squared equals one. I thought you’d like that. Care to take a guess what it is?”

“Okay,” Stan said, sitting up and taking another sip of tea. After thoughtfully swallowing, he said, “Is the equation x over 1-x or something similar?”

“Very dang close, Stanley old bean,” Herb said, not smirking but instead involuntarily smiling.  He enjoyed quizzing Stan and getting him to think, and he mostly enjoyed the superior feeling he got when he posed a question Stan couldn’t answer quickly. Sitting up straight, and picking up his coffee cup, Herb said just before he took a sip: “But that’s not an equation.”

“You are an ass,” Stan said in a good-natured tone of voice.

“That’s irrelevant,” Herb immediately said, as if mimicking a judge overruling an objection.

“Well, it’s more like irrevelant, which probably isn’t even a word,” said Stan in response. “But I can come up with an equation.” Stan takes a pen out of his pocket and does some scribbling on his napkin. “Okay, here is the equation,” he said, looking Herb in the eye and sliding the napkin over to him.

Herb took a sip of his coffee, put the cup down, and picked up the napkin. On the it was written “x²/(1-x) = 1”.

Herb pulled a pen from his shirt pocket and scribbled on the napkin. “Not quite, but actually only off by a minus sign,” he said. “Changing the minus to a plus and kicking the 1+x over to the right-hand side gives the right quadratic equation. But this is what I thought was so cool.” He handed the napkin back to Stan. “Check out the rearrangement I did.”

Stan looked at the scribbled equations on the napkin. The last equation on the napkin, written on the back of the napkin, was “x-1 = 1/x”.

Sitting up straight in his chair, his back not supported by the back of the chair, Stan held the napkin in front of him and stared at it intently. He kept staring, lost in thought. Herb let him think without bothering him.

He noticed the young couple they’d seen earlier get up from their table, talking to each other now that she’d put away her book. Apparently, they’d arranged to meet at the coffee shop, and were about to go their separate ways. Hearing the girl’s voice, Herb realized she was daughter of his ex-girlfriend Denise. Her name was Rachel.

“Did you see anything in there about us?” Rachel said as the guy put the Austin Chronicle into his army-green knapsack.

“I wasn’t really looking for anything about us,” said the guy, glancing up at Rachel in a distracted sort of way that was nevertheless intense. Herb recognized him then as Jacob Josephson, the movie star son of Jason Josephson, a local film director who had recently made it big. In the issue of the Chronicle that Jacob had been looking through so quickly, Herb had earlier read a short article about a movie with Jacob in it that was being filmed in Austin.

“Us?” Herb said quietly to himself.

“What?” Stan said quickly, his reverie with the napkin now broken.

Herb leaned over the table toward Stan. “It’s Rachel,” he said in a low whisper, nodding toward the young couple’s table.

Stan turned around to look at Rachel as she was shouldering her backpack. For the first time, she looked toward Stan and Herb’s table. At first she looked puzzled, then she smiled and unselfconsciously said, “Hi Herb, hi Stan!”

“Hey, Rachel,” Herb and Stan said simultaneously, causing the three of them to smile briefly. Then Herb added, “How are you?”

“Great. Good to see you guys.” As Rachel spoke, Jacob looked at Stan and Herb, then back at Rachel. Before Jacob could speak, Rachel stepped toward Stan and Herb’s table. “Jacob, these are some friends of mine."

“How ya doin’” Jacob said, with a friendly nod and a quick smile. He and Rachel stepped over to Stan and Herb's table.

“This is Herb, and this is Stan,” Rachel said, motioning with her left hand toward Herb and then Stan. Looking at Jacob, she added, “And this is Jacob.”

Being the closest to Jacob, Stan put his hand out. “I thought you looked familiar,” he said to Jacob as they shook hands. “How’s the shooting going?”  Stan had read that "shooting" was the proper word among film people, not "filming".

“Well, we don’t start shooting till tomorrow,” said Jacob. After a brief pause, he added quickly, “At sunrise.”

Herb stood up and shook hands with Jacob across the table. “Good to meet you,” Herb said. Rachel spoke next. “It’s been, like, four or five years since we’ve seen each other.”

“Yep, five years.” Herb said quickly. “You look really great. How’re your mom and Luke doin’?”

