Kert Rats meets Red Frawd title

the previous generation
These are the voyages of the Starhip Boobyprize,
her continuing mission:
to deplore strange new worlds,
to shriek at strange new life and new civilizations,
to cautiously come from where
no one(3) will come from again.

Ian Stewart

2 Something Strange About Light

Captain’s log: stardate Libra. Mr Outa-Data and Lieutenant Woof, two of the ‘Kert Rats’ that form the crew of Federation Starship Boobyprize, have found a wandering Time-Traveller from the era of H.G.Wells, and in the manner of this kind of writing, are explaining Special Relativity to him. The Time Traveller has suggested that the Michelson-Morley Experiment is inconclusive, because the Earth might carry the luminiferous ether along with it. A subspatial energy being has invaded the matter/antimatter reactor core, but Pox believes he can flush it out with some Matter/antimatter reactor core Duck. A space hurricane is forecast. Situation normal.

While Outa-Data was trying to puzzle that one out, he and Lieutenant Woof conveyed the Time Traveller to the bridge, to meet Captains Kert and Pickup. The diversion gave Outa-Data enough time to think of an answer.

"That is a fascinating theory," he said, as they arrived on the bridge. "But you woud expect to see strange effects in the light from distant stars if the ether was swirling around in such a manner."

"Discussing ancient technological history, Mr Outa-Data?"

"Yes, Captain Pickup. That is very perceptive of you. Good day, Captain Emeritus Kert. What a nice garotte."

"We are in an important discussion with the Tritovoglian Ambassador," said Pickup, gesturing towards a tiny alien that had adhered itself to his console. It looked for all the world like a piece of chewing-gum, but in actual fact it was the spiritual leader of one of the most powerful and most easily annoyed races in the galaxy.

"Good day, Ambassador. How are the budlings? Excellent. Captain, I am sure that the Ambassador will not object while you interrogate the humanoid alien that Lieutenant Woof has reported to you."

"Ah. He does look quite humanoid," said Kert. "Let’s shoot "

"Not as humanoid as me," said The Cat, who belonged in a different programme altogether but had drifted into the wrong lane of the information superhighway, towing a trailerload of suits along behind him. "And look at that dreadful cloak! Black doesn’t even match its teeth!"

"Undeniably alien," said Pickup.

"In a humanoid kind of way," Kert added after a moment.

"True," said Pickup.

"I was explaining the Michelson-Morley experiment," said Outa-Data.

"Mr Outa-Data, your interest in facts is extensive, but you are very insensitive with people," interrupted Cleverly Blusher, the Boobyprize’s doctor. "First I must give this being a thorough medical examination." She waved an instrument vaguely in the Time Traveller’s direction. "He’s fine. Carry on."

Outa-Data collected his thoughts. "Michelson and Morley came to the conclusion that either there is no ether at all, or the Earth is not moving relative to it which is not very credible "

"Why not?" said the Cat. "Maybe they like each other’s company."

" or that there is something rather strange about light."

"There is. Look at what it’s done to this necktie! Faded!"

"Cat, why don’t you go back to Red Frawd where you belong?" snarled Woof. "And take that smarmy Harold Strimmer with you."

"Now, now, Mr. Woof," Pickup remonstrated. "It is our duty as citizens of the Federation to offer aid to any distressed lifeform. Even one as distressed as Harold Strimmer."

"A physicist," Outa-Data bravely continued,"called Albert Einstein, is generally credited with the theory called Special Relativity, as I said that there is something rather strange about light. He published it in 1905. But a lot of other people among them the physicist Hendrik Lorentz and the mathematician Henri Poincar were working on the same idea, because it was widely recognised that Maxwell’s equations for electromagnetism were not entirely consistent with Newtonian mechanics."

"The problem was one of ‘moving frames of reference’, as I recall," said Renault Pickup. He smiled knowingly at the Tritovoglian Ambassador, which bobbed its tiny snout in agreement.

"You mean ‘How do the equations change when the observer is moving?’ " amplified Mr. Pox, whose button nose and shaggy ears revealed that he came from the planet Vulgaria. In fact his father was from Vulgaria and his mother from Wapping, so Pox was only half-Vulgar.

"There are formulas that answer this question," said Outa-Data. "They are known as coordinate transformations."

"Yes," interrupted Pox. "In Newtonian mechanics "

"Pox, this is my series," said Outa-Data. "Kindly do not interrupt unless the script calls for two irritating know-alls. In Newtonian mechanics, for example, velocities measured by (or relative to) a moving observer change by subtracting the motion of the observer. But Newtonian transformations mess up Maxwell’s equations. The answer is to use different formulas, called Lorentz transformations. They keep the speed of light constant, but have spin-off effects on space, time, and mass. Objects shrink as they approach the speed of light, time slows down to a crawl, and mass becomes infinite."

"It is difficult to credit such a strange tale," said the Time Traveller.

"You arrive in the middle of this building in what you claim is a Time Machine, and you say that he’s telling an incredible story?" queried The Cat. Woof howled with mirth. This Cat was OK, maybe, after all.

