Holograms at Halloween

I don't suppose that there's any reason why holograms have to be created on a flat sheet.
It's traditional to do it that way, and it probably makes the optics easier, but there doesn't seem to be an especial reason why all of the sheet has to be at the same angle. If you created a hologram on a curved sheet that surrounded an object, then as long as the sheet kept the same shape, it should presumably look as if the object is inside the volume (rather than appearing to be in front of or behind a flat "window"). There's also no obvious reason why you can't produce cheap printed lenticular holograms on curved sheets either, other than that it'd make the initial processing more difficult.

So, Halloween. Once we're set up for manufacturing curved holograms, the obvious application (at this time of year) is the creation of the world's most scary Halloween masks.

Put a hologram of a human skull onto a curved transparent sheet, use the sheet as a visor, fitted inside the cowl of a black cloak, and make the inside smoked or semi-mirrored, and you have a "Death" Halloween costume, where, if anyone gets too close and peers under the cowl though the sheet to try to see who's face is behind the visor, they get a rather nasty shock!

Okay, on reflection, maybe not such a great idea after all. :(
You don't want people dropping dead of heart attacks when they realise that "the death guy" appears to be wandering about with what seems to be a real, genuine, gaping skull on the top of his neck. I mean, realism is all very well, but to be striding around town leaving a trail of traffic accidents and screaming people and dead bodies in your wake would probably be taking authenticity a bit too far. Oh well.

The Villarceau Coil

Sometimes it's fun to try to take the most ludicrously-abstract and pointless geometrical results and to try to turn them into something useful. It's a fun game, and the more abstract the thing is, the higher the chance that nobody's actually brainstormed it properly before you. The "square-cutting" exercise ended up as a possible idea for new storage media for hydrogen-powered cars, so after uploading the "Cutting up Doughnuts" post, I was scratching my head to try to think of some real-world application for the "Villarceau Circle" result, that might turn the pastry-cutting exercise into something with actual physics applications.

The best I could come up with was a variable-geometry magnetic containment device.


If we take our two interlocking Villarceau circles, and delete one of them, we're left with a simple ring that wraps once around the torus limb and its central void. This counts as a special-case toroidal winding. We can interleave a series of these single angled rings around the torus, intersecting, without any of them clashing or colliding. If current is circulated around each ring (perhaps by "breaking" the rings and wiring them in series), you have yourself a rather unusual toroidal coil.

What's unusual about it that it has variable geometry. Each circular ring-segment can be a rigid wound coil, and by tilting the angle of these coils we can create a larger torus with arbitrary proportions (major axis radius fixed, minor axis radius variable). Okay, so there's a limit to how fat or thin we'd be able to go due to the finite thickness of the rings that we're using to construct it, but essentially, we have something that looks like a toroidal accelerator and containment device, that can actually change shape while it's running.
Provided that the "open" configurations of the resulting toroidal coil aren't too open, this might let you prototype a device without having to calculate the ideal proportions beforehand - you'd be able to adjust the torus shape while the device was actually operating.

Now, suppose for the sake of argument that you wanted a containment device that allowed you to open it out, fire high-energy particles into it in low-energy mode, then close the coils, squash the plasma density to encourage some sort of reaction, and then open the coils again to allow the reaction products to spill out into the surrounding coolant. You could have a system that "breathes", and holds different shapes for different parts of its cycle.

Okay, I'm trying not to be too glib here – because nuclear physics is NOT my specialist field – but this thing would look awfully like a cross between the "cage fusor devices" and the "tokamak" configurations that people use for nuclear fusion. When it's closed you have something that looks like a tokamak, and when it's open you have something that looks (superficially) more like a fusor cage. One of the annoyances of the tokamak designs is that once you've built them, they're usually locked into particular configuration – with a Villarceau coil, the variable geometry means that you should be able to get some pretty significant changes in internal volume and field strength without having to vary the current flow to the coils. And if the internal pressure gets too great, the thing's going to have a tendency to self-adjust by opening out like a flower-bud, reducing internal pressure and temperature, and releasing excess plasma into the surrounding coolant in a semi-controlled way (rather than being all bottled up until things go more badly wrong).

Anyhow ... bottom line is, that even if this configuration is no damned use at all for conventional nuclear fusion, it'd still look damned cool as a piece of hardware.

Designers and art directors for science fiction movies take note. Remember how cool people though the Big Scary Spinny Machine was in Contact (1997)? Well, this configuration would be a really nice thing to use next time you have to design a cool fictional device for a spaceship reactor or engine pod. Shiny silver interlocking steel circles that tilt and swivel, with a whizzy blue plasma glow inside. Mmmm.

I want to see this cool thing in a movie NOW ! :) Who's going to be first?

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PS: I did spent the last couple of weeks seriously consider building one of these as a toy, sticking it in a small vacuum chamber and whacking a high-tension voltage into it, as a version of those plasma balls that you find in gadget shops. I figured that with that, plus a set of circular coil units, and I might have a cool little device that could spin plasma (or bits of shiny silver paper) in an amusing way. I got as far as looking up coil formers. But sanity prevailed. Plus, I think my current landlord might take a dim view of his tenants trying to build small prototype nuclear fusion reactors on the premises.

Holographic Diamonds

Back in January 2000, the Millennium Dome exhibition opened to what was supposed to be a display of the best of British achievement. Unfortunately the people in charge of setting it up didn't seem to have a clue how to run this sort of exhibition or what to put inside the dome, and it ended up as a bit of a national embarrassment.

One of the last-minute additions to the show was the Millennium Star diamond.

To see the diamond, you had to walk though an angled passageway that was completely pitch black apart from some slightly odd (monochromatic?) blue light, and there, in the middle, you'd see a case walled with bulletproof glass, containing the blue-lit diamond. You walk past it, perhaps pause, and then make your way out. No loitering, no photography.

Something struck me when I was in there. The thickness of the cabinet's glass meant that the diamond appeared be in different places, depending on which pane you viewed it through - that's completely normal, you usually see a similar effect with fishtanks. But the blue light confused me, because normally you only see blue-lit rooms when someone's trying to hide something. Okay, so it was a blue diamond, but still ...
The human eye is pretty bad at seeing sharp details in blue light, which is why Windows has traditionally had a blue-themed startup screen - the old splash screen used crude dithering to recreate the effect of a smooth variation in tone using the default 16-colour VGA pallette, and by doing this in blue, the eye was fooled into not noticing the effect too much. If Windows 3.1 had tried that trick in red or green or yellow, the result would have been bitty and grainy and would have looked awful. In blue, you can't see the fine detail that gives the trick away.