“Oh, about the same, doing okay,” Rachel said, moving a loose strand of blond hair out of her face and tucking it behind her ear. “They’re gonna be in the movie, too.”

“So you’re in it with Jacob,” Herb said, realizing why Rachel had used the word “us” earlier. “Congratulations on that!”

“Thanks. I’m a little nervous. I’ve been in lots of plays, but never a movie”

Jacob looked at Herb and Stan and said, “Well, I better get going. Good to meet you guys.” He shook hands with Herb, who was still standing, then with Stan, who was still seated.

“Break a leg!” Stan said as Jacob turned to leave. After he said it, he immediately wondered if it was appropriate to movie making.

“Yeh, thanks, I’ll do my best—it’s actually part of the story!” Jacob said with a quick smile as he turned to walk away.  Stan, Herb and Rachel all laughed together, then Herb sat down in his chair, picked up his coffee cup and leaned back.

Seizing the moment to keep the conversation going, and thinking Herb and Rachel probably felt awkward after not having seen each other for so long, Stan said to Rachel, “Herb and I are exploring the romantic implications of the golden ratio.”

Rachel giggled a comfortable and confident sort of giggle, then asked, “What romantic implications?”

Herb grinned, remembering how good Rachel was at asking direct questions.

“Well, we haven’t figured that out yet,” Herb said, looking at Rachel then at Stan. “The philosopher here is working on a theory.”

“Actually, I just now worked out the theory,” said Stan, looking from Rachel to Herb and raising his eyebrows while smiling a bigger than normal tight-lipped smile. “This equation for the golden ratio, x minus one equals one over x, says when you become an ex, you have to let go of the other person—that's the minus one part—if you are going to turn your life around—that's the reciprocal part. And I re-wrote the reciprocal to make the equation have a more symmetric appearance.”

Stan pushed his scribbled-on napkin to the center of the table where Rachel and Herb could both see it. The final equation on the napkin was x-1 = x^-1.

“Stan, you’re a genius!” Herb exclaimed as he picked up the napkin. “That’s a beautiful equation!”  Stan looked up at Rachel, who shifted her weight and adjusted her backpack on her shoulder, smiling at Stan as she did so.

“You guys ought to write a book.” Rachel said, looking back at Herb. “A book about the algebra of breaking up.”

“If only it were that simple,” said Herb, taking a sip of his coffee but finding it too cold to be enjoyable.

“Well, I better get going,” Rachel said, looking at her watch. “Mom wants me to get some stuff at the store for dinner. It’s going to be a crazy night around our house—everybody’s nerves are jangled.”  

“Yeh, I’d be nervous too,” Herb said, standing up and looking Rachel.. “Congratulations again on being in the movie. Tell your mom and Luke I said hi.”

Stan then stood up, and said, “Tell them hi for me, too.”

“I will!” Rachel said as she smiled and hugged Herb. “Good to see you guys.”

“Good to see you,” said Herb as Rachel stepped over to hug Stan.

“Take care,” Stan said as he and Rachel hugged.

“You too,” said Rachel.  Then as she was turning away, she said “Bye guys.”

“Bye!” Stan said as Herb simultaneously said, “Bye-bye!”

 “She’s grown into quite a beautiful young lady,” Stan said as he and Herb sat down.

“She always was one,” Herb said, before taking his last bite of cranberry bread and washing it down with a sip of cold coffee. “Now I need to get going,” he said, looking at his watch. “I’m teaching a class at 6 tonight, you know. But first, I’ll show you something. “

Herb picked up a fresh napkin and began writing on it. "You have x, which is really x to the first power," he said as he briefly glanced up at Stan. "Then you have one, the loneliest number, which can be written as x to the zeroth power, right? Then, lastly, you have x to the minus one power."

Herb pushed the napkin over to Stan, who looked at it and saw: “x¹- x⁰- x^-1= 0.”

“Well…” Stan said, pausing as he looked down at the napkin. “I still like my way of writing it better."

"Well, naturally," said Herb as he stood up and picked up his book bag. "And it is definitely a beautiful way of writing it. But having dealt with sequences and series in math, I couldn't resist the temptation to write it as consecutive powers of x, like it's part of an infinite series. Now I gotta go, bro, so take care and get in touch about that thought experiment we’ve been talking about."

