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The Adventures of Yin and Yang

             

 

 To Vincent and his Peers

 Contents

Chapter 1. Yin and Yang Discover New Symmetries

Chapter 2. Yin and Yang Take a Trip

Chapter 3. Yin and Yang Adjust to Length Contraction and Time Dilation

Chapter 4. Vanishing Point?

Chapter 5. Yin and Yang Learn Something New About Mass

Chapter 6. Yang Demonstrates Length Contraction and Time Dilation

Chapter 7. Yin Transforms Force

Chapter 8. Light Drops

Chapter 9. Position Counts!

Chapter 10. The Gap

Chapter 11. Red Shift

Chapter 12. Newton/Lorentz

Chapter 13. No Escape

Chapter 14. Imagine

Chapter 15. No Matter?

Chapter 16. One Wish

Chapter 17. Into the Box

Chapter 18. Space Zero

Chapter 19. Back to Square One

Epilog

 

Chapter 1

Yin and Yang Discover New Symmetries

 Yin and Yang were sitting at the origins of respective inertial reference frames K and K'. The x'-axis of frame K' was parallel to and just above the x-axis of frame K and was traveling in the positive x direction of K at constant speed v. Yin knew this to be the case because he had measured Yang's velocity in the following way: (1) When Yang was precisely above point x = 0 in frame K, the clock there read t = 0. (2) When Yang was precisely above the point x = v, that clock read t = 1. Since all the clocks on the x-axis of K were synchronized, Yin concluded that Yang's speed was v/1.

How did Yin synchronize the clocks on the x-axis? Well, careful experiments always corroborated Maxwell's prediction that the speed of light was a constant c in all directions. So when the clock at x = 0 read T, Yin sent out a flash in the positive and negative x directions, and had the clocks at x = +X set to T + X/c as the flash passed by.

Now Yang had synchronized the clocks in K' the same way, and found that Yin traveled in the negative x' direction of frame K', also at speed v. So for quite a long time Yin and Yang coexisted peacefully, each content that the world was nicely symmetric. Each did experiments, using his own meter sticks and clocks, and each found that the laws of physics corresponded with physical measurements, etc.

One day Yin decided to measure the length of a meter stick at rest on the x’-axis of frame K’. Yin did this by noting what points, on his x-axis, the left and right ends of the moving stick coincided with at the same moment. He was amazed to find that the moving meter stick was only (1 - v2/c2)1/2 meters long! Spurred on by this unexpected result, Yin measured the rate at which a clock at rest in K' ran. He did this by noting what the clock read at time t1, and then again what it read at time t2, where the times t1 and t2 were obtained from the two clocks in K that the clock in K' coincided with at those two moments. And behold, Yin found that Yang's clock advanced only (1 - v2/c2)1/2 (t2 - t1). Evidently clocks at rest in K' were running slowly!

As Yin pondered these two strange findings, it occurred to him that the clocks in K’ could not in fact be synchronized. After all, how could they be? Light traveled with the one speed c in all directions relative to frame K. It followed that light traveling in the positive x direction had a velocity of only c-v relative to K’. And of course light traveling in the negative x direction had a speed of c+v relative to K’. Since Yang had undertaken to synchronize the clocks in K’, using the same procedure as had been used in K, it was clear that the clocks in K’ were not synchronized. Of course this could easily be verified. So Yin noted what the clocks (a) at O’ (the origin of K’), (b) to the left of O’, and (c) to the right of O’ read at the same moment. And sure enough, the clock to the left of O’ read ahead of the clock at O’, and the clock to the right read behind the clock at O’.

With deep bows and in the most polite possible voice, Yin informed Yang of three facts: (1) the clocks at rest in K' were not synchronized, (2) the meter sticks at rest in K' were only (1 - v2/c2)1/2 meters long, and (3) the clocks at rest in K' ran slowly by a factor of (1 - v2/c2)1/2.

Now Yang, knowing Yin to be scrupulously honest, smiled incredulously (but politely), and asked how that could be. The meter sticks and clocks at rest in K' had been obtained from the same stock room as the meter sticks and clocks at rest in K.

"I don't know," Yin answered, "it's very strange."

"And why do you say that the clocks at rest in frame K’ are not synchronized?" Yang continued. "I synchronized them the same way you synchronized the clocks at rest in K."

"But that is just the problem," Yin suggested. "You assumed that light travels at the one speed c in all directions relative to frame K’, when in fact it doesn't."

"But I can do experiments that show it does," Yang protested.

"Yes, I watched you do them," Yin answered apologetically. "Evidently the shortening of your meter sticks, and the slowed rate of your clocks, are just what is required to produce the illusion that light travels with the constant speed c, in all directions, relative to frame K’. But in fact it does not."

Being a scientist, Yang was not particularly biased about the objects at rest in K', and he suggested that they determine how much out of synch the clocks at rest in K' were. So Yang joined Yin, at rest in frame K, and together they derived the following results:

From the perspective of frame K, the K’ clock at x’ = L runs behind the K’ clock at x’ = 0 by the amount

(L/c)(c + v)1/2/(c – v)1/2 – (L/c),

and the K’ clock at x’ = -L runs ahead of the clock at x’ = 0 by the amount

(L/c) – (L/c)(c – v)1/2/(c + v)1/2.

Although he was too polite to say anything, Yang was not convinced. So he hopped back into frame K’ and resolved to measure a meter stick at rest in K, using essentially the same method Yin had used. That is, despite what Yin had said about the clocks in K' being out of synch, Yang decided to proceed on the assumption that they were in fact synchronized. And of course Yang proceeded on the assumption that the meter sticks at rest in K' were in fact 1 meter long, and that the clocks at rest in K' ran at the correct rate. And behold, when Yang measured the length of a meter stick at rest in K, he found it to be only (1 - v2/c2)1/2 meters long! Furthermore, clocks at rest in K were found to run slowly by a factor of (1 - v2/c2)1/2! Yang then did some calculating. Assuming that light actually traveled at the constant speed c in all directions relative to K', Yang concluded that it was the clocks at rest in K that were out of synch. And they were out of synch quite as Yin had avowed the clocks at rest in K' to be out of synch. That is, from the perspective of frame K’ the K clock at x = -L ran behind the clock at x = 0 by the amount (L/c)(c + v)1/2/(c – v)1/2 – L/c. And the clock at x = L was found to run ahead of the clock at x = 0 by the amount (L/c) – (L/c)(c – v)1/2/(c + v)1/2.

"Hm-m-m," Yang thought, "Yin might not like this. But science is science."

"Ah-h-h, Yin," Yang murmured, stepping back into frame K, "I just did some measurements, and guess what. I'm finding in K’ that the meter sticks at rest here in frame K are only (1 - v2/c2)1/2 meters long, and the clocks run slowly by a factor of(1 - v2/c2)1/2, and they're out of synch."

"But how can that be?" Yin protested.

"I don't know," Yang said. "I measured the length of one of the frame K meter sticks the same way you measured the length of one of the K’ sticks. And it's only (1 - v2/c2)1/2 meters long."

"And the rate of one of my clocks?" Yin inquired.

"Same deal," Yang answered. "I measured its rate just as you measured the rate of one of the K’ clocks. Viewed from frame K’, clocks at rest here in K are definitely running slowly."

"I get it!" Yin shouted. "You got those results because the K’ clocks are out of synch and the K’ meter sticks are foreshortened!"

"Is that possible?" Yang exclaimed. "Each frame measures the other's meter sticks to be foreshortened by the same identical factor, and so forth?"

"Yes, yes, that's it!" Yin said excitedly. "In fact, if you do the math, you'll find that's why you conclude that light propagates with the one speed c, in all directions relative to K’, when in fact it doesn't!"

"But wait a minute," Yang answered, "how do we know that it's the K’ clocks that are out of synch? How do we know that it isn't the K clocks that are out of synch?"

"Yes, I see your point," Yin conceded somberly. "In fact, clocks in both frames might be out of synch."

"As measured in what ... some third frame ... say one that’s at rest relative to the universe's center of mass?" Yang wondered.

"Maybe," Yin answered. "I mean, think about it. I say a charge at rest in frame K’ has a magnetic field, but one at rest in frame K has only an electrostatic field."

"And K’ says just the opposite," Yang rejoined.

"Quite so," Yin continued. "Am I right, and is an observer at rest in K’ wrong? Is an observer at rest in frame K’ right and is frame K wrong? Are they both wrong?"

There was a deafening silence as Yin and Yang pondered this question. At length Yang quietly said,

"They’re both right."

"How can that be?" Yin asked.

"All of the laws of physics work in every inertial frame of reference."

"Including the fact that light is measured to propagate with the one constant speed c, in all directions, relative to every inertial frame?"

"Right! And I find that the results obtained in frame K, using its clocks and meter sticks, are consistent with what the laws of physics say must be true in frame K’ because from frame K’ the K clocks are observed to be out of synch and to run slowly, and so forth," Yang marveled.

"And I find the same things," Yin added.

"And both findings may be wrong," Yang murmured.

"Or both right!" Yin concluded.

"Wow, will we ever know which frame, of all the inertial frames, is the one true frame?" Yang wondered.

"The frame where a resting charge really has no magnetic field, for example?" Yin probed.

"Yes," Yang answered, "the one true frame ... the primary frame ..."

"The frame in which the universe's center of mass is at rest?" Yin asked.

"Perhaps," Yang answered.

"Or perhaps not," Yin suggested. "Maybe the only thing that matters is that the laws of physics work for all inertial observers. Let's face it, there's no way to determine which frame the universe's center of mass is at rest in!"

"How extraordinary! How elegant!" Yang rejoiced. "Even Maxwell didn't know about this!"

"It's a safe bet he didn't," Yin agreed.

"Let’s push this to the max and see what the meter sticks and clocks in frame K look like when they’re traveling at a speed close to the speed of light," Yang suggested.

"Using the space pod?" Yin queried.

"Sure. We could leave this Saturday."

"How long would we be gone?" Yin wondered.

"Well, if you’re agreeable, let’s make a real trip out of it. We’ve never visited any of the outposts far out on the x-axis. What say we travel out for a month, check things out way out there, and then return home."

"Hm-m-m, that could be interesting," Yin answered. "I’m game if you are."

So the two friends agreed to embark on an extended trip out along the x-axis of frame K. That evening, as he lay waiting for sleep to claim him, Yin had a nagging sense of foreboding. But he couldn’t figure out why. The space pod was proven technology. It could support two passengers for well over a year. He didn’t see what could go wrong. Still ...

 

Chapter 2

Yin and Yang Take a Trip

At 9 a.m. on Saturday Yin and Yang met at the space pod. The plan was to rapidly accelerate up to a speed of .9999995c, and to ascertain that the speed of light was c in all directions, relative to that new rest frame. If this turned out to be the case, they would synchronize nearby clocks in their new rest frame, and (a) measure the length of meter sticks at rest in frame K, (b) measure the rates of clocks at rest in frame K, and (c) determine whether or not clocks at rest in frame K were synchronized (and if not, how much out of synch they were). Having performed these experiments, they would then coast along until they had spent a total of one month in their new rest frame. They would then decelerate back to a state of rest in frame K, spend a day visiting with the folks manning the outpost there, and then return to the origin of frame K at the same speed. All told they planned to be away for about two months. Along the way there would be frame K clocks, evenly spaced, to guide them on their way.

On departure day Yang kissed his wife and baby girl goodbye, assuring his wife that he'd see her again in about nine weeks. Yin, a confirmed bachelor, was already on board programming the pod’s thrusters.

At last Yang boarded and they were on their way. Once up to speed, the two voyagers quickly determined that light traveled with the one speed c in all directions inside the pod. So they synchronized the clocks in the pod in the usual way. As the x-axis clocks went whizzing by they were able to verify that the clocks at rest in frame K were running more slowly than their onboard clocks, quite as expected. On the other hand, the x-axis clocks kept reading later and later than the onboard clock.

"How can that be?" Yang wondered. "They’re running more slowly than our clocks."

"It's because the clocks on the x-axis aren't synchronized," Yin answered. "We thought they were before starting out. But in our current rest frame it's clear that the clocks out here on the positive x-axis are set progressively more and more ahead of the origin clock."

"Oh yeah, we already derived that," Yang recalled.

"Note how frequently the K clocks pass," Yin observed.

"Yes," Yang answered. "I suppose it’s because, from our present perspective, they’re much more closely spaced then we thought they were when we were at rest in frame K."

"Of course," Yin agreed. "The meter sticks along the x-axis are contracted from the perspective of this frame, so naturally the K clocks arrayed along those meter sticks are more closely spaced. You know, if we keep this up for a month, we’re going to be a lot further out on the x-axis than I had originally envisioned."

"You’re right," Yang said. "But that isn’t a problem, is it?"

"No, I can’t think why it would be," Yin replied. "After all, we did want to make a real trip of this."

Yang looked out of one of the pod’s side windows.

"I wonder where all the stars have gone?" he mused. "Everything looks black out there."

"Yes," Yin agreed. "That is strange. We’ll have to give that some thought."

Yin and Yang had brought along lots of books to read, and could play a variety of games with the onboard computers. The month passed quickly, and they soon decelerated to a state of rest again in frame K. The signpost said they were nearly 85 light years from the origin!

"Wow! We've been traveling at something less than the speed of light for only a month, and yet we’re 85 light years away from the origin!" Yang marveled.

"Yes, it’s been a month according to our onboard clock. But remember, all that time the x-axis was greatly contracted from our onboard perspective," Yin reminded.

Both men noticed that the clock at the outpost read 85 years later than their onboard clock ... the cumulative effect of all that progressively increasing degree of non-synchronization with the origin clock.

The folks at the outpost wanted to hear all the news from the origin, and Yin and Yang tried to bring them up to date. But much of the information was already known by the locals, who had heard it on recent TV newscasts from the origin.

That evening as Yin and Yang lay waiting for sleep to claim them, Yin made an interesting observation.

"Did you notice the date/time stamp on the TV signal from the origin?" he asked.

"No," Yang answered, "why?"

"It was 85 years earlier than what the local clock reads," Yin said.

"Well ... sure, doesn't that make sense?" Yang answered. "We're 85 light years from the origin.

"Yes," Yin continued, "but here's my point: in our current rest frame, the clock at the origin is synchronized with the clock here."

"So?" Yang pressed, unsure of what Yin was getting at.

"So right now, at this moment, back at the origin it's 85 years later than when we set out."

"Wha-a-a-at?" Yang cried. "How can that be?"

"Because in our current rest frame it's clear that, while under way, we were aging much more slowly than we now are," Yin explained.

"That's crazy!" Yang snorted. "That would mean my whole family has probably died of old age!"

Yin lay quietly in the darkness, not answering. At length Yang asked nervously if that was really possible.

"I don't know," Yin said softly. "I'm sorry I didn't work this out in my head before we left. Perhaps you shouldn't have come."

Yang felt like all the blood drained from his head. Could this possibly be? If true, then what despair might his wife have felt when they failed to return? Would she have remarried in time? And what would have become of his baby girl? Heaving a great sigh, Yang tried to fall asleep, and eventually he did. But it was a sleep punctuated with dreadful dreams.

The next morning a sober Yin and Yang bade farewell to the outpost inhabitants and started the trip back home. Again, once up to speed, their onboard clock was noted to run faster than any clock on the x-axis. Yet the x-axis clocks kept reading successively later.

"They're out of synch again, aren't they?" Yang asked despondently.

"Yes, from our present perspective that is again the case," Yin answered.

Yin was determined to figure out how this unfolding tragedy could have occurred. Their speed in both directions was .9999995c, very close to 3 x 108meters per second, the speed of light. At that speed, in 30 days they traveled about 7.776 x 1014 meters. But that distance amounted to 7.776 x 1017 meters in frame K, owing to its length-contracted meter sticks. One light year was approximately 9.46 x 1015 meters. Thus the K distance did indeed come out to about 85 light years!

"So we were at rest in frame K’, traveling in the positive x direction of frame K with a speed of .9999995c for 30 days. And in that time we covered 85 light years according to frame K," Yin marveled silently.

A month later they were approaching the origin of frame K and applied the deceleration thrusters. As the ship decelerated, the rate of the x-axis clocks picked up. By the time they came to rest at the origin, the origin clock and their onboard clocks were running at the same rate. But ... the origin clock read 170 years later than the onboard clocks.

The people who greeted Yin and Yang when they disembarked had no idea who they were, and Yin and Yang were taken into custody. They told their story to an incredulous police chief, who recounted it at home that evening to his family, which included his 90-year-old grandfather.

"It's true," the old man rasped. "I remember reading about those guys when I was a boy. They had left, oh, nearly 100 years before then and never returned. I read about it in a science article ... something about time dilation. The author said they might still be out there ... still young men. Everybody thought he was crazy."

The next day the police chief got some people to research the matter, and found that two men named Yin and Yang had indeed left on a trip 170 years earlier. The faded photographs were perfect matches with the men in custody. With profuse apologies and wondering eyes, the chief released Yin and Yang.

The two adventurers headed immediately for the city library, which was hard to find since nothing looked the same. The library was in a completely different building from the one they remembered. Indeed everything was different ... strange and amazing vehicles, holographic TV ... everything!

Once in the library, Yin and Yang pulled up the news from 170 years ago. Sure enough, there they were on their departure day! An anguished sob exploded from Yang’s body. Yin put his arm around his friend's shoulders.

"Let's search personal records," he suggested gently.

They pulled up the records on Yang's wife. She had never remarried, but had died of old age 120 years earlier. Yang's daughter had grown up to distinguish herself as a concert pianist and had married a famous composer. She too had died nearly 90 years ago. Yang's sobs ebbed away as he read with growing wonder the saga of his family.

Two days later the Secretary of the Interior received two young men at the prodding of the President's Secretary of Science.

"What's it about?" the Interior Secretary had asked when he got the call from The White House.

"You'll see. Just meet with them," the Science Secretary had said with obvious glee.

As the two young men entered his expansive office, the middle aged Secretary rose and approached them with outstretched hand.

"Good morning," he smiled uncertainly. "I'm Jonathan. And you must be ...?"

"Yin!" Yin smiled, grasping the proffered hand.

"And I'm Yang," the other said gently, beaming affectionately at the older man. The Secretary looked into Yang's eyes curiously and then stiffened when the younger man grasped his hand and said,

"I'm your great great great grandfather."

Chapter 3

Yin and Yang Adjust to Length Contraction and Time Dilation

Yang was uneasy with the fact that moving meter sticks are foreshortened in their direction of motion, and moving clocks run slower than resting ones. On top of that there was the lack of clock synchronization among reference frames in motion relative to one another. Life had once been so simple! Everyone had believed that all identical clocks run at the same rate regardless of their state of motion or rest, that all meter sticks are one meter long, and that the clocks in all reference frames can be synchronized. But now ...

"Another fine mess," Yang thought. "But ... nature is calling the shots. How could we have been so wrong for so long?" After thinking about it, Yang realized that the ubiquitous term (1 - v2/c2)1/2 differs significantly from unity only when v approaches c. "And men have only recently studied such cases," he concluded.

Yang voiced his anxiety to Yin, who at first nodded, but then said,

"Things might not be all that bad."

"How so?" Yang pressed.

"Well," Yin continued, "in the old world view we had the Galilean transformation among reference frames. According to it, if you're in frame K', moving in the positive x direction of my rest frame K with speed v, and if I note an object to be at x at time t, then you'll observe it to be at x' = x - vt."

"You mean x - vt', don't you?" Yang replied.

"Well ... yes of course ... but t = t' in the old view of things."

"Oh yeah," Yang answered, "because in the old regime everybody agrees that all clocks can be synchronized and that they all run at the same rate. But wait a minute ... suppose that at time t = 0 my origin is already at x. If the object is at rest relative to you, then obviously at some later time it's going to be on my negative x’-axis. But if vt is less than x, then x' = x - vt is positive ... the Galilean transformation fails!"