Now, the glass.
Bulletproof glass uses a "sandwich" of alternating toughened glass and shock-absorbing plastic sheets, so that even if you shatter every layer of glass, the shatter-patterns are different, and the pieces stay stuck together by the plastic. If someone had simply added an additional sheet of plastic film with a with a hologram of a diamond ... then how would you be able to tell? You couldn't look for alignment errors between the sheets on different panes, because the diamond woudl appear at differtent positions when viewed through the different panes anyway, due the the thickness of the glass.
Does a holographic diamond appear to refract light in the same way as a real diamond? I don't know, but if someone wanted to look for an "anomalous" spectrum effect that didn't correspond to real diamond, the use of monochromatic blue light might be a good way to stop them. And with single-colour light source, we'd also find it difficult to see any interference fringes due to misregistration of the holographic films. Optical theory says that to see those coloured fringes, the colours already have to be present in the original lightsource, andf in our "blue room", that light wouldn't be there.

Of course, for all this to work, de Beers would have to have their own in-house holography R&D department aligned with their security people, which sounds pretty unlikely. But in fact, deBeers do have very strong links to holographic reseach: They have laser systems for checking diamonds, and for laser-etching holographic security marks onto them, and slightly more peripherally, Lucent have been researching diamond as a potential holographic storage medium. DeBeers also have a holographic diamond passport scheme. So diamonds and security and holography research and lasers and de Beers all have a pretty strong overlap. There probably aren't that many companies that know more about certain sorts of holography than de Beers do.


So here's a fun, harmless little conspiracy theory to ponder that's worthy of Sherlock Holmes or Jonathan Creek: What if this diamond, which thieves tried to steal from the Dome in November 2000 in a ram-raid using a mechanical digger, nailguns and a getaway speedboat, was protected by the ultimate "stage magic-based" security system? What if the diamond, that perhaps many thousands of people would swear on oath to having seen in person ...
... was never actually there?

Cosmological Hawking Radiation, and the failure of Einstein's General Theory

Cosmological horizons are rather arbitrary. The cosmological limit to direct observation is at different places for different observers, and if you change position, your horizon position changes to match. In that respect, a cosmological horizon is a little bit like a planetary horizon - it's different for everyone, and every physical location can be considered as being at a horizon boundary for someone.

With a cosmological horizon, we can mark out a region of space that we reckon should be directly visible, and another region beyond that shouldn't be, and try to draw a dividing line between the two that represents the horizon. The unseen region doesn't exist in an observerspace map even as space, which (in an observerspace projection) seems to fizzle out and come to a stop at the horizon limit.
As we try to look at regions further and further away, we're seeing larger and larger cosmological redshifts, and seeing further and further back in time, until we approach a theoretical limit where the redshift is total, time doesn't appear to have moved on at all since the Big Bang, and events apparently frozen into the horizon correspond to those in the vicinity of Time Zero.
In an idealised model, trying to see any further away than this means that we'd be expecting to be seeing spacetime events that originated before the Big Bang, which – in our usual models – don't exist. So the cosmological horizon is the rough analogue of a censoring surface surrounding a notional black hole singularity under general relativity. It kinda ties into the cosmic censorship hypothesis that, if any physical singularities do exist anywhere in Nature, Nature will always make physics work nicely and politely helpfully hiding the nasty singularities from view.

HOWEVER ... with a cosmological horizon, there are logical arguments that insist that we can receive signals though it.

Suppose that we have two star systems, A and B, whose spatial positions are on different sides of our drawn cosmological horizon, a couple of hundred lightyears away from each other. Let's say that B's the closer star to us – 100 ly inside our nominal horizon – and A's 100 ly outside. In an observerspace projection, we'll eventually be able to see the formation of the nearer star B (if we wait a few bazillion years) but A is off-limits.

But the nearer star B is quite capable of seeing events generated by A, and then helpfully relaying their information on to us. If A goes supernova, we should (eventually) be able to see a cloud of gas near B being illuminated by the flash. B can pass A's signals on, just as an observer at a planetary horizon can see things beyond our horizon and describe them to us, or hold up a carefully-angled mirror to let us see for ourselves.

So technically, Star A, under QM definitions, is a virtual object. It doesn't exist for us according to direct observation, but it's real for nearby observers and we can see the secondary result of those observations. B radiates indirectly through the horizon, so not only does the supposed Big Bang singularity have a masking horizon, the horizon emits Hawking radiation. If we'd bee a bit brighter back in the 1950's, we'd have been able to predict Hawking radiation by taking the "cosmological horizon" case and generalising over to the gravitational case. What stopped us from doing this was an incompatibility with the way that GR1915 was constructed.

The cosmological horizon is an acoustic horizon. It fluctuates and jumps about in response to events both in front of it and behind it. If someone near star A lobs a baseball at star B, we'll eventually see that baseball appear, apparently from nowhere, as a Hawking radiation event. And depending on how close the thrower is to the horizon, and how hard they throw the ball, we might even get a glimpse of their shoulder, as the physical acceleration of their arm warps spacetime (accelerative gravitomagnetism, Einstein 1921) making the nominal horizon position jump backwards.

For this sort of acoustic horizon to work, the acceleration and velocity of an object has to affect local optics (if the ball had been thrown in the opposite direction, we'd never have seen it).
If the local physics at a cosmological horizon generates an acoustic horizon, then that physics is going to correspond to that of an acoustic metric. NOT a static Minkowski metric. The presence, velocity and acceleration of objects must change the local signal-carrying properties of a region. Since the operating characteristics of an acoustic metric are different to those of the Minkowski metric that defines the relationships of special relativity, the local physics then has to operate according to a different set of laws to those of special relativity – the velocity-dependent geometry of an acoustic metric makes the basic equations of motion come out differently. For cosmological horizons to work as we expect, the local light-geometry for a patch of horizon has to be something other than simple SR flat spacetime, and the local physics has to obey a different set of rules to those of special relativity.

Now, the punchline: Since our own region of spacetime will in turn lie on the horizon of some distant far-future observer, this means that if we buy into the previous arguments, our own local "baseball physics", here on Earth, shouldn't be that of special relativity either.


The good news is that if we eliminate special relativity from GR, to force cosmological horizons to make sense, GR's predictions for gravitational horizons would also change. The revised general theory would predict indirect radiation effects through gravitational horizons, bringing the theory in line with quantum mechanics. Which would be a Good Thing, because we've been trying to solve THAT problem for most of the last 35 years.

The bad news is that there doesn't seem to be any polite way to do it. Disassembling and reconstructing general relativity to address its major architectural problems involves going back to basics and starting from scratch, questioning every assumption and decision that was made the first time around, and being pretty ruthless about which parts get to stay on in the final theory.

I find this sort of work kinda fun, but apparently I'm in a minority.

Cutting up Doughnuts

A cool thing that I didn't know about doughnuts until someone pointed it out a few months back: no matter what proportions a doughnut has, there's always an angle that you can slice though it to produce a perfect pair of interlocking circles.

