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 ...

The Principle of Relativity

The principle of relativity is pretty straightforward: it's essentially that "nothing is nailed down" The locations and properties of our universe's contents are defined by their relationships to other things in the same universe: there is no absolute sheet of "universal graph-paper" that's overlaid on the universe from outside that defines where everything "really" is, and which dictates the laws of physics in some occult manner.

If we think about the problem logically, we find that there's another aspect to the idea: if there were such a sheet of universal graphpaper, and that sheet did force physics to operate in such a way that we could identify an objects absolute motion relative to it, then that hypothetical sheet of graphpaper would (in a sense) have to exist within our universe, and the motion of bodies could once again be described using the principle of relativity, by treating our absolute frame as another (rather special) physical "thing". But it's perhaps slightly perverse to decide that the universe exactly one of these special things, with nothing else like it, so Occam's Razor pretty much demands that we reject the idea of a single absolute reference frame, unless there's compelling supporting evidence for it.

A more serious problem with the idea of an absolute, inviolable aetheric medium is that such a thing would appear to break some basic principles concerning cause and effect. Normally we assume that when a thing acts, it knows that it's acted ... that is, that there is a back-reaction for every reaction. We assume that if Object A exerts power over Object B, that A's ability to influence is somehow reduced, or at least altered in some way. There is no “something from nothing”, no expenditure of influence without a corresponding lessening of the bank account, and no free lunch. If A's ability to affect B was absolute and without consequence for A, then we could say that A's stock of influence appeared to be infinitely large. And if we are talking about an identifiable physical and quantifiable influence, it leads to some nasty mathematical results if we say that anything has an infinite quantity of a real physical thing. A further problem with these infinities is that they break accounting rules and the chain of causality. When asked where this influence comes from, we can't reverse the sequence of events and extrapolate any further back than the dictatorial rulings of our infinitely-strong metric, which then acts as a limit for any further logical analysis. It becomes a prior cause, a thing that can't be politely incorporated into a larger, fluid, mutually self-contained logical structure, but has its own separate anchor-point that doesnt relate to anthing else inside the structure, and allows it to dictate terms to everything else without retribution.

This sort of “absolute aether” is a way of saying that things simply happen in a certain way because they do, with no further analysis possible, and from a theoretical-analytical point of view, it's a dead end.

It was partly Einstein's appreciation of this problem that led him to the conviction that spacetime itself had to be a stressable, flexible, malleable thing. The “medium” of Einstein's general theory was the background gravitational field (which also defined distances and times), but the assumed properties of this field were no longer absolute, but were affected by the properties of the physics that played out within it. Spacetime was an interactive, integrated part of physics. The “fabric of spacetime” deformed gravitomagnetically as objects passed through it, and spacetime itself was the medium by which masses communicated with and connected causally to other masses. There was an interplay between the properties of spacetime and the properties of matter and energy – as John Wheeler put it, “Matter tells space how to bend, space tells matter how to move”.

The more static, "fixed" spacetime of special relativity, Einstein later decided, was a somewhat distasteful creature. Certainly special relativity had done away with the idea of there being any absolute reference for location, and even for absolute independent values of distance and time, but the overall spacetime structure still had an “absolute” quality to it, in that the geometry of Minkowski spacetime was meant to control and define inertial physics, without its own properties being in any way affected (a slightly abstract version of "action without reaction"). Minkowski spacetime was still "absolute" in the geometrical sense.

To quote Einstein ("The Meaning of Relativity", Princeton University Press):

... from the standpoint of the special theory of relativity we must say, continuum spatii et temporis est absolutum. In this latter statement absolutum means not only "physically real", but also "independent in its physical properties, having a physical effect, but not itself influenced by physical conditions".
...

