Cosmology

If the Starship Enterprise dipped into the Earth’s atmosphere at a sub-warp speed, would we see it? And if the craft were visible, would it look like the object we’re familiar with from TV, with its saucer section and two nacelles? Well, if the Enterprise were travelling fast enough, then – bright physicists that we are – we’d expect the craft to experience the length contraction dictated by special relativity.

According to this famous principle, a body moving relative to an observer will appear slightly shorter in the direction the body’s travelling in. Specifically, its observed length will have been reduced by the Lorentz factor (1–v²/c²)^1/2, where v is the relative velocity of the moving object and c is the speed of light in a vacuum. However, the Enterprise won’t be seen as shorter despite zipping along so fast. In fact, it will appear to be the same length, but rotated.

You might not have heard of this phenomenon before, but it’s often called the “Terrell effect” or “Terrell rotation”. It’s named after James Terrell – a physicist at the Los Alamos National Laboratory in the US, who first came up with the idea in 1957. The apparent rotation of an object moving near the speed of light is, in essence, a consequence of the time it takes light rays to travel from various points on the moving body to an observer’s eyes.

Terrell’s insight was that even if those rays leave different parts of the moving body at the same time, they won’t reach the observer’s eyes at the same time. The net result is a distorted view. Amazingly, for an object that subtends a small solid angle, the Terrell effect will cancel out the Lorentz contraction, making the object appear to have been rotated. Indeed, it allows you to see partly around the object and observe its reverse side.

Of course, the rapidly moving Enterprise would also be subject to the Doppler effect, which would render it a different colour. Indeed, the frequency shift might even make the craft totally invisible to human eyes. The Enterprise would also appear brighter or dimmer due to relativity’s “headlight effect” (which I’ll come back to later). All in all, these effects will make a rapidly moving object, such as the Enterprise, appear nothing like it does at rest.

Now, if you think that’s confusing – don’t worry, you’re not alone.

Length contraction: the short story

The idea of length contraction was postulated by the Irish theoretical physicist George FitzGerald in 1889 and by the Dutch theorist Hendrik Lorentz in 1892. Also known as Lorentz–Fitzgerald contraction or just Lorentz contraction, it was invoked to account for the negative outcome of Albert Michelson and Edward Morley’s memorable experiment of 1887. The two Americans had famously tried – and failed – to detect the “aether”, a hypothetical stationary medium that was supposed to carry electromagnetic waves, much as the surface of water carries water waves.

Keen to rescue the hypothesis of a stationary aether, Lorentz and Fitzgerald’s solution to Michelson and Morley’s null result was to suggest that objects travelling at a substantial fraction of the speed of light will – when viewed by another observer – appear shortened in the direction of their travel. Based on a calculation by Oliver Heaviside of how the magnetic vector potential in Maxwell’s equation transformed between reference frames, their idea was, in truth, just one of a few proposed “solutions”. None was completely satisfactory, however, and for many years, length contraction remained an ad hoc hypothesis.

It was only in 1905 that Albert Einstein cut through the confusion when he published his special theory of relativity. It did away with the aether altogether and proposed two postulates. The first is that the laws of nature are the same in all “inertial” reference frames – i.e. those frames moving at a constant velocity with respect to one another, in which any object with no net forces on it will appear at rest. The second postulate states that an observer in any inertial reference frame will measure the speed of light in a vacuum to be the same.

From his theory, Einstein derived length contraction as well as time dilation, the equivalence of mass and energy, and more. In particular, he realized that an observer at rest on a high-speed train will measure the length of a passing high-speed train as shortened in the direction of its motion, by the factor (1–v²/c²)^1/2 where v is the relative velocity between the observer and the other train. (Paradoxically, an observer on that train will measure the first as shortened by the same factor. Isn’t special relativity fun?)

Length contraction has never been directly measured. But its effects show up in the magnetic force that acts between parallel, current-carrying wires. Bizarrely, this force, which is purely magnetostatic, appears in one wire due to length contraction as experienced by the charge carriers in the other wire’s frame. (It’s complicated. You can find more information in two papers by Paul van Kampen from Dublin City University – Eur. J. Phys 29 879 and Eur. J. Phys 31 L29 – and also in the chapter “Current-carrying wires and special relativity” in the book Trends in Electromagnetism). Current-carrying wires aside, there is little doubt that Lorentz contraction occurs if length is measured – that is, when different points on a moving object are all measured at the same instant of time in the observer’s stationary frame of reference.

But when you view an object with your eyes or a camera, it’s a different story. You’re then recording the photons the object emits – and these photons arrive at your eyeball or camera lens at the same time. What both Einstein and Lorentz overlooked, however, is the fact that the photons may have been emitted by the object at a different time, especially if the object is large. In fact, in 1922 Lorentz erroneously claimed, in the Dutch version of his Lectures of Theoretical Physics, that the contraction could be photographed.

Lost in history

Over the years, few people paid much attention to observing Lorentz contraction. Anton Lampa, an Austrian physicist who had once helped Einstein get his first university professorship, did publish a paper on this topic in 1924 (Zeitschrift für Physik 27 138). Concerning the appearance of a moving rod to an observer, his work was unfortunately largely overlooked. Indeed, in his famous 1940 children’s book Mr Tompkins in Wonderland, the Russian-born physicist George Gamow got length contraction all wrong. He showed bicycles as simply shortened in the direction they are travelling, instead of being distorted and elongated when approaching an observer, or contracting as they receded into the distance.

