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.”

Coffee, tea, and a dream

Herb and Stan meet for coffee later in the day.  Besides his part-time physics teaching, Herb also is trying to write fiction.  Stan brings up that subject after they sit down with their coffee—or in Stan’s case, hot tea.

“How’s the writing going?” Stan asks, after swallowing a sip of tea.

“Pretty much, it’s not going.  It’s stalled. Stuck.  I figure I’m not cut out for it.”  Herb takes a sip from his coffee and a bite of pumpkin bread and shakes his head.

“I figure everybody’s cut out for it, Herb,” Stan says with a tight-lipped smile, looking directly at Herb.   “It’s just a matter of whether you want to badly enough.”

“Well, okay,” Herb retorts, with exasperation in his voice.  “Maybe I don’t want to badly enough.  It’s all about betraying people.  If you want to write a good novel, you have to betray people, and the first person you betray is yourself, because you never thought you’d betray anybody!”

Stan calmly looks off into the distance, or actually at a good-looking girl sitting near the window, and raises one eyebrow as he takes another sip of tea.  After he sips, he says quietly, “I don’t care if you write about me, Herb.  And nobody else would care either, I bet.  They’re not even going to recognize themselves.  And what does it matter if they do?”

After thirty seconds of silent munching and not so silent sipping, Herb says, “All right, that’s enough about the writing thing.  I can take a class or something if I want to keep it going.  What about your space and time investigations?”

“Stalled like your novel.”

Herb laughs out loud.  “Well, I guess I shouldn’t feel too bad, then.  I’m not the only one who’s stuck.”

“Maybe I should take a class.  Or something.”

Herb chuckles at the joke, then sits up straight in his chair and holds his arms out at a 45-degree angle from his body.  “I’m here to help you!” he says boldly, with a big smile, causing a few concerned looks from the adjacent tables.  “I can give you the standard scenario. And actually you’ve helped me.  All I need is a little outside encouragement, and I can keep at it, so thanks.”  Herb takes the last bite of his pumpkin bread and pulls a legal pad out of his book bag.  “Did you do a drawing yet?”

“I started one, but I haven’t labeled it yet.”  Stan pulls two pieces of paper out of his notebook.  “Here’s the drawing, and the calculation I already did for the stationary streetlight case. I didn’t understand what you meant about the imaginary observer.”  Herb takes a look at the drawings, and points to the place where the imaginary observer is stationed.

“Well, you marked the spot, or at least you have an arrowhead there.  Here, let me mark it with an X, and label the times represented by the drawings.” Herb writes on the drawing 

  then says, “What you see first are identical scenes in the different rest frames, but with the arrows reversed.  So far so good, but nothing relativistic has happened yet.”  Herb pulls his chair up closer to the table, takes a sip of coffee, and says, “Now.  A rest frame is like a railroad track with the railroad ties evenly spaced, giving a measurable distance from wherever you are.  Okay so far?”

“So far as establishing spatial coordinates in each case, yes,” Stan says, looking at Herb, “But I still don’t see why you put the X where you did.  The light came from the streetlight, and the X is not at the streetlight.”

“Ahh,” croons Herb, putting his elbows on the table and looking Stan in the eye, “But where was the streetlight when it flashed?”

Stan looks down at the drawing, his lips pursed slightly.   Then he slowly smiles, although his lips still don’t part to show his teeth, and looks back up at Herb.  “X marks the spot,” he says, nodding in appreciation of what he’s just learned.

Herb nods along with Stan.  “Yep, you got it, Stanislaw, old bean.  That’s where the streetlight was when it flashed.  Then it moved the additional distance to the left while the light was traveling to the car.  In the street rest frame, it’s the car that travels that distance while the light is on its way from the streetlight.  But, according to the street rest frame, how far did the light travel to get to the car?”

“Farther, or further, than the car rest frame.  The car is moving away and light has to catch up with it.”  Stan sits back, takes a sip of tea, which he finds to be still warm enough to be enjoyable.  Herb is nodding and writing on the diagram.