"Sorry to say I'm losing interest in it," Stan said, looking in Herb's direction, but looking past him rather than at him. "But, yeah, I'd like to come up with an answer to the original question, about the time of the streetlight flash in the two rest frames."

"Let's do that, then," Herb said, starting to walk toward his nearby bicycle.

"Hey!" Stan said suddenly, as if having a delayed insight into the thought experiment. 

“What?” said Herb, turning around quickly.

"Remember, stay away from the Golden Arches."

Herb laughed and said, "I'll do my golden best." His bike had been leaning unlocked against one of the city beautification trees on the sidewalk. He swung himself onto the seat and rode into the bike lane, heading south on Congress Avenue.

One of the café staff was cleaning off Jacob and Rachel's table. Stan figured he’d go ahead and leave so his table could be cleaned and he wouldn't have to worry about whether to take his and Herb's cups and saucers inside.

After walking about a block south on Congress he passed the homeless man dressed in dirty black clothes who'd been sitting on the bench across the street earlier. Stan was about to nod and say hi to him, but the man glared so ferociously at him that Stan just looked away.

As he was about to turn the corner to head west on 8^th Street, Stan looked back and saw the homeless man holding up the hinged top to a green metal trash container while poking his head inside and reaching down into it.  After turning the corner, Stan didn’t see anyone on the sidewalks nearby. Instead of whistling as he often did in such situations, he began to sing the theme from George of the Jungle in a quiet but energetic voice. 

Schnecken

The next morning at 10:30, the café was crowded with people from the office buildings nearby. Herb arrived first and got in line. The café staff consisted of young guys and girls who wore artistic-looking clothes and had various body piercings and tattoos. Herb guessed they were musicians or actors or artists of some kind, or just Austin slackers temporarily working to pay the bills. Original artwork and photos were displayed on the café walls. The pieces were for sale at what Herb considered to be rather steep prices. Occasionally he saw something he liked, but usually not.

The staff was mostly polite and helpful, but a particular girl—tall, tattooed and with jet-black dyed hair and always plenty of mid-section skin showing—seemed unfriendly and short-tempered. Not that Herb had seen her lose her temper. He merely judged from her curt demeanor that she probably could become angry easily, plus he’d never seen her smile.

She was one of the two people behind the counter taking orders when Herb reached the head of the line. Pen in hand, she looked up at him and raised her eyebrows.

“How’re you?” Herb said, wishing immediately that he hadn’t. He wondered fleetingly why he was trying to be friendly with this girl he didn’t like.

“Sir?” the girl said with a slight frown. Had she heard him and just pretended not to? People were standing in line behind Herb and also standing beside him waiting on their orders. Herb felt his face start to turn red.

“Uhhh,” Herb said, realizing it would be ridiculous to say ‘how are you’ again. “A large coffee and large Darjeeling tea…a pumpkin bread and a walnut-apricot scone.”

Before he could add “to go,” she said flatly, “We’re OUT of pumpkin bread.”

“Okaay,” Herb said as he glanced at the counter to see what was available. He was getting over his embarrassment. An idea occurred to him.

“Schnecken,” he said quickly and fairly quietly.

“I’m sorry?” the girl replied, with irritation evident in her voice.

“Schnecken,” said Herb, in the same quiet tone of voice.

The girl then said, in a loud, exasperated voice, “I can’t hear you!”

“A fucking schnecken!” Herb yelled back, defiantly.

Nearby conversations stopped. With her mouth open, the girl turned wordlessly toward a skinny guy with blond bangs who was taking an order from another customer. Before she could speak, the skinny guy looked at Herb and said nonchalantly, “We don’t carry schnecken, sir.”

Herb had figured that to be the case. “All right. Sorry,” he said in a normal, level tone. The girl looked back at him and as their eyes met Herb said quickly and coolly in his best teacher’s voice, “Cranberry bread then, and make the tea a hot tea, please, and it’s all to go.” The girl simply turned and began putting his order together.

Behind him and to his right, Herb heard someone say in a loud, disapproving whisper, “Typical passive-aggressive personality.”

