Mitchell’s playful story contains a number of impossibilities. But the impossibilities are not all of the same order.
Take Mitchell’s explanation of the Dun Suppressor. (A dun, incidentally, is an archaic term for a debt collector.) The construction of such a device is entirely impossible for present-day engineers. Nothing in the laws of physics, however, rules out the fabrication of a tube linking two antipodal points. If humankind possessed greater technical capacity (incredibly strong, heat-resistant materials for the tunnel walls, for example); plus a means of alleviating the Coriolis effect, a by-product of Earth’s rotation that would cause the traveller to be slammed into the tunnel walls; plus the political desire and economic wherewithal to drill down to the centre of the Earth and then straight out again … well, I suppose a Dun Suppressor could be made. And yes, after being pushed into the tube, the poor victim would indeed oscillate between opposite points on the Earth’s surface like a mass oscillates on the end of a spring. So the Dun Suppressor is a
practical impossibility but it’s not a fundamental impossibility.
In the story, however, Professor Surd considers setting the unwelcome suitor one of three challenges—each of which we now know to be
The construction of a perpetual motion machine—that’s one challenge Surd ponders whether to set. As far back as the 15th century, Leonardo da Vinci was scathing about the possibility of perpetual motion. He wrote: “Oh ye seekers after perpetual motion, how many vain chimeras have you pursued? Go and take your place with the alchemists.” Modern scientists understand that perpetual motion in an isolated system violates either the first law of thermodynamics, the second law of thermodynamics, or both. The first law is essentially the conservation of energy: you can’t create energy from nothing. The second law is essentially the observation that heat flows spontaneously from a hot place to a colder place. (In the previous chapter I mentioned how Asimov’s Multivac computer wrestled with the question of whether the second law of thermodynamics could be reversed.) Thus the laws of physics rule out the construction of a perpetual motion machine. Building such a device is a
fundamental impossibility. Mitchell might not have kept abreast of contemporary developments in thermodynamics, but he probably knew that a perpetual motion machine was impossible.
Another challenge Surd considers—squaring the circle—is often used as a synonym for impossibility. Ancient geometers first posed the question: if you possess only a compass and a straight edge can you, in a finite number of steps, construct a square that has the same area as a given circle? In 1882, a German mathematician proved the answer is: no. So this is a
mathematical impossibility—and although the proof did not appear until eight years after publication of his story, Mitchell probably suspected the challenge could not be met.
And then there’s the third of Surd’s challenges: show mathematically how to propel an object to an infinite speed in a finite time. For a practical demonstration Surd merely wants to see a method for travelling at 60 miles per minute. Such a speed must have seemed effectively infinite for Mitchell, who was writing at a time when steam trains provided the fastest method of travel: a stately 1 mile per minute, flat out. Nowadays, of course, space rockets can go much faster than Mitchell would have dreamed. So, since it’s eminently possible to accelerate a vehicle to a speed of 60 miles per minute, let’s stick with the
spirit of Surd’s challenge. Is it possible to get an object to move at infinite speed?
Mitchell seems to consider this challenge to be along the same lines as the construction of a Dun Suppressor: difficult to build in practice but possible in principle. However, as Albert Einstein showed 31 years after the publication of Mitchell’s story, the laws of physics make the task impossible. The suitor’s tachypomp could not possibly work. Here’s why.
Imagine a physics laboratory floating freely out in space, far away from the gravitational pull of planets, the particle wind from stars, and the magnetic fields from the galaxy. In this ideal situation physicists inside the lab don’t have to worry about any complicating effects from outside the lab when they measure something. Suppose physicists inside such a lab decide to make a measurement of some fundamental physical quantity (the speed of light, say). Here’s an interesting fact about how our universe is put together: the result of a measurement made
now will be the same as the result of a measurement made tomorrow. It doesn’t matter when they make the measurement—in these ideal circumstances the outcome is always the same. In other words, the laws of physics don’t depend upon time. Similarly, if they make a measurement here and repeat the measurement there then they’ll obtain the same result. It doesn’t matter where they make the measurement. In other words, the laws of physics don’t depend upon location. And if they make a measurement with the lab pointing in one direction then they’ll get the same result with the lab pointing some other direction. It doesn’t matter how they are oriented when they make the measurement. In other words, the laws of physics don’t depend upon direction.
In 1905, Einstein extended this idea regarding the universality of the laws of physics. As long as you are moving smoothly, at a constant velocity, it doesn’t matter how
fast you are moving when you make a measurement of some fundamental physical quantity. You’ll measure the same laws of physics no matter what your speed happens to be.