"No, no," Yin answered, "it's taken for granted that the origins of the two frames coincide when the clocks there read to = t'o = 0."

"Oh ... you didn't tell me that," Yang murmured defensively.

"Sorry," Yin said.

Yang thought about it, and quickly realized that x' = x - vt would then be correct.

"So what's the story in the new regime?" Yang wondered.

"Well, let's agree for openers that the origins of K and K' coincide when the clocks there mutually read to = t'o = 0," Yin suggested.

"Okay, but I'm going to find that the clocks along your x-axis are more closely spaced than the ones along my x’-axis, and that your clocks are all running more slowly than mine. And on top of that, I'm going to think that your clocks are out of synch, right?" Yang asked.

"Yes, and I'm going to find the same thing to be true about things at rest in your reference frame," Yin rejoined.

"Amazing!" Yang exclaimed. "And all because of the way we've synchronized our respective sets of clocks!"

"Yes ... that along with the fact that the moving meter sticks are length contracted, and the moving clocks run slowly."

"And those three facts produce this beautiful symmetry!" Yang cried. "Whoever would have thought it possible? But where do we go from here?"

"Let's forge ahead," Yin replied. "To keep things simple, let's say that at some time t greater than zero I find an object to be at x."

"Where t is ...?" Yang pressed.

"Strictly speaking, t is what my clock at x reads when the object coincides with it," Yin answered. "But since I believe all my clocks to be synchronized, I would contend it's what every clock in K reads when the object coincides with the clock at x."

"But at that moment the K' clock that coincides with the object isn't going to be the one at x' = x - vt," Yang said.

"Exactly," Yin agreed. "And the K' clock that coincides with the object isn't going to read t. Our job is to figure out what the x' coordinate of that clock is, and what time it reads."

"So that we can come up with a new transformation!" Yang exclaimed, guessing where Yin was headed.

"Right!" Yin affirmed. "Let's start with x'. In my opinion, at time t your origin will be at xo = vt. So as far as I'm concerned, the distance between your origin and the object is x - vt."

"The Galilean transformation," Yang mused.

"Yes. But then again, I perceive your meter sticks to be contracted. So the x' coordinate of the K' clock coinciding with the object is going to be (x - vt)/ (1 - v2/c2)1/2."

"I agree," Yang stated. "So our transformation from x to x' is x' = (x - vt)/ (1 - v2/c2)1/2."

"Correct!" Yin agreed. "That's not so bad, is it? Now let's see what t' is."

"Where t' is what ... the time shown by that particular K' clock?" Yang queried.

"Yes, and in my opinion only by that clock, since I don't believe that all the K' clocks are synchronized."

"Agreed," Yang said. "Actually we know that the clock at O' reads only t(1 - v2/c2)1/2 at the moment in question, don't we?"

"Very good," Yang praised. "At least that's what I would say, since I perceive all the K' clocks to run slowly and the O' clock read to' = 0 when my clocks all read t = 0."

"And you believe that clocks on my positive x’-axis run behind the clock at O'," Yang continued.

"Correct," Yin said, "owing to the way you synchronized them."

"And we’ve already worked out how much that clock lags behind my origin clock ..." Yang said.

"Indeed we have," Yin affirmed, thumbing through his notes. "And the answer is ... the clock at x’ reads behind your origin clock by (x’/c)(c + v)1/2/(c – v)1/2– x’/c."

"So t’ = t(1 – v2/c2)1/2 - (x’/c)(c + v)1/2/(c – v)1/2 + x’/c, which simplifies to-o-o ..." Yang did some quick scribbling on a note pad ... "t’ = (t –xv/c2)/ (1 – v2/c2)1/2 ."

Yin was busy calculating things in his own notebook. Being somewhat more methodical than Yang, it took him longer. But at length he agreed.

"So we're done, aren't we?" Yang asked.

"Almost," Yin replied. "We really ought to see how y transforms into y' and z transforms into z'."

"But that's a no-brainer, isn't it?" Yang asked. "Measuring rods are foreshortened only along their direction of motion."

"So y' = y and z' = z?" Yin continued.

"Right," Yang said. "So our complete transformation, in the brave new regime, is x' =(x - vt)/ (1 - v2/c2)1/2, y' = y, z' = z, and t' = (t - xv/c2)/ (1 - v2/c2)1/2."

"By George, I think we've got it!" Yin agreed.

"Now let's work things out from the K’ point of view," Yang said.

"Sure. We're saying that you observe an object to be at x', y', z' at time t'. And we want to know what I'll observe x, y, z and t to be, right?"

"Right!" Yang said.

The two friends went through much the same reasoning as above, the difference now being that K' was assumed to be at rest, with K moving in the negative x' direction at speed v. And they found in this case that x = (x' + vt')/ (1 - v2/c2)1/2, y = y', z = z', and t = (t' + x'v/c2)/ (1 - v2/c2)1/2. Pleased with this accomplishment, they called it a night.

The next morning Yang mentioned what they had derived to a physics professor with whom he spoke frequently.

"Oh, yes," The Professor answered. "That is called the Lorentz transformation."

"Lorentz?" Yang pressed.

"After Lorentz, who first suggested it to explain the null results obtained by Michelson and Morley."

"Michelson and Morley?" Yang asked sheepishly.

"Yes. Back in those days everyone believed that the Earth must be moving at least part of the time relative to the luminiferous ether."

"Lu ... luminiferous ether?" Yang asked, now acutely embarrassed.

"Yes, that was the frame relative to which it was believed light really traveled with the one speed c, in all directions, as Maxwell had predicted."

"The Primary Frame!" Yang cried out. The Professor looked at him quizzically.

"Never mind," Yang apologized. "We were talking about Michelson and Morley?"

"Yes," The Professor continued. "They, like everyone at the time, thought the Earth must move at least part of the time relative to the lu ... relative to the primary frame. So they devised a clever interferometer to detect the differences in the speed of light, in different directions and at different times of the year, relative to the Earth's rest frame."

"And?" Yang asked.

"Nada. Zilch. All indications were that light always propagates with the one speed c, in all directions, relative to the Earth's rest frame."

"Which would be just what one would expect them to observe if their interferometer was foreshortened along the Earth's trajectory," Yang concluded.

"Exactly," The Professor said. "It was at least partly to explain that null effect that Lorentz deduced the transformation that bears his name."

"So the Lorentz transformation was originally sort of ad hoc," Yang observed.

"To some extent it was," The Professor agreed. "It wasn't until years later that time dilation ... the slowing down of processes in moving systems ... was actually observed."

"And the foreshortening of moving measuring rods?" Yang pressed.

"Length contraction," The Professor amended. "That's even harder to observe. We've only recently managed to do so."

"Fascinating information," Yin said gratefully. "Yin will be interested to learn about this."

Later that day, Yin and Yang had lunch together, and Yang related to Yin what he had learned about the transformation they'd derived the day before.

"So we did all that brain storming for nothing," Yang concluded.

"Oh, I don't think so," Yin rejoined. "You know, it's always good to work these things out for yourself."

"You are truly a skeptic," Yang teased affectionately.

"Yes, I suppose I am," Yin answered. "I consider it to be one of my better traits!"

"To skepticism!" Yang toasted, raising his glass.

 

Chapter 4

Vanishing Point?

It took a few days for Yin and Yang to get used to the Lorentz ideas that x' = (x - vt)/ (1 - v2/c2)1/2 and t' = (t - vx/c2)/ (1 - v2/c2)1/2. (As Yang had pointed out, y' = y and z' = z couldn't be simpler.) It soon became evident, however, that the new transformation explained such seeming paradoxes as how each person could literally measure the other's meter sticks to be shorter than his own. For example, happily at rest in frame K', Yang could watch in mild disgust as Yin thought he measured the length of a meter stick at rest on the x’-axis, using the clocks and meter sticks of frame K. Yin might note that the right end of the stick coincided with his origin at some time t. He would then also note where the left end of the stick was at time t. Now since the meter sticks along the x-axis of frame K were, in the opinion of K', all contracted, one might at first think that Yin would find Yang's meter stick to be even longer than one of his own. But there was another factor at work: the clocks on the negative x-axis of K were (in the opinion of K') running behind the clock at the origin of K. And by the time the left end of Yang's meter stick coincided with a K clock that also read time t, the stick had moved to the right just far enough that Yin "measured" its length to be only (1-v2/c2)1/2 meters. The lack of clock synchronization more than compensated for the fact that (in the opinion of K') the x-axis of K was contracted.

Of course the situation was entirely symmetric. Yin (at rest in K) thought the x’-axis of K' was contracted. But in his opinion the lack of clock synchronization more than compensated for that fact, and Yang would (erroneously, of course) conclude that one of the K meter sticks was foreshortened, when "in fact" (in the K observer's opinion) it was just the other way around!

Similar considerations explained how each observer could literally "measure" the other's clocks to run more slowly than his own. The more the two friends played with such ideas, the more they became convinced of the Lorentz transformation's validity. But ... for the most part such considerations only kicked in when the relative motion of two frames approached c. So the older Galilean transformation, which was clearly simpler, still had its practical uses.

One day Yin observed that, knowing how x, y, z and t transformed, it should be a simple matter to determine how kinematic quantities like velocity and acceleration transform.

"I mean, how does one measure velocity?" he asked. "You note where something is at time ti and then again where it is at time tf, right?"

"Where ti and tf are arbitrarily close together," Yang added.

"Right!" Yin continued. "Let's say an object is moving parallel to my x-axis. At time ti it's at xi and at time tf it's at xf."

"So its velocity is (xf- xi)/ (tf- ti)," Yang said.

"By definition." Yin agreed.

"And if we know xi, xf, ti and tf, then we can transform to xi', xf', ti' and tf''."

"Bingo!" Yin exclaimed. "We know what the velocity of the same object relative to K' has to be."

As usual, Yin and Yang started out by reviewing the old beliefs. According to the Galilean transformation, if an object had velocity components ux, uy and uzin frame K, and if frame K' moved in the positive x direction of K with speed v, then the velocity components relative to K' would be ux' = ux - v, uy' = uy, and uz' = uz.

But under the Lorentz transformation it turned out that ux' = (ux - v)/(1 - vux/c2), uy' = uy(1 - v2/c2)1/2 /(1 - vux/c2), and uz' =uz(1 - v2/c2)1/2 /(1 - vux/c2). And all the signs, except for the one in (1 - v2/c2)1/2, were reversed when transforming ux to ux', etc.

"Once again a tad tougher to remember than the Galilean transformation," Yang sighed.

"Such is life," Yin agreed. "Nature is as nature does."

"Amen to that," Yang said.

The Galilean transformation of acceleration was as simple as one could wish: ax' = ax, ay' = ay, az' = az. Yin pointed out that things had to be that way in order for Newton's laws of mechanics to be covariant.

"Covariant?" Yang asked tentatively.

"Yes ... that's a fancy way of saying that the laws work equally well in all inertial frames."

"So if an object is experiencing a force, it has an acceleration of F/m in frame K ..." Yang began.

"And exactly the same acceleration in K'," Yin finished.

"Of course, how could it be any other way?" Yang exclaimed. "But the velocities will be different!"

"Oh, to be sure," Yin agreed. "A planet may orbit the Sun in a closed ellipse, viewed from the Solar System's center of mass rest frame."

"But viewed by a space traveler passing by the Solar System ..."

"Definitely not a closed trajectory." Yin observed.

"And of course the Sun itself is not practically at rest in such a traveler's rest frame," Yang continued.

"Right. The Sun moves nearly with a constant velocity, assuming the traveler is at rest in an inertial frame," Yin agreed.

"Wow, the classical view of things was really elegant, wasn't it?" Yang exclaimed.

"Yes, it truly was," Yin concurred. "But ..."

"Nature is calling the shots," Yang murmured.

"Precisely," Yin answered.

"So let's get on with this, and see how acceleration transforms under Lorentz," Yang said.

Each began scribbling furiously in his notebook. At length, after some debate, they agreed that the Lorentz transformation for acceleration is

,

 

,

.

"Ugly ... truly tougher than the Galilean, " Yang rued.

"Yes, but nature is ..."

"Yeah, yeah, I know," Yang groused.

"The good part is that once again you can go from ax' to ax, and so forth, simply by changing a sign," Yin observed.

"Yes, that is a small happiness," Yang agreed. Yang stared at the transformation

.

"Wow," he murmured, "that's really interesting."

"What is?" Yin asked.

"Look at the transformation for ax," Yang continued. "Note what happens if K' is the object's rest frame ..."

"So that ux' equals zero?" Yin interjected.

"Yes ... look what happens as v approaches c," Yang went on.

"Yes, I see what you mean," Yin said. "As v approaches c ..."

"Or as ux approaches c ..." Yang added.

"Yes, as ux and v approach c, ax approaches zero."

"No matter what ax', the acceleration in the object's momentary rest frame, might be," Yang added.

"So what are we saying ... there's no way to accelerate something traveling at the speed of light?" Yin wondered.

"Or even up to the speed of light," Yang amended, "since ax approaches zero asymptotically as ux approaches c."

"So we can never actually observe anything being accelerated all the way up to speed c", Yin mused.

"Right!" Yang agreed. "But wait a minute ... How about light itself?"

"It never accelerates at all," Yin pointed out.

"You're right," Yang agreed. "So evidently everything we can observe falls into two categories ..."

"It would seem so," Yin said. "There are things like light, that travel relative to every inertial frame with the constant speed c in every direction ..."

"And the rest of the stuff, that can't ever be pushed up to speed c relative to any inertial frame," Yang continued.

"And as a sub-light speed object's speed is pushed closer and closer to c, the object starts to vanish," Yin supposed.

"Because of length contraction?" Yang queried.

"Yes. But then again it might make its presence known in other ways."

"What do you suppose the length of something traveling faster than c would be?" Yang asked.

"If such things exist," Yin cautioned.

"Yeah, and if they could be observed by us!" Yang added.

"The Lorentz transformation certainly has some novel implications, doesn't it," Yin murmured.

"Novel? I'd call them WEIRD!" Yang shouted.

 

Chapter 5

Yin and Yang Learn Something New About Mass

Yang had an interesting dream the night after he and Yin had worked out the Lorentz transformation for velocity and acceleration. In the dream he was sitting on a tennis referee's chair at the origin of frame K, and Isaac Newton stepped from K into frame K', pulling a weight from K into K' in the process. Sir Isaac did this by hooking the weight with a hand-held scale, which clearly showed that a force was exerted on the weight. After doing this, Sir Isaac stepped into the next frame (which moved even faster relative to K) and repeated this act. The process continued repeatedly until eventually Newton's speed got very close to c. At this point there was no discernible difference in the reference frames' speeds relative to K. Yet each time Sir Isaac moved the weight from one frame to the next, he had to exert a force.

When Yang awoke, he wondered what it all meant. As he saw it, the idea seemed to be that F = ma began to fail as an object's speed approached c. For evidently a nonzero force produced no measurable acceleration at such speeds. Rubbing the sleep from his eyes, Yang shuffled out to his bookshelf and opened an old copy of Newton's Principia. In that venerable work Newton spoke of "quantity of motion," which Yang concluded was the same thing as modern day momentum, mv. And Newton had written that force is proportional to the rate at which the quantity of motion changes: F = d(mv)/dt. This was of course the same thing as F = ma, provided mass is constant.

"But where is it written," Yang wondered, "that an object's mass has to be constant? What if mass, like length, depends on an object's speed?"

At lunch that day, Yang asked Yin if he was happy with the idea that no object can be accelerated up to the speed c, relative to any inertial observer.

Yin looked up, with a fork of food halfway to his lips. "Am I happy?" he asked quizzically.

"Yes," Yang answered. "Don't you get the feeling there has to be some physics behind all this?"

"M-m-m-m, possibly," Yang agreed, chewing thoughtfully. "Like maybe a force can't be exerted on something going that fast?"

"Yes, I guess that's one way of looking at it," Yang answered. "But what would that amount to ... an object subjected to a force in its rest frame would experience no force to speak of, viewed from another frame in which the object's speed was practically c?"

"That's what I had in mind," Yin said. "Isn't the whole idea that laws like F = ma work in all inertial frames?"

"But here's an interesting point," Yang exclaimed. "Newton didn't say that F = ma."

Yin looked sharply at his friend. "Of course he did," he insisted with a trace of indignation.

"No, actually he said that F = d(mv)/dt," Yang continued.

"So," Yin pressed, "isn't that the same thing as F = ma?"

"Not if mass, like length, varies somehow with an object's speed."

"Yes, I see your point," Yin answered, blinking and placing another forkful of food into his mouth. "In that case F would equal ma + v(dm/dt)."

"Right!" Yang agreed. "And when v is much less than c, dm/dt is practically zero."

"Which is just another way of saying that the mass appears to be practically constant when v is much less than c."

"So where do we go from here," Yang wondered.

"Well, there appear to be two possibilities," Yin answered. "(1) the force on an object approaches zero as the object's speed approaches c, or (2) the object's mass begins to increase as v approaches c."

"Yes, and either case leaves the door open for Newton's Second Law, in the form F = d(mv)/dt, to work in all inertial frames."

"But there's always the possibility that the law simply doesn't work for all inertial observers. How shall we find out what's what?"

"I'm going to a physics seminar at the university this afternoon," Yang said. "Why don't you come along. We can ask my friend, The Professor."

"Thanks," Yin replied. "As a matter of fact, my afternoon is free." They agreed to meet at the university physics building at 2 p.m.

Yin and Yang enjoyed the seminar presented by a visiting scholar, although they failed to grasp every point. Afterward Yang introduced Yin to The Professor, and following the usual idle chat, Yang related their concerns.

"All indications are that your idea of speed-dependent mass is the right one," The Professor said. "Newton's law, in the form F = d(mv)/dt, seems to work under Lorentz transformations ... a remarkable fact, since Sir Isaac no doubt didn't know about the dependence of mass upon speed, and so forth."

"It was certainly an inspired moment," Yin mused.

"What was?" Yang asked.

"When Newton penned his three laws of mechanics."

"Yes, it was," The Professor agreed.

"But how do we know that mass depends on speed?" Yang asked.

"And what is the relationship?" Yin chimed in.

"Well, the relationship is very simple," The Professor answered, fishing out his pipe and lighting it. "If you symbolize the so-called rest mass ... the mass of an object at rest ... as mo, then its mass at speed u is m = mo/(1 - u2/c2)1/2."

"Ah hah!" Yang exclaimed. "Enter our old friend, (1 - v2/c2)1/2."

"Yes," Yin agreed. "And note how the mass is practically constant at speeds much less than c, but begins to increase without bound as u approaches c."

"Um-m-m hm-m-m, good eye," The Professor said, making little sucking sounds with his pipe. "And to answer your question about how we know this equation is correct, there are actually many ways. One of the simplest involves a LINAC."

"A LINAC?" Yang asked in a small voice.

"Yes ... a Linear Accelerator," The Professor answered. "We can inject electrons, traveling at practically the speed of light, into such a device and subject them to a constant force while they travel down the device's interior."

"But at that speed they don't accelerate ..." Yin objected politely.

"A point well taken," The Professor chuckled. "Calling it an accelerator is actually something of a misnomer."

"So how do we know the mass increases?" Yang pressed.

"Quite simple," The Professor answered. "As you know, energy comes in many forms, one of them being kinetic energy and another being heat. We can, for openers, let the electrons smash into a target just as they enter the LINAC. Their kinetic energy mostly gets converted to heat energy, and we get a measure of their kinetic energy by noting how much the target heats up."

"And then you do it again, with the target at the end of the LINAC!" Yin guessed.

"Exactly," The Professor corroborated. "And the kinetic energy is greater by the force the LINAC exerts on the electrons, times the distance over which that force acts."