Researchers only properly started to take notice of the practicalities of observing Lorentz contraction when Terrell published an internal Los Alamos article in 1957 about the effect, followed two years later by a paper in the Physical Review (116 4). His work was spotted by the theorist Victor Weisskopf, who, while serving as president of the American Physical Society, wrote an article for Physics Today presenting Terrell’s findings in a simpler form.

Like Terrell, Weisskopf also noted an earlier 1959 article by the British mathematical physicist Roger Penrose, in which he analysed the appearance of a relativistically moving sphere. In his 1905 paper on special relativity, Einstein had said that such a sphere would look like an ellipsoid, contracted in the direction of motion. But Penrose reckoned that the object would still be spherical, albeit rotated. Indeed, thanks to Penrose’s insights, the Terrell effect is sometimes referred to as the “Penrose–Terrell effect” (though the two worked independently).

Terrell’s and Penrose’s simple insight – overlooked by nearly every physicist before them – was that light rays that simultaneously leave a moving object do not necessarily strike the eye or camera film at the same time. The eye or lens sees images from photons that strike it simultaneously, but these photons – especially for a rapidly moving object – do not leave the object simultaneously. Surprisingly enough, in certain circumstances this difference exactly cancels the Lorentz length contraction and the moving object appears rotated.

Rotation, not contraction

To appreciate why Lorentz contraction disappears for an object moving at relativistic speeds, Peter Signell from Michigan State University has considered the simple case of a cube moving left to right when viewed head-on (see figure 1, above). Amazingly, the “front” of the cube nearly disappears – and the observer can see almost the entire “left” side. Indeed, it’s easy to extend this thinking to other shapes (such as Penrose did with spheres) and to other viewing angles.

Of course, to fully understand the appearance of a rapidly moving object, you have to calculate the geometry not only for arbitrary viewing angles but also for arbitrary distances, velocities and sizes (i.e. subtended angles). You also have to include two purely relativistic effects: the relativistic Doppler effect and relativistic aberration – the “headlight” effect mentioned earlier whereby light emitted isotropically by a moving object is seen by a stationary viewer as bunched in the direction of motion.

Do all that and a “small” object moving from left to right will appear to change colour from high-frequency blues to low-frequency reds – and may even be invisible if its apparent colours are Doppler-shifted outside the visual range of your eye or recording medium. The object will also become dimmer, while its apparent angle of rotation will change from near zero far away to nearly 90°. And because the left-hand side of the object becomes visible, the object will appear to be receding before it has even reached the observer.

As for a sphere, it will – as Penrose showed – retain a circular outline no matter what its speed and subtended solid angle, albeit rotated and partly distorted. It will also undergo all the colour and intensity changes noted above. The bottom line is that the observed shape of any object will not depend on its Lorentz transformation. Yes, length contraction occurs – and can be detected by careful measurement. But if you try to observe length contraction, you can’t because your view of it is compensated for by the finite speed of light.

Cool findings

Over the years, only a few hardy researchers have studied the geometrical appearance of large objects moving at relativistic speeds. In 1961 Mary Boas of DePaul University in Chicago showed that straight lines will appear curved (Am. J. Phys. 26 283), while four years later David Scott and M R Viner of the University of Toronto showed that extended objects – those not meeting Terrell’s assumption of a small subtended angle – are distorted in shape too (Am. J. Phys. 33 534).

“Their grossly altered appearance,” Scott and Viner wrote, “might be described as a non-uniform shear deformation in each of the two perpendicular planes through the line of motion.” Weirdly, the non-central parts of such bigger objects will, at larger subtended angles, appear rotated and behind the centre along a hyperbola.

Strange things also happen to the apparent speed of a relativistically moving object. In 2005 Robert Deissler, who is now at Case Western Reserve University in Ohio, showed that an object with a measured speed v will appear to be travelling faster than it actually is (Am. J. Phys. 73 7). That’s because the object’s apparent position is always behind the actual position as light takes a finite amount of time to travel to the eye or lens.

According to Deissler’s calculations, the apparent velocity of a distant object approaching an observer is v/(1–v/c), which means that if v ≥ c/2 the object will appear to be moving faster than the speed of light (you can do the sum to check). And if the object’s speed starts getting close to c, it’ll seem to be travelling infinitely fast. Indeed, this effect, which is purely geometrical, has been observed in some stars and galaxies that seem to be moving faster than the speed of light.

Considering all these effects – geometrical plus relativistic – is a challenge, but one rendered easier by computer graphics, as described by Zachary Sherin, Ryan Cheu and Philip Tan from the Game Lab at the Massachusetts Institute of Technology and Gerd Kortemeyer from Michigan State University in 2016 (Am. J. Phys. 84 369). Many such graphics and videos have also been collected at the website spacetimetravel.org, a translation of a German website created by Ute Kraus and Corvin Zahn from the University of Hildesheim in Germany (see figure 2, above). (Kraus has written more about her work in “First-person visualizations of the special and general theories of relativity” Eur. J. Phys. 29 1).

Some six decades after Penrose and Terrell’s publications, the Terrell effect is still not widely known – Terrell died in 2009 after a lifetime at Los Alamos – and many textbooks and science presenters still get length contraction wrong. But when you realize what Penrose and Terrell had to say, you’ll surely wonder why you never thought about Lorentz contraction in this way before. As the German philosopher Arthur Schopenhauer once wrote, in a remark that’s often erroneously attributed to Erwin Schrödinger: “The task is…not so much to see what no-one has yet seen; but to think what nobody has yet thought, about that which everybody sees.”