Herb turned and saw Stan at the end of the line smiling his tight-lipped smile. Strands of his wispy blondish hair were standing up high above his head. He appeared to have just gotten out of bed. Herb erupted with a loud laugh at Stan’s comment and his disheveled look. It was a standard joke between them that they were both passive-aggressive types.

“Good morning to you, too,” Herb said. “If you can help me carry this food, I won’t yell at you.”

Stan stepped ahead of the people waiting in line, politely saying “excuse me” several times. As he stepped up to the counter beside Herb he said, “Where do you want to sit?”

“Let’s go up to the capitol grounds and find some shade,” said Herb. The tall girl behind the counter had finished getting their order together and ringing it up.

“Seven dollars and eighty-seven cents,” she said to Herb, in an even voice without emotion.

Herb said, “Okay,” and handed her a ten-dollar bill. Stan picked up his part of the order and began to walk behind Herb toward the door. As the girl handed back his change, Herb said to her, “I apologize for the angry words.”

“Apology accepted,” said the girl, looking at Herb and nodding her head slightly.

“Thank you,” Herb said, putting all the change in the tip jar.

“Thank you,” the girl said as Herb turned to follow Stan toward the door. “Have a good day.”

Stan and the Doppler Shift, with drawing & calculations

The astute industrious reader will have noticed no math appendix has been added to this little relativity drama yet. We can now leave that to our hero, Stan O'Stanley, who after his late night call from Herb is in his kitchen making a cup of hot tea and toasting two pieces of whole wheat bread while singing the theme to The Beverly Hillbillies. He is confused but happy, because for a philosopher being confused is merely a prelude to discovery.

Stan wants to start from scratch in finding the equations to use for his car experiment, but after talking to Herb he feels there is really nothing to discover. After all, his car clock, like any car clock, goes from one rest frame to another whenever it accelerates or decelerates, so how can he possibly find a difference in the time of the streetlight flash as seen in his frame and as seen in the street rest frame? 

"What I want to do," Stan says to himself, putting the toast on a plate and buttering it, "is to start with the most basic measurements possible." Dribbling honey onto his toast and into his cup of tea he continues, "Which would have to be a measurement of velocity using a clock and meter stick. It's a relative velocity measurement, but relativity doesn't seem to be involved." He takes the teabag out of the hot tea and squeezes the bag so as to extract every milliliter of liquid from it. After pitching the bag in the trash, he takes his tea and toast into his well-ordered living room, which is filled with antique furniture, most of it inherited. Besides being a philosopher and gadget nut, Stan is also something of a neat-freak, quite the opposite of Herb.

"All right,” says Stan, "starting with speed-equals-distance-traveled-divided-by-time-of-travel, how the heck do I figure out the time of the streetlight flash, as measured in my rest frame? The streetlight is what's moving, from my point of view. It flashes sometime after it passes me, but I don't detect the flash until later." Stan puts the tea cup and matching saucer down on the dark oak coffee table he inherited from his father's side of the family, then he sits down on the beige claw foot couch inherited from his mother's side of the family. Balancing the plate of toast in his lap, he suddenly laughs out loud and says, "This is exactly the kind of problem I hated so much in algebra in high school—a word problem!"

He takes a couple of bites from his toast and carefully but loudly sips from his cup of tea. Then he picks up his spiral notebook and writes two equations. He smiles and whistles some of the theme from Mr. Ed. "Sorry Herb, old bean," Stan says, "but I have to do this my way. I'm really only trying to find the time interval between the flashing of the light and the moment the light first hits my rearview mirror. I'm just going to call this time interval t. In this time interval, the light is headed toward my car at speed c, and the streetlight itself is continuing to move away at speed v. The two speed equations are therefore..." he looks down at the equations and the drawing in his spiral notebook:

c = x/t

v = (d-x)/t = d/t -x/t.

Stan begins musing about the two equations.  "Now, if I had the value of x, I could use the first equation to calculate t.  Since I don’t, I can substitute c for x/t in the second equation, and voila!, I’ve gotten rid of x." Stan writes that down

v = d/t - c,

v + c = d/t, or

t = d/(v + c).
 