"And their speed doesn't change at all?" Yang asked.

"Not measurably," The Professor said.

"But the mass has increased, so the kinetic energy, mu2/2, is greater," Yin suggested.

"Actually that familiar old formula for kinetic energy isn't correct at near-light speeds," The Professor said. "Einstein saw that mass and energy are two sides of the same coin. Mass can be converted to energy, and in some cases energy can be converted to mass."

"As in nuclear fission!" Yang guessed.

"Yes, that is one important case where mass is converted to energy," The Professor agreed.

"And the relation between mass and energy is E = mc2?" Yin added.

"Correct," The Professor said, "where m is whatever is appropriate ... mo if the converted mass is at rest, and mo/(1 - u2/c2)1/2 if it's moving."

"But what is the correct formula for kinetic energy?" Yang asked.

"Well, when an object is at rest, it has an intrinsic energy of moc2," The Professor hinted.

"And when it's moving its energy is moc2/(1-u2/c2)1/2," Yin guessed.

"Um hum," The Professor agreed.

"So the kinetic energy is really (m-mo)c2," Yang exclaimed.

"Just so," The Professor approved.

And moc2(1/(1 - u2/c2)1/2 - 1) is essentially the same as mou2/2 when u is much less than c," Yang observed.

"Very good," The Professor praised.

Yin envied how quickly Yang deduced such mathematical truths, but said nothing. They invited The Professor to join them for dinner, but he politely declined. So after thanking The Professor and telling him how happy he was to have met him, Yin joined Yang for dinner at a favorite Italian restaurant. Over a plate of pasta Yang said that there ought to be some way that length contraction could be shown to be a physical effect. After all, a meter stick was just a bunch of atoms, wasn't it?

"You mean use physical law to demonstrate how a moving atom is length contracted?" Yin asked anxiously.

"Sure," Yang replied. "There must be some physical reasons for length contraction and time dilation, mustn't there?"

"Well-l-l," Yin winced, "perhaps there is. But shouldn't quantum theory be invoked to show that?" Yin was not particularly comfortable with the mathematical trappings of quantum theory.

"Yes, strictly speaking quantum theory should be invoked," Yang agreed. "But still ... there ought to be some way to demonstrate length contraction, and for that matter time dilation, using Maxwell. I'll have to think about it."

The two friends let go of the problem for the time being, and enjoyed the small restaurant's ambiance. But Yang quietly resolved to demonstrate length contraction and time dilation using the classical ideas ... subject of course to what they had learned that day about the dependence of mass upon speed.

Yin was also quietly working up an agenda. The fact that mass was a function of speed meant that they already knew everything they needed to work out how force ... or what amounted to the same thing, how d(mv)/dt ... transformed from one frame to another. He would be out of town for five days on business, and relished the idea of working on the problem on his own, away from Yang's quick mind.

So the two friends said goodbye, with Yang wishing Yin a safe trip. The next morning, as the jet that would whisk him halfway around the world climbed smoothly into the sky, Yin took out his notebook and wrote

"Given:

(1) x' = (x - vt)/ (1 - v2/c2)1/2.

(2) t' = (t - vx/c2)/ (1 - v2/c2)1/2.

(3) m = mo/(1 - u2/c2)1/2.

(4) Fx = max + ux(dm/dt).

Fx' = ???"

 

Chapter 6

Yang Demonstrates Length Contraction and Time Dilation

The morning after Yin left on his business trip, Yang awoke and immediately knew the manner in which he would try to demonstrate how length contraction and time dilation are a natural consequence of physical law. He would model a macroscopic version of Bohr's hydrogen atom on his PC. Specifically, he would compute the motion of a 1 coulomb negative charge, orbiting a 1 coulomb positive charge in a circle of radius R = 1 meter. The positive charge would be assumed to have an arbitrarily large mass, so that it remained practically at rest at the origin of inertial frame K. And the negative charge's rest mass, mo, would be such that its constant speed around the positive charge was .9c.

Yang knew that a circling charge radiates energy, and that some agent must exert a tangential force in order to maintain the circular motion. But ... like Bohr, he decided to ignore this complication, at least for the present. Excitedly Yang went to his desk and sketched things at time t = 0.

Yang quickly saw how the classical law, F = ma, needed to be modified to account for the dependence of mass on speed. Since u, the negative charge's speed, was constant, dm/dt was zero and Newton's Second Law became

.

And, since the positive charge was at rest, it generated no magnetic field. The force on the negative charge was therefore simply coulombic:

.

Substituting q = 1 coulomb, etc., Yang calculated that the negative charge's rest mass would have to be

.

Yang rose from his desk and went into the kitchen. While he absently fixed some breakfast, his mind swam with the ideas of where he was headed. Thus far the situation resembled a Sun/planet scenario (although the planet's orbital speed would be considerably less than .9c). Yang remembered his and Yin's conversation about covariance under a Galilean transformation. Viewed from frame K', moving in the positive x direction of the Sun's rest frame with speed v, the Sun would move in the negative x' direction of frame K' at constant speed v. More interestingly, the orbiting planet would trace out a cycloid in K', always remaining the constant distance R from the Sun. The beauty ... the covariance of it all was that F = ma would produce the cycloidal trajectory in K', quite as it produced the circular motion in K.

"But," Yang thought as he munched on a slice of toast, "the electromagnetic force, experienced by the orbiting charge in K', probably doesn't equal the electrostatic force in K." For in K' the positive charge would generate a magnetic field, and in K’ the force on the negative charge, moving with velocityu’, would be

.

In order to solve for F', Yang needed to know what the electric and magnetic fields of a positive charge are, when said charge moves with constant velocity. As he thought about it, it made sense that E' would not equal E, since in K' there was a time varying magnetic field. Yang knew that it was a two way street in Maxwell's equations: a time varying E’ field induces a component of B’, and a time varying B’ field induces a component of E’.

"That's how light propagates through charge-free space," he thought. In the present case the induced component of electric field, added to the positive charge's divergent electric field, would quite likely produce an E' that differed from E.

Beyond this, Yang needed help. Placing the breakfast dishes in the dishwasher and rinsing his hands, he retreated to his study and pulled a favorite text in electromagnetism from the bookshelf. He quickly determined what E' and B' are in the case of a positive charge moving with constant velocity, and began designing his modeling program.

Starting with the conditions he had sketched for t = 0, Yang saw that if he set v equal to u, then initially in K' the negative charge would momentarily be at rest on the y' axis. And of course the positive charge would momentarily be at the origin of K', moving in the negative x' direction with constant speed v.

"What should the negative charge's trajectory look like in K'," he asked himself. It seemed clear that it should not be a cycloid. If length contraction occurred, then the negative charge should cut the x’-axis, in front of and in back of the positive charge, at a distance of only R(1 - v2/c2)1/2 from the positive charge. But since y = y' under a Lorentz transformation, the negative charge should still be a distance R from the positive charge when directly above and below it.

"And how about time dilation?" he wondered. If all went well, he should find that the elapsed time, between any two successive moments when the negative charge was directly above the positive charge, would be

.

"And of course," Yang thought, "I could just as easily start out with the positive charge at rest in K', and compute the same length contraction and time dilation in K." Was it possible? Did all this amazing stuff lie hidden in Maxwell and Newton, provided only that one make the simple adjustment for speed-dependent mass?

"Only one way to find out," he thought. But this could take a while. So reluctantly he got into his running clothes, went out and jogged around the park five times, and came home and showered.

While toweling off, an interesting thought occurred to Yang. As far as the K' observer was concerned ... indeed as far as any inertial observer is concerned ... the Galilean transformation is the correct transformation. The Lorentz transformation is needed to accommodate the contracted meter sticks and the time dilated, unsynchronized clocks in other inertial frames.

"What happens if the K' observer applies the Galilean transformation to the computed positive and negative particle positions in K'?" Yang asked himself. That would place the positive charge at rest at the origin of K', and the negative charge would travel around the origin in a closed path. The K' observer would "see" what the moving "atom" looked like! Yang decided to add this final transformation to his modeling program. He spent the rest of the morning and most of the afternoon working on the program. Shortly before dinner, the program was finished and he ran it. With wondering eyes he gazed at the plotted picture of what the moving "atom" would look like in K'.

 

Yang's moving "atom" was indeed a foreshortened circle. The oval had a diameter of precisely 2R(1 - v2/c2)1/2 along the x’-axis, and a diameter of 2R along the y' axis. And although not shown in the plot, the program revealed that T', the time needed by the negative charge for one trip around the oval in K', was precisely T/(1 - v2/c2)1/2, where T was the time for a trip around the circular orbit in K.

"Wait until Yin sees this!" he thought excitedly. Yang ran the program for other orbital speeds in K. At .99c the oval in K' was extremely narrow in the x' direction. And at speeds less than .1c the shape in K' was for most practical purposes a circle, quite as it was in K.

That evening, as Yang lay waiting for sleep to come, a disquieting thought occurred to him.

"What about gravity?" he wondered. Suppose that some space traveler passed by the solar system and traced Earth's trajectory relative to his own rest frame. Assuming the Earth's orbit around the Sun was circular in the Sun's rest frame (and this is nearly the case), then its trajectory relative to the space traveler's rest frame should be similar to the computed trajectory of his modeled atom's negative particle relative to K'. In brief, the motion in the Sun's rest frame should transform to the space traveler's rest frame according to the Lorentz transformation. Among other things, the time for one cycle of the quasi-cycloid should be (1 year)/ (1 - v2/c2)1/2, where v is the space traveler's speed relative to the Sun.

"Does the gravitational field transform like the electric field?" Yang wondered. "And is there a field analogous to the magnetic field, in the case of uncharged body interactions?"

Pondering such possibilities, Yang slipped into a fitful sleep. Some time during the night he dreamed that Newton and Maxwell were seated at a small table amidst the ruins of an ancient castle in the English countryside. The table was spread with a checkered tablecloth, and the two men were enjoying a bottle of red wine. They were having an animated conversation, and at one point appeared to toast some idea. Try as he might, however, Yang could not make out what they were saying. It was rather like when he was an infant ... he could hear the voices and laughter, but it was only meaningless noise. And like when he was an infant he tried to join the conversation by yelling "Ar-r-r Ar-r-r Ar-r-r!" But the men at the table took no notice, and the sound of his voice aroused Yang from his slumber.

 

Chapter 7

Yin Transforms Force

Once he had reached his destination, it took Yin a couple of days to adjust to the new time zone and to recover from the long flight. By the second evening, however, he felt up to investigating how d(mu)/dt transforms from K to K'. (He had decided to use the symbol "u" for a particle's velocity in K, reserving "v" for the speed of frame K' relative to K.) Since a particle's inertial mass, m = mo/(1 - u2/c2)1/2, was essentially a kinematic variable, he concluded that working out the transformation of d(mu)/dt would for the most part be a problem in algebra.

Like Yang, he began by expanding d(mu)/dt, and wrote down

 (1)

And, like Yang he quickly deduced that

. (2)

And of course since

, (3)

it followed that

. (4)

Thus

. (5)

A component of force ... the x component, for example ... therefore became

. (6)

Yin and Yang had already worked out how the components of velocity and acceleration transform, and so after an abundance of painstaking algebra, Yin was able to deduce that

, (7)

, (8)

. (9)

By the time Yin had derived these results, it was late and he reluctantly retired. As he waited for sleep to come, he mentally compared the results he had derived with the much simpler Galilean transformation F' = F. As usual, life was more complicated in the Lorentz scheme of things. But ... Yin remembered Yang's frustration at his repeated mantra that "Nature is as Nature does." Smiling, he heaved a sigh and drifted off.

The next evening, back in his hotel room, Yin quickly checked his algebra and everything appeared to be in order. It was his habit always to try to understand the physics in an equation, and so he stared at the transformation for Fx'.

"What happens if the particle's velocity is momentarily zero In K?" he wondered. "Then Fx' = Fx. What does that imply?" If u was zero in K, then u' was in the negative x' direction of K' and had the magnitude v. According to Lorentz, an x-component of force in K had the same x'-component of force in K'. But when v was close to c, the physics was distinctly different for observers in the two frames. For a K observer, Fx resulted purely in an x-component of acceleration equal to Fx/mo. But for a K' observer, Fx' resulted almost exclusively in a change in the particle's mass. Yin recalled the Professor's discussion about near-light speed electrons, acted on by a constant force in a linear accelerator. Their acceleration was immeasurably small, but their increase in mass was dramatic.

"And now we know why," Yin mused. He noted that, in the electrons' momentary rest frame, they had the acceleration ax' = Fx'/mo = Fx/mo. In the LINAC's rest frame, on the other hand, the electrons' acceleration was practically zero, but their mass increased.

"What you perceive depends upon your frame of reference," Yin concluded. "But what about Fy' and Fz'?" According to Lorentz, Fy' is less than Fy by a factor of (1 - v2/c2)1/2 when u equals zero. Does that make sense? He decided to test that idea using the electromagnetic force law of Maxwell and Lorentz. (Maxwell showed how to calculate the electric and magnetic fields in any given frame, and the Lorentz force law specified what force a charged particle would experience in those fields.) Yin began by drawing the following sketch of two charged particles in his notebook:

 

 

In their momentary rest frame K, each charged particle would experience a repulsive force away from the other, and hence each particle could be expected to accelerate away from the other.

In K', when v was nearly equal to c, the two charges initially moved in the negative x' direction at speed v.

"What would their motions in K' look like?" Yin wondered. In K' the masses would be greater than mo. But then again, Yin knew that in K' the electric fields "to the sides of" each charge would be greater by the factor 1/(1 - v2/c2)1/2. But wait ... wouldn't that make Fy' greater than Fy??? Had he made a mistake?

"No!" Yin barked aloud, looking around the empty room furtively, as if to see if anyone had heard him. "Because there is also an attractive magnetic force in K'." He knew that the magnetic field was related to E' by

. (10)

So in K' there was a repulsive electric force of magnitude

 (11)

and an attractive magnetic force of magnitude

. (12)

"And the total force?" Yin asked himself.

. (13)

"Which is exactly how Fy transforms into Fy'," he realized.

"Positively amazing!" Yin thought. Was it possible that the electromagnetic force law of Maxwell and Lorentz, as applied in arbitrary inertial frames of reference, generally produced results in agreement with the Lorentz transformation of d(mu)/dt?

"That must be the case!" he thought. "How else could F = d(mu)/dt work in every frame? It was elegant almost beyond belief how Newton and Maxwell came together under the Lorentz transformation.

"And the key ... at least part of the key ... is the way inertial mass depends upon a particle's speed," he concluded.

In a state bordering on reverence, Yin closed his notebook and got ready for bed. After he had turned out the light, he began to think about the other forces in nature ... gravity, the nuclear force ...

"Must not all of the force laws, when applied in different frames of reference, produce results consistent with the Lorentz transformation of d(mu)/dt?" he wondered. If true, then what did that imply about the force laws in general? Newton's original gravitational force law was coulombic. But to produce the right ... the Lorentz results in various frames of reference, there would have to be more to it. And the nuclear force ... that was quite complex, and not even fully understood yet. How would that one need to be modified? And the elastic force ... Hooke's force law ... were moving springs less stiff when oriented a certain way?

"So much to think about," Yin sighed. The good part seemed to be that there was only one unique transformation for d(mu)/dt. To the extent every force law, applied in various frames of reference, had to produce results in agreement with that transformation, there were at least some solid guidelines for how the force laws needed to be modified.

With a tired sigh, Yin wondered what Yang was up to. The two of them would have some interesting philosophical issues to discuss when he returned home. Not to mention ... mathe ... matical ...

 

Chapter 8

Light Drops

On the Saturday following his return from overseas, Yin invited Yang to lunch. Yang brought along his laptop computer, and after lunch he demonstrated his "atom" modeling program. Yin was impressed and eagerly experimented, giving the orbiting "electron" various speeds in frame K. In every case the atom was length contracted and time dilated in frame K' quite as predicted by Lorentz.

"So in every case the Maxwell fields E' and B', coupled with the Lorentz force law and Newton's second law, produce a trajectory in K' that's consistent with the Lorentz transformation out of K," Yin mused.

"Provided the dependence of mass on speed is taken into account," Yang added.

"And when you add vt' to the computed satellite positions in K' ..." Yin continued.

"You get a picture of what the moving atom's shape is, according to a K' observer."

"Fascinating!" Yin said. "You certainly appear to have demonstrated that length contraction and time dilation are real effects, with a basis in physical law. At least you've done so for this particular system."

"Yes, and I have little doubt that the same would be true for real atoms," Yang replied.

"Using the quantum laws?" Yin asked.

"Right. But that is no doubt a more challenging problem," Yang stated.

"Oh, to be sure," Yin hastily agreed. "But let me show you what I've been up to."

Yin opened his notebook and showed Yang how the components of d(mv)/dt in frame K transform to components of d(m'v')/dt' in frame K'.

"As usual, more complicated than the good old Galilean transformation, F' = F," Yang observed wryly.

"Yes," Yin agreed, refraining from further comment.

Yang flipped back a few pages in Yin's notes and scanned through the force transformation's derivation. All appeared to be in order.

"You know," he murmured, "I'm just thinking ... these results should be one and the same as the Lorentz force in K'."

"You mean that Fx' should not only equal the transformation of d(mvx)/dt, but it should also equal the x'-component of q(E' + u' x B'), and so forth?" Yin asked.

"Precisely," Yang affirmed. "In fact, haven't I sort of demonstrated that with my model of the Bohr hydrogen atom?"

"Yes, I would say so," Yin answered. "But is that generally true? Given E andB in frame K, wouldn't we have to know in general what E' and B' are in K', in order to show that?"

"Yes, we would. And we do know that!" Yang exclaimed. "While looking up the fields of a point charge moving with constant velocity, I came across the general transformations of E and B into E' and B'."

Yin and Yang went back to Yang's place, and Yang pulled a text from his bookshelf and showed the general field transformations to Yin. Yin studied the transformations for some time in silence.

At length Yang asked, "Is there a problem?"

"We have to be careful here, don't we?" Yin answered softly.

"How so?" Yang queried.

"Well, let's say we have a snapshot of the electric and magnetic fields in frame K."

"Where by 'snapshot' you mean that we know what E and B are at all points in space, at some given instant?"

"Right," Yin answered. "We mustn't think that these transformations give a snapshot of the electric and magnetic fields in K'."

"Why not?" Yang asked.

"Well ... consider a plane wave, traveling in the negative y-direction of frame K," Yin began.

"With, say, Ez = Eo cos(wt) at points in the xz-plane?" Yang suggested.

"Sure, and with Bx = -(Eo/c) cos(wt)," Yin agreed.

"OK. But I still don't see why we have to be careful," Yang continued.

"Well, let's look at things through the eyes of a K' observer," Yin said.

"Where K' moves in the positive x-direction of K with constant speed v?" Yang clarified.

"Right," Yin conceded. "Consider the situation at t = 0."

"OK," Yang said. "According to a K observer, at time t = 0 the wave has maximum positive Ez and maximum negative Bx at all points in the xz-plane."

"And specifically at all points on the x-axis," Yin added. "But notice something: the K' observer agrees that these maxima occur at the origin of K' at time t' = 0. But since the clocks on the negative x’-axis run ahead of the clock at O', he would find that the maximum E' and B' occur after t' = 0 at points to the left of his origin."

"Wow! You're right!" Yang exclaimed. "And they occur before t' = 0 out on the positive x’-axis. What does that mean?"

"I would say that, from the K' perspective, the wave fronts are not parallel to the x'z'-plane, nor are they propagating strictly in the negative y'-direction. Evidently they also have a negative x'-component of velocity."

"I agree," Yang said. "But that means ... what ... that, in K', B' must have a y'-component as well as an x'-component."

"And according to these transformations, it does!" Yin exclaimed, pointing to the equations in Yang's book.