"Well, when I started out this starship didn’t exist," said the Time Traveller. "Despite which, I am here."

"Yes. And so is Special Relativity. Now, I admit it is not easy to think about this kind of thing using just the formulas, and the idea did not really become popular or ‘take off’ as I believe you humans say until 1908. In that year the mathematician Hermann Minkowski provided a good geometric model for relativity a simple way to visualise it now called Minkowski (or flat) spacetime.

"Precisely because relativity is about the non-relative behaviour of light, everything in it depends heavily upon which ‘frame of reference’ used by an observer. Moving and static observers see the same events in different ways."

"That I understand. The Time Machine works on just such a principle."

"Mathematically, a frame of reference is a coordinate system. Newtonian physics provides space with three fixed coordinates (x,y,z). The structure of space was thought to be independent of time, and it was not traditional to represent time as a coordinate at all. Minkowski introduced time as an explicit extra coordinate. We can draw two-dimensional Minkowski spacetime as a plane (Fig.2a). The horizontal coordinate, x, determines a particle’s position in space; the vertical coordinate, t, determines its position in time."

fig 2

Figure 2 Minkowski spacetime.

"But that is what I told you!" the Time Traveller said excitedly. "Time is just a fourth dimension!"

"Yes, but there is an extra wrinkle that you will not have realised. Let me continue. In full-blooded Minkowski spacetime x is three-dimensional; but for convenience let us pretend it is one-dimensional. Later on I may also have to represent space as being twoÑdimensional. The problem is that four dimensions of spacetime do not fit conveniently on a two-dimensional screen, so a lot of the mathematics involves tricks for cutting down the number of dimensions of space. The simplest trick is to ignore a few dimensions.

"As the particle moves, it traces out a curve in space-time called its world-line. If the velocity is constant, then the world-line is straight, and its slope depends on the speed. Particles that move very slowly cover a small amount of space in a lot of time, so their world-lines are close to the vertical; particles that move very fast cover a lot of space in very little time, so their world-lines are nearly horizontal. In between, at an angle of 45°, are the world-lines of particles that cover a given amount of space in the same amount of time measured in the right units. Those units are chosen to correspond via the speed of light say years for time and light-years for space. What covers one light-year of space in one year of time?"

"Um light?" the Cat hazarded.

"Exactly." The Cat preened himself. "So 45° world-lines correspond to particles of light light rays or photons or anything else that can move at the same speed."

"Particles of light?" asked the Time Traveller.

"Look, just accept it as an image, OK? Think of light rays, if it makes you feel more comfortable," said Kert, jiggling his garotte.

"As you wish. My head is starting to ache."

"That’s not where I’m aching," said the Cat. "This guy is a real pain in the "

"But "

"You gottit."

"But, relativity forbids bodies to move faster than light. The mathematical reason is that their lengths would become imaginary involving the number i=ÿ-1 as would masses and the local passage of time. So the world-line of a real particle can never slope more than 45° away from the vertical. Such a world-line is called a timelike curve (Fig.2b). Any event point in space-time has associated with it a light cone, formed by the two diagonal lines at 45° inclinations that pass through it."

"Why is it called a cone," asked Counsellor Sweetly Coi, "when it looks like a triangle?" Coi was half Zetaboid and had the amazing power to intuit answers to questions just after asking them. "Ohhhh, it’s called a cone because when space has two dimensions, the corresponding surface really is a (double) cone."

"Indeed, Counsellor. The forward region contains the future of the event, all the points in space-time that it could possibly influence; the backward region is its past, the events that could possibly influence it. Everything else is forbidden territory, elsewheres and elsewhens that have no possible causal connections with the chosen event.

"Now, Pythagoras "

"Whose ass? Is it as cute as mine?"

"Pythagoras’s Theorem tells us that in ordinary space, the distance between two points with coordinates (x,y,z) and (X,Y,Z) is the square root of the quantity

(x-X)2 + (y-Y)2 + (z-Z)2.

In Special Relativity, there is an analogous quantity "

"Called the interval between events (x,t) and (X,T); it is "

"Pox! Shut up! It is given by the formula

(x-X)2 - (t-T)2.

Note the minus sign: time is special. That is where H.G.Wells went wrong. Time is another dimension, but it is not like the spatial dimensions. Though it can get mixed up with them, to some extent, as I will explain in a moment. The main point is that along the lines of 45° slope where (x-X)2=(t-T)2, the interval is zero. Those 45° lines are called null curves."

"I see that. But what does this ‘interval’ represent?"

"The interval is related to the apparent rate of passage of time for a moving observer. The faster an object moves, the slower time on it appears to pass. This effect is called time dilation. As you approach a null curve that is, travel closer and closer to the speed of light the passage of time that you experience slows down towards zero. If you could travel at the speed of light, time would be frozen. No time passes on a photon.

"It seems to me that time is somewhat mutable in this theory," said the Time Traveller thoughtfully.

"Not as mutable as it is in this series," said Renault Pickup.

To Be Continued in Episode 3: Rosencrantz and Guildenstern

Ian Stewart

2 Something Strange About Light

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