"Hmmm," hums Stan, switching from whistling the Mr. Ed theme to humming it. "Mmmm-mm-m-m-m-mm-m-mm-m!" he hums, and then sips tea and munches toast while staring across the room, thinking about the fact that speed, v, is measured using his car clock and the rolling tires of his car (not a meter stick). "This is something Herb objected to, and I can't say that I blame him," Stan says, his mouth partially full of chewed-up toast. "And I don't seem to be able to get rid of v without measuring the distance x directly. That's why everybody doing this thought experiment lets the flashing light leave a mark so the distance can be measured. You have to know the relative speed or know the distance to the event, the flash. At this point, I won't worry about my measurement of v. But I will have to go check the car computer and see what v is, after all." 

Stan takes another sip of tea, but almost coughs it into his lap as he suddenly sits up straight, swallows the tea, and shouts, "Wait a minute--the Doppler shift! Of course! The velocity can be found from the Doppler shift! Why didn't I think of that before? Why didn't Herb think of that before!?" Stan reaches for his cell phone, and in his excitement doesn't remember until the phone is already ringing that it's four o'clock in the morning. The receiver is picked up on the other end but there is only silence until Stan interrupts it by blurting, "I'm sorry Herb—I'm sorry!  I –"

"I was dreaming about a naked woman," Herb says as if speaking in a trance. "A naked, beautiful woman. We were about to kiss. . ."

"Herb--"

"Okay, Stan. I can hear the excitement in your voice. I just hope it's justified."

"The Doppler shift! My computer data can give us the Doppler shift in the streetlight spectrum. I dialed the phone without thinking about the time. We can use your formula and my formula and compare them, and maybe even publish a paper together, maybe not this thought experiment exactly, but--"

"Wow," says Herb, with uncharacteristic mellowness.  "Why didn't I think of that?"

 

__________________________
 

Now for a review of the time-of-flash formulas.   Comparing Stan’s original speed-of-light formula, c=d/t, to his last one, c = x/t, and looking at the car-rest-frame drawing below, we see that the original formula doesn’t use the right distance:  x is the distance traveled by the light, not d.  So Stan’s original formula doesn’t give the correct time for the flash.

Stan’s revised formula does give the correct time, but we need to have a value of velocity (or a value of x) in order to use it.  So let's say v is 51.23749458 meters per second, found from the Doppler shift in the streetlight spectrum.  (Stan’s rearview mirror contains a diffraction grating that breaks the light into its spectral components, like a prism, and Stan’s computer has the stationary streetlight spectrum stored in it.  The shift can be found by comparing the two spectra.)

Using Stan's formula with d = 644.000002 meters gives

t = d/(v + c) = 644.000002 / (51.23749458 + 299,792,458) = 2.14815241 microseconds.

The time of the streetlight flash in his stationary-car reference frame is given by subtracting 2.14815241 microseconds from his car clock reading at the moment the flash reaches the rearview mirror.

How does this compare with the formula Stan used incorrectly when he didn’t know any better?  He calculated t = 0.0000002148 = 2.148 microseconds.  Given the precision of the distance measurement, however, the exact value is

t = 644.000002 / 299,792,458  =  2.14815278 microseconds. 

His car clock is not precise enough to be able to distinguish between these two time values—the difference is a few thousandths of a nanosecond—because of the relatively slow speed he was traveling.  (Actually, he was speeding rather extremely:  51 meters per second is about 115 mph!  But this is still slow relative to the 186,000 miles per second speed of light.) 

Using Herb’s formula and Herb’s definition of t as the time the car takes to get from the streetlight to the place where the light hits the rearview mirror, which is, by the way,

 t = d/v = 12.56892062 seconds,

then the time of the flash is

 t_f’ = 12.56892062/(1 + 51.23749458/299,792,458) = 12.56892062/(1 + 0.709098852X10^-7)

=  12.56891847 seconds.


The difference in these two times is 2.1482 microseconds.  That agrees with either of the above time values, because of the limited precision. In order to actually compare it to the value given by Stan’s correct formula, we’d need more precision (I’m working on that).    What about the other formula, Herb’s formula for the time of flash in the streetlight-at-rest frame?  We don’t have a value of time-light-reaches-rearview-mirror for that frame!  How could we obtain it?…(to be continued).

Not Herb's Way, Says Stan

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)

t_f = 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.”