"This all makes sense, doesn't it?" Yang exclaimed. "I mean, we already know that a particle ... say a rain drop ... traveling down the y-axis will have a negative x'-component of velocity relative to K'."

"And the same can be said for a light drop," Yin added.

"A light dr ... you mean a photon?" Yang clarified.

"Sure," Yin said. "Here, let's sketch a typical case. Suppose the K observer is viewing the plane wave through the tube of a telescope, like this."

Yin sketched the following figure in his notebook.

 

"If v is large enough, and if the K' observer has his own scope, then it looks like he won't receive the signal if his tube is oriented straight up," Yang mused.

"Exactly," Yin confirmed. "Once a wave is in the tube of his telescope, the back side of the tube will intercept the wave before it reaches the detector."

"But if the K' observer tilts his tube just the right amount, toward positive x' ..." Yang continued.

"Then his detector could receive the waves too," Yin added excitedly. "From the perspective of the K observer we'd have something like this:"

Yin quickly drew the following in his notebook.

 

"Bingo! When the tilt is right, the waves reach the detector," Yang said.

"Yes. But notice that a wave front doesn't hit the K' detector all at once," Yin added with a troubled frown. "Does that distort the image?"

"But a wave front does impinge on the detector all at once ... at least according to the K' observer!" Yang exclaimed. "The clocks on the trailing part of the detector run ahead of those on the leading part."

"You're right!" Yin cried, smacking his forehead with the palm of his hand. "So from the K' perspective, things would look like this:"

Yin sketched yet a third drawing:

 

"Right!" Yang agreed, studying the drawing. "So let's say the K and K' observer are both looking at a star."

"Yes. And let's say that the K observer sees the star to be at rest on his y-axis."

"If the two telescopes coincide at the mutual origins at t = t' = 0, then the K' observer sees the star to be off the y'-axis, at some positive value of x'," Yang concluded.

Yin was silent. At length Yang asked, "You don’t agree?"

"No ... Yes! I agree with what you said," Yin replied. "The problem is that the star lies on the y-axis of frame K ... everyone must agree that this is the case."

"Hm-m-m, I see your point," Yang murmured. "So in general, the K’ observer visually observers the y-axis of frame K curving away in the positive x’ direction."

"Yes, that is what he must see with his eyes, although according to his clocks on the y’-axis all points on the y-axis coincide with his y’-axis at t’ = 0."

"It sort of makes sense, doesn’t it?" Yang said. "Relative to the K’ observer, frame K, and specifically the y-axis, are moving in the negative x’ direction."

"Yes, and the greater the value of y, the further out in the positive x’-direction that point on the y-axis was when the light left for the K’ observer’s eye."

"So in general what we observe, with one set of eyes, doesn’t necessarily agree with what we measure over extended space, using our distributed meter sticks and clocks."

"Evidently not. Shall we put it to the test?" Yin wondered.

"How?" Yang asked.

"Well ... suppose we take a quick trip around the Sun, watching some particular star all the way."

"If the star is at rest relative to the Sun ..." Yang murmured.

"Then we should see it go in a circle during our trip!" Yin said.

"Because we're constantly changing inertial frames as we go in a circle," Yang added.

"And thus constantly swinging our tilted telescope around to keep the star in view!"

"But wait ... if our space pod is always oriented tangent to our orbit ..." Yang frowned.

"Hm-m-m, I see your point," Yin answered. "We need to keep the pod's orientation constant in inertial space. Can we do that?"

"Oh sure," Yang said. "It's not the usual way to fly, but in space the pod can go sideways or even backwards as easily as forward."

"So shall we give it a try?" Yin asked.

"Let's do it!" Yang replied.

Yin and Yang boarded their space pod, traveled to a point between Earth and Mars, and commenced circling the Sun at a constant speed of .8c, being careful to keep the pod's orientation constant relative to the fixed stars. They trained the space pod's onboard viewing scope on a star that lay on the axis of their circular orbit. Because of the star's enormous distance from the Sun, they could rest assured that the wave fronts from the star were practically plane at all times.

 

 

Sure enough, in the course of one orbit they had to swing the telescope around in order to keep the star in view. The star appeared to move in a small circle.

When the two space travelers returned to Earth, they noted that their onboard clock had again lost time relative to Earth clocks.

"We're a little bit younger than we would have been, had we remained on Earth!" Yin exclaimed.

Yang nodded somberly. Realizing his gaffe, Yin laid his hand on Yang's shoulder and apologized.

"Oh, it's OK," Yang said. "I've long since accepted the consequences of our last trip."

"Do you suppose The Professor knows about this tilting business?" Yin wondered.

"I would bet he does," Yang answered. "Let's ask him! He said he'd be working in his lab this afternoon."

The two adventurers hastened over to the university and searched out their learned friend. They recounted what they had observed during their quick trip around the Sun.

"Oh, yes," The Professor said, adjusting an instrument he was working on. "Astronomers observe the same thing through Earth-bound telescopes. Of course the Earth's speed around the Sun is much less than the speed you traveled at. So the telescope tilts are less ... the circle apparently traced by a star has a smaller radius than what you saw."

"And it takes longer for the apparent motion to complete the circle," Yin suggested.

"Yes, to be precise it takes a year," The Professor said, looking up and smiling at Yin.

"So we really haven't discovered anything new, have we?" Yang lamented.

"Not this time," The Professor commiserated. "Scientists have known about this particular effect for quite a while. We call it stellar aberration."

Yin and Yang thanked The Professor for his time. Later, while walking off campus, Yang said, "You know, this stellar aberration is kind of interesting. Let's say that the visible stars are all more or less at rest in frame K."

"I think I see where you're headed," Yin answered. "If we were to switch from that frame to a frame K’, where v is close to c ..."

"Then the positions of the stars should appear, to our eyes, to shift out in front of us."

The two friends walked on in silence. At length Yang blurted, "Shall we do it?"

Yin glanced at Yang. "I have no objection," he said, "as long as we don't leave our present frame for too long. I'm sort of comfortable in the times we now live in."

"Me too," Yang answered. "I'd hate to take another long trip and find out that mankind had become extinct when we returned."

"An unsettling thought," Yin murmured.

Yin and Yang decided to test the "ultimate aberration" effect the next day. At 9 a.m. they boarded the space pod and programmed it to accelerate very rapidly to a speed of .98c. Within seconds of on-board time they had accelerated to the targeted speed, and gazed with wondering eyes out of the pod’s hemispherical viewing bubble. Everywhere except directly in front of the pod the universe appeared to be void and black. The expanse of stars that they had observed before embarking had merged forward to a single dazzling light directly in front of them.

The space travelers immediately decelerated to a state of rest relative to frame K. As they did so the circle of light appeared to their eyes to expand with wonderful symmetry, stars streaming out in all directions. By the time they had returned to a state of rest in frame K, the sky was again filled with stars.

"Wow, that was something else!" Yin exclaimed.

"Yes," Yang agreed. "It reminded me of an experience I once had in the dentist’s chair."

"Really! Tell me about it," Yin begged.

"Well, I was administered anesthesia," Yang continued, "and I dreamed I was in pure blackness, staring at an intense circle of light."

"A near death experience ..." Yin murmured.

"What?" Yang asked sharply.

"I’ve read that people who have experienced clinical death, and were then revived, very often have experienced the same thing. Sometimes they perceive the light to be at the end of a tunnel."

"Fascinating!" Yang replied. "Do you think there’s a connection?"

Yin looked down at his plodding feet.

"We don’t know, do we?"

The two friends had some early dinner and took leave of one another. That night Yang dreamed he was in a small classroom with ten or twelve other students. Evidently a course on the philosophy of religion was being given by a famous teacher, and the day’s topic was the kingdom of heaven.

At one point one of the students asked, "Where is the kingdom of heaven?"

The teacher answered, "According to Scripture, the kingdom of heaven is everywhere."

The puzzled students looked around the room and out the window, but saw nothing particularly heavenly.

"And how do we enter the kingdom of heaven?" the student pressed.

With a kind smile the teacher replied, "While alive, by emulating The Master."

"And afterward?" another student asked.

"By accelerating to the speed of light," the teacher whispered.

 

Chapter 9

Position Counts!

Three days following their observation of "the ultimate aberration effect," Yin called Yang with a complaint.

"Almost everything we’ve observed thus far has been while we were at rest in one inertial frame or another," he pointed out.

"Yes, our periods of acceleration have been brief and intense, with little time for observation," Yang agreed.

"I’ve been wondering what things might look like during an actual period of acceleration," Yin continued.

"So have I," Yang replied. "But what do you mean by ‘look like’? What we see with our own eyes doesn’t necessarily agree with what the distributed meter sticks and clocks in our rest frame tell us."

"Yes, that’s true enough," Yin concurred, recalling how the y- and z-axes of frame K appeared to the eye to curve away and out in front of them when they accelerated along the x-axis, whereas, according to the grids of frames K and K’, the axes were parallel at all times. "Actually we could check both aspects out."

"You mean we would actually accelerate a grid and its set of distributed clocks, and construct a scenario after all the results from distant points had been gathered?" Yang asked in a somewhat skeptical tone.

"Hm-m-m, you sound doubtful and I think I see why," Yin mused. "At best we could only hope to accelerate a very limited grid."

"Perhaps we don’t have to accelerate anything but ourselves," Yang suggested. "What if we define acceleration simply to be the transitioning from inertial frame to inertial frame at an accelerating speed?"

"But what inertial frame’s clocks and measuring rods would we use to construct a view of things from our accelerated perspective?" Yin pressed.

"Those of our initial rest frame," Yang answered.

There was a brief pause while Yin considered that idea.

"That’s an interesting approach," he said at length. "But in that case all we really need do is measure things using that initial rest frame."

"Yes, I see your point," Yang concurred. "We could then use the Lorentz transformation to construct a view from all our other instantaneous rest frames during our period of acceleration."

"Actually we don’t really even need to take a trip to accomplish this, do we?" Yin continued. "We can calculate all of this while at rest here in K."

"True," Yang answered tentatively. "But I’d still like to see what things look like to my own eyes, while accelerating."

"So would I," Yin said. "But I suggest that for openers we do some ground work."

"Sounds good!" Yang exclaimed. "Your place or mine?"

"Well, if you’re free today, we could have lunch and do some work here," Yin suggested.

Yang agreed, and the two met for lunch at a small bistro on Yin’s street. They decided to concentrate for openers on plane electromagnetic waves. They had already qualitatively seen how such waves change their direction of propagation as an observer shifts from one inertial frame to another. That effect was what the Professor referred to as stellar aberration. And they had determined that a plane wave in one inertial frame is plane in all other inertial frames. It remained to determine how the curvature of a given ray depended on the magnitude of their acceleration. Agreed on an afternoon’s agenda, the two walked to Yin’s place and settled into his study.

"So let us say that at t = 0 a photon is traveling down the y-axis and has a velocity of uy = -c in frame K," Yin began.

"And at time t = 0 we’re momentarily at the origin of K and have an acceleration of magnitude ‘a’ in the positive x-direction," Yang added.

"Right. We know that some short time later ... say at time t = dt ... we’ll be momentarily at rest in frame K’, traveling in the positive x-direction of frame K with speed a(dt)."

"Yes, assuming our acceleration relative to K is constant," Yang clarified.

"Of course," Yin answered. "And relative to K’ the photon has a small negative x’-component of velocity, as well as a negative y’-component," Yin said.

"Right. Relative to K’ we have ux’ = -a(dt)," Yang proclaimed.

Yin felt his scalp tighten. Was it really that simple? He thumbed through his notes and found the transformation ux’ = (ux – v)/(1 – vux/c2). Since the photon had no x-component of velocity in K, Yang was right.

"Evidently so," he answered lamely. "And," he added in a brighter tone while staring at the transformation uy’ = uy(1 – v2/c2)1/2/(1 – vux/c2), "relative to K’ the photon’s y’-component of velocity will be uy’ = -c(1 – a2(dt)2/c2)1/2."

"So u’2 = ux2 + uy2 = c2," Yang added with an impish grin.

Yin scribbled furiously in his notebook.

"Correct," he agreed. "So life is beautiful, isn’t it? The photon’s speed is c in both frames."

"But it has a negative x’-component of velocity in K’."

"Yes. So while accelerating during the brief period dt, we perceive the photon to curve slightly toward negative x."

Yang nodded, but he was frowning.

"What is it?" Yin asked.

"It isn’t clear to me that, from our accelerated perspective, all parts of a plane wave front propagate at a common speed," he muttered.

"Because?" Yin pressed.

"Well, instead of a photon, consider the extended plane wave front traveling relative to K in the negative y-direction."

"Okay-y-y," Yin answered, still not sure what Yang was getting at.

"My point is that a plane wave front in K is also a plane wave front in K’," Yang continued.

"Yipes!" Yin cried. "So at t = 0 we find the wave front parallel to the xz plane. And a short time dt later, the part of the wave front in contact with the y-axis has only moved down a distance of approximately c(dt) ..."

"But from our accelerating perspective, at a point well out on the x-axis, a point on the same wave front has moved down quite a bit more than c(dt)."

"And at large negative values of x ..."

"From our accelerating perspective, a point on the wave front may actually have displaced upward, in the positive y-direction!" Yang exclaimed.

"Hm-m-m, can it be that the Lorentz velocity transformation for light doesn’t apply to phase velocities, but strictly speaking only to photon velocities?" Yin wondered.

"Evidently ... or perhaps to what is classically referred to as group velocities," Yang agreed.

Both men were silent as they pondered this strange new development. At length Yang spoke again.

"Here’s an interesting thought. There really isn’t anything sacred about the y-axis of frame K, is there? Suppose that at time t = 0 you’re at some large positive value of x, accelerating in the positive x-direction just as I am."

"I think I see what you’re getting at," Yin answered. "A short time dt later, when at rest in K’, I’ll see the part of a wave front directly above me to have come down only approximately c(dt)."

"Exactly!" Yang cried. "But if I’m initially at the origin of K, also accelerating, then I’ll see that part of the wave front to have come down a lot further than c(dt)."

"And how can that be?" Yin wondered.

"We don’t both start accelerating in K’ at the same moment?" Yang suggested meekly.

"But we do, don’t we?" Yin complained. "At t = 0 we’re mutually at rest in K, with identical accelerations in the positive x-direction. Surely we come to rest in K’ at the same moment!"

"According to the K clocks ... " Yang said.

"Of course!" Yin shouted. "But not according to the K’ clocks! According to those clocks, I’ll start accelerating into K’ before you do!"

"Exactly!" Yang rejoined. "So naturally by the time we both come to rest in K’, that piece of wave front directly above you will have traveled further than c(dt)."

Once again Yin grew silent.

"You don’t agree?" Yang asked.

"No ... yes, I agree," Yin answered. "But consider this: from the K’ perspective I start accelerating out of K before you do."

"So?" Yang pressed.

"So between the time I start accelerating and we both come to rest relative to K’, I'll have traveled further than you have."

"So from the K’ perspective, your onboard clock will be running slower than mine ... " Yang thought aloud.

"Exactly! So if I’m way out at some positive value of x at time t = 0, and if we both start accelerating at that moment, then more time will have elapsed for me by the time we’ve both come to rest in K’!"

"Good grief!" Yang exclaimed in a dumbfounded tone. "Are we saying that I’ll have aged less than you, once we’ve both come to rest in K’, simply because of our positions on the x-axis?"

"Evidently so," Yin answered.

"Even though we both were at rest at the same moment in K, with identical accelerations ... "

"Yes, it appears that if we accelerate in separate space pods, and if we each cut our thrusters the moment we come to rest in K’, then you’ll have aged less than I."

"Wow!" Yang exclaimed. "This I’ve got to see! Shall we try it?"

"Sure," Yin answered. "But it’s too late to do it today. How about tomorrow morning?"

"Works for me!" Yang answered, and the two agreed to meet at the spaceport the next morning at 8 a.m.

Once asleep that night, Yang had another unsettling dream. As they had discussed, Yin was leading the way in transitioning from frame K to K’, with Yang following in a second pod. Once at rest in frame K’, each traveler cut his thrusters. It had been agreed that they would walk to the midpoint between their respective craft and compare their watches, which had been synchronized prior to accelerating out of frame K.

As Yang stepped out of his pod, he saw Yin’s pod ahead in the distance, also at rest in K’. Yang began to walk toward the midpoint, but it seemed to him that Yin was walking very slowly. Yang lessened his gait so that they would still meet at the midpoint. As the two drew closer, Yang noted that Yin was bent over and seemed to be hobbling along on a cane. Finally, when they were only a few meters apart, Yang gasped in disbelief. Yin had aged an entire lifetime! Only a wisp of white hair remained on his wart-covered head. A crazy laugh exploded from his toothless mouth when he beheld Yang’s horrified expression.

"Still want to lead on the way back?" he cackled, dancing a little jig.

 

Chapter 10

The Gap

After taking leave of Yang, and hours before Yang had his disquieting dream, Yin retired to his study and opened his notebook. He had long since resolved not to embark on another trip without doing some quantitative analysis. Tonight he would try to determine exactly how much more he would have aged than Yang had, once they were mutually at rest in frame K’.

He began by deciding that target frame K’ would move in the positive x-direction of their present rest frame, K, at speed v = .9c. They would embark in separate pods at 9 a.m. the next morning. Yang would initially be at the origin, and he would be at, oh, say x = c seconds (i.e. one light second). He decided to denote 9 a.m. as t = 0. At that time they would start their respective stop watches and turn their thrusters on.

Now since the objective was to determine the difference in the amount they aged, Yin realized that the effect (if any) acceleration had on the rate of clocks would be irrelevant. For with their pod engines producing equal thrusts, they would move identically during their respective periods of acceleration.

"From the perspective of frame K’," he thought, tapping his pencil on the desk, "any differences must occur when one of us is accelerating and the other isn’t."

It was clear that, from the perspective of K’, he would start accelerating first, at time t’ = -(c sec)(.9c)/[c2(1 - .92)1/2] ~ -2.06 sec. Yang would start accelerating at time t’ = 0. Thus for a period of 2.06 seconds Yang would (relative to K’) be moving at a speed of .9c, and hence would age only (2.06 sec)(1 - .92)1/2 = .9 seconds.

It was also clear that (from the perspective of K’), Yin would come to a state of rest before Yang would. Indeed the problem was symmetric, and for a period of 2.06 seconds he would be at rest while Yang was still decelerating. Since he was at rest, he would age the entire 2.06 seconds. The difference in how much they had aged, by the time Yang came to rest in K’, would therefore be (2.06 - .9) = 1.07 seconds.

"Nothing to obsess about there," Yin thought. "But definitely measurable using our stopwatches."

The procedure the next morning would be for Yang to radio Yin once he had come to rest in K’. They would then agree to read their stopwatches at some mutual future time (using the local K’ clocks to tell the time), and report the results to each other. They would then again start their thrusters simultaneously (using the K’ clocks to determine simultaneity) and return to a state of rest in frame K. Finally, once Yin had come to rest in K, he would radio Yang and the two would again simultaneously note the reading of their stopwatches (now using the local clocks of K to determine simultaneity). If Yin understood the problem correctly, the stopwatches should now be back in synch.

With a feeling of confidence, Yin greeted Yang the next morning at the spaceport. Yang was clearly agitated, and Yin asked what was troubling him. In sheepish tones Yang recounted the gist of his dream. Yin laughed and reassured his friend that it was, after all, only a dream. Opening his notebook, he showed Yang what they could in fact expect. Yang studied Yin’s work carefully and at length, with a relieved look, agreed that they should proceed as planned.

Yin boarded a space pod and sped away in the positive x-direction, promising to radio Yang as soon as he had reached (and come to rest at) point x = c seconds. This he did well in advance of 9 a.m., and each traveler programmed his thrusters to engage at precisely 9 a.m. When 9 a.m. arrived, each man reset his stopwatch to zero, and away they sped in the positive x-direction at ever increasing speeds.