Someone mentioned this on sci.math, and pointed to "Doughnut Slicing", a webpage by John Banks and Jeff Brooks, and then, someone else pointed out that there was already a a Wikipedia article on it, under the name "Villarceau Circles" ... at which point I scooted off and tried to work the thing out from scratch, before reading anyone else's "spoilers". There's only a certain number of cool results like this, and if you read other people's work before you've had a crack at a problem yourself, that's an opportunity that you never get back.

Anyhow, it turns out that if a torus has major radius R (distance from central axis to centre of limb) and minor radius r (radius of solid limb), the magic angle A that you have to cut at to get to see the double-circle is simply

SIN A = r/R

Going back and looking at the other two webpages, it seems that, unless I missed it, the authors don't seem to have actually written that down explicitly anywhere (although they do seem to have included some more involved math).

So, one quickie download of GFA BASIC 32 (and some quickie trig) later, and the relationship's obviously right. One quick program run while my tea was cooking, generating a few hundred images of tori with radius ratios from zero to one, tilted by the appropriate angles, and I now have a sequence of pretty Villarceau images sitting on my harddrive that I'll probably string together as a YouTube animation at some point.


If you want to cut up a doughnut or bagel purchased at your local bakery to see the Villarceau circles, thread a thin stick or skewer all the way through through the central hole, and then tilt it to a maximum so that your pointy-stick is touching two different parts of the surface. That line gives you the plane that you need to cut along, and the two points where your stick touches the doughnut are the two points where the pair of circles intersect.

Gulliver's Travels, Isaac Newton, and Flying Saucers

Jonathan Swift (1667-1745) anonymously published his four-part novel, "Gulliver's Travels" in 1726, at the end of a visit to London.

Most people know it for the chapters set in Liliput (where Gulliver is a giant compared to the natives), and maybe also Brobdingnag (where the natives are giants, and it's Gulliver who's considered tiny). It's a scathing social and political farce, where Gulliver's visits to other societies show different systems of government and different social orders. While in Liliput, Gulliver is considered a dangerous giant, and treats the tiny locals with callous indifference. In Brobdingnag it's Gulliver who's overlooked and considered unimportant, so the extent that he's caged and treated as a pet.

But there's also a chapter (at the end) where he visits the Houyhnhnms, a race of talking horses that Gulliver considers entirely superior to humans, who regard the local ape-decended species (the "Yahoos") as loud, primitive, warlike and violent. After living with the Houyhnhnms, Gulliver comes to see all humans as Yahoos.

And for the science fiction fans, there's a chapter about a giant flying saucer.

Really, there is. The third section of the story has Gulliver being rescued by a scientifically advanced society, based on the flying island of Laputa. The city is built on a four-and-a-half-mile-wide concave circular plate topped by buildings, along with four lakes for collecting rainwater, surrounding an astronomical observatory built into a central shaft, that also includes the levitating mechanism. It is, quite literally, a "castle in the air" inhabited by scientists.

Laputa rules over a kingdom (Balnibari), whose borders are defined by the limits of a naturally-occurring geological magnetic anomaly, and the flying city is held aloft by a giant tiltable magnet, held in place by unbreakable "adamant" cage that is of a single piece with the city's baseplate. The city rises and falls and gets sideways propulsion by adjusting the alignment of the magnet.
The flying city is a local scientific superpower, and the king's response to rogue cities below is to steer the saucer above the rebel stronghold and set it down, crushing them.

Unfortunately, I'm not aware of any illustrated editions of "Gulliver's Travels" where the illustrators tackled Laputa. Perhaps the idea was just too freaky for them. If they had, they'd have probably ended up drawing something that looked like the mothership in Spielberg's 1977 "Close Encounters" film.

The other notable thing about Swift's flying city of Laputa is that although it is ordered along entirely scientific principles, its (highly quotable) math-and-music obsessed inhabitants at the Academy of Lagado are buffoons, working on crazy and expensive projects such as the extraction of sunlight from cucumbers, constantly begging for more money for their projects as the society below them decays – it's a fairly small step to suspect that Swift was taking the mickey out of the esteemed Royal Society (then headed by Isaac Newton), and it's even been claimed that Swift emphasised this by basing all of the Lagado projects on specific Royal Society papers.

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This raises an intriguing question: did Swift actually meet Newton?
It seems that when Swift had been in London in 1710, he'd been visiting a woman called Catherine Barton. Barton was Newton's half-niece, and one of the few people that Newton was close to. Barton wasn't just some peripheral nominal relative of Newton, she'd actually moved to London and moved in with Newton in about 1696 (about the time he got his job at the Mint), and kept house for him.

If Catherine Barton was living with Isaac Newton and being visited by Jonathan Swift, then Isaac Newton would have cast a rather large shadow over Swift's consciousness, even if he hadn't been /the/ Isaac Newton.

And if that wasn't enough, there was also the subject of Money.
Immediately before "Gulliver's Travels", Swift's celebrity was based on his having anonymously written and published the Drapier's Letters in ~1724-25, a series of pamphlets railing against the coining of copper currency for Ireland, which led to a widespread boycott of the new coins in Ireland and their withdrawal. One of Swift's (many) objections was an allegation that the coins were of poor quality - Newton, as Master of the Mint since the mid-1690's, had to get involved and do an assay, and reported that the allegation wasn't true.
Newton was known for his tetchiness, but Swift in particular had a reputation for being gratuitously and grossly offensive. I've got an old C19th copy of "Gulliver's Travels" that describes Swift as having "more than any other man who ever wrote in English, a liking for saying nasty things", and blames this for Swift's repeated ruination of his own career prospects. Apparently Swift wanted to be a bishop, but even as a returning hero of the Irish people, when the people in charge actually met him, it became clear that this wasn't going to happen. That edition of "Gulliver's" mentions "the deadly agitations of his private life" as being something that the C19th reader might want to enquire about in later life – but whatever this unmentionable personal train wreck was, it doesn't seem to have made it as far as his Wikipedia page.

So perhaps the two wouldn't have wanted to meet each other, especially since they both cared about the same woman. Having the the brittle, acidic, reserved Newton in the same room as the extrovert, scandalous, offensive Swift might not have been a good idea, and the fact that they both had strong ideas about currency would probably just have made things worse.

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The young, exiled Francois Marie Arouet ("Voltaire") was also in London around this time, and seems to have been rather keen on Catherine, too.
Voltaire later went on to write "Micromegas" (1752), a short satire that appears to have been partly inspired by "Gulliver", in which a pair of giant aliens from Sirius and Saturn arrive on Earth and meet up with and ridicule a bunch of tiny Earth philosophers (with the exception of one guy who is a follower of John Locke). That's John Locke, the guy whose writings seem to have influenced the American Declaration of Independence, not John Locke, the character from "Lost" (a TV series about a strange island with a natural magnetic anomaly).