The theory Einstein built around this idea goes by the name of relativity, even though the theory at its heart is about an invariance: the invariance of the laws of physics with respect to velocity. The reason for the term “relativity” is the following. If you’re inside a space-based laboratory that’s moving smoothly then you can’t perform an experiment inside the lab to tell whether you are moving. You can consider yourself to be at rest. If you look out the lab windows and see a similar lab whizz smoothly past then you can say the other object is moving
relative to you; scientists in the other lab can say you are moving relative to them. Nevertheless, physicists inside labs that are moving smoothly relative to one another will measure the same value of fundamental quantities. In particular: all observers measure the same value for the speed of light.
From Einstein’s realisation—namely, that all smoothly moving observers measure the same value for the speed of light—it follows that observers who move at different velocities must make different measurements of space and time (because, by definition, to measure the velocity of something you must measure the space traversed in a given time). In our everyday world people agree on
when an event happens and they agree on the distance between events—but that’s only because we only ever move slowly. When velocities are small in comparison with the speed of light (and remember that the fastest aircraft moves at only a tiny fraction of light speed) the effects predicted by Einstein’s relativity are unobservably small. But when velocities are a significant fraction of light speed, observers disagree about the passage of time and they disagree about distances in space. This conclusion might seem deeply counterintuitive—we tend to think in terms of time and space being absolute; they are supposed to be quantities we can all agree upon—but it happens to be how the universe is put together. Since 1905, countless experiments have confirmed Einstein’s insights.
What has all this got to do with the impossibility of a tachypomp? Well, the tachypomp works by adding velocities—and another counterintuitive aspect of relativity is the way in which velocities add.
Suppose you are in one of those smoothly moving space-based laboratories, far away from any complicating effects, and you see another lab move smoothly past. In other words, the two labs are moving relative to one another, and occupants of both labs are entitled to believe they are at rest and it’s the
other lab that’s moving. So far, so good. Now, suppose you see the other lab move past you at an even pace of 5 km/s, and you observer that an experimenter in the other lab fires a ball out of a front-facing cannon at a speed of 10 km/s. Unless you have exceptionally accurate equipment you, in your lab, will measure the speed of the ball to be 5 + 10 = 15 km/s. That’s how velocities combine in our everyday world: they simply add.
Mitchell, in his story, writes that if a train travels past telegraph poles at 30 mph, and a passenger walks along the train corridor at 4 mph, then the passenger moves past the telegraph poles at 30 mph + 4 mph = 34 mph. This is the basis of the tachypomp (see Fig.
), and it assumes that the rule for adding small velocities holds when large velocities are involved.
But the rule doesn’t hold! Fig. 8.1
A sketch of the tachypomp, which appeared alongside an early re-publication of Mitchell’s story. In a tower of vehicles, each moving relative to the one below, will the topmost vehicle be able to exceed the speed of light? Mitchell’s character Rivarol said yes; Einstein told us no (Credit: Reginald Birch)
Returning to our space lab, suppose the other lab zooms past you at 0.5
c. (Here, c is the standard symbol for the speed of light. So we’re supposing that the lab moves past you at half the speed of light.) And instead of firing a ball, the experimenter in the other lab uses a torch to fire a beam of light out in front. According to the normal rule for adding velocities, you should see the light beam moving at 0.5 c + c = 1.5 c. But that’s clearly wrong. It must be wrong because everyone always agrees about the speed of light: you see the light beam travel at c not at 1.5 c. Light travels at light speed, irrespective of how fast the original source of the light happens to be moving. Clearly, then, there must be a different rule for adding velocities when large speeds are involved. This isn’t a textbook, so I shan’t go through the formula for adding relativistic velocities; you can easily look it up if you are interested. It’s enough to give some examples.
If a spacecraft moves past you travelling at 0.5
c and it fires a projectile forward at 0.5 c then you as a stationary observer see the projectile moving at 0.8 c; if the spacecraft is moving at 0.75 c when it fires a projectile at 0.5 c then you observe the projectile travel at a speed 0.909 c; if the spacecraft moves at 0.8 c and the projectile is fired forward at 0.8 c then you observe the projectile to be moving at 0.976 c. The formula is such that when you add together any two sub- c velocities the result is always less than c.
Had Mitchell been able to apply the correct formula for the relativistic addition of velocities to his tachypomp he would have discovered that all observers conclude that the device
doesn’t reach an infinite speed in a finite time. Indeed, it doesn’t—and it never can—even reach light speed.
Einstein’s picture of a relativistic universe is contrary to our common, everyday experience. Einstein told us that clocks run slow when they move; that rods contract along the direction of travel; that, no matter how much you try, you can’t accelerate an object to faster-than-light speed. We don’t observe these effects in our low-speed world, but relativistic effects always come into play whenever velocities are large. We can’t avoid those effects. They are an inevitable consequence of the way the universe is put together. A tachypomp
can’t move faster than the speed of light. It’s an impossibility.
You can no more build a tachypomp than you can a perpetual motion machine: the laws of physics rule them both out. Chapter
examines yet another phenomenon that runs foul of universal laws: antigravity. 9