The instant his pod came to rest in K’, Yang’s thrusters shut down as scheduled.

"Hello, Yin, I’m at rest in K’. Do you read me?" he spoke into his microphone.

"I read you loud and clear," Yin’s voice came back. "I have a K’ time of 11:20. Shall we read our stopwatches at 11:30?"

"Sounds good to me," Yang answered. At precisely 11:30 K’ time each man read his stopwatch and radioed the result to the other. And just as Yin had figured, Yang’s stopwatch read approximately 1 second less than Yin’s.

"OK, let’s return to a state of rest in K," Yin radioed. "Shall we restart our thrusters at 11:45?"

"Will do," Yang answered, and the two decelerated back to a state of rest in K. Once his pod had come to rest in K, Yin’s thrusters shut down and he radioed Yang. Again they agreed to read their stopwatches at an agreed upon time, and this time the watches were back in synch. In a state of mutual excitement, the two returned to the origin of K.

"Well, it’s been quite a day!" Yang exclaimed, once they were back at the spaceport. "Shall we have supper in the snack bar?"

"Sounds good," Yin agreed.

Once settled at a table, the two rehashed the day’s events.

"Clearly the greater our separation, the greater the difference in our stopwatches, once we’re both at rest in K’," Yang mused.

"Yes, and it’s going to be the same difference, no matter what our acceleration may be," Yin added.

"Hm-m-m, that’s true," Yang murmured, chewing thoughtfully. "But the rate at which the age difference develops depends on the magnitude of our acceleration."

"I agree," Yin replied. "The greater our acceleration, the less time it takes for us to reach a state of rest in K’."

"And all this because we start out at different points on the acceleration axis," Yang marveled.

"I wonder if The Professor knows about this?" Yin sighed.

Yang had been wondering the same thing. Deciding to chance catching The Professor still at his office, he took out his cell phone and dialed the number. His face brightened when The Professor answered.

"Hello, Professor," he greeted. "This is Yang. My friend, Yin, and I have just had an interesting experience and we thought you might like to hear about it. Would it be convenient for us to call on you sometime?"

Yang listened for a moment, and then asked Yin if he could join him at The Professor’s office the next day at 2 p.m. Yin assured Yang that he could, and Yang told The Professor that they’d see him the next day.

After finishing their dinner, the two adventurers bade each other good evening in the spaceport parking lot. That night Yang slept peacefully, with no troubling dreams. The next morning, while brushing his teeth, he recalled the dream of two nights before.

"Silly!" he thought, scrubbing his molars vigorously. "Still … what if K’ had a speed of .999c? Or what if Yin had started out at x = 1 light year instead of 1 light second?" All things considered, Yang decided that he would not dismiss future dreams lightly. Too often his dreams had contained a grain of truth. How fascinating the human mind was! And what further surprises might the Lorentz transformation have in store for them?

 

Chapter 11

Red Shift

"That is just as interesting as can be," The Professor mused after Yang had related their experience of the day before. Tilting back in his chair and clasping his hands behind his head, he continued.

"You know, this isn’t my area of specialization, but I would bet that you two are the first to actually demonstrate this uniform field effect."

Yin and Yang exchanged puzzled glances.

"Uniform field?" Yang inquired.

"Oh, yes," The Professor smiled. "Are you familiar with Einstein’s Equivalence Principle?"

"Doesn’t that state that phenomena viewed from an accelerating frame, and viewed from a state of rest in a gravitational field, are indistinguishable?" Yin asked.

"Correct," The Professor confirmed, fishing out his pipe. "One of the misconceptions that I’ve read about lately is the belief that the aging rates of two observers, at rest in a gravitational field, differ only if the field strengths at their locations differ. But you have clearly demonstrated that the aging rates are different even when the field is uniform."

"Because we both accelerated identically?" Yin asked.

"Precisely," The Professor agreed. While he tamped tobacco into his pipe and lit it, Yang voiced a concern.

"I’ve read that one of the first tests of the Equivalence Principle was when the apparent positions of certain stars appeared to shift during a solar eclipse."

Rising from his chair, Yang stepped to an empty section of the blackboard.

"May I?" he asked, picking up a piece of chalk.

"Of course," The Professor agreed, drawing flame into the pipe’s bowl.

Yang drew the following diagram on the blackboard.

 

"As you can see," he said, gesturing at the diagram, "the observer’s line of sight is almost directly at the star. Let’s assume that the observer, the Sun and the star are all at rest relative to one another. According to the Equivalence Principle, what he observes should be the same as if he was accelerating away from the star. So why the curvature?"

"Ah-h-h," The Professor exclaimed. "I see your problem. The thing is that the Equivalence Principle applies all along the light beam’s path. At each point in a photon’s path as it grazes the Sun, for example, its path experiences a slight curvature toward the Sun’s center, quite as it would relative to a frame accelerating away from the Sun’s center."

"And any observer sees the integrated sum of such effects," Yin interjected.

"That is my understanding," The Professor nodded.

Yang stared at the drawing.

"So the photon’s path is affected somewhat like a material particle’s," he thought aloud. "There is radial acceleration as it grazes the Sun, regardless of whether there is an observer there or not."

"Um-m-m Hm-m-m, I would call that a fair analogy," The Professor said.

"And what about linear accelerations?" Yin asked.

"Yes, how does the analogy go then?" Yang added, tracing the portion of the ray path from the star toward the Sun. "Here the Sun’s field and the light path are practically collinear. What’s the story here?"

"Well," The Professor answered, again clasping his hands behind his head, "let’s consider the acceleration you two underwent yesterday. Suppose we view things from the perspective of frame K, while you were both accelerating into K’. You, Yang, were trailing on the way out. Let’s say that you were emitting a beam of monochromatic light at Yin. If you were both traveling at a constant velocity, then from the perspective of K your light would be Doppler shifted toward the blue. But would that be what Yin observed?"

Yin and Yang pondered this question, and Yin’s face soon brightened.

"How could it be?" he challenged. "We’d both be at rest in some common inertial frame!"

"Very good," The Professor smiled. "So from the K perspective, the light is Doppler shifted down again at your eye, because of your motion."

"But when we’re both accelerating …" Yang murmured.

"Ah! Different story!" The Professor exclaimed. "Again from the perspective of frame K, by the time a blue-shifted wave from Yang’s lamp got to you, Yin, your speed would have increased!"

"So I’d actually observe a net red shift," Yin thought aloud.

"I would think so," The Professor agreed. "The effect is often referred to as gravitational red shift."

"And I’d observe a blue shift in the light from Yin’s lamp," Yang exclaimed.

"Um-m-m Hm-m-m," The Professor agreed, sucking on his pipe.

"So let’s see if I understand this," Yang continued. "If our accelerations are constant and identical, then what we observe is the same as what we’d observe if we were both at rest at different points in a uniform gravitational field.

"And if I’m further out in the field than you are, then I observe light from your lamp to be red shifted, even though we’re at rest relative to one another."

"And I observe light from yours to be blue shifted."

"And," The Professor interjected, "the amount of shift is a function of the amount of your separation and the strength of the field."

"And here again, the effects can be viewed as the integral sum of incremental shifts in the wavelengths, as the waves travel from a source to an observer’s eye," Yang suggested.

"I would agree with that," The Professor answered.

"So in the pseudo-linear acceleration case … the case where a photon’s path and the field are collinear … it’s the wavelength that changes, and not necessarily the speed of the signal," Yin murmured.

"Or, what amounts to the same thing, thanks to Planck and Einstein, it’s the photon energy which increases or decreases, depending on whether the photon is traveling with or against the field," The Professor pointed out.

"Quite as a material particle’s kinetic energy increases or decreases," Yang added.

"Yes," The Professor said, "I would agree that there lies the analogy when photon path and gravitational field are collinear."

"Let me pose … let me run a thought pass you two," Yin said. "Consider the gravitational field of a star. It asymptotically approaches zero as the distance from the star increases. But of course it never goes to zero. In effect, it approaches a small, practically constant value at great distances from the star."

"So?" Yang pressed.

"So, consider yourself to be a distant observer, at rest relative to the star but at several different, vast distances from the star at different times."

"I think I see where you’re headed," Yang said, returning to his seat. "The light from the star shifts toward the red as your distance from the star increases."

"But isn’t that just what astronomers observe, in the case of light from distant sources?" Yin asked The Professor.

"It is," The Professor answered, arching his eyebrows somberly.

"But if we’re at rest in the Earth’s field, the light is inbound and therefore blue shifted," Yang objected.

"Yes, to a relatively small extent," The Professor agreed. "But bear in mind that the star’s mass and field are thousands upon thousands of times greater than that of the Earth."

"So the net effect adds up to a red shift … one that increases with our distance from the star," Yang murmured.

"But isn’t the red shift attributed to recession? Isn’t the prevailing wisdom that the Universe is expanding?" Yin complained.

Yin and Yang both looked at The Professor. He had placed the pipe in a large ashtray on his desk. Without saying anything he nodded affirmatively at Yin.

"So what is it? Recession, or energy loss attributable to a climb out of the star’s gravitational field?" Yin muttered. "Is there any way to tell?"

"Maybe!" Yang chirped. "Does the light from distant sources grow dimmer with time?" he asked The Professor. "It seems to me that if the red shift is ascribable to recession, then that would be the case. But if it’s due to the kind of energy loss we’ve been discussing here, then the brightness should remain constant."

"I don’t know the answer to that question for sure," The Professor confessed. "But you can bet I’ll look into it if you wish. It seems to me that I read somewhere about the brightness of a Quasar being measured at different times."

"So the red shift we observe in the light from very distant sources could actually be a combination of gravitational red shift and Doppler shift, due to recession," Yin murmured.

"Yes, there is that possibility," The Professor agreed. His face then took on an impish look.

"How would you two like to share these thoughts with a larger audience?" he asked.

Yin and Yang gazed speechlessly at him.

"I could set up a colloquium," he continued. I’m sure there are others who would be interested in hearing about these ideas, not to mention some of the results you two have obtained."

"I don’t know," Yang stammered, shifting nervously in his chair. "We’re rank amateurs, compared to you and your colleagues …"

"Oh, I don’t think that’s a problem," The Professor reassured. "We rarely give our guest speakers a hard time."

"Well … I guess …" Yang said, glancing at Yin.

"Sure, why not?" Yin added. "It could be interesting. But I must agree with what Yang said," he added. "We really are amateurs. Anything we say will mostly be of a qualitative nature. Your colleagues may be left wanting, quantitatively speaking."

"Oh, I shouldn’t worry about that," The Professor replied. "Most ideas start out qualitatively. There will be no lack of people … many better versed than I … who will no doubt be willing to address the quantitative aspects with relish if they think your thoughts on distance-dependent red shift and so forth have merit."

So Yin and Yang agreed to be the presenters at a future colloquium and took leave of The Professor. Later, while walking across campus, Yang broke the silence.

"Are you as tied up in a knot as I am?" he blurted, glancing at Yin.

"If having an acute anxiety attack qualifies, then my guts are twisted into the mother of all granny knots!" Yin chortled in return.

 

Chapter 12

Newton/Lorentz

Two days after their meeting with The Professor, Yin and Yang agreed to have dinner at a favorite German restaurant. The place had a woodsy German décor and the piped in music and German beer put them in a festive mood.

"So, have you given any thought to what we’ll say at the colloquium?" Yin asked.

"No, I’m afraid I got sidetracked," Yang admitted. "I visited a web site that features monographs on math and physics, and an article on equilibrium got me thinking about how the gravitational force law has to be modified in order to produce results, in different inertial frames, that are consistent with the Lorentz transformation of d(mv)/dt."

"Yes, I wondered about that myself when I first transformed d(mv)/dt," Yin answered. "But isn’t it the contention of General Relativity that there is no gravitational force … that the behavior of neutral matter, in the presence of other neutral matter, is due to space-time geometry?"

"I know, I know," Yang said. "But the mathematics of General Relativity is so abstract, and some of the ideas that I toyed with are so simple, that I just couldn’t resist."

Now mathematics was decidedly not one of Yin’s stronger suits, so he encouraged Yang to continue. Yang began by outlining what he had read on the web site.

"Imagine that you have two resting, small and equal charges, like so," he began, drawing the following diagram.

 

"Each charge experiences a repulsive electric force in the other’s electrostatic field."

"Um-m-m Hm-m-m," Yin acknowledged.

"If some agent doesn’t counteract the electric forces, then the charge’s will accelerate away from each other. But let’s do this: let’s overlay each charge with some neutral matter, in an amount such that the attractive gravitational forces counteract the repulsive electrostatic forces."

"Which, assuming the matter and charge are linked, places the system in equilibrium," Yin observed, "regardless of whether there’s actually a gravitational force or there’s a distortion of space-time."

"Precisely," Yang said. "And here’s the interesting thing: if we view the system from any other inertial frame, then the system must still be in equilibrium."

"Hm-m-m, yes I would agree with that," Yin affirmed. "If this is inertial frame K that the charges are at rest in, then relative to any other inertial frame they must move with a common, constant velocity."

"Right! But what does Maxwell/Lorentz say? Relative to other frames each charge may well experience a magnetic force in addition to the electric force. And for that matter the electric force may not be the same in other frames, since the electric field of a moving charge is different from the electrostatic field of a charge at rest."

"Yes, yes we discussed that," Yin exclaimed. "The elegant thing is that the electromagnetic force of Maxwell/Lorentz, in some other frame K’, is exactly equal to the transformation of d(mv)/dt out of K into K’."

"Right!" Yang agreed. "But with the neutral matter factored in, the total force on either sphere must be zero in every imaginable inertial frame!"

"But of course!" Yin cried. "So the interactive force between neutral bodies … assuming such a thing exists … must transform in the same way as the electromagnetic force does!"

"Or like d(mv)/dt does" Yang exclaimed. "That presumably has to be the case with every force. We could, for example, have held the charges at rest in frame K using a spring."

Yin was quiet for a moment as he turned these ideas over in his mind.

"Well," he said at length, "Newton’s gravitational force law is mathematically the same as the Coulombic electrostatic force law …"

"Exactly!" Yang answered excitedly. "So why shouldn’t we hypothesize that the gravitational field transforms in the same way as the electric field?"

"OK, but what about the magnetic forces in other frames?" Yin protested.

"I would suggest that there must be a force analogous to the magnetic force in the case of uncharged body interactions!"

"Interesting," Yin said. "But why have we never head of such a force?"

"Well, let’s look at that," Yang retorted. "The magnetic force is ordinarily much smaller than the electric force."

"True …" Yin murmured.

"And the gravitational force is ordinarily minuscule compared to the electric force. For example, compare the gravitational force of attraction and the electric force of repulsion between two protons."

"So this new force … this force that’s analogous to the magnetic force … would be really really small," Yin suggested.

"Practically undetectable in most cases," Yang affirmed. "But essential in order to make Newton’s inverse square force transform, under Lorentz, like d(mv)/dt does."

"Assuming, that is, that the gravitational field transforms like the electric field. I must admit it’s an interesting idea," Yin said. "Let’s take a really big mass … say the Sun. I’m pretty sure it rotates."

"Yes, it does," Yang confirmed. "Actually I read once … I can’t remember where … that different parts of it rotate at different rates. But I’m pretty sure that they all rotate around a common axis, and in the same direction."

"So at points in the Sun’s equatorial plane, this hypothetical magnetic analogue field would point perpendicular to the plane."

"Right," Yin responded. "So a planet, orbiting in a circle in the equatorial plane, would feel a small magnetic-like force toward or away from the Sun, depending on whether its orbit went with or against the Sun’s rotation."

"And that force would modify the gravitational force slightly. But none of the planets have a perfectly circular orbit. So …"

"So the magnetic analogue force is always perpendicular to the orbiting planet’s velocity. I’m betting that when you factor this subtle magnetic-like force into elliptic orbits, that the orbits precess," Yang added.

"You know of course that the elliptic orbits precess even without this extra force," Yin pointed out.

"No … no, I didn’t know that," Yang responded. "Where did you read that?"

"I didn’t … I modeled it on my PC after we learned about the dependence of inertial mass on speed."

"Ah! So the orbiting planet’s speed varies in the course of an elliptic orbit, and when you factor that into d(mv)/dt, the orbit isn’t a closed one," Yang surmised.

"That’s right," Yin replied.

"But what did you use when computing the force on the planet … its rest mass or its speed-dependent mass?"

"You mean in the force law, F = GmSunmplanet / r2 ?" Yin asked. "That’s a good question. To tell the truth I used the planet’s rest mass."

"Which is exactly what you’d do if you were modeling a very light negative charge, in an elliptic orbit around a very massive positive charge!"

"Yes, of course you would!" Yin agreed. "Charge is invariant and not dependent on speed. That’s what you did in your ‘atom’ modeling program!"

"So to make this magnetic analogy thing work correctly, we’d have to recognize two kinds of mass: the inertial mass, say, in d(mv)/dt, which is speed-dependent, and the nonspeed-dependent gravitational mass, let us say, in the Lorentz-type force law."

"Yes, I would think so," Yin agreed. "Think of your ‘atom’. You’d have to use the same approach to make the trajectory of a planet, circling the Sun, transform according to Lorentz in other frames."

Yang was silent for a few moments.

"It’s certainly a relatively painless approach," he murmured at length. "I’m surprised we haven’t heard about it, especially in cases where the interacting masses are small."

"So that the geometry of space-time is Euclidean for most practical purposes?" Yin clarified.

"Right. But let’s get back to our rotating Sun and elliptically orbiting planet."

"Well … I can’t really say, off the top of my head, whether the magnetic-like force would lessen or increase the precession attributable to speed-dependent mass."

"I can’t either," Yang answered. "I guess we’ll just have to factor the new force into your modeling program and run a few orbits."

"Shall we swing by my place after we’ve finished eating?"

"Sounds good," Yang agreed, smiling as the waitress set two plates of steaming German food in front of them. "But for now, Zum Wohl!" he toasted, raising his stein of beer.

"Prosit!" Yin answered, lifting his own stein.

Yin and Yang finished their German dinners and hurried back to Yin’s place.

"Here, you sit in the driver’s seat," Yin suggested after they had entered his study. "I’ll ride shotgun." Once Yang had settled into the chair in front of Yin’s PC, Yin started his orbiting particle program … the one with speed-dependent mass factored in.

"So how does this work?" Yang asked, when the first prompt appeared on the screen.

"Well, the first thing you’ll be asked to provide is an orbital speed in the case of a circular orbit," Yin answered. "Then you can change the circular orbit to an elliptic one by specifying some fraction of that speed to create an elliptic orbit."

"Neat-O!" Yang exclaimed. "What do you suggest for the circular orbit speed?"

"I would try .1c," Yin answered. "This isn’t a very sophisticated program, and if you specify too great a speed the thing will go haywire due to numerical errors."

Yang specified a speed of .1c, and then cut that by a factor of one half for the apogee speed. The program immediately began to plot the trajectory of an orbiting satellite. Yang stared with admiring eyes at the unfolding trace on the PC screen.

"I couldn’t have done better myself," he thought.

 

"Wow, look at that!" he exclaimed. "And the amount that each orbit precesses depends on the particle’s speed at apogee."

"Correct! Feel free to experiment with other speeds," Yin encouraged. Yang tried a speed of .001c, and there was almost no precession at all. For all practical purposes the particle acted like its mass was independent of speed. At .2c the precession was much more pronounced.

"Very interesting!" Yang said, smiling at Yin, who by now was beaming. "So I suppose that now we ought to try factoring in a Magnetic Analogue field."