Another link between Voltaire's story and Swift's is that both throw in a little casual detail (known to the fictional Laputans and alien scientists) that the Mars was "known" to have two moons, and it seems natural to assume that Voltaire probably borrowed this detail from Swift. In fact, Mars has got two moons – Phobos and Deimos – but they didn't get discovered for real until 1877. That earned both writers an astronomical "credit": the only two named features on the smaller of the two satellites are a pair of adjacent craters, named "Voltaire" and "Swift".

After he'd given up on the brilliant Catherine and snuck back to France, Voltaire shacked up, long-term, with another brilliant woman obsessed with Newton, Émilie du Châtelet, who as well as being a serious respected researcher in her own right, translated, produced and reworked (with her comments) the French edition of Newton's Principia.

To the C18th coffee-house intelligentsia, a mix of physicists, philosophers and political theorists, this was a time of revolution and restructuring (not to mention a certain amount of fluidity over people's living arrangements). England had recently undergone a rapid turnover of rulers, flip-flopping from Monarchy to Republic, and back to Monarchy again, then Monarchy chosen by Parliament. Cromwell had kicked out Charles I, Charles II and James II had taken over from Cromwell, and the Glorious Revolution had then given Paliament the right to choose the monarch, which brought in William and Mary, and which they then exercised again in the Act of Settlement to shunt the succession to Anne, who'd then died, too. The Acts of Union in 1707 had then finally united England and Scotland as a single kingdom. In the politics of 1726 England there were various entrenched factions with specific ideas about how the country ought to be run, and by whom, but there was no guarantee that any one particular group would obtain ultimate control.

There was a sense that this was where we decided what the future was going to look like. Was it going to be run by royalists or republicans? Theologians or scientists? Committees or street campaigners? "Gulliver's Travels" tapped into an appetite for exploring possibilities, and showing how different systems failed. Voltaire's later story got a charge out of lampooning philosophers because at the time, philosophy was reckoned to matter. These guys were potentially the architects of the new society.

"Gulliver" can be seen as parody of how people brought up in different political and philosophical systems can believe that their own way of seeing the world and the correct order of things is right and proper, even when outsiders can see that it's ludicrous, and was, in a real sense, revolutionary. Together with a surrounding body of other philosophical and campaign literature, it helped to set up the context for debate that made the French Revolution and the American War of Independence seem possible to the people who risked their lives to make those things happen.

It's not just a kid's story about a shipwrecked guy being tied down with string by little tiny people.

The Michael Jackson Continuum

Here's what happens if you take three single images of Michael Jackson - one as a kid (probably some time around "Rockin' Robin"), one during his "Thriller" period, and one when he was doing odd stuff on balconies – turn each one into a FaceGen head, and then use the program's ageing and tweening settings to generate a set of intermediate heads.

The three original heads are top left, centre, and bottom right. The rest are tweened and age-tweeked extrapolations, courtesy of FaceGen.

You should be able to get much better results than this with a more representative set of photos. A more useful source picture taken between the "Jackson Five" years and "Thriller", would have been handy, but the otherwise-usable ones that I turned up on the net all seemed to be in monochrome. :(

"Bottom left" is what FaceGen extrapolates for Michael Jackson as a fifty-something-yearold with no plastic surgery, top right runs the process in reverse, working backwards from the last picture. The rest are intermediates.

Water on the Moon

In today's Times, there's a front page story saying the the Indian Chandrayaan-1 probe, carrying NASA's Moon Mineralogy Mapper has now found signs of what might be significant amounts of (presumably frozen) water on our Moon.

For anyone who wants bullet points to explain why this is potentially a game-changer, here they are:

• Water + electricity = life support
  Humans need water and air to survive (along with temperature control). With enough solar cells, the Moon's not short of electrical power – no pesky atmosphere to get in the way – but water and air are biggies. If the water's already there, we can tick one box, and using electricity to electrolyse water gives us hydrogen and oxygen. Oxygen lets us tick the second box. Normally we breath atmospheric-pressure air, with 20% oxygen and nearly 80% nitrogen, but we can use pure oxygen at a lower pressure, if we can deal with the additional fire risk associated with pure O2. It'd be nice to have a decent local supply of nitrogen, too, but not strictly necessary.
  
  
• Water + heat + rock = building materials?
  Use solar furnaces to roast moondust, or break moonrock into pulverised dust and drive off the more volatile elements, then add water ... and we might just have ourselves a form of locally-sourced readymix concrete.
  
  You know how in films where moonbases are often all shiny white metal? To start with, they'd probably look more like adobe mud huts, or holes in the ground, with all the shiny stuff on the inside (apart from the solar panels). What you'd ideally want is big thick walls at least ten or so feet thick, on all sides, to buffer the temperature changes and block some of the radiation when the sun does annoying things with solar flares. Perhaps you'd want to maximise your protection from flare radiation without tunnelling, by by building in the bottom of a deep crater, near one of the poles ... which is also where we're hoping that some of surviving "accessible" ice might be found.
  
  Our building materials don't have to be incredibly strong, or even airtight, we could build a crude hollow blocky mesa as our surface structure and inflate a pressurised mylar balloon inside or below for living quarters. But it'd be nice to be able to pour a bit of concrete around the balloon to minimise accidents, and it'd be handy to turn moondust into something more manageable. Other than that, we're stuck trying to stack up rocks and fill sandbags with dust. In a vacuum. Not good. Quite how you're supposed to work with concrete in a vacuum without the water immediately boiling off, I don't know, but I'm sure that some clever concrete technologists are working on it. Supercooling, perhaps?
  
  One problem with building at the bottom of a polar crater is that having a few kilometres of rock in a straight line between you and the Sun isn't so good for solar power. So you'd probably want an array of thin foil mirrors around set up around part of the crater rim, redirecting and focusing concentrated sunlight down onto your generators. Luckily, your mirrors can be ultra-lightweight, there's no weather to damage them, and no intervening air to soak up the transmitted energy. Using reflectors minimises the amount of heavy power cabling, and also the number of solar generators, and depending on the shape of the ice formation that you're trying to exploit, an aimable solar furnace might also be handy for mining.
  
  
• Hydrogen + Oxygen = rocket fuel
  Hydrogen and oxygen burn rather well together to turn back into water, giving a nice roaring flame. That's the reaction that drives the shuttle's main engines. Given a solar farm and enough time, it'd be nice to have a local fuel production plant on the Moon, making rocket fuel simply from local materials. We'd probably need a robotic refueller to pick up H2 + O2 from the plant, fly back to Earth orbit, find the satellite and fill up its tanks (or swap a standardised empty satellite launch tank with a nice pre-refilled one).
  
  
• H2 + O2 + fuel cell = mobile power
  Fuel cells have a capacity that's only limited by the amount of hydrogen and oxygen you have to feed them. If you're building a water-splitting plant anyway, you might want to send along a spare set of empty fuel cells.
  