It is contrary to the mode of thinking in science to conceive of a thing (the space-time continuum) which acts itself, but which cannot be acted upon. This is the reason why E. Mach was led to make the attempt to eliminate space as an active cause in the system of mechanics. According to him, a material particle does not move in unaccelerated motion relatively to space, but relatively to the centre of all the other masses in the universe; in this way the series of causes of mechanical phenomena was closed, in contrast to the mechanics of Newton and Galileo. In order to develop this idea within the limits of the modern theory of action through a medium, the properties of the space-time continuum which determine inertia must be regarded as field properties of space, analogous to the electromagnetic field.
...
... the gravitational field influences and even determines the metrical laws of the space-time continuum."

Because the word "relativity" is often equated with the predictions of specific theoretical implementations of the principle, it comes with a certain amount of historical baggage that isn't always useful when one wants to discuss a problem more generally. Sometimes it's more convenient to start from scratch and use a different form of words when trying to explain a relativistic principle without getting bogged down in historical implementational specifics. John Wheeler used the term "democratic principle" to refer to the idea that there's no single overriding cause that determines the forces on a particle, and another way of describing it might be to refer to the principle of mutuality, in that everything in the universe might be expected to not only have a vote in influencing anything that happens (subject to signal-propagation times), but also to be influenced itself in return.

So really, the principle of relativity in its broadest sense is just about going back to classical first principles: there's no action without origin and/or consequences, causality is A Good Idea, and nothing happens for no reason. These are somewhat pragmatic assumptions if we want to analyse the pattern of rules that the universe obeys – the first step is to assume that there IS a pattern.

There are, of course, more specific definitions of what the principle of relativity "says", which are tailored to the contexts of specific theories (usually Einstein's special and general theories). But we aren't obliged to use those existing definitions, and if we want a chance of discovering broader and deeper theories, we probably shouldn't.

Special Relativity is not Compulsory

One of the foundations of Twentieth Century relativity theory was the idea that Einstein's early "special-case" theory of relativity ("Special Relativity", or "SR") had to appear as a complete subset of any larger and more sophisticated models.

At first glance, this seemed unavoidable.

Einstein's later and more sophisticated general theory was at its heart a geometrical theory of curved spacetime... it described gravitational fields in terms of how they warp lightbeam geometry, and then used the principle of equivalence to argue that the effects associated with accelerations and rotations must also follow the same set of rules. We could then model all three classes of effect as an exercise in curved-spacetime geometry, and go on to extend the model to include more sophisticated gravitomagnetic effects.

But Einstein's general theory didn't attempt to apply these new curvature principles to simpler problems involving basic relative motion, because his earlier special theory had already dealt with those cases by assuming flat spacetime. Instead of going over the same ground a second time, Einstein simply said that, just as classically-curved surfaces reduced over sufficiently small regions to apparent flatness, so the geometry and physics of general relativity, if we zoomed in sufficiently far, ought to reduce to flat spacetime and the "flat-spacetime" version of physics described by the special theory.

There were good pragmatic reasons for Einstein's adoption of special relativity as a foundation for GR, but geometrical necessity wasn't one of them. Here's why:
... It's true that if we zoom in on a GR-type model sufficiently far, we end up with effectively-flat spacetime, but this doesn't automatically mean that we then have flat-spacetime physics. It might instead mean that we've zoomed in so far that there's no longer any meaningful classical physics to be had. We have to accept at least the logical possibility that real physical particles (and their interactions) might be unavoidably associated with spacetime curvature, and in that scenario, we can't derive their relationships by presuming absolutely flat spacetime, because that condition would only be met if our particles didn't physically exist.

Allow any form of velocity-dependent curvature at all around moving particles, and SR's flat-spacetime derivations fracture and fail. This is especially unfortunate since the experimental evidence suggests that moving particles do seem to disturb the surrounding lightbeam geometry, just as we'd expect if curvature effects were a fundamental part of physics, and if the flat-spacetime basis of special relativity was wrong.