"Right!" Yin agreed. "But since we don’t know the specifics about the Sun’s mass distribution and the different rotation rates of its parts, what say we just imagine for now that this field, which is perpendicular to the xy-plane, is the same as that of a spinning loop of matter."

"Analogous to the magnetic field of a current loop whose dipole moment lies on the z-axis?" Yang clarified.

"Right. I think I can find the expression for such a field, as a function of distance from the loop center."

Yin went to his bookcase and after a minute or so found what he was looking for.

"Here it is," he said. "The magnetic field at points on a line bisecting a magnetic dipole amounts to a constant divided by the cube of the distance. So it would be the same in the case of our Magnetic Analogue case, wouldn’t you agree?"

"Yes," Yang answered, rising from his seat. "Here, it’s your program. Why don’t you program the fix in, and we’ll see what happens."

Yin modified the program so the new field … the one that was perpendicular at all points in the satellite’s orbital plane … was large enough to produce obvious changes to the satellite’s orbit, but not so large as to cause the plot to ‘blow up’. Both men studied the new satellite path.

"It looks to me like the Magnetic Analogue force adds to the precession when the planet orbits in the same direction as the Sun’s spin," Yin mused.

"I would concur with that," Yang answered. "Isn’t this interesting? I wonder what the case with Mercury orbiting the Sun will turn out to be."

"So do I," Yin said. "I’ll see what kind of data I can dig up on the Sun and Mercury, and we’ll revisit the program."

"How will you calculate the Magnetic Analogue field in the case of a spinning sphere?" Yang asked.

"I think I’ll just model the sun as a set of quasi-current loops, and integrate numerically," Yin answered.

"That ought to work. And on that note," Yang said, rising and yawning, "it’s past my bedtime. Keep me posted, OK?"

"Absolutely," Yin promised. "And let me know when you hear something from The Professor."

Not much later Yang’s head was sinking into the pillows of his bed, visions of precessing orbits dancing before him in the darkness. It would be interesting, to say the least, if this magnetic analogue effect produced the same precessions as General Relativity theory. Were General Relativity and the electrodynamics of Maxwell/Lorentz (suitably modified) two paradigms for describing the same phenomena … somewhat like Lagrangian and Newtonian mechanics were? And what about photons? If they were radially deflected when propagating at right angles to a gravitational field, would they also be radially deflected by the magnetic analogue field?

"Now there’s an interesting thought," he murmured into the darkness just before slipping into a pleasant dream.

 

Chapter 13

No Escape

Yang often awoke in the middle of the night and lay awake, turning one problem or another over in his mind. This turned out to be one of those evenings. He had always been intrigued with the concept of escape velocities. Intuitively it seemed to him that a particle, with a radial velocity away from some massive "parent" body like the Earth, must sooner or later come to a halt and fall back again. Yet owing to the way the parent body’s gravitational field falls off as the square of the distance, there was undeniably a velocity at which the particle would never come to rest. At that velocity (and at all greater ones) the particle would recede forever. In effect all such particles were "unbound."

But what effect might a magnetic analogue field have on such particles? As he lay in the darkness visualizing things in his mind’s eye, Yang realized that a particle, with an initial radial velocity outward in the parent body’s equatorial plane, would experience a sideways deflecting force. And the moment the particle acquired a tangential component of velocity, the parent body’s gravitational force would augment such a deflecting force. If the magnetic analogue field were uniform, then alone it would make the particle go in a circle. But in the case of a dipole field, the magnetic analogue field fell off as the cube of the distance from the parent body. And gravity further complicated matters.

Yang realized that, given a particle with an initial escape velocity, he could at best only guess what its motion would be when a magnetic analogue dipole field augmented the gravitational field. The exception was when the particle’s initial velocity was along the dipole axis. In that special case it would experience no magnetic analogue force and would in fact escape. A simple program would provide an answer to other initial conditions, and he resolved to write one in the morning.

Still not ready to go back to sleep, Yang’s thoughts drifted to his wife and baby girl. There was a part of him that could not accept the cruel trap that time dilation had sprung on Yin and him. Surely, he thought, his family still awaited his return somewhere in time. Would he find his way back to them? Would he ever see them again, other than in a dream?

"Marie, Marie …" he heard his soul whisper into the darkness. Like a mantra, over and over again the name floated up from his unconscious mind and settled heavily on his heart. The laws of physics had allowed him to vault ahead, losing in the mists of time all that was dear to him. Like a diode, however, the door back seemed to be locked. But ... diodes could be broken down! Could the same be true of a portal into the past?

"There must be a way … there must be a way," he sighed. But how?

 

*****

 

The next morning Yang brewed a cup of coffee and went into his study. His objective was to write a program that computed the motion of a particle in a combined gravitational and dipolar magnetic analogue field. The force experienced by such a particle would in general consist of a gravitational component and a magnetic analogue component. The gravitational component … always radially inward toward the parent body … would be given by F=GMm/R2. And the magnetic analogue component would be a variation on the Lorentz force law: F=m(v x O), where O symbolized the magnetic analogue field.

Having written the program, Yang experimented first with cases where the Ofield was zero. In such cases the particle should escape when its outward initial velocity equaled or exceeded the classical escape velocity. Assuming the potential and kinetic energies were zero at infinity, Yang readily deduced that the escape velocity, at distance R from the parent body, was (2GM/R)1/2. As expected, the particle invariably escaped when there was no O field.

It was quite another story, however, when an adequately sized magnetic analogue field was introduced. Experimenting first with cases where the particle’s initial velocity was in the equatorial plane, Yang found that even particles with initial velocities greater than the escape velocity might remain bound. It was a simple matter to plot the particle’s path when its motion was confined to the equatorial plane. When this was done, Yang typically obtained a picture like the one below.

 

Thinking that Yin might find these results interesting, he dialed Yin’s cell phone number.

"Hello," Yin answered in a hushed tone.

"Hi," Yang answered. "Is this a bad time?"

"No, no it’s fine. I’m in the library."

"Ah! Digging up data on the Sun and planets, I trust."

"Well … that was the initial intent. But I’ve gotten sidetracked."

"As do I all the time," Yang chuckled. "But listen: I’ve written a program that produces some results you might find interesting. Can you swing by some time?"

"Sure," Yin answered. "It’s … 10:45. If I come over now, you could show me what you’ve got, and we could have some lunch if you’re free."

"That sounds good," Yang agreed. "This shouldn’t take long. And you can show me what you’ve gotten sidetracked on."

"I should be there by 11:15," Yin promised. Not long after, the two were settled in Yang’s study and Yin was experimenting with various initial particle velocities.

"This is an amazing coincidence," Yin said at length. "It relates directly to what I got sidetracked on this morning."

"Which is?" Yang pressed.

"Black holes," Yin answered, opening a library book he had brought with him. "Look at this … these are actual photographs of jets, streaming out of what are believed to be black holes."

"Yes … yes, I see what you mean," Yang exclaimed excitedly. "The jets would presumably fix the black hole’s magnetic analogue dipole moment in space. Only along the dipole moment do particles not experience a deflecting force."

"Precisely," Yin concurred. "And presumably we can’t see the hole itself because photons, like material particles, are deflected by a powerful magnetic analogue field."

"I agree," Yang murmured thoughtfully. "It occurred to me last night that that might be the case."

"I’d say this is fair game for the symposium," Yin grinned. "Any word on when it will be?"

"Nothing definite yet," Yang answered. "But I expect to hear from the Professor soon."

"Great! I guess we’re done here for now. What do you feel like eating?"

"Well … there’s a new Chinese Buffet a couple of doors down the street. I haven’t tried it yet."

"Let’s do it!" Yin said. The two hurried into the street, and were delighted to find the air redolent with the smell of garlic and frying onions.

"China’s gift to the world," Yin thought with a smile. "FOOD!"

 

Chapter 14

Imagine

Yin loaded his plate up with a little bit of everything from the Chinese buffet.

"How do they do it?" he mused aloud as he savored various tidbits. "I’ve tried for years to duplicate these flavors at home, with little success."

"Secret sauces, I expect," Yang suggested.

"Yes, and closely guarded secrets at that," Yin agreed. "These people can go anywhere in the world and start a successful restaurant business."

Yin and Yang grew silent as they enjoyed the Chinese cuisine. Suddenly Yin slammed the palm of his hand on the table. Yang looked up at him questioningly.

"I just thought of a problem with the parallels we’ve been drawing between charge/charge and matter/matter interactions," he announced.

"Really!" Yang retorted. "Tell me."

"Well, have you ever read about the energy in the electric field?"

"Yes, it’s proportional to E2, isn’t it?"

"Right!" Yin affirmed. "And it’s really quite elegant. If you compute the field energy of two like-signed spheres of charge, then push them closer together and compute the field energy again, the work done to push them closer together exactly equals the gain in field energy."

"Yes, that is incredibly elegant," Yang answered. "But where lies the problem?"

"Well, consider two masses," Yin continued. "If you allow them to move closer together, they do work on you."

"Hm-m-m, but their gravitational fields must look like the electric field of two like-signed charges," Yang murmured.

"Exactly! So work is done on you; yet there’s an increase in field energy!"

"Assuming gravitational field energy is proportional to the square of the field."

"As it presumably would be if our parallels are valid."

Yang grew silent. What a disappointment! He had been so sure they were on to something. But how could the volume integral of g2 decrease when the two masses were moved closer together?

Yin was silent too. Both continued to pick away at their food, but no longer with gusto. Suddenly Yang carefully laid his fork on his plate.

"I’ve got it," he announced quietly.

"Tell me!" Yin answered.

"The gravitational field is imaginary."

Yin’s face took on a stupefied look.

"And the field energy is thus always negative," he marveled.

"Right. And when you allow two masses to move closer together, the field energy becomes more negative."

"And the difference obviously equates to the work the masses do on you," Yin rejoiced. "What a neat idea!"

Yang beamed, and both men began to enjoy their food again.

"So what does this imply?" Yin murmured at length. "That the mass is imaginary?"

Yang pondered that question, and surmised, "I’d say the mass is complex."

"Really! With real and imaginary parts?"

"Yes," Yang continued. "The real part corresponds to the inertial mass. Obviously in F=ma, m has to be real if F and a are."

"True," Yin agreed, munching on a bite of spicy chicken.

"And the imaginary part corresponds to the gravitational mass," Yang continued.

"So an imaginary gravitational mass generates an imaginary field," Yin added.

"Right," Yang exclaimed. "And if the imaginary field vector points away from the source …"

"Then another mass, multiplied by the first particle’s field vector, experiences a real force toward the first mass," Yin finished.

"And the two masses accelerate toward one another according to F=ma."

"Assuming the real and imaginary parts of the mass are linked."

"Well … sure. They are presumably two faces of the same coin."

"What an elegant idea," Yin mused. "We really should try to work out the equations that parallel Maxwell’s equations."

"Yes, we should," Yang agreed. "Are you free this afternoon? We could go back to my place and see what we come up with."

"That sounds good," Yin agreed, smiling over Yang’s shoulder at a very pretty Eurasian woman approaching their table.

"Finished?" she asked in a feminine voice, as she gathered up their emptied plates. Yin looked at Yang and allowed that he was stuffed. Yang said that he was too, and the young woman laid their check and two fortune cookies on the table. It seemed to Yang that she smiled especially warmly at Yin. After she had left, Yang spoke again.

"Wow! I’d say a smile like that warrants a handsome tip!"

Yin gave a nervous little laugh.

"Absolutely!" he said, fishing his wallet out of his pocket.

"So … what does the future hold?" Yang asked rhetorically, cracking open his fortune cookie. "Your wish will be granted."

Yin had read his fortune cookie’s contents also, and looked at Yang in disbelief.

"That’s what mine says!" he exclaimed, passing the scrap of paper across to Yang.

"Well I’ll be!" Yang said, comparing the identical messages. "What are the odds of that happening?"

 

*****

 

Once the two men were settled back in Yang’s study, Yang pulled a text from the bookshelf. At length he found what he was looking for.

"Here," he said, "check this out. It’s pretty clear you can make a case that a moving charge’s momentum resides in its electromagnetic field."

Yin studied the text.

"Because the field momentum density is proportional to B," he murmured. "Yes, I remember reading about this years ago when I was an undergraduate."

"So let’s think about the equilibrium experiment again," Yang suggested, drawing the following on a sheet of paper.

 

"So the idea is that the particles are in equilibrium … the gravitational forces of attraction exactly cancel the electrostatic forces of repulsion, right?" Yin asked.

"Right. But now view things from an inertial frame moving to the left with constant speed v." Yang said.

"OK. Relative to that frame the particles move to the right with a constant velocity."

"So from the perspective of that frame, the total force on each particle must also be zero," Yang pointed out.

"But now there’s a downward magnetic force on the top particle, and the total, upward pointing electromagnetic force is less than the electrostatic force in the rest frame."

"That’s the idea," Yang exclaimed. "And thus there must be an upward pointing magnetic analogue force on the upper particle."

"And assuming we use the imaginary part of the mass in the Lorentz force law equivalent, the magnetic analogue vector should point … what … out of the plane of the diagram, just as the magnetic field vector does, right?"

"Right, assuming the magnetic analogue field vector is imaginary, like the gravitational field vector hypothetically is."

Yin frowned and stared at the diagram.

"Is there a problem?" Yang asked.

"Perhaps. Is the field momentum proportional to the cross product of the gravitational and magnetic analogue field vectors, like the cross product of the electric and magnetic field vectors here?" Yin asked, alluding to the passage in the text.

Yang made a motion in the air with his right hand.

"Rats, I see what you’re getting at," he answered ruefully. "If the field vectors are both imaginary, then the field momentum points to the left."

"But if the cross product is formed by cross multiplying one field vector by the other’s conjugate, then the momentum points to the right," Yin added.

Yang nodded, but was not altogether happy with this somewhat ad hoc solution. Yin, meanwhile, was again studying the text.

"Here’s another possible problem," he muttered. "When you calculate the momentum in the electromagnetic field of a moving charge, you find it’s proportional to the velocity of the charge all right, but the constant of proportionality contains q2."

The look of frustration on Yang’s face deepened.

"But the momentum in the gravitational/magnetic analogue field must be proportional to the real part of m, and not to m2," he said quietly.

Yin nodded gravely. "I think we’ve got a major problem here," he murmured.

"I agree," Yang answered. "Should we see if the audience has any ideas at the symposium?"

"Yes … but I think we have to be careful," Yin concurred. "I think we should lay the ideas out, along with the problems, but not actually solicit any feedback. If anyone wants to volunteer anything, that will be fine. But mainly we should just sow the seeds."

"Yes … yes, that would be best," Yang agreed.

Yin looked at his watch.

"And on that note, I’ve got to get going," he announced. "I’ve got a dinner date this evening."

"Really!" Yang exclaimed, smiling quizzically. "Anyone I know?"

"Well … probably not," Yin answered guardedly. "It’s with the waitress at the buffet."

Yang stared at Yin, momentarily at a loss for words.

"When did you manage to set that up?" he asked weakly.

"While you were getting seconds," Yin answered.

"Yin, you never cease to amaze me!" Yang exclaimed, smiling with a trace of envy and rising to walk him to the door. He promised to call Yin as soon as he heard from The Professor, and wished him a fun evening.

Yang turned back to his study and settled again in front of his PC. He suddenly felt awash with feelings of loneliness.

"With that gorgeous young woman," he muttered. "Yin, after all these years you still manage to surprise me!"

 

Chapter 15

No Matter?

Yang was saved from an altogether desolate evening by a phone call from The Professor.

"Hi," The Professor greeted. "I wanted to get word to you before I left for the weekend. I’ve tentatively set things up for your symposium next Wednesday at 4 p.m. Will that be convenient?"

Yang assured The Professor that the date and time would be fine, and agreed that he and Yin would be at the Physics Building’s main lecture hall at 3:45 on the scheduled day. Recharged with a fresh sense of purpose, Yang briefly considered calling Yin, but decided to wait until the next morning.

That evening Yang went to an early movie and was in bed by 9. Once again he awoke around 1 a.m. and his thoughts more or less automatically began turning over the problem of inertial mass. As Yin had rightly pointed out, the inertia (or so-called electromagnetic mass) of a sphere of charge was proportional to the square of the charge. Yang remembered reading about the underlying physics on the same web site that featured the mass/charge equilibrium problem. It was readily shown that the time-varying fields of an accelerated charge result in an induced component of electric field right at the charge. The magnitude of this field is proportional to the charge, q. Thus the charge … or more specifically the agent causing the charge to accelerate … experiences a force of qEinducedwhich always opposes the acceleration. This was posited to be the inertial reaction force to the force exerted by the driving agent. And since Einduced was proportional to q, the reaction force, qEinduced, was proportional to q2. In effect the charge had an electromagnetic inertia proportional to q2.

In a parallel fashion it seemed that the time-varying magnetic analogue field of an accelerated mass should induce a component of gravitational field right at the mass. But this would mean that the reaction force, mginduced, would be proportional to m2. And if one used the usual signs in Maxwell’s equations, it appeared that the inertial reaction force would point in the same direction as the accelerating force!

"And that’s crazy!" Yin snapped aloud in the darkness.

There was of course always the possibility that Maxwell’s equations carried over into matter/matter interactions, but only with certain modifications. For example, perhaps in the case of matter/matter interactions Yin’s idea of using complex conjugates was of key importance.

On the other hand, what was neutral matter? In most practical cases it was clusters of atoms. And Yang had little doubt that the momentum of a moving atom closely approximated the electromagnetic field momentum "inside" the atom. Of course the fields "outside" the atom were practically zero. In a sense, a moving atom was a tiny capsule of time-varying electromagnetic fields.

This idea, that the internal parts (and specifically the nuclei) of atoms were largely insulated from the internal parts of other atoms, explained an important Newtonian fact: the inertial mass of two atoms, each of mass m, was just 2m. In contrast, the induced "inertial" electric field of a naked charge existed not only right at the charge, but also in the surrounding space. And a second charge, arbitrarily close to the first, would experience not only its own "inertial" force, but also a force in the other charge’s induced "inertial" field. The total reactive force from 2 such closely juxtaposed charges would thus be practically 4 times that of either charge alone.

"But in the case of atoms," Yang thought, "the induced fields of each atom go practically to zero at the atom’s boundaries. Thus the nuclei of two closely juxtaposed atoms did not feel the induced inertial electric field of their neighbor, and the effective mass was just twice that of either atom alone.

But what about neutrons and other "uncharged" elementary particles? Yang was not well versed in particle physics, but he was under the impression that even "neutral" particles consisted of charged parts called quarks. Were similar "insulating" mechanisms at work here, so that the mass of two neutrons was just twice that of one? Was it possible that there was no such thing at all as pure neutral matter? Could it be that all "neutral" particles were in fact intimate mixes of positive and negative charge, with only fading electromagnetic fields at points "outside" the particle? If this were the case, then what was gravity? Was it nothing more than a relatively minuscule whisper of the fields inside atoms … a whisper that could never be totally silenced, owing to leakage from the atom?

Yang realized that these were questions that an amateur might pose, but that only a guru could answer. Perhaps, time permitting, he would toss some of these thoughts out into the audience at next Wednesday’s symposium. Who could say whether such seeds might take root, and if they did what they would blossom into?

With a sigh, Yang rolled onto his side and socked his pillow into a soft nest. Thankfully his thoughts this evening did not stray from physics as he succumbed to the irresistible pull of Morpheus. The last conscious thought that rippled over his cerebral hemispheres, before slipping into a dream, was "All is charge … All is charge …"

The next morning sometime after 10 a.m. Yang called Yin and informed him of the symposium’s date and time. He was curious about how Yin’s date had gone, but refrained from prying.

"Wednesday at 4. I can make it … I can make that," Yin answered. "But it’s going to be tight. I’m flying out of here this afternoon, and won’t be back until Wednesday noon."

"Yipes!" Yang exclaimed. "That doesn’t leave us much time to prepare!"