  
• Water+ electricity + rock + atmosphere = food
  Sure, we can set up a hydroponics lab to grow our own veggies in space, recycle biomass, and use the plants help remove CO2 and other nasties from the air ... and in theory we can get pretty damned close to a sealed self-perpetuating system. But in practice, you need topups, and safety margins, and an awful lot of water to get the thing started (as the name "hydroponics" kinda suggests). If you're going to be growing algae or fungus or plants to eat, there's a lot of water locked up in the system while they're going through their cycle. Industrial biological reactors usually need whole tanks of the stuff, and water's actually pretty heavy. If water's costing you thousands of dollars per kilo to ship from Earth, it's not cheap stuff. It's probably not quite as expensive as gold, but with current shuttle per-kilo launch costs, it's in the ball-park.
  

With water, the moon becomes a solar-powered robotically-constructed and remotely-operated gas station and hydroponics plant, remote-controllable from the Earth, with a mild gravity penalty. It can have its own fleet of little refuelling craft, powered by locally-produced lunar rocket fuel.

Without water, its just a big chunk of rock with some handy boulders to hide behind when there's a bad solar storm.

Anyone whose job involves thinking a decade or two ahead about future lunar, manned or deep space payload missions will be watching this story very carefully.

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see also: Ice Splat on Mars

Black Holes, Coordinate Reversals, and r=3M

Coordinate projections sometimes have a habit of going wierd when you try to project them past a gravitational horizon. Sometimes you can do it, sometimes you can't, and sometimes the attempt turns various things inside out.
A cool physical inversion that happens outside the horizon was used as the March 1993 cover story for Scientific American: Black Holes and the Centrifugal Force Paradox (by Marek Artur Abramowicz).

The effect isn't really paradoxical, but it's counter-intuitive until you think it through. Normally, if you orbit a body, you can break free of that body by firing up your spaceship's engines and going faster – too fast to be able to orbit at your current distance.
What the BHCFP says is that if you're skimming too close to a black hole event horizon, and you fire up your engines, then the faster you try to circle, the more that your trajectory is deflected inwards, towards the hole. The centrifugal forces that would normally throw you away from the body, now seem to be inverted, pointing inwards rather than outwards.

The critical threshold beyond which this effect appears is the distance r=3M, exactly one-and-a-half times the radius of the horizon surface (which is at r=2M).

It turns out that the r=3M radius is the photon orbit. It's the critical distance at which light aimed at 90 degrees to the mass will be deflected enough by gravity to perform a complete orbit and end up at its starting-point. The SciAm article has some nice computer graphics showing what a circular self-supporting scaffolding tube constructed around the hole at r=3M would look like to an observer standing inside it ... it'd appear to be straight, and if the observer pulled out a telescope and looked far enough along the tube, they'd expect to see the back of their own head.

So r=3M is special. From the perspective of the observer at r=3M who's hovering with the aid of rocket engines, or standing in our circular tube up above the hole, the universe seems to be divided into two regions. On one side they see the black hole and its immediate surroundings, and on the other, they see the starfield that represents the outside universe. Topologically, both regions can be thought of as solid spheres, with their external parallel surfaces meeting at r=3M. Both regions are trying to impose their will on the observer's local geometry, but at r=3M, a stationary observer feels the geometrical competition between the effect of the two spheres as being in balance (although in order to maintain their position hovering above the hole, they're feeling rather a strong gravitational pull!). Spin either one of the two spheres, and the observer will be pulled towards it – spin both at exactly the same rotational rate – the effect that we'd see if we passed along the tube at high speed – and the radial gravitomagnetic effects of both spheres cancel.

So if you built an electric train to run around the interior of the tube, it'd feel the black hole's conventional gravitational attraction pulling it against one side of the tube ... but that pull would seem to be exactly the same no matter how quickly it circled the hole.

The author's moral is that if you're in a spaceship close to a black hole, and you want to escape, don't just throttle up your engines, actually point your ship away from the damned thing, or you're liable to get a nasty crashy surprise.

"Observerspace" Description:

When we think about the optics of the situation, though, perhaps the hypothetical spaceship captain wouldn't be all that surprised:

See, if we imagine standing on a suspended non-orbiting platform at r=3m, we find ourselves looking along the r=3M surface in any (perpendicular) direction. The surface appears to us to be a flat plane cutting through our location. And because our view along r=3M circles around the hole indefinitely, our view along this apparent plane repeats indefinitely, too – the plane appears extend indefinitely far in all directions, showing us older and older views of the surface at greater distances, right back to the time that the black hole originally formed. So logically, anything that we see to one side of the plane corresponds to the interior of the r=3M sphere, and everything we see to the other corresponds to the contents of the "rest-of-the-universe" sphere.
The outside universe only seems to exist on one side of this plane. On the other, gravitational lensing effects make the black hole's r=2M surface beneath us appear to be opened out into a second indefinitely-repeating surface, at some distance below the 3M plane.

Once we're at the 3M surface, there are two ways that we can go.
If we slowly winch ourselves upwards away from the hole, then we see the flat 3M boundary of the outside universe curving itself back into a more normal-looking inward-facing enclosing sphere. But if we allow ourselves to be lowered further towards the black hole, to less than r=3M, then the 3M surface continues to distort past being a flat plane, to becoming a concave surface that curves above us, away from the hole. Instead of the universe surrounding the black hole, it now seems to us that the black hole (and the r=3M surface) is surrounding the universe!
The region that we know ought to be just above the 2M surface appears visually to us to be part of a concave shell, apparently wrapped around a ball representing the remaining universe. The abstract, "topological" idea that our location can affect the choice of which sphere is "really" on the inside or outside now appears to us, visually, to be concrete reality!

The further we descend (slowly) towards 2M, the more pronounced the effect becomes, the more sharply the 2M surface appears to be curved around us, and the more that the outside starfield above appears to shrink to something that looks like a little bright ball suspended somewhere above, in the enveloping black-holey gloom directly above us, like a tiny planet or star.

So if we're hovering too close to r=2M, (or flying past in a spaceship) we shouldn't really be surprised if increasing our forward speed results in our colliding with part of the hole, because that's exactly what our forward view tells us is directly in front of us (and on every side, and directly behind us). If we want to escape from the hole's influence and get back to normal space, then we have to aim our spaceship at the little shrunken blob of compacted blueshifted starfield directly above us. All other directions point at the black hole.

So the rule-of-thumb for navigating within r=3M would seem to be: forget about your ship's fancy gyroscopic navigation systems, just look out of your window and make sure that the ship's nose appears to be pointed approximately at the part of the universe that you want to go to. But don't take your eyes off the forward view, because the harder your engines fire on your way out, the the stronger those distortion effects are going to become.

