Einstein’s twin paradox explained

One of my favourite Robert A. Heinlein novels is Time for the Stars, a realisation of Albert Einstein’s ‘twin paradox’ that Heinlein wrote in the mid-1950s as part of his so-called ‘juvenile’ series. It was an outcome of the slightly berserk nature of Einstein’s 1905 theory of Special Relativity, which – with his theory of General Relativity of 1915 – made clear that the underlying reality of our universe is nothing like the Euclidian/Newtonian-style perception we regard as normal from our personal everyday experiences.

Einstein envisaged the twin paradox as a demonstration of Special Relativity, his theory explaining what happens as a result of light-speed in a vacuum being a constant. The outcome, basically, is that if light-speed in a vacuum is a constant, making it independent of any given inertial reference frame (the point of observation) then mass, length and rate of time as seen from any given inertial reference frame, when looking at another frame, must vary.

You can see from this why Einstein’s theory (the first of two) was called ‘relativity’. Everything it described was to do with relative observation. This renders the ‘twin paradox’ even cooler because it is, of course, a pun – at least in English. Take an identical pair of twins. One remains on Earth (the inertial reference frame) and the other travels very close to light-speed to another star system, then returns. When the two meet again, the one who stayed on Earth is older due to time dilation experienced (relative to the inertial reference frame) on the spacecraft moving close to the speed of light. That’s right. The rate at which time passes, relative to an external inertial reference frame, slows down as the spacecraft approaches light-speed.

The way all this works was devised by Einstein in 1905 on the basis of a remarkable analysis a few years earlier by Hendrik Lorentz; but Einstein extended that into a more encompassing understanding. If you measure the rate at which time passes on any moving object, relative to another object, time will always travel more slowly. The effect is only noticeable, however, at close to the speed of light because it’s logarithmic – and at light-speed itself, time stops. This is one of the reasons why you cannot reach light-speed, although it is possible to approach it increasingly closely, if you pour enough energy into doing so (the amount goes up exponentially, hitting infinity at lightspeed).

There is a notion that the dilation difference somehow reverses as the moving object returns to the original reference frame – that the travelling twin might not have an age difference from the non-travelling twin after all. This is a false understanding, though, because what counts isn’t place or vector but acceleration relative to the inertial reference frame. If we take the Earth as one reference frame, time on an object moving at (say) 99.99 percent the speed of light, if observed from Earth, will appear to be moving about 1.414 percent the speed it is on Earth. This is true whether the object is moving away or towards Earth.

But what say we reverse the idea? Let’s consider the spaceship as the fixed reference frame. Now, Earth is moving away from the spaceship (even though the spaceship did the acceleration), and when it reaches 99.99 percent light-speed the rate of time on Earth, as measured from the spaceship, is just 1.414 percent speed. But it doesn’t invalidate the observation from Earth, which is the exact reverse. Both can be true; but because of the acceleration to get from one frame to another, the two are operating in different inertial reference frames. The result is asymmetric and means that the twin who has travelled (relative to Earth) at relativistic velocities has aged less when measured from that original (Earth) reference-frame. The direction of flight (out and back) is irrelevant; what counts is the inertial reference frame. It’s counter-intuitive, but that’s physics for you.

Albert Einstein lecturing in 1921 – after he’d published both the Special and General Theories of Relativity. Public domain, via Wikimedia Commons.

I should add that when I refer to ‘relativistic velocities’ I mean a big chunk of the speed of light. The dilation curves are logarithmic. To make the relativistic effects into something meaningful at human perception level, you have to be moving at least at half light-speed. At that velocity, if viewed from the Earth frame, you’d be experiencing a rate of time 86.602 percent that of the Earth frame. It rises sharply; at 90 percent light-speed it’s 43.588 percent, and at 99 percent it’s 14.106 percent. And, as we’ve seen, at 99.99 percent it’s 1.414 percent. (Just for the record, at 99.999999999 percent light-speed, time runs at 0.0004472148552665697 percent relative to an external inertial reference frame).

For me the cool part is that Heinlein wrote this into a novel. His hero returned home, 70-odd years after leaving Earth, and still in his early 20s, to meet his twin brother as a very old man. Almost all of it was straight Einstein, although of course Heinlein did a certain amount of hand-waving to make the plot possible, notably in his ‘torch drive’ starships which were propelled by engines capable of directly converting mass to energy.

This was a quite subtle-hand-wave that invoked the same Einsteinian theory. According to Special Relativity, mass and energy are aspects of the same thing – defined by the equation E = MC^2, where e is energy, m is mass and c^2 is the speed of light, squared. This is one of the few equations I can think of that has become embedded into pop-culture. Although, of course, I’d prefer this one:

Go on, I double-dog dare you to put THIS equation on a T-shirt.

The real-world problem with the mass-energy exchange is the actual conversion from mass to energy (or vice-versa). It is possible for a photon (energy) to split into mass consisting of a particle and its anti-particle (electron and positron). They usually then re-combine and annihilate each other, producing energy (a photon). Weirdly, most of the mass we experience (and are comprised of) exists in the nucleus of the component atoms. This mass primarily comes from the binding energy of the quarks that make up those particles (proton and neutron). But I digress.

The positron-electron collision also shows how you can turn mass to energy: it happens in a matter-antimatter reaction. However, the only other way to convert mass to energy is through violently altering the atomic structure; but even nuclear fusion reactions aren’t great at it. The really violent mashing of hydrogen atoms in the core of the sun into helium, for instance, produces only about 0.7 percent conversion efficiency. So Heinlein’s ‘torch drive’, with its 80 percent-plus conversion of mass into energy (in some kind of reactor), wasn’t physically possible. That figure, incidentally, wasn’t mentioned in Time for the Stars, but it was in his torch-ship rescue story Sky Lift.

Naturally Heinlein knew there was no process that simply ‘converted’ matter into energy, still less any reasonable way of controlling the resulting forces – effectively those of a continuous matter-antimatter explosion – to run a rocket motor. But accelerating to near-light-speed takes a ridiculous amount of energy, and for plot purposes he had to get his twin character and a turnip-shaped ‘torch ship’ with 200 other crew there somehow. The mass-to-energy conversion factor of Special Relativity offered a plausible hand-wave, validated in his book by the fictional ‘Kilgore equations’ which allowed Einstein’s theory to be practicably harnessed. As an aside, the sci-fi trope that your star-drive exhaust is also a deadly weapon wasn’t first invented by Larry Niven for his Kzin stories – it was Heinlein, who mentioned it as a throw-away line in Time for the Stars in 1956, then showed it in action, briefly. But I digress again.

So where does all this leave us? Special Relativity isn’t as counter-intuitive as quantum mechanics, but it’s still up there for weird in an everyday Earth sense. I should add that thanks to the Lorentz transformations that Einstein built into it, Special Relativity also offers ways of getting objects to fit into spaces shorter than the object is, admittedly only while they are passing through that space at close on light-speed. So it’s not a solution to awkward inner-city car-parking problems. But we’ll go into that another time.

Copyright © Matthew Wright 2019