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.
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.
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
tf’ = 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).
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).