My Chocolate Tree is Unhappy

I keep chocolate trees. They're not too difficult to grow (if you set up an incubator), but keeping the things alive as houseplants without a controlled environment can be tricky. They generally do okay until you have One Bad Day with light levels that are too bright, or too dim, or the humidity's too low, or the temperature is too hot or too cold, and the things panic and drop all their leaves and turn into ugly bare sticks. And when that happens, it seems to take about eight months to coax the things into producing more proper leaves, and get back into the swing of things. Maybe it's a way of outliving predators - if any beasties have eaten the last set of leaves, the tree waits until they and their offspring have all starved to death before growing any more. Dunno.

I had two gorgeous bushy indoor trees last year, sitting by the back window, and moved them to the front of the house where the light levels were slightly lower. One day later, all the leaves had gone sickly. A day or so later they all fell off. A couple of earlier trees got trashed by a few hours of unusually harsh UV light on one clear winter's morning.

After a number of house-moves, I'm now down to just one small tree, which is only about a year old. It had a nice cluster of healthy dark-green leaves. But after just one hour's car journey (on a fairly hot day), the thing had virtually turned albino. The leaves went almost white, apart from the veins, and it's been struggling ever since. Once a leaf loses its "green", it's one short step away from dying completely, and going brown and falling off, and when all the leaves fall off, you're in trouble.

So what I have to do now is coddle the thing so that the existing leaves hopefully last until the plant has decided to try cautiously growing some new ones.
Maybe I should switch to growing something less challenging. If I used a set of mirrors to catch and redirect daylight around the room, indoor climbing roses would be nice ...

Dark Stars and Hawking Radiation

Some people have trouble getting used to the idea of Hawking radiation outside the context of strict quantum mechanics. For those people, I'd suggest that they consider the mechanics of a crusty old Nineteenth-Century “Dark Star” model.

The Dark Star was the predecessor to the modern black hole, and the basic properties of the object were worked up and published by John Michell back in 1784. Michell worked out many of the “modern” Twentieth-Century black hole properties from Newtonian principles, including the r=2M event horizon radius, gravitational spectral shifts, and a method of calculating the number of these “invisible” gravitationally-cloaked objects by finding the proportion of unseen “companion stars” in binary star systems, and then using statistics to extrapolate that proportion to the larger stellar population.

The main difference between an old “dark star” and John Archibald Wheeler's 1950's-era “black hole” was that dark stars could emit faint traces of indirect radiation. In theory, signals and particles could still migrate upstream out of the dark star's gravitational trap by using local objects as accelerational stepping-stones, whereas under GR1915, this mechanism couldn't exist – objects smaller than their r=2M event horizon radius weren't just incredibly dark, but totally black. Their signals and radiation-pressure signature weren't just absurdly faint, but entirely missing. The thing really was, as Wheeler memorably described it, a truly black "hole" in the surrounding landscape.


From the perspective of the Twenty-First Century, we can describe the difference in another way: dark stars emit classical Hawking radiation and GR1915 black holes don't.
Some people will take issue with that statement. They'll say that a hypothetical dark star's radiation-pattern is about acceleration effects rather than QM, and that Hawking radiation is all about particle-pair-production, a completely different mechanism.

So here's the sanity-check exercise. Suppose that the GR1915 description of horizon behaviour was wrong, and that a more "dark-starry" description was right … but that we still believed in GR1915. More general approaches (like statistical mechanics) would have to insist that the radiation effect was real, even though GR1915 disagreed. So how would we explain the reappearance of our naughty radiation effect?

There are number of stages we'd have to go through:

1. In a thought-experiment, catch an escaped particle and measure its trajectory.
2. Extrapolate that trajectory back to the originating body as a smooth ballistic trajectory. In our "dark star" scenario, this extrapolated trajectory is wrong – the particle only escaped by being "bumped" out of the gravitational pit by interactions with other bodies or radiation – but in our GR1915 description there's no self-supporting atmosphere outside the black hole to allow this sort of acceleration mechanism, so we have to (wrongly) assume an unaccelerated path.
3. Notice that the earliest part of this (fictional!) escape-path is superluminal. In order to escape along a ballistic trajectory, a particle would have to have started out travelling at more than the speed of light (!).
4. Apply coordinate systems. Using a distant stationary observer's coordinates, we break the fictitious trajectory into two parts, an initial superluminal section, and the later, legal, sub-lightspeed part of the calculated path. The first section appears to be off-limits in our coordinate system, and an orderly transition between the two, as the particle supposedly jumps down through the lightspeed barrier seems impossible, but …
5. … then we then notice that in a very idealised description of a superluminally-approaching particle, the particle ends up described as time-reversed ("tachyonic" behaviour). If an (over-idealised) particle approaches at more than the speed of its own light (which shouldn't normally happen, but ...), we'd end up describing it as being seen to arrive before it was seen to set out. Our artificial coordinate system approach then describes the particle as being seen to originate at the nearest part of its path, and to be apparently moving away from us at sub-light speeds, as its earlier signals eventually arrive at our location in reverse order.
6. Time-reversal counts as a reversal of one dimension, which flips a left-handed object into its right-handed twin, and vice versa (chiral reversal). So if our particle was an electron, this artificial approach would describe the earlier part of its supposed path as belonging to a positron, instead.
   
7. Our final description would then say that a particle and its antiparticle both appeared to pop into existence together outside the horizon (from nowhere) and moved in opposite directions, with the "matter" particle escaping and being captured by our detector, and its "antimatter" twin moving towards the black hole to be swallowed.
   

And this is, essentially, the 1970's QM description of Hawking radiation.

The Moon, considered as a Flat Disc

Mathematics doesn't always translate directly to physics.
That statement might sound odd to a mathematician, but consider this: even if you believe that physics is nothing but mathematics, that makes physics a subset of mathematics ... which means that there'll be other mathematics that lies outside that subset, that doesn't correspond cleanly to real-world physical theory. The key (for a physicist) is to know which is which.

That's not to say that "beauty equals truth" isn't a good working assumption in mathematical physics – it is – the problem is that the aesthetics of the two subjects are different, and mathematical beauty doesn't necessarily correspond well to physical truth. The physicist's concept of beauty is often different to that of the mathematician.

The "beauty equals truth" idea is often used as an argument for special relativity. SR uses the Lorentz relationships, and to a mathematician, it can sometimes seem that these are such beautiful equations that a system of physics that incorporates them has to be correct.

But the Lorentz relationships can also appear in bad theories, as a consequence of rotten initial starting assumptions:
Our Moon is tidally locked to the rotation of the Earth, so that it always shows the same face to us, and we always see the same circular image, with the same mappable features. Now suppose that a 1600's mathematician has a funny turn and decides that it's so outrageously statistically improbable that the moon would just coincidentally just happen to have an orbit that results in it presenting the same face to us at all times, that something else is going on. Our hypothetical "crazy mathematician" might decide that since we always see the same disc-image of the Moon, that perhaps, (mis)applying Occam's Razor, it really IS a flat disc.

