Lorentz
Abstract. Two intrepid explorers rediscover the Lorentz Transformations.
Oliver Hardy was sitting at the origin of inertial frame K. The x axis was regularly marked off in arbitrarily tiny increments of a meter. And at each tick mark there was an arbitrarily small (but accurate) clock. He noted with satisfaction that the clocks were all synchronized.
How did he know that? Well, sometime in the past he had performed a Michelson-Morley type experiment and found that light travels at the single speed c in all directions relative to K. And he noted that he perceived the clock at +x to run |x|/c behind his own. He even took the trouble to ascertain that an observer at +x perceived the origin clock to read |x|/c behind his own.
Now just above the x axis of K, the x’ axis of frame K’ slid by to the right at speed v. Its x’ axis was also marked off in increments, and there were clocks at each of those marks too. Oliver decided, on an impulse, to compare the rate of the clocks at rest in K’ to the rate of his own K clocks. Knowing that the speed of light is finite, he decided that the best way to do this would be (1) to note what a K’ clock read when it momentarily coincided with a K clock that read t; (2) to wait a bit and then ascertain what the same K’ clock read when it coincided with a different K clock that read t+Dt. Signifying the change of time on the K’ clock to be Dt’, Oliver found (much to his surprise) that
Dt’=(1-v2/c2)1/2Dt.
“Well, well,” he mused, “the K’ clocks run more slowly than my own K clocks do. I wonder why.”
Undaunted, Oliver decided next to measure the distance between successive tick marks on the x’ axis. He did this by noting what the x positions of two K clocks were (a) when the two x’ tick marks coincided with them, and (b) when both of the K clocks read the same thing. He dubbed the interval between the K’ marks Dx’, and the difference between the x positions of the two K clocks Dx. And he found, again to his puzzlement, that
Dx’=(1-v2/c2)1/2Dx.
“Ver-r-y interesting,” Oliver thought. “The x’ axis appears to be contracted, compared to my own x axis.”
Now Stan Laurel, Oliver’s friend, was at rest at the origin of K’ and had been watching Oliver make these measurements. Quite naturally he decided to make an identical set of measurements of the objects at rest in K, using the measuring rods and clocks at rest in K’. Stan believed that the clocks at rest in K’ were all synchronized for the same reasons Oliver believed the clocks in K to be synchronized. That is, sometime in the past Stan had performed a Michelson-Morley type experiment and had convinced himself that light propagates in all directions, relative to K’, at the one speed c. And he had perceived the clocks at +x’ to read |x’|/c behind his own origin clock.
Well, to cut to the chase, Stan made the following measurements:
Dt=(1-v’2/c2)1/2Dt’,
Dx=(1-v’2/c2)1/2Dx’.
But Stan and Oliver agreed that the relative velocities of their two reference frames were equal in magnitude:
v’=-v.
Thus the two concluded the same things about the axes and clocks at rest in the other’s reference system.
Oliver had of course been watching as Stan made his own set of measurements. And he guessed what Stan’s conclusions were.
“Another fine mess,” he exclaimed.
“What is?” Stan asked.
“You’re measuring my meter rods to be shorter than your own. And you’re measuring my clocks to run more slowly than your own. But the truth is precisely the opposite!”
Stan wrinkled his brow and fidgeted nervously.
“But how can that be?” he asked at length. “Did I make a mistake while I was measuring?”
“No, you didn’t make any mistakes,” Oliver muttered. “I was watching you like a hawk. The problem is that your measuring rods are contracted, and your clocks run slowly and aren’t synchronized. You see, the truth is that light doesn’t really travel with the one speed c in all directions relative to your reference frame.”
“But I’ve shown experimentally that it does,” Stan complained.
“I KNOW that!” Oliver shouted. “But your conclusions are based on the belief that the length of an object at rest in your frame is a constant attribute of the object. The reality, however, is that the lengths of things at rest in your frame vary as you alter their orientations. This includes the dimensions of your Michelson-Morley apparatus.”
“Well,” Stan sniffed. “I think it’s the lengths of objects at rest in your frame that vary as you alter their orientation. I think that you’re the one who’s delusional … about that and about everything else!”
At this point Oliver flushed red and considered giving Stan a swift kick in the seat of his pants. But in the next instant a disturbing thought creased through his well-oiled mind. Stan could be right! Perhaps it really was the clocks at rest in K that were out of synch, etc.
“But what a fantastic symmetry that would be!” Oliver marveled privately. “Indeed, the measured speed of light might be a constant c in every direction relative to every inertial frame of reference, and not just in K and K’.”
“I’m coming over,” he told Stan, and Stan made room for his portly friend.