“We now need labels for all the quantities in the diagrams, then the final step will be to relate them to each other—the times and the distances, in both rest frames.  How about if I give you some labels for the diagram and you see if you can find how they relate?”  Herb looks up at Stan and adjusts his black-framed glasses with his right hand.  His glasses are relics of the late 1960s that have come back in style again.  His hair, partially gray but still mostly black, is long and thick and pulled back in a ponytail..  His body is trim from bike riding and walking instead of driving a car. 

Stan, who has thin wispy blonde hair, no glasses, and is tall and lanky naturally, without exercising, asks Herb about his earlier experiment in the car, where he thought he determined the time of the flash of the streetlight:  “What about the time I measured, or calculated actually, early this morning for the flashing streetlight?  Is it right?”

Herb now purses his lips slightly, but stops when he realizes he’s imitating Stan.  “Well, actually, no.  Your time on your car clock is not synchronized with the street rest frame, and the distance you used is the street rest frame distance, too, instead of the X-marks-the-spot distance for your rest frame.  And the car odometer, using rolling tires as a way of measuring distance, is not something I’ve ever seen analyzed in relativity, by the way. It’s a tough thought experiment you’ve presented me with!”  Herb now looks toward the window where he, too, had noticed the good-looking girl, and is slightly disappointed to see she isn’t there anymore. He takes a final sip of his coffee, which he finds to be too cold to be enjoyable, and says, “At this point don’t worry about anything except the standard time you can find by doing the calculations for the car rest frame, using the diagram here and the labels I’m putting on it.  I’ll leave the distances as variables, so you don’t have to worry yet about how they’re measured.”

After he glances at the diagram, Stan looks up at Herb quickly, as he remembers something from earlier in the day.  “What about that dream you were having when I called you this morning?”

“Oh, yeh,” says Herb, with , “the one you interrupted at a crucial moment.”

“Sorry about that, but I hope you wrote it down.”

“I did, but I don’t have my journal, and I can’t recall specifics at the moment.  Except I was in a small motel room, and I was going outside and coming back in to look at myself in the mirror, and a small passenger train backed into the yard outside, and men and women dressed like postmen were passing by me in a hurry, not answering any of my questions, maybe because I didn’t have shoes on and was standing on the wet grass, trying to talk to them.”

“That’s pretty good for not recalling specifics.”

“Funny, isn’t it, how you think you can’t remember something, but once you start talking about it, it starts to come back.  Except that usually doesn’t happen with dreams.”

“What was happening when I woke you up?”

“I’d gone to another motel room, where there were two girls about 10 years old dressed up like women, in long dresses and wearing beaded necklaces.  The mother of one of the girls was standing at a mirror, dressed but still making some final adjustments.  She didn’t look at me.”

“Somebody you know?”

“Oh, yeh.  You know her too.”

“Your ex?”

“My most recent ex, yep. I hadn’t seen her or the girls for a long time, but it was like the time hadn’t passed, like we were all on a trip together.  I felt tongue-tied, as usual—”

“Not a problem you’ve ever had with me,” Stan interrupted.

 “Yeh, well, you like to talk about philosophical and physics questions, so that’s different.  You know small talk has never been my strong point.”

“I never gave it much thought …”

  “Anyway, I remember one of the girls saying something like ‘If I could have brought my dog’ but otherwise I can’t recall any conversation.  I put my hand on Denise’s shoulder, and felt the chiffon, or whatever it’s called, of her blouse, and then you woke me up.”

“Well, maybe it ended when it should have,” Stan said, putting his notebook back into his backpack on the table in front of him.

“You would take the positive view of your not very well timed call,” said Herb, smiling in spite of the serious tone of his voice.  After putting his legal pad back in his black book bag, he stands up and says, “Please wait until later in the morning to call me next time okay?”

“Sure, I promise I won’t call before 7 a.m. next time,” Stan says, getting up to go.

“How about if you make it 8 a.m.?” Herb says as they walk out the café door.

                “Eight a.m. it is—no earlier, I promise!”

                “Later!”  says Herb.

                “Later!”  Stan responds as they head off in opposite directions along the sidewalk.  A moment later Herb hears Stan mumble “not earlier” and then hears him start whistling “We’re Off to See the Wizard” as he turns a corner and moves out of hearing range.

Stan O'Stanley & the flashing street light

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.”

Herb gets his journal out and writes down the dream as best he can remember it. Then he writes a synopsis of the subject he’s been discussing with Stan: 

“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.”