Our mathematician could start examining the features on the Moon's surface, and discover a trend whereby circular craters appear progressively more squashed towards the disc's perimeter. We'd say that this shows that we're looking at one half of a sphere, but our mathematician could analyse the shapes and come up with another explanation. It turns out that, in "disc-world" the distortion corresponds to an apparent radial coordinate-system contraction within the disc surface. For any feature placed at a distance r from the disc centre, where R is the disc radius, this radial contraction comes out as a ratio of 1 : SQRT[1 - rr/RR ] .

In other words, by treating the Moon as a flat disc, we'd have derived the equivalent of the Lorentz factor as a ruler-contraction effect! :)
Our crazy mathematician could then go on and use that Lorentz relationship as the basis of a slew of good results in group theory and so on. They could argue that local physics works the same way at all points on the disc surface, because the disc's inhabitants can't "see" their own contraction, because their own local reference-rulers are contracted, too. Our mathematician could arguably have advanced faster and made better progress by starting with a bad theory! So "bad physics" sometimes generates "good" math, and sometimes the worse the physics is, the prettier the results.

The reason for this is that, sometimes, real physics is a bit ... boring. If we screw physics up, the dancing pattern of recursive error corrections sometimes generates more fascinating structures than the more mundane results that we'd have gotten if we simply got the physics right in the first place.

Sometimes these errors are self-correcting and sometimes they aren't.
If we considered the Earth as flat, then, because it's possible to map a flat surface onto a sphere (the Riemann projection), it'd still be theoretically possible to come up with a complete description of physics that worked correctly in the context of an infinite rescaled Flat Earth. We'd lose the inverse square law for gravity, but we'd gain some truly beautiful results, that would allow, say, a lightbeam aimed parallel to one part of the surface to appear to veer away. We'd end up with a more subtle, more sophisticated concept of gravitation than we'd tend to get using more "sane" approaches, and all of those new insights would have to be correct. In fact, studying flat-Earth gravity might be a good idea! We'd eventually end up deriving a mathematical description that was functionally identical to the physics that we'd get by assuming a sphericial(ish) Earth ... it'd just take us longer. Once our description was sufficiently advanced, the decision whether to treat the Earth as "really" flat or "really" spherical would simply be a matter of convenience.

But with the "moon-disc" exercise, we don't have a 1:1 relationship between the physics and the dataset that we're working with, and as a result, although the moon-disc description gets a number of things exactly right, the model fails when we try to extend it, and we have to start applying additional layers of externally-derived theory to bring things back on track.
For instance, the "disc" description breaks down at (and towards) the Moon's apparent horizon. For the disc, the surface stops at a distance R from the centre, and there's a causal cutoff. Events beyond R can't affect the physics of the disk, because there's no more space for those events to happen in. The horizon represents an apparent causal limit to surface physics. But in real life, if the Moon was a busier place, we'd see things happening in the visible region that were the result of events beyond the horizon, and observers wandering about near our horizon would see things that occur outside our map. So if we were to use statistical mechanics to model Moon activity, and were to say that the event-density and event-pressure have to be uniform (after normalisation) at all parts of the surface, then statistical mechanics would force us to put back the missing trans-horizon signals by giving us "virtual" events whose density increased towards the horizon, and whose mathematical purpose was to restore the original event-density equilibrium. In disc-world, we'd have to say that the near-edge observer sees events in all directions, not because information was passing through (or around) the horizon, but because of the disc-world equivalent of Hawking radiation.

So in the disc description, the telltale sign that we're dealing with a bad model is that it generates over-idealised horizon behaviour that can't describe trans-horizon effects, and which needs an additional layer of statistical theory to make things right again. In the "moon-disc" model, we don't have a default agreement with statistical mechanics, and we have to assume that SM is correct, divide physics artificially into "classical" and "quantum" systems, and retrofit the difference between the two predictions back onto the bad classical model – as a separate QM effect, as the result of particle pair-production somewhere in front of the horizon limit – to explain how information seems to appear "from nowhere" just inside the visible edge of the disc.

Clearly, in the Moon-disc exercise this extreme level of retrofitting ought to tell our hypothetical crazy mathematician that things have gone too far, and suggest that the starting assumption of a flat surface was simply bad ...
... but in our physics, based on the early assumption of flat spacetime, and generating the same basic mathematical patterns, we ran into a version of exactly the same problem: Special relativity avoided the subject of signal transfer across velocity-horizons by arguing that the amount of velocity-space within the horizon was effectively infinite (you could never reach v=c), but when we added gravitational and cosmological layers to the theory, the "incompleteness problem" with SR-based physics showed up again. GR1915 horizons were too sharp and clean, and didn't allow outward flow of information, so to force the physics to obey more general rules, we had to reinvent an observable counterpart to old-fashioned transhorizon radiation as a separate quantum-mechanical effect.

So the result of this sanity-check exercise is a little humbling. We can demonstrate to our hypothetical 1600's "crazy mathematician" that the Moon is NOT flat, no matter how much pretty Lorentz math that generates, and we can use the horizon exercise to show them that their approach is incomplete. By assuming that their model is wrong, we correctly anticipate the corrections that they'd have to make from other theories in order to fix things up. That ability to predict where a theory fails and needs outside help is the mark of a superior system, and shows that the "Flat-Moon" exercise isn't just incomplete, it generates results that are physically wrong, and that don't self-correct. It's faulty physics.

But the same characteristic failure-pattern also shows up in our own system, based on special relativity. So have we made a similar mistake?

On Catching Rainbows

I saw a nice rainbow yesterday.

I was out to do some shopping but took a random detour, following my feet. The detour just happened to take me to a suitable road junction, at exactly the right time. By rights, I shouldn't have been there to take the picture.

But "lucky catches" aren't just about accidentally being in the right place at the right time by nothing but dumb good luck, or about preserving a certain random element in your approach (although that certainly helps). If you want to be able to catch something that other people miss, you have to expect to spend at least some of your time in places where they aren't, and looking at things that don't always seem to be immediately necessary to the job in hand.
You also have to be prepared for the possibility of success (I try to keep a camera with me, and it had just enough juice left in the batteries to fire off a few shots for the critical sixty or seventy seconds), you have to be able to recognise the preliminary signs of something interesting (I saw a faint 'bow forming, realised what was coming, and was able to fish the camera out and find something to shield it from the rain, in time) and you also have to be prepared to look stupid (standing in the rain with a plastic folder over your camera, taking photos of the sky, at an angle where most of the people who can see you have no idea what you're doing).

But the main thing is to have your eyes open. If you're absolutely sure that nothing interesting is going to happen, then on the occasions when it does happen, you're liable to miss it.

