Stan&Herb’s Big Time Adventure, cont’d.      © 2005 David W. Trulock

 

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 whistling 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 in his spiral notebook:

c = x/t

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

 

(a labeled drawing of the car-at-rest scene)

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 instead 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 afterwards. You have to know the relative speed or know the distance to the event, the flash. At this point, I won't worry about the rolling tires 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, then suddenly sits up straight and almost coughs the tea into his lap before he manages to swallow it.  He stands up with the cup and saucer in his hand and shouts "Wait a minute--the Doppler shift! 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 puts down the cup and saucer and reaches for his cell phone.  In his excitement he 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, very uncharacteristically. More in character he says, "Now why didn't I think of that!?"

(to be continued, one more time)