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This suggestion that "all physics is curvature" was put forward at the end of the Nineteenth Century by a mathematician called William Kingdon Clifford, who's usually remembered for having his name on Clifford Algebra. The critical thing about a "Cliffordian" model in this context is that when we implement the principle of relativity within it, we find that the resulting physics doesn't reduce to special relativity and the relationships of Minkowski spacetime. Instead of a Minkowski metric, it reduces in the presence of moving particles to something that looks more like a relativistic acoustic metric, and which appears to be much more compatible with quantum mechanics than our current classical models.

So the perfect, unbreakable geometrical proofs of SR's inevitability as physics aren't complete. In order to complete them, we have to be able to show that Cliffordian models can't work ... and that seems to be difficult, because the results of taking a Cliffordian approach seem to be pretty damned good.

To date, nobody seems to have been able to come up with a convincing disproof of this class of curvature-based solution, and until that happens we have to accept the possibility that special relativity might not be a part of our final system of physics.

Isaac Newton and E=mc²

The history of the idea of mass-energy conversion is a slightly murky one. Textbooks and lecturers find it convenient to say that Albert Einstein was the first person to suggest that mass and energy were interchangeable, but really ... he wasn't. That's a handy piece of educational fiction. It ain't so.

By 1905, a number of researchers were reckoned to be close to the E=mc² result. The basic argument went something like this: imagine a mirrored cavity embedded in a piece of material, containing a trapped light-complex, in equilibrium with its container. The radiation pressure of the trapped light within the container is the same in all directions. But if the container and its trapped electromagnetic (EM) energy are now viewed by a different observer who reckons that the container is "moving", then that observer will assign different Doppler-shifted energies and radiation pressures to different parts of the light-complex: The forward-aimed components now get assigned greater energy and momentum than the rearward-aimed components, and the overall momentum of the complex no longer cancels out - the container's nominal motion gives the trapped light an overall momentum that points in the direction of motion.
So the EM contents of the moving container appear to contribute additional momentum to it, as if it contains a speck of matter rather than EM energy. If we aren't allowed to look inside the container, we might not be able to tell whether it contained EM energy or real matter, and by working out how much energy it takes to reproduce the external effects associated with a given amount of mass, we end up with a very short equation for the conversion factor between rest mass and rest energy. That (if we calculate it correctly) is E=mc².

However, it seems that Einstein's competitors either didn't calculate the conversion ratio properly, or failed to come out and suggest in print that this wasn't merely an apparent conversion of mass and energy, but The Real Thing. Einstein did both, and earned the credit.

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If we want to go back further, to find an older example of the idea of "interconvertibility" in a major English-language physics text by a famous author, all we have to do is open a copy of Isaac Newton's "Opticks" [Babson archives]/[1717 edition.pdf], and flip to the "Queries" section at the back. The relevant section is Query 30:

Qu.30: Are not gross Bodies and Light convertible into one another, and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition?...
The changing of Bodies into Light, and Light into Bodies, is very conformable to the Course of Nature, which seems delighted with Transmutations.

I've quoted this at the start of Chapter 2 of the "Relativity..." book ("Gravity, Energy and Mass"), which goes through some of these arguments in more detail (with the help of some pictures).

Traditionally, at this point in the discussion, a physicist will interrupt and say something like,

"Okay, perhaps Newton had the idea, but we weren't able to calculate the specific relationship until we had special relativity. Einstein used Lorentz's relationships in his calculations rather than Newtonian physics, so so E=mc² is clearly specific to Einstein's physics."