The same thing goes for theoretical physics. If you want to catch things that have eluded other people (whether it's math, or theory, or experimental research), you don't always have to be so much smarter than everyone else, or to have better equipment. Sometimes it's enough just to be prepared for the possibility of being surprised. If you're too rigid about what you're trying to find, you miss out. In my case, I was popping out for a plank of wood for some shelving, and I came back with a plank of wood and a bloggable photograph of a rainbow. If I'd been more singleminded in my shopping, I'd have only come back with the bit of wood.

M.C. Escher's "Relativity", Intransitivity, and the Pussycat Dolls

There's a nice example of intransitive geometry in the latest Pussycat Dolls video ("Hush hush").
No, really, there is. It's the bit where the girls are on four staircases attached to the sides of a cube, that each have a different local direction of "down". The "stairwell" section of the video starts at about 58 seconds in and goes on until about a minute thirty. While you're waiting for it to start you'll have to put up with the sight of Nicole Scherzinger nekked in a bathtub making "ooo, yeah" noises for nearly a minute, though. Sometimes doing research for this blog is really tough.

The video seems to be inspired by the famous "Relativity" lithograph by M. C. Escher, which had three intersecting sets of stairs and platforms set into three perpendicular walls, as a piece of "impossible" architecture (physically you could build it, but you wouldn't be able to walk on all the surfaces as the people do in the illustration).Escher's illustration was incredibly influential, and as well as the Pussycat Dolls video (!), there are some more literal tributes online, including Andrew Lipson's recreation of the scene using Lego, part of the 1986 movie Labyrinth, and a funny short video called Relativity 2.0, that has people trapped in a nightmarish Escherian shopping mall.





If you know of any other especially good ones, please add them to the end of this post as a comment!

Next, we need a Beyonce video illustrating the event horizon behavour of acoustic metrics ...

Special Relativity is an Average

Textbooks tend to present special relativity's physical predictions as if they're somehow "out on a limb", and totally distinct from the predictions of earlier models, but SR's numerical predictions aren't as different to those of Nineteenth-Century models as you might think.

One of the little nuggets of wisdom that the books usually forget to mention is that most of special relativity's raw predictions aren't just qualitatively not particularly novel, they're actually a type of mathematical average (more exactly, the geometric mean) of two earlier major sets of predictions. So, in the diagram above, if the yellow box on the left represents the set of predictions associated with the speed of light being fixed in the observer's frame (fixed, stationary aether), and the red box on the right represents the set of physical predictions for Newtonian optics (traditionally associated with ballistic emission theory), then the box in the middle represents the corresponding (intermediate) set of predictions for special relativity.

If we know the physical predictions for a simple "linear" quantity (visible frequency, apparent length, distance, time, wavelength and so on) in the two "side" boxes, then all we normally have to do to find the corresponding central "SR" prediction is to multiply the two original "flanking" predictions together and square root the result. This can be a really useful method if you're doing SR calculations and you want an independent method of double-checking your results.


This usually works with equations as well as with individual values.
F'rinstance, if the "linear" parameter that we were working with was observed frequency, and we assumed that the speed of light was fixed in our own frame ("yellow" box), we'd normally predict a recession Doppler shift due to simple propagation effects on an object of

frequency(seen) / frequency(emitted) = c / (c+v)

, whereas if we instead believed that lightspeed was fixed with reference to the emitter's frame, we'd get the "red box" result, of

frequency(seen) / frequency(emitted) = (c-v) / c

If there was really an absolute frame for the propagation of light, we could then tell how fast we were moving with respect to it by measuring these frequency-shifts.

The "geometric mean" approach eliminated this difference by replacing the two starting predictions with a single "merged" prediction that we could get by multiplying the two "parent" results together and square-rooting. This gave

frequency(seen) / frequency(emitted) = SQRT[ (c-v) / (c+v) ]

, which is what turned up in Einstein's 1905 electrodynamics paper.

The averaging technique gave us a way of generating a new prediction that "missed" both propagation-based predictions by the same ratio. Since the numbers in the "red" and "yellow" blocks already disagreed by the ratio 1: (1- vv/cc), the new intermediate, "relativised" theory diverged from both of these by the square root of that difference, SQRT[ 1 - vv/cc ]. And that's where the Fitzgerald-Lorentz factor originally came from.

---==---

Why is it important to know this?

Well, apart from the fact that it's useful to be able to calculate the same results in different ways, the "geometric mean" approach also has important implications for how we go about testing special relativity.
Our usual approach to testing SR is to compare just the the "yellow" and "orange" predictions, identify the difference, say that the resulting differential Lorentz redshift/contraction component is something unique to SR and totally separate from any propagation effects, and then set out to measure the strength of this relative redshift/contraction component, in the range "zero-to-Lorentz". Having convinced ourselves that these effects are unique to SR, we usually don't then bother to check whether the data might actually make a better match to a point somewhere to the right of the diagram.
Since the "yellow box" predictions are so awful, special relativity comes out of this comparison pretty well.

But once you know the averaging method, you'll understand that this is only half the story -- these "derivative" effects that appear under SR but not "Classical Theory" ("orange" but not "yellow") must have counterparts under Newtonian optics ("red"), and these are usually stronger than the SR versions. So any experimental procedure or calculation that appears to support the idea of time dilation or length-contraction in an object with simple constant-velocity motion under SR would also generate an apparent positive result for those effects if SR was wrong and the older "Newtonian optics" relationships were the correct set (or if some other intermediate set of relationships was in play). We can say that special relativity's concept of velocity-based time dilation didn't exist under NO, but hardware doesn't care about concepts or interpretations, only results ... and the result of performing an SR-designed test in an "NO universe" would be that the test would throw up a "false positive" result apparently supporting SR (with an overshoot that'd then have to be calibrated out).

And, actually, the situation is worse than this.
... Since the "yellow" and "red" blocks represent the two extremal predictions for theories that allow linkage between the velocity of a light-signal and the motion of a body ("yellow" = zero dependency, "red" = full dependency), they also seem to represent the cutoff-limits for a whole slew of old Nineteenth-Century "dragged aether" models, all of which would be expected to produce similar physical effects to special relativity, differing only in their scaling and strength. So typical test procedures designed to isolate the "new" SR effects should be able to generate "false positive" results with almost all of these old theories and models.

While some of special relativity's concepts might have been new, its testable numerical predictions lie right in the middle of a pre-existing range. Any time you see a claimed experimental verification of SR that forgets to take this into account, treat it with caution.

Fibonacci Kitchenware (well, almost)

I popped into Habitat yesterday, and they're selling a range of five pseudo-Fibonacci nesting trays (four smaller trays plus a bigger one to hold them). It's just a shame that they chose such and awful selection of colours for them (who the heck decided on yellow, brown and navy blue??!?).