The strange thing about Einstein’s special relativity is that the theory was spot on – even though an essential assumption was way off-beam. Mark Buchanan discovers the dark side of E = mc2

IMAGINE you are on a bicycle, pedalling across the cosmos. A beam of light – perhaps sent off by a distant collapsing star – zings past you. How fast are you and the light approaching each other? You are travelling at hardly any speed, so the answer will be more or less exactly light^{’}s speed through the interstellar vacuum, around 300 million metres a second.

Now imagine you abandon pedal power for the day. Bowling along in your spaceship at half light speed, you meet another light pulse head-on. What is your speed of approach now? Surely it is just your speed plus that of the light: in total, one and a half times light speed.

Wrong. Your speed of approach will be the speed of light, no more – and that^{’}s true however fast you are travelling. Welcome to the weird world of Einstein^{’}s special relativity, where as things move faster they shrink, and where time gets so distorted that even talking about events being simultaneous is pointless. That all follows, as Albert Einstein showed, from the fact that light always travels at the same speed, however you look at it.

Really? Mitchell Feigenbaum, a physicist at The Rockefeller University in New York, begs to differ. He^{’}s the latest and most prominent in a line of researchers insisting that Einstein^{’}s theory has nothing to do with light – whatever history and the textbooks might say. ^{“}Not only is it not necessary,^{”} he says, ^{“}but there^{’}s absolutely no room in the theory for it.^{”}

What^{’}s more, Feigenbaum claims in a paper on the arXiv preprint server that has yet to be peer-reviewed, if only the father of relativity, Galileo Galilei, had known a little more modern mathematics back in the 17th century, he could have got as far as Einstein did (www.arxiv.org/abs/0806.1234). ^{“}Galileo^{’}s thoughts are almost 400 years old,^{”} he says. ^{“}But they^{’}re still extraordinarily potent. They^{’}re enough on their own to give Einstein^{’}s relativity, without any additional knowledge.^{”}

The claim has got other physicists thinking. Take Feigenbaum^{’}s argument a step further, some say, and we might long ago have seen our way not only to Einstein’s relativity but also to the idea of an expanding universe –even one whose expansion is accelerating – without the intellectual upheavals that have led us to those conclusions today.

The discussion centres on two assumptions that Einstein made when formulating his special theory of relativity in 1905. The first is uncontroversial: that the laws of physics should look the same to anyone at rest or moving steadily. Say I am standing motionless and you are moving past on a train travelling at a constant speed in a straight line – in other words, at a constant velocity. To you on the train, I am the one who seems to be moving. But it does not actually matter who is ^{“}really^{”}moving relative to whom: although perceived velocities depend on one^{’}s point of view, the physical laws governing motion stay the same.

This is the principle of relativity proposed by Galileo in *A Dialogue Concerning the Two Chief World Systems, *his treatise of 1632 that got him into hot water with the Catholic church for discussing Copernicus^{’}s idea that Earth goes round the sun. Galileo writes of a passenger inside a ship who cannot tell if it is moving or standing still ^{“}so long as the motion is uniform and not fluctuating this way and that^{”}. The analogy was aimed at those sceptics who believed that Earth could not be moving because they could not feel it.

Galileo^{’}s relativity served well for almost 250 years. But when Scottish physicist James Clerk Maxwell derived his theory of electricity and magnetism in the late 19th century, it hit a snag. Maxwell^{’}s equations make clear that light is a wave travelling at a constant speed. But oddly, they do not mention from whose point of view this speed is measured.

This was a problem if Maxwell^{’}s theory, like all good physical theories, was to follow Galileo^{’}s rule and apply for everyone. If we do not know who measures the speed of light in the equations, how can we modify them to apply from other perspectives? Einstein^{’}s workaround was that we don’t have to. Faced with the success of Maxwell^{’}s theory, he simply added a second assumption to Galileo^{’}s first: that, relative to any observer, light always travels at the same speed.

This ^{“}second postulate^{”} is the source of all Einstein^{’}s eccentric physics of shrinking space and haywire clocks. And with a little further thought, it leads to the equivalence of mass and energy embodied in the iconic equation *E *= *mc*^{2}*. *The argument is not about the physics, which countless experiments have confirmed. It is about whether we can reach the same conclusions without hoisting light onto its highly irregular pedestal.