But that's not true either. It's correct that Einstein originally presented E=mc² in the context of his new "special" theory, but if he'd done the momentum calculations with the same degree of care using "olde" Newtonian emission theory, he'd have gotten the same result (with slightly less working). In fact, we can construct a continuum of hypothetical theories whose relationships differ by Lorentzlike ratios, and all of them generate E=mc². Turns out, E=mc² is a general result. I've put the details of the "Newtonian optics" argument into the book's "Appendices" section, as "Calculations 2"

So, while some physics histories present Einstein's discovery of E=mc² in 1905 as a triumph of the scientific method, the reality seems to be that the equation's discovery is marked by a sequence of earlier human failures going back two hundred years.
To start with, Newton couldn't calculate E=mc² because he'd gotten the relationship between energy and frequency upside down, and assumed (reasonably but wrongly) that the "bigger", redder wavelengths of light carried more energy and momentum for a given amplitude, rather than less ("The Newtonian Catastrophe", chapter 3). Newton lived 'til 1727, and then his his successors still couldn't calculate E=mc², because they trusted Newton to have gotten it right. If you were an English physicist, suggesting that Newton might have made a mistake was heresy. Towards the end of the century (1783), John Michell used Newton's arguments to calculate the gravitational wavelength-shifting of light, but he was still citing Newton's writing and using the old bad "inverted" relationships. Defending Newton from criticism was now a matter of national pride, and in 1772, Joseph Priestley's History of Optics had been cheerfully ridiculing the mental capacity of those poor retards in Europe who were so behind the times that they actually still thought that light was a wave! Antagonism between the two sets of researchers meant that the Newtonian group couldn't admit the possibility of major error.

The next couple of decades saw Europe shaken up by the French Revolution, and then Continental physics really began to hit its stride. Newton's mistake had generated a bad prediction that light should travel more quickly through glass than air, and when Continental experimenters started using new technology to measure lightspeeds, they were able to show, quite conclusively (and perhaps slightly gleefully), that this wasn't the case. As we got to the mid-C19th, work by Christian Doppler and others meant that we were now quite sure how to calculate the effect of velocity on light for any given model, but instead of going back and correcting Newton's error, Newton's supporters slunk off with their tails between their legs, and did their best to rewrite physics history so that later English-speaking physics students hopefully wouldn't realise just how dumb they'd been.

The latter part of the C19th was then "lost", too. Although we now had plenty of expert wave theorists, lightwaves were now generally reckoned to propagate through some sort of aetheric medium, and there was no agreed set of governing principles defining what that medium's properties ought to be. The credibility of the older Newtonian principles concerning the behaviour of light (such as the idea that the behaviour of matter and light ought to obey a single set of underlying rules) were now widely considered to be "damaged goods", and the proliferation of aether models meant that we now had a bewildering array of competing predictions for exactly how the properties of light ought to be affected by motion. There were just too many damned versions for us to be able to do these sorts of calculations confidently, and be sure that our results meant anything.

That state of affairs lasted until the early Twentieth Century.

This is where Einstein came onto the scene. Einstein had three advantages over most other contemporary theorists when it came to deriving E=mc² - he was a fan of the idea that the principle of relativity should apply to light, he was definite about the set of equations that he wanted to use, and he was (apparently) blissfully unaware of almost all of the previous two centuries of political bickering on the subject (probably helped in part by his habit, as a student, of not bothering to turn up for lectures). So Einstein was able to come to the problem "fresh", without a lot of preconceptions. He'd already tinkered with emission theory, recognised some of the problems, and had then latched onto Lorentzian electrodynamics, and decided that this was The Future.

In 1905, he published his "reimagining" of Lorentzian electrodynamics , which took the characteristics of Lorentz's relativistic aether and deleted the "physical medium" aspect as unnecessary. According to Einstein in 1905, aether was irrelevant to the problem - all that was required to generate the Lorentzian relationships was the principle of relativity and an assumption about lightspeeds. These two postulates were then sufficient to generate all of Lorentz's important math.

And then (as a very short followup paper) ... if the Lorentzian relationships in the previous paper were correct, internal energy imparted mass to bodies, according to the relationship E=mc².
At this point, Einstein was on a roll, and he was looking forwards rather than backwards ... he didn't really have much motivation to point out that, if the relationships in his earlier paper were wrong, and we reverted to the previous relativistic calculations for light, we still got E=mc². Pointing that out was a job for peer review and outside commentators, but almost no-one noticed.