Once at rest in K’, Oliver could see right away that the clocks in K “really” ran more slowly than those at rest in K’, etc.
“And if I jump back into K, I’ll be obliged to change my mind again,” he thought aloud.
“What’s that?” Stan asked. “Change your mind about what?”
“What do you know about the ether?” Oliver asked, ignoring Stan’s question.
“Isn’t that the medium that oscillates as light waves pass through it?” Stan offered.
“Exactly!” Oliver confirmed. “And what is the ether’s rest frame?”
Stan pondered that for a moment, and then brightened.
“Why it’s this frame … my frame!” he crowed proudly. “Isn’t that a coincidence!”
Oliver glowered at his slim friend.
“You still don’t get it, do you?” he demanded. “Every inertial frame can serve as the ether rest frame!”
Stan’s expression seemed to suggest that Oliver had lost his mind.
“REALLY!” Oliver persisted. “We can measure the speed of light to be c in every direction here in frame K’. But we’ll get the same result over in K. We’ll get the same result in every inertial frame!”
“We will?” Stan frowned, beginning to suspect that his rotund friend might be right. But then his eyes narrowed.
“What about magnetic fields?” he asked smugly.
“What about them?” Oliver rejoined innocently.
“Ah Ha!” Stan exulted silently. “I’ve got him!”
“Well,” he continued, “here’s an electric charge at rest in K’. It has no magnetic field, right?”
“Yes, yes,” Oliver agreed impatiently.
“But in your original frame K, the charge moves. So it should have a magnetic field. But surely there can’t both be and not be a magnetic field!”
For a brief moment Oliver was dumbstruck. He wasn’t accustomed to being caught off balance by Stan.
“Maybe. Then again, maybe there can be,” he thought aloud. “Let’s find out.”
“Find out … what? Whether or not there’s a magnetic field?” Stan asked.
“Exactly,” Oliver concurred cautiously, being cognizant of the fact that they were treading on unfamiliar ground.
“Sure, no problem,” Stan exclaimed. “Here’s another charge. Look! I’m moving it near the first one. No lateral force. No magnetic force. The resting charge has no magnetic field.”
“Yes, I can see that,” Oliver agreed. “But let me view things from K.”
“Be my guest,” Stan answered, personally seeing no reason to abandon K’.
Oliver jumped back into K. And much to his amazement, he found that the second charge, moving relative to both K and K’, did experience a magnetic force, quite as would be expected since the charge at rest in K’ was moving, from the perspective of K, and had a nonzero magnetic field. The only exception to the rule was when he brought the second charge to rest in K. Then there was only an electric force.
“Do you see what’s going down here?” he asked Stan. “A charge’s magnetic field is as relative a matter as its velocity is! From the perspective of K’, the charge you’re holding in your hand is at rest and thus has no magnetic field. But from my perspective here in K the charge is moving and does have a magnetic field.”
“But … how can that be?” Stan murmured. “The magnetic field is determined by Maxwell’s equations. It’s determined by physical law!”
“Yes, yes, you’re right,” Oliver agreed. “As is the rule that light travels in all directions at the one, constant speed c.”
“So those laws appear to work in every inertial frame of reference,” Stan exclaimed. “Maybe all physical laws do!”
Oliver frowned, now lost in thought. Something whispered deep within his mind that Stan might be right. But if true, then that would change everything! They’d have to rethink everything!
“Another fine mess,” he repeated, this time to no one in particular.
“Perhaps not,” his friend answered gently. “Come back into K’. I’ll brew a pot of coffee and we’ll do some brainstorming.”
“Where shall we start?” Oliver asked, stepping back into K’.
“Well, I’m at rest here at the origin of K’,” Stan thought aloud. “If I tap the K’ clock at rest here, it will read a unique time.”
“Yes, everyone will have to agree with that,” Oliver averred. “It’s a perfectly definite event.”
“The question is, what will the clocks and measuring rods of K say the coordinates of the event are?”
“Yes, what are the transformation rules that map an event’s space and time coordinates from one frame to another?” Oliver added.
“They probably won’t be what we thought they’d be before discovering that the clocks in different frames run at different rates,” Stan suggested.
“Or that moving measuring rods are contracted,” Oliver added thoughtfully.
“But I bet we know enough, now, to figure out what the correct transformations are,” Stan said brightly.
Oliver nodded in agreement. Both men, their competitive spirits now aroused, began scribbling in their notepads. In due course they would come up with identical formulas.
After all that work they discovered, much to their disappointment, that the Dutch physicist, Hendrik Lorentz, had already worked out the same transformations a century earlier.
But that is another story.