Yin agreed, but there was nothing to be done about it. So the two agreed to independently generate a list of candidate topics, and to hone the list down and divide the topics up Wednesday afternoon prior to the symposium.

Wishing Yin a safe trip, Yang rang off and got into his running togs. Nearly four days … should he try the Chinese Buffet again some evening? All things considered, he thought not. There were plenty of restaurants. And he could always fix stir-fry at home. Perhaps this time he’d hit upon the unique flavors that had eluded himself and Yin through years of culinary experimentation. But … he doubted it.

 

Chapter 16

One Wish

Yin called Yang as soon as he returned from his trip, and the two planned to meet at Yin’s place. Yang agreed to pick up some take-out lunches on the way over. By the time he arrived, Yin had showered and was dressed casually for the symposium.

Without any time to spare, the two wolfed down the sandwiches Yang had brought along and retired to Yin’s study. Each had prepared a list of topics, and with few exceptions the lists were essentially identical. It was decided that Yin would start first, briefly recounting what the world had been like nearly 200 years ago, and rehashing their disastrous experience with time dilation. Yang would, in the meantime, be writing on the blackboard various Lorentz transformations. (Although the audience was well acquainted with such relations, who could remember them without a little help?)

Yang would then take over and talk about his "atom" modeling program … the one that provided an electrodynamic basis for length contraction and time dilation. And so it would go. They estimated that, without interruptions, they would talk for approximately 40 minutes. And they anticipated that questions would follow.

All too soon it was time to leave for the university.

"Nervous?" Yang asked, as they drove through the city in his car.

"Not really," Yin answered. "I’m expecting a friendly crowd. Actually, I think we’re going to have fun!"

Yang smiled, negotiating a turn. He suspected that Yin was right.

Yang parked in the science campus parking lot, and the two made their way to the physics building’s main lecture hall. The Professor was already down in the front of the room, setting up projection equipment.

"Welcome!" he beamed. "Are you ready to let us in on all your secrets?"

Yin and Yang laughed and returned The Professor’s greeting. He suggested that they be seated in the front row of seats, just to one side of the podium.

"We’ll wait until a few minutes after 4, for stragglers to find a seat, and then I’ll introduce you to the crowd," he said.

"I hope your colleagues aren’t bored with our non-technical discussion," Yin reiterated.

"You’ll do fine," The Professor reassured. "Remember, you’re among friends."

Not a minute later the first attendees began to arrive. Yin and Yang glanced shyly at them as they filed into the hall. Their glances were met with broad smiles, filled with wonder and respect. The simple fact was that these two visitors from a bygone era were celebrated personalities the world over, and particularly in the physics community. In the two years or so since their return from the long trip … the one during which they had jumped ahead in earth time nearly two centuries … they had given numerous interviews to the media, etc.

By the time the big clock on the front wall clicked to 4:04, the hall was filled to overflowing, with students sitting on stairs and standing in back. Yang glanced back at the crowded amphitheater and arched his eyebrows at Yin.

"Relax," Yin whispered. "We’re here to have a good time."

The Professor stepped up to the podium and told the crowd that their featured speakers needed no introduction. After asking that the guests not be barraged with too much math, he nodded to Yin and Yang and surrendered the lectern to them. The room erupted into protracted applause, bringing broad smiles to the faces of Yin and Yang.

Yin leaned over the lectern and said, "Good afternoon. My name is Yin, and chances are your ancestors read about my disappearance nearly 200 years ago."

The crowd laughed with delight when Yin recounted their incarceration upon returning from "the long trip," and grew silent when he told of their time in the library. The atmosphere lightened again when he recounted their meeting with the Secretary of the Interior (who still served in the nation’s capitol).

And so it went. There was applause when Yang demonstrated length contraction and time dilation using the classical theory (with the dependence of inertial mass on speed factored in). And everyone was intrigued with the way Yang had aged slightly less than Yin when they accelerated into another reference frame in separate space pods.

After they had concluded their remarks , several comments followed from the floor. A student wondered why, in view of the gravitational red shift, the astronomers seemed to insist that quasars were so tremendously distant and, in view of their brightness, that they were cranking out the power of entire galaxies.

"Might they not be much closer and on the verge of becoming black holes?" he asked. No one answered him.

A very old, retired professor emeritus commented on the analogies between charge/charge and matter/matter interactions. He told how as a boy he had discovered, in an attic trunk, his great grandfather’s notebooks from his undergraduate days at MIT. Evidently Yin and Yang’s thoughts on charge/charge and mass/mass analogies were discussed way back then, and a Professor Morrison had suggested at that time that a possible solution to the field energy problem was for the gravitational field to be imaginary.

"But as you have rightly pointed out," the ancient gentleman concluded, "there are other problems."

"I was wondering whether matter even really exists," Yang replied, catching Yin off guard. Yang briefly stated his thoughts on the encapsulated electromagnetic fields inside atoms, and possibly also inside neutrons and other "uncharged" elementary particles.

"Well, that is still one of the great questions of our age," a middle aged professor commented. "Is gravity force or geometry?"

"We … I was wondering whether the analogies to charge/charge interactions might be an alternative way of looking at things, a la Lagrangian versus Newtonian mechanics," Yin suggested.

"It’s probably more like Newtonian and quantum mechanics," the professor answered. "Where gravity … or the bending of space-time, if you will … isn’t too severe, the simple analogies are probably useful. After all, the mathematics of General Relativity is a stretch even for most professional physicists. But … when the effects are intense, as they are for example at the surface of stars, then General Relativity theory predicts things beyond the scope of the classical theories and analogies thereto."

"Such as gravitational red shift?" the student who had complained about quasars asked.

"Among other things," the professor agreed, glancing back over his shoulder.

At 5 p.m. The Professor stepped up to the lectern and allowed that there were probably many more questions but that it was time to bring things to a close.

"Just one more question?" a feminine voice called from the back of the hall.

The Professor squinted up over the heads of the crowd and noted that the question came from someone he had never seen before. Who was she, and who was the man accompanying her?

"OK, one more for the road," he granted.

"I’d like to ask each of your guests what he would want, if one wish could be granted," the young woman said.

Several heads turned and looked back at her. The Professor concluded that she must be a media type, and even considered admonishing that this was a scientific gathering. But this was a question that Yin and Yang were perfectly comfortable answering.

Yang stepped up to the microphone first and quietly said, "In my case the answer is simple. I would wish to see my family again."

The hall grew very quiet. Many … especially those old enough to have families of their own … clearly empathized with Yang.

"And your wish, sir," the young woman asked Yin.

"Well," Yin began, stepping up to the microphone, "as illogical as my friend’s wish might seem, mine would be an even more grandiose version. I’ve always been interested in the natural history of our planet."

Yang smiled and nodded knowingly. One entire bookshelf in Yin’s study was packed with books on paleontology.

"If I could have one wish … and mind you, I know how illogical this is … I would like to be able to go back to the beginning … see the world as it was then, and see the changes from then ‘til now, in time lapse fashion."

"Why is that so illogical?" a young voice asked innocently.

"Well … I suppose because, as the fathers of quantum mechanics first pointed out, we can’t observe the world without changing it," Yin answered. "And who can say what the consequences of those changes might be. In replay, we might not even be here today, having this discussion!"

The Professor laughed with the rest of the crowd.

"And on that note, I think we really must call it a day," he ruled. "I’d like to thank our guests for sharing some of their unique experiences and interesting ideas with us."

The room again erupted with applause, and several faculty members stopped to chat with Yin and Yang as the hall emptied. Finally there were again only the three of them.

"I think it went very well," The Professor beamed. "Did you have fun?"

"Yes, I really did," Yang smiled back.

"Me too," Yin chimed in. After a few more words, Yin and Yang took leave of The Professor, who again thanked them for coming while he dismantled the presentation equipment.

The two friends wended their way back to Yang’s car. As they entered the parking lot they were surprised to see the young woman, who had asked what they would wish for, approaching with her male companion. Yin and Yang had also concluded that these two were from the media. They had always made themselves freely available and smiled now as the two drew near.

Without hesitation or explanation, the man commanded, "Go to Space Pod 1," and pressed a scrap of paper into Yin’s hand.

"Yes, go to Pod 1," the woman repeated, also handing a paper strip to Yang. Without further comment the two withdrew and were soon out of sight.

"What was that all about?" Yang exclaimed, looking at Yin with wide eyes.

Yin shook his head, as if to say that Yang’s guess was as good as his own. He absently held his palm up to examine the scrap of paper the stranger had given him.

"Your wish will be granted," he murmured. By now utterly confused, he looked at Yang, who was unfolding the strip of paper the young woman had handed him.

"Your wish will be granted," he echoed in a shaky voice. With gaping jaws the two friends examined the pieces of paper more carefully. The small strips appeared to be the same ones they had extracted from fortune cookies several days earlier.

"What do you think?" Yin asked tentatively. "To the Space Pod?"

"To the Space Pod!" Yang shouted in agreement, lunging into his car. The car’s engine roared to life and the two adventurers sped off toward the spacedrome where the pods were kept under tight security.

 

Chapter 17

Into The Box

Yin and Yang had full access to the space pods, thanks to a directive from the President of the United States. When they arrived at the spaceport they were quickly passed through to a secure hangar where the pods were parked.

"Everything in order? No visitors to the pods?" Yin asked the guard.

"All quiet, sir," the guard answered. "No one’s been near those pods since your last flight."

Yin and Yang hurried over to Pod Number 1. Yang placed his eye over the opening in a scanner, and the pod door hissed open. The two men entered the pod and began to make their way to the flight deck. All appeared to be in order.

"I don’t get it," Yang muttered as they stepped up onto the flight deck. "Do you see anything out of the ordinary?"

Yin was silent, so Yang glanced at him. He was shocked to see that Yin’s face had gone white, and his mouth was working but no sound was coming out.

"Are you all right?" he asked. "What is it?"

Yin raised a trembling finger and pointed at the craft’s front window.

"Something outside!" Yang thought, as he wheeled and followed Yin’s gaze. What he saw momentarily took his breath away. The Earth was a small, cloud-enshrouded sphere, receding rapidly in the blackness of space. But Yang had felt no movement, and certainly neither of them had touched the controls.

"A trick? A hologram?" Yang wondered. He wheeled and flipped switches on the navigation console. The instruments indicated that they were 46 million miles from Earth and receding fast.

"Any thoughts?" he asked Yin through clenched teeth.

Yin had partially regained his composure, and shook his head no.

"Not a clue," he half whispered. "We can’t really be out here where the instruments say we are, can we?"

"Of course not," Yang scoffed. "I haven’t felt a thing. Have you?"

Yin gasped and Yang again swung his gaze to the window. The craft appeared to have rotated around, with the Sun now centered in the window. But again neither man had felt any motion.

"It’s got to be a hologram," Yang muttered. "But who? And why?"

As he spoke, the Sun spun in front of them and the Earth … now a tiny speck … appeared in the field of view off to one side. It was as if they were going around to the far side of the Sun at tremendous speed. In the next instant the Earth dipped below the Sun’s horizon.

"For some reason, someone’s simulating a trip to the far side of the Sun," Yang observed. Just then it seemed as if his brain exploded in a blinding flash of light. The feeling passed in an instant.

"What was that? Did you feel anything?" he asked Yin. Yin looked at him with terrified eyes, his head nodding yes.

"You sound a little different," Yin said in a frightened voice.

"So do you," Yang answered. Yang’s eye noticed something strange. "Hello," he murmured, "what’s this." Yang kneeled and gazed at a strange black box resting on the floor.

"How did you get here?" he wondered.

"We brought it in," a familiar voice spoke from the pilot’s seat. Yin and Yang spun around and looked with disbelieving eyes at the young woman from the symposium and the parking lot. Seated beside her, in the copilot’s seat, was her male companion.

Yin’s eyes looked imploringly at Yang. They seemed to ask how he and Yang could possibly not have noticed these two occupants when they first entered the flight deck.

"Who are you, and how did you get in here?" Yang growled, crouching slightly in a fighting stance.

Yin, not being quite the warrior, wished fleetingly to distance himself from the unexpected apparitions, and found himself floating to the back of the compartment!

"We came in with you!" the woman answered. "My name is Emma, and my companion’s name is Sean."

"And to answer the rest of your question," Sean added, "You couldn’t see us until you were dematerialized. That was the white flash you experienced."

"Demat … What is this?" Yang snarled.

"You are no longer interacting with the real world," Sean expanded. "You and the entire space pod have been separated into real and imaginary parts."

"Oh really?" Yang sneered. "And what are we interacting with now? The imaginary world?"

"Not a bad way of putting it," Sean answered.

"Can you believe this," Yang said in an aside, turning to where he had last seen Yin. But Yin was now up against the back wall of the compartment! He wanted to reach out to Yin, and immediately he began to float back himself. The two friends stared at each other. Traces of confusion mingled with astonished belief began to steal across their faces.

"And our real parts?" Yang asked more meekly. "Where are they?"

"In the box," Emma answered, nodding to the mysterious box on the floor.

Yin cleared his throat and ventured to speak.

"What is the purpose of all this?"

"Your wishes are going to be granted," Sean said in a congratulatory tone.

"Our wishes are … we’re going back in time?" Yang asked excitedly.

"Yes you are," Emma answered. "You’re going way back in time."

"To the beginning," Yin breathed, his eyes glazing over with wonder.

"To the beginning of life on Earth," Sean clarified.

"But how … How is that possible?" Yang pressed.

"Patience," Emma admonished. "All will be revealed in due course. For now, it’s time for a trial run, to let you see what’s ahead and to reassure you that you’re in no danger."

"Look," Sean said, gesturing toward the front window. Yin and Yang gasped. The pod was plunging headlong into the Sun! Huge solar flares shot into space around and above them. Just below and coming up fast lay the plasma of a living star, seething and boiling like the lava pools of a thousand volcanoes!

Yang looked at Yin. This was it, they both thought. In seconds they’d be vaporized. Yin smiled weakly. His eyes seemed to say, "So long … it’s been fun."

And then they were into the plasma! The light was blinding and neither could see anything else … yet they felt nothing and evidently continued to exist. In moments they were out again, and the solar star was receding rapidly.

"How … is that … possible?" Yang asked, looking at the two strangers with beseeching eyes.

"We told you … you’re not in a form that can interact with the real world," Emma reminded him.

"But … I could see …" Yin argued.

"That’s because light has real and imaginary parts. You’re now interacting, more or less, with the imaginary parts."

"But not with the real parts, and thus not affecting the real world in any way," Yang mused, beginning to understand.

"Correct. That’s why you’ll be able to go across in time without altering the present," Sean answered.

"But … we can hear you … we can hear each other," Yin said.

"True, we can all hear one another because phonons also have real and imaginary parts," Sean stated. "But again, there’s no interaction with the real world."

Yin and Yang were momentarily at a loss for words. They gazed bug-eyed at the receding Sun. A troubling thought occurred to Yang.

"How will my wish be granted … to see my family again, if we’re going so far back in time?"

"In due course you’ll skip forward to 172 years ago, by taking many trips out to the stars and back," Emma explained.

"Yes … we know how that works," Yang murmured. "But why us … why me? Why is my wish being granted?"

"Because you could have wished for anything, but you chose love," Emma answered with a radiant smile.

"And me? Why my wish?" Yin added.

"Because above all else, you wished for knowledge," Sean replied.

"Yes, but that was just playing to the audience," Yin objected. "How do you know that I really meant it?"

"I know," Sean answered flatly. Yin nodded his head. In fact he had meant it. But still … how could they know that?

"How long have you … been with us?" Yang asked, beginning to get an inkling of who these two might be.

"Since your births," Emma smiled.

"Since our bir … you’ve been with us constantly?" Yin gasped.

"Yes," Sean answered.

Yin looked at Yang and both men flushed red.

"Don’t be embarrassed," Sean smiled. "We were once real ourselves."

"Are you our guardian … are you angels?" Yin asked timidly.

"We have been called that," Sean said.

"But look, Ma, no wings!" Emma joked, pirouetting in front of them. Yin and Yang smiled wanly. It seemed to them that this was no time for levity.

"You see," Sean continued, "there’s a plan … a master plan. When any of us are in our real forms we’re in harm’s way, so to speak. There really are forces that try to subvert the plan. Our job is to protect you, not from yourselves but from these forces.

Yang was still dwelling upon his own wish.

"And when we have advanced adequately in time, and I see Marie and my daughter again, will I be rejoining them? Will all that’s transpired since I lost them be wiped out for me?"

"Sadly, no," Emma said softly. "You cannot change the past. "But you’ll be able to see and hear them."

"What happens if we come back to the time before we left on that fateful trip," Yin challenged. "Will Yang be able to see himself?"

"Yes, that is a possibility," Sean acknowledged.

"And you … will we see you there too?" Yang asked.

"Oh yes, I’ll be there and Sean will be with Yin," Emma replied.

Yang’s jaw dropped. This was bizarre beyond belief! Was he dreaming?

"No, but I suppose you could say you’re imagining," Emma giggled. Yang stiffened. He’d have to be careful. These two could evidently read his and Yin’s thoughts!

Yin wondered if he and Yang were in fact now dead.

"Will we ever be real again," he asked tremulously.

"Yes, you are scheduled to be reconstituted when you’ve returned to the present," Sean answered.

"Reconstituted … you mean recombined with our real parts?" Yin pressed.

"Correct," Sean said.

"This is incredible!" Yang exploded. "Will we remember anything after we’ve been … reconstituted?"

"You’ll remember everything," Emma answered. "And what’s more, you’ll have a graphic record to show others what you’ve observed, once you’ve returned to the present!"

Yin wondered how that would be possible.

"Look in the storage locker," Sean suggested, seeming to read Yin’s mind.

Yin willed to move to the storage locker, and like magic he floated over to it.

"Amazing!" he thought. "In our present state we go wherever we will to go!"

"Yes, that is how you get around in your present form," Sean agreed, again seeming to read Yin’s thoughts.

"Why do we need this pod, then?" Yin asked slyly.

"To keep you from getting lost. The box will navigate for you on your trips away from Earth and back again. And it will give you a familiar sense of being enclosed."

Yin opened the storage locker and took out what appeared to be a camera.

"This will be reconstituted along with us, once we’ve returned to the present?" he wondered.

"Its memory will be reconstituted," Sean clarified, "and you’ll be able to play all that’s stored in that memory, on any current holographic projection system."

"So we’ll be able to show others everything that we’ve seen?" Yin mused excitedly. "That’s going to be some show!"

"Yes," Sean affirmed. "That will be your special legacy to humankind."

"Dinosaurs … ice ages … the great extinctions …" Yin marveled.

"And who gets all this information … these filled in gaps in human knowledge … once we’ve returned?" Yang asked.

"I suppose in due course many will have access to it," Emma answered. "But you might want to start with your friend, The Professor."

"Yes, of course," Yang agreed, glancing at Yin, who appeared to have no objection.

"And what becomes of us, once we’re made whole … once we’ve been reconstituted?" Yin asked.

"Whatever you wish," Sean replied.

"Will we eat … sleep in our present state?" Yang asked.

"No, nor will you age," Emma answered.

With a start, Yin realized he wasn’t breathing.

"Are you breathing?" he asked Yang in a hushed tone.

Yang stiffened.

"No!" he answered.

"And a good thing that is," Emma continued. "When you arrive at your initial destination in the past, there will be no oxygen to speak of in the Earth’s atmosphere."

Yang wondered about their not aging.

"Will we age after we’ve been reconstituted?" he asked.

"Yes, once you’ve been reconstituted you will begin aging again," Emma answered. This reply raised a host of new questions in Yang’s mind, but he decided not to get into death and other issues just now.

"Where …" he asked instead, "or rather how do we go back in time?"