We then coasted though another century, without much to suggest that anyone had connected the dots and understood the broader context for what Einstein had done and how it really related to Newton's earlier work. Right into the 1990s, students were still being told that E=mc² was unique to special relativity, and that the fact that atom bombs worked was ample evidence that no other system of equations could be right. Those claims weren't scientifically or mathematically correct, and weren't researched, but everyone seemed to believe them. Some people wrote research papers and entire books on the history of E=mc², and still somehow managed not to mention the Newtonian connection.

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Not everybody missed it. The Sandman series by Neil Gaiman quotes and cites the key section in "Opticks" and points out its its significance. But Sandman isn't a book on theoretical physics, it's an illustrated fantasy graphic novel. So what we appear to have here is a subject where some people who write university textbooks seem to be doing rather less background research and fact-checking than some people who write comicbooks.

I feel that this is an unhappy situation. But it seems to explain a lot about why theoretical physics is in its current state.

John Milton, 'Paradise Lost', and General Relativity

John Milton (1608-1674) was a linguist, pamphleteer and poet, nowadays best remembered for having written Paradise Lost, first published in 1667.

England in 1667 had been experiencing decades of social upheaval, and an accelerating succession of crises. Oliver Cromwell's side had won the civil war, abolishing the monarchy and executing Charles I in 1649, but Cromwell had then died in 1658, and without Cromwell as a driving force, Parliament had decided to restore the Monarchy, with Charles II being appointed the new king in 1660. Milton had campaigned for religious reform, written campaign material for Cromwell and the Republic, and had held a post in the new regime before going blind. With the Restoration, Milton briefly became a wanted man, and his books were publicly burnt. As England was coming to terms with the abrupt political reversion, it got hit first by the Great Plague of 1665, and then by the Great Fire of London [*] [*] in 1666.

These were, as the saying goes, interesting times.

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Paradise Lost, Milton's masterwork, was an epic poem, originally in ten sections, that outlined the rebellion and fall of Lucifer and the subsequent fall of Adam and Eve, in the contemporary poetic equivalent of high-definition widescreen. Milton had something of a talent for visual imagery and a good turn of phrase, and some commentators later ruefully pointed out that Milton seemed to have been influenced by his experiences with the ill-fated English revolution, and not only made the rebellious Lucifer a sympathetic character, but given him some of the the best lines. "Better to reign in hell than serve in heaven", indeed!

The mind is its own place, and in itself
Can make a heaven of hell, a hell of heaven

There are some great phrases in the poem – when we talk about "the fabric" of spacetime, we're arguably borrowing from Milton – but to a physicist, one of the most surprising sections (along with his name-dropping of Galileo) is the bit where Milton writes :

... whether heavn move or earth
Imports not, it thou reckon right.

To a historian, those lines might be taken as an assertation of independence, and a rejection of the notion of centralised supreme power. There's no single authority that decides what is really "moving" and what isn't. No church, no Pope, no religious leader, no monarch.

But to a physicist, they set out the general principle of relativity. When the Earth spins on its axis and loops along its orbit around the Sun, it's convenient to think of the Earth as moving and the background starfield as fixed. But in reality, there's no "special" status accorded to those stars. They're just stars, each following their own line of least resistance as they drift in the wash of their own individual surrounding gravitational tides and currents. It doesn't matter whether we say that the Earth rotates beneath Heaven, or that Heaven rotates around the Earth. If you calculate properly, (said Milton), the answers should be the same in either case.

And if they're not, you've done it wrong.

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While Milton was getting his piece finalised and published in 1665-67, the plague had other consequences. Cambridge University sent its students home in the fall of '65, and one these, a previously unremarkable and undistinguished student, chose to make use of the two years of enforced comparative isolation back at his family's farm in the village of Woolsthorpe to work through some ideas on gravity, optics and mathematics. His name was Isaac Newton.