According to David Mermin, who has been teaching relativity at Cornell University in Ithaca, New York, for 30 years, a consensus has emerged that we can, although this shift has yet to filter through to a wider audience. ^{“}All the textbooks teach relativity based on Einstein^{’}s principles,^{”} he says. ^{“}And there^{’}s an extremely widespread misunderstanding that relativity is somehow tied up with light.^{”}

Two years ago, Feigenbaum^{’}s puzzlement with relativity^{’}s logic led him to Galileo^{’}s *Dialogue. *^{“}The book is quite a knockout,^{”} he says. ^{“}When I finished reading, I wondered, if you take what he says seriously, what can you produce?^{”} So he sat down and started calculating as Galileo might have, but using today^{’}s more sophisticated mathematics.

He starts with a simple scenario. You are standing watching a friend, Frank, moving past you on a train at 50 kilometres per hour towards the east. Frank, on the other hand, has his eyes on Kate, whom he sees receding from him at 50 km/h towards the north. Feigenbaum asks a simple question: how do you see Kate moving?

It seems natural that Kate^{’}s velocity relative to you should in some sense be the sum of Frank^{’}s velocity relative to you and Kate^{’}s relative to Frank. The fact that Frank sees Kate both receding to the north and keeping up with his eastbound motion implies that, from your stationary point of view, her motion is towards the north-east.

But now swap Frank and Kate^{’}s motions. Frank is travelling at 50 km/h northwards relative to you, and Kate at 50 km/h eastwards relative to Frank. This should not affect how Kate is travelling relative to you – you will still see her heading off towards the north-east.

Galileo would certainly have said so. Only with Einstein^{’}s introduction of a space-time warped, as he thought, by a universal speed of light did it become clear that the rules of adding motions were not quite so simple as all that. But in fact, says Feigenbaum, both Galileo and Einstein missed a surprising subtlety in the maths – one that renders Einstein’s second postulate superfluous.

It is this: if Frank^{’}s world is aligned with yours – if the north and east of both you and Frank point in the same direction – and Kate^{’}s world is similarly aligned with Frank^{’}s, you might think that Kate^{’}s is aligned with yours. The problem is, mathematical logic alone does not permit that conclusion. Strange as it may seem, it in fact allows a distinct possibility that Kate^{’}s world could be rotated with respect to yours – even if she is perfectly aligned with Frank and Frank is perfectly aligned with you.

This means that, while still seeing Kate careering off towards the north-east, you might also see her skewed slightly to the left or right relative to her direction of motion (see diagram, right). The direction of the rotation, and thus Kate^{’}s motion as seen by you, would depend on what the relative motions of you and Frank and of Frank and Kate are.

^{“}**Not only is light not necessary in relativity – ****there**^{’}**s no room in the theory for it**^{”}

The possibility of such rotations turns out to have far-reaching consequences. Ignore them, and Galileo^{’}s relativity pops out. Allow them, and the algebra works out very differently: the mangled space-time of Einstein^{’}s relativity emerges, complete with a definite but unspecified maximum speed that the sum of individual relative speeds cannot exceed. ^{“}These rotations are hard to understand,^{”} Feigenbaum says, ^{“}but they^{’}re the wellspring of physics.^{”}

Feigenbaum emphasises that he is not the first person to question Einstein^{’}s second postulate or arrive at the idea of such bizarre rotations. Even so, Mermin is impressed.

^{“}Mitch^{’}s way of deriving the theory is quite complicated,” he says, “but the rotations come up in a very natural and beautiful way.^{”}

The result turns the historical logic of Einstein^{’}s relativity on its head. Those contortions of space and time that Einstein derived from the properties of light actually emerge from even more basic, purely mathematical considerations. Light^{’}s special position in relativity is a historical accident: it was just the first (and is still the most obvious) phenomenon we have encountered that travels at the universal maximum speed.

The idea that Einstein^{’}s relativity has nothing to do with light could actually come in rather handy. For one thing, it rules out a nasty shock if anyone were ever to prove that photons, the particles of light, have mass. We know that the photon^{’}s mass is very small – less than 10^{-49} grams. A photon with any mass at all would imply that our understanding of electricity and magnetism is wrong, and that electric charge might not be conserved. That would be problem enough, but a massive photon would also spell deep trouble for the second postulate, as a photon with mass would not necessarily always travel at the same speed. Feigenbaum^{’}s work shows how, contrary to many physicists^{’} beliefs, this need not be a problem for relativity.