"It will occur at the galaxy’s center. There is a black hole there, and you will pass through the singularity at its center," Emma explained.

"The galactic center … that’s a long way from here," Yang observed.

"Yes, it is. But at the speed we’ll be traveling, we’ll be there in a few minutes of onboard time," Emma pointed out matter-of-factly.

"Will we … be traveling at light speed?" Yang wondered.

"Almost," Emma smiled. "But not quite. None of us, even in this imaginary form, can attain that."

Yang remembered his dream about the classroom and the question about the kingdom of heaven. He looked sharply at Emma. Had she read this thought too? Emma only smiled and nodded her head imperceptibly.

"Even at light speed it will take us thousands of years, in the present frame’s time, to reach the galactic center," Yin mused.

"Yes," Sean agreed. "By the time we reach galactic center, nearly 15,000 years will have passed here."

"And The Professor will be long gone," Yang murmured.

"Mankind might even be extinct," Yin added. He looked at Sean to see if he or Emma knew what the future held, but Sean’s expression revealed nothing.

"Shall we get underway?" Sean said, and instantly the panoply of stars out the front window converged to a dazzling point. Again Yin and Yang felt no acceleration.

"We’re rockin’ and rollin’," Yang whispered in an aside to Yin. "Am I … are we dreaming?"

"I don’t think so," Yin whispered back. "Color us lucky! My hunch is that we’re about to embark on one of the greatest adventures of all time!"

 

Chapter 18

Space Zero

"Shall we step up into the viewing hemisphere?" Sean suggested.

"Good idea," Yin agreed, and the four climbed the three steps and seated themselves under the large, transparent bubble. Ordinarily the view would have been dazzling … a black sky filled with innumerable stars as never seen through the blanket of Earth’s atmosphere. But now, quite as Yin and Yang had come to expect, there was only an intense circle of light directly in front of them.

Suddenly there was blinding light for an instant! Yang’s first thought was that something had gone wrong, and that they had been reconstituted. As if reading his thoughts, Emma smiled across at him.

"A star," she said.

"A star?" Yang asked in a puzzled tone.

"Yes, we just passed through a star."

Yin swung his chair around slightly and looked at Yang. Yang’s eyes opened wide and he puffed his cheeks out in amazement. Yin shook his head imperceptibly with a look bordering on disbelief.

"I would think we’d encounter them more often," Yin murmured, staring at the bright disc of light in front of them.

"Well, bear in mind that they only appear to our eyes to be concentrated out in front of us," Sean reminded. "In actuality they’re still all over the sky."

"Of course … aberration," Yin replied, slightly mortified.

"But," Sean continued, "the density will increase rapidly as we approach galactic center, and we can expect to encounter them at a greater frequency."

"And we pass right through them," Yang marveled.

"Or, from the perspective of our present frame, they pass right over us," Yin added. Emma and Sean smiled and nodded.

"I’m amazed we were able to make our long trip … the trip where I lost my family … without colliding with anything at all," Yang said. "I guess space must be fairly empty out where the Solar System is situated."

"Relatively speaking that is true," Emma confirmed. "But make no mistake, you had help. The odds of traveling 85 light years, as you did each way, and not colliding with anything at all, are very slim."

"We had help?" Yang pressed, looking at Emma curiously. "I thought in our present … in your form that interaction with the real world isn’t possible."

"For us that’s true," Emma smiled. "But the box has special powers."

Yang looked at Emma.

"And exactly how did the box keep us from colliding with anything?" he asked at length.

"By sweeping the path in front of you free of all matter, using its power to alter the gravitational field," Emma replied.

"Yes, you may have noticed that you felt no sensation of acceleration during our trip out to the far side of the sun," Sean added. "Even though you were still in your real forms."

"Yes … yes we did notice that," Yin answered. "It felt like we were still at rest in Earth’s gravity."

"The box at work," Yang murmured.

"Precisely," Emma said. "In fact, the box is our means of propulsion right now. It’s why our speed, relative to the nominal galactic rest frame, can so nearly approach the speed of light."

In the next few minutes the bright flashes came more and more frequently. And then, suddenly, the circle of light before them again exploded, stars streaming out and around them in every direction. In a fraction of a second the sky was filled with stars as Yin and Yang had never seen them before. The density was dazzling, and many stars appeared not as points of light, but as bright orbs of various sizes.

"And here we are," Sean stated matter-of-factly.

"At the Milky Way’s core," Yin marveled. Yin and Yang contemplated the sky directly in front of the pod. It was devoid of stars, but why? The answer occurred to Yang as quickly as his mind had framed the question: A black hole!

"Yes," Emma said, again seeming to read his mind. "It’s the largest black hole in the Milky Way."

Yin’s gaze swung out into the space around the hole. Beyond its periphery, where the faint, nearly infrared light from a few large stars flickered through, two diametrically opposed jets streamed away into the void.

"The box will navigate you down one of the jets, through the central singularity and into the past," Sean stated, again seeming to read Yin’s thoughts.

"And how will we … find our way back home, once we’ve arrived in the past?" Yin inquired anxiously.

"The box will take you back to Earth," Sean reassured. "In fact, it will be your navigator from here on out. You say what you want to do, or where you want to go, and the box will comply."

"And you … Emma and you?" Yin asked.

"Oh, you won’t be needing us in your imaginary forms," Emma answered gently. "We’ll be leaving you until you’ve been reconstituted."

"Leaving? What will you … do?" Yin cried in alarm.

"You’ll be fine," Sean soothed. "If you like, you can think of the box as looking out for you."

"And as for us … we’re not certainly what we’ll be doing while you’re working your way back to when you were dematerialized," Emma said. "Maybe we’ll get a vacation!"

"Working our way back to when … but of course!" Yang blurted. "Right now it’s nearly 15,000 years later…"

"But we’re going back hundreds of millions of years before now," Yin continued.

"So we’ll be able to hopscotch back to the pres … back to where … back to when we were dematerialized, by making side trips like this one," Yang mused.

"Correct," Emma answered. "But of course you needn’t always come here to the galaxy’s center. There are plenty of other worlds to drop in on."

Yang’s eyes glazed over momentarily.

"This is unreal," he muttered.

"Yes, it is!" Emma giggled.

"Could we … I’d like to record a short sequence before we shoot into the past," Yang said.

"That presents no problem," Emma agreed. "It will give you an opportunity to gain some hands-on experience with the camera."

"The only constraint is that you not record us," Sean added.

"No problem," Yang replied.

Yang took a seat and began to marshal his thoughts while Yin retrieved the camera from the storage locker and browsed through the instructions in the camera’s help file.

"Once we’ve returned to where … whence we came from," Yang asked Emma, "will we still be protected from collisions, should we decide to explore the future?"

Yin looked up sharply from the camera. Yang seemed to be considering moving on from the times they had just come from, once their exploration of the past had been completed. And why not? It would be interesting to vault ahead 15,000 years from then … to where the Earth was now, at this moment … and to see what had become of mankind.

"You will be protected if that is what you choose to do," Emma answered.

"What do you think?" Yang asked Yin. "Should we move on after we’ve explored the past?"

"Works for me," Yin smiled. "We’ve always gone for broke!"

Yang relaxed visibly, secure in the knowledge that their partnership would not be broken up.

"And will you be there with us?" Yang pressed, looking again at Emma.

"We’ll be there," Emma smiled gently. "But of course once you’ve been reconstituted, you won’t be able to see us."

Yang shook his head in wonder. Who could have guessed, in their wildest imagination…

"Ready?" Yin asked, lifting the camera and focusing on Yang.

Yang shook himself out of his reverie and again thought of The Professor. His learned friend was now dust … dead for more than 14,000 years. Yet if one could believe the fantastic scenario that Emma and Sean had painted for them, The Professor would see and hear what Yin was about to record!

Sitting up erect in his seat, Yang smiled into the camera.

"Roll it!" he commanded.

 

Chapter 19

Back To Square One

"That’s a wrap!" Yin exclaimed, lowering the camera. Yang rose from his seat, taking the device from Yin and examining it.

"So this has both telescopic and macroscopic optics," he noted.

"Yes, and for all practical purposes its memory is unlimited," Emma answered. "You could film over a century, non-stop."

Yang smiled in amusement.

"That would be quite a documentary! I expect we’ll film a lot less than that."

"Perhaps, perhaps not," Emma pointed out. "Remember: you’re not aging in your present form, so there’s no rush to get back to when you were dematerialized."

"How exactly is this all going to work out?" Yin asked. "Where … or when in time will we emerge from the black hole?"

"Well, our perception is that your main interest is with terrestrial life, so we tentatively instructed the Navigator to take you back to the beginning of the Precambrian era … about 4 billion years ago," Sean suggested. "But if you wish, you can push it back to 5 billion years, when the Solar System formed."

Yin looked at Yang nervously. Did they really want to go back to a time before the Earth itself had taken shape?

"What do you think?" he asked Yang.

Part of Yang yearned, of course, to see his wife and baby girl again. But he realized now that he would not be able to interact with them in any way, nor would they be able to see him. As a result, he felt less of a sense of urgency for that time to come.

"Oh, I don’t know," he mused tentatively. "I know on the one hand that the mechanism is a matter of some debate…"

"The mechanism of what? Of how the Solar System formed?" Yin pressed.

"Right. It’s probably safe to say that there would be a lot of disappointed astrophysicists if we pass up the opportunity to settle that question."

"And," Yin added, "we needn’t spend a lot of time there … we can take some pictures, skip ahead 50 or 100 million years, take some more, and so on."

"True," Yang agreed. "I wonder how far from Earth we’d have to range in order to skip ahead that far in time."

"With that much time to burn, you could visit a neighboring galaxy," Emma observed.

"Another galaxy!" Yang exclaimed. "And the box … the Navigator would still be able to get us back to Earth?"

"I wouldn’t worry about that at all," she assured. "In fact, you can specify how much time you want to skip ahead, and the Navigator will give you options of places to visit."

"How does that work? How do we communicate with it?" Yin asked, stepping down to the flight deck and staring at the smooth cubical box.

"Ask it a question," Sean suggested. "Just start the question with the word ‘Navigator’."

Yin paused uncertainly.

"Na … Navigator," he stammered, "when in time are we scheduled to emerge from the black hole?"

Yin and Yang stiffened when the answer was tendered. They heard nothing! But in their minds they sensed a gentle voice that said, "Four billion years before the present."

Yang looked at Yin with eyes that seemed to beg, "Did you hear … did you sense that?"

Yin nodded wordlessly. But something was still bothering him.

"How will we be able to evaluate where next to visit in time?" he asked. "Neither of us is very well versed in what’s known and what’s not … what’s a matter of conjecture and debate."

"If you look in the storage locker you’ll find an electronic book that contains that kind of information," Sean replied. "Think of it as your book of knowledge. The Navigator is also familiar with such issues, and will provide meaningful options when the need arises."

"So he … it is programmed with some very sophisticated artificial intelligence," Yang mused.

"Oh yes," Emma answered. "Actually, a good deal more than that. The Navigator is in communication with … an entity or entities that even Sean and I can only speculate about."

"And you said that you’d be rejoining us when we work our way back to the time we were dematerialized?" Yin asked.

"Yes," Sean replied. "When you return to that time, it will take place on the far side of the Sun, a few minutes before you were dematerialized. We’ll rejoin you in the pod at that time, and we’ll be there when you and the pod are reconstituted."

"But you, Emma and all this gear will disappear at that time?" Yang asked.

"Almost," Emma smiled. "The camera’s memory will be reconstituted along with you. You’ll be able to play all you’ve recorded, on any real world holographic system, for others to see."

Yin and Yang nodded that they understood. Neither could think of any more questions at the moment.

"And now," Emma said, "it’s time for you to embark. Remember that you can go anywhere you wish in your present form. You can plumb the depths of ancient seas as well as the vacuum of outer space. But we suggest you do the latter always in the pod, so as not to get lost."

Yin and Yang nodded soberly.

"But the pod will be reconstituted along with us, right?" Yang reiterated anxiously.

"Yes," Emma assured. "Everything will be just as it was when you were dematerialized, except that you’ll have your memories and the camera’s record of all you observe in the past."

Emma laid her hand on Sean’s forearm.

"Shall we go?" she asked.

"Yes, it’s time," Sean answered.

"But wait a minute!" Yang cried. "Where are you going?"

Emma moved to the wall of the flight deck. She pushed her arm through it as though it were nothing more than a hologram.

"But … how will you find your way back to our rendezvous point?" Yang fretted. "And won’t it be nearly 30 thousand years later than when we were dematerialized?"

"Don’t worry about us," Emma smiled. "We’ll catch a ride. And once we’re back in the Solar System’s environs, we too will regress back in time by going through a local black hole there.

"There are … multiple black holes?" Yin asked.

"Oh yes, there are many throughout the galaxy," Sean replied. "This one at G.C. … at galactic center … only happens to be the largest."

"Here," Emma said, motioning for Yin and Yang to settle into the pilot and copilot chairs. "Let’s get you situated so that you can enjoy the show!"

Yin and Yang moved to the chairs. The sky, ablaze with stars, swung swiftly in front of them, and in a moment they found themselves squarely situated in one of the jets streaming out of the G.C. black hole.

"Have a fruitful trip!" Emma bade them

"And remember: you can completely trust the Navigator," Sean added. "You’ll still be watched over, even though we’re away from you for a time."

And with that, Emma and Sean floated through the ceiling of the flight deck and were gone. Yang jumped up and pushed his head through the ceiling. When he pulled it back into the compartment, Yin was staring at him speechlessly.

"Gone," Yang stated flatly.

Yin nodded, still dumbfounded and at a loss of words. This new form of theirs was obviously going to take some getting used to!

"What did it look like?" he asked Yang.

"Outside?" Yang retorted, casting a puzzled look out of the front window.

"No … in the actual wall of the pod."

"Ah! Sort of gray … sort of brown … it didn’t look like much of anything," Yang answered.

"But not black…" Yin added.

"No, not completely. Evidently there are photons flitting around among the atoms in there," Yang surmised.

Yang moved back to his chair and the two friends stared silently into the oncoming jet of particles and light, punctuated occasionally with bright flashes.

"Well, shall we do this thing?" Yang asked.

Yin looked across at his friend and reached out to him. Yang grasped the outstretched hand and shook it.

"Let’s go for it," Yin said quietly, swinging his gaze back out into the jet. "Let’s see what lies beyond the looking glass."

"Navigator," Yang uttered nervously, "Take us back 5 billion years."

 

*****

 

Well, dear reader, we shall take leave of Yin and Yang during their adventures in the past. It is of course a fascinating story. But it is a story more about natural and human history than it is about space and time, and perhaps best left to tell another time. Suffice it to say here that, after a great many stops on the way back from when the Solar System was formed, and after many visits to other stars and planets … some with life forms of their own … Yin and Yang worked their way back to the time when they had been dematerialized. By then they had filmed an astounding 75 years in actual viewing time … more than any one human being could view in an entire lifetime. But, as Sean and Emma had said, Yin and Yang aged not at all in their imaginary forms. And, with the help of the book of knowledge, the two novice documentary makers created a vast table of contents, so that later viewers would be able to find episodes of interest.

Emma and Sean reappeared on schedule when the two explorers returned to the far side of the Sun. Yin informed them that he and Yang had definitely decided to move on into the future, after leaving the record of their journeys with The Professor.

"Of course you could stay in your present form," Emma said. "But that would preclude leaving any record of your observations with living human beings."

"And if we opt to be reconstituted…" Yin wondered.

"Then you begin to breathe again … to age … you are once again living souls."

"So if we choose that course, then we grow old ... eventually we die," Yang murmured.

Emma and Sean nodded affirmatively.

"And after we die … do we become like you?" Yin asked hesitantly.

"We don’t know," Sean answered. "Quite simply, that isn’t for us to decide."

"Were you ever alive … were you ever real?" Yang asked.

Emma glanced at Sean. His answering gaze seemed to indicate that there was nothing wrong with answering that question.

"I was your great great grandmother," Emma said quietly to Yang.

Yin and Yang gasped in unison.

"And you…" Yin asked Sean.

"One of your ancestors," Sean smiled.

"But enough!" Emma exclaimed. "It’s almost time for your reconstitution. You must decide whether to become real again, or to stay in your present forms."

Yang looked at Yin.

"This … everything we’ve been through doesn’t make much sense unless we pass the record along to living people, does it?"

Yin pursed his lips, nodding in agreement.

"Yes," he answered, "even though there are no guarantees of what becomes of us, once we die."

"Shall we do it then?" Yang asked.

"I think we have to," Yin agreed.

"We expected you would," Emma said with an admiring smile. "Are you ready?"

And then … the bright flash, and Emma and Sean were gone from view. Yang stepped to the wall of the compartment and pressed his hand against it. Solid!

Yin opened the storage locker. The camera and electronic book of knowledge were gone. But, a small metallic box remained where the camera had been.

"Here it is," he said to Yang. "The record of all we’ve seen and filmed."

Yang grinned and gave the thumbs up sign.

"Shall we see what The Professor’s been up to?" he asked.

"Absolutely!" Yin answered. "Hey! Guess what! I’m hungry!"

"Me too," Yang rejoined. "And I know where there’s a good Chinese buffet!"

"Let’s hit it!" Yin shouted. "Navigator, take us home!"

 

Epilog

Yin and Yang arrived back at the spaceport early in the evening, and decided to wait until morning to contact The Professor. They treated themselves to a sumptuous feast at the Chinese Buffet on Yang’s street. Yin did not spot the pretty young woman he had dated.

"Just as well," he thought as he contemplated their impending departure into the future. The two felt a common anxiety about what they might find there, and decided to hop along only a century or two at a time.

The next morning they met The Professor in his office.

"What happened?" their learned friend asked. "I got word that the pod flew around to the far side of the Sun after the symposium."

Yang grinned at Yin.

"It’s … a long story. This will tell you all about it," he smiled, setting the camera’s memory on The Professor’s desk.

"We suggest that you watch the Introduction with your colleagues, and then make copies for other scientists around the world," Yin added.

The Professor pulled the box toward him. The ports were there to hook into his holographic viewing system. But what might it contain?

"Will you be joining us?" he asked quietly, somehow already knowing what the answer would be.

"No, we’ve decided to move on into the future," Yang answered softly.

The Professor’s face darkened. For a time he grew silent.

"So, I won’t be seeing you again?" he asked at length.

"Possibly not. But in view of what we’ve been through," Yang answered, pointing to the camera’s memory, "we can’t rule anything out."

The Professor frowned. They were moving into the future; yet they might see him again? All he knew about physics suggested that this was not possible! Yet he had learned to expect anything from these two.

"Well," Yang said, reaching across the desk, "I’ll say goodbye, at least for now. I’ll … miss you, kind sir."

The Professor sprang to his feet and pumped Yang’s hand speechlessly.

"Professor, it’s been a pleasure," Yin added, also offering his hand.

As Yin and Yang walked out of the office, Yang turned.

"Enjoy!" he smiled, pointing to the camera’s memory. "It’s quite a show! But it’s a long one. You can pick out the parts that interest you from the table of contents."

And then they were gone.

That afternoon, in response to an emergency meeting called by The Professor, virtually the entire science faculty convened in the main science lecture hall. Several federal personnel had also flown in and were present. The Professor stepped to the lectern and addressed the crowd.

"I have no idea what we’re about to see," he said. "This was given to me by yesterday’s symposium guests, and they suggested that we all have a preliminary look together."

The Professor flipped a switch and took a seat in the front row. With rapt eyes he and the others watched as a hologram of Yang took form in the space before them.

"Hello, Professor," the image greeted. "My friend, Yin, is filming this from the center of the Milky Way galaxy. At this moment, you and all watching this with you will have been dead for nearly 15 thousand years. How, then, can you be watching this now? If you’ll bear with us, we will try to both explain and entertain…"

 

The End