^{“}Feigenbaum’s ideas could be very helpful in correcting this misconception,^{”} says physicist Sergio Cacciatori of the University of Insubria in Como, Italy. He suggests that further thinking along similar lines could reveal much more about the universe. Together with his colleague Vittorio Gorini, and Alexander Kamenshchik of the Landau Institute for Theoretical Physics in Moscow, Russia, he has explored what would happen if you took Feigenbaum^{’}s conclusions about adding motions and applied them to changes in position (www.arxiv.org/abs/0807.3009). What if where you ended up after two consecutive displacements depended on the order of their occurrence?

In the world around us, it is obviously a pretty good approximation to reality that, if you take 20 paces forward and 10 to the left, you end up in the same place as if you had taken 10 equally sized places to the left and then 20 forward. On the vast scales of the cosmos, however, the same assumption might be dangerously misleading. It amounts to requiring the universe to have a flat, Euclidean geometry – one like that in our immediate environment, in which parallel lines never cross and the inside angles of triangles add up to 180 degrees.

Abandon that assumption, Cacciatori and his colleagues show, and things look very different. The universe is not flat and Euclidean, but curved in on itself, creating a geometry akin to that of the surface of a sphere such as Earth – where the parallel lines of longitude converge at the poles and the internal angles of triangles add up to more than 180 degrees.

That discovery is significant because it feeds into a long-running debate about the shape – and fate – of the universe. Back in 1916, Einstein fused special relativity with Newton^{’}s ideas of gravitation to create a universal theory of gravity, known as the general theory of relativity. General relativity predicts that mass and energy warp space and time, and that the distribution of mass and energy determines the universe^{’}s geometry.

**Predictable fate**

When Einstein applied these ideas to calculate the dynamics of the universe, the outcome was decidedly odd: the gravity of the universe warps its fabric so much that it becomes unstable and collapses in on itself. To avoid this dispiriting and apparently nonsensical conclusion, Einstein added to his general relativistic brew a new quantity, the ^{“}cosmological constant^{”}, to counteract gravity and create a stable, static universe. This universe^{’}s geometry was curved and closed in on itself – rather like the three-dimensional surface of a four-dimensional sphere.

Einstein^{’}s constant was short-lived. In 1929, Edwin Hubble found evidence that distant galaxies were receding from the Earth, implying that the universe was expanding dynamically. Faced with this evidence against a static universe, Einstein famously decried the constant as his ^{“}greatest blunder^{”}.

Just recently, however, the cosmological constant has come back into fashion. The reason is the evidence accumulated by astronomers over the past decade – in the unexpected dimness of some extremely distant supernovae, and in the cosmic microwave background, the still-reverberating echo of the big bang – that the expansion of the universe is accelerating. That acceleration seems to require just the kind of antigravitational effect that motivated Einstein^{’}s constant in the first place.

The irony, says Gorini, is that we could have seen all along that something like the cosmological constant makes sense. Follow the mathematics of relativity through to its logical conclusion, allowing for displacements that add up in different ways, and you find that space must have just the curvature that a cosmological constant helps to produce. That has nothing to do with the distribution of mass, but follows from mathematical logic alone.

So had we put our faith in mathematical reasoning, might we have been able to predict the dynamics of the universe much sooner? Gorini thinks so. He claims that the German mathematician Hermann Minkowski came close to exploring these displacement effects in lectures on relativity he delivered in 1908. ^{“}Had he done it,^{”} Gorini says, ^{“}physicists would have known that the universe expands, and that its expansion is accelerated, well before the development of general relativity or even Hubble^{’}s observation of the recession of galaxies.^{”}

As it happens, in 1968, physicists Henri Bacry and Jean-Marc Levy-LeBlond of the University of Nice in France predicted the existence of a cosmological constant from first principles *(Journal of Mathematical* *Physics, *vol 9, p 1605). Their work presages that of Feigenbaum and of Gorini and colleagues, but remained unnoticed – largely because the constant was out of fashion at the time.

Against that prevailing scientific wind, it would have been a bold leap for those researchers to have predicted the dynamics of the universe; for Galileo, centuries before, even more so. Einstein, the ultimate physics revolutionary, probably would have afforded himself a wry smile at the picture that is now emerging. The startling edifice of the new physics he built remains undisturbed, even as its logical foundations are being greatly strengthened. Meanwhile, the power of mathematical reasoning in unlocking the secrets of the universe continues to amaze: what physicist Eugene Wigner once called ^{“}the unreasonable effectiveness of mathematics^{”} is one of the deepest mysteries of them all. ~

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