Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction
Page 19
Ultimately the problem with Heinlein’s novel comes from the fact that the twins are communicating with each other at speeds faster than light, which violates the precepts of the special theory of relativity. If we put v > c into the equation for the gamma factor we get the square root of a negative number, that is, an imaginary quantity—indicating that we can’t do it. Some people have tried to come up with clever ways around this problem, but most physicists think that the speed of light is the ultimate limiting speed in the universe.
This does lead to another issue: why is it that at the end of the voyage, Tom is younger than Pat? If the effect is symmetric, why should either of them be younger than the other?
This has been discussed to death in the past. The answer is straight-forward: yes, time dilation is symmetric, but there is something that isn’t. That is their accelerations. Pat, sitting still on Earth, is not being accelerated. Tom, on the other hand, when he climbs aboard the ship and takes off for the stars, is. Even if the acceleration is for a very short period of time, this difference is what makes Tom younger than Pat. Let me work through an example to show why this is true. Here are the assumptions I’m going to make. This example is based on the discussion in Wolfgang Rindler’s book, Special Relativity [198, pp. 30–31].
• Tom is going to make a trip to the Alpha Centauri star system, 4.3 light-years away.
• His spacecraft is capable of traveling at 86% of the speed of light after a very rapid period of acceleration. (We’ll ignore the problem of keeping Tom from getting smashed to jelly during the acceleration.)
At 86% of the speed of light it takes Tom five years to get there and five years to get back as measured by Pat’s clocks. The gamma factor is almost exactly 2, however, so to Tom it should take only 2.5 years out and back, meaning he should be five years younger than Pat on his return. This is from Pat’s perspective. How about from Tom’s perspective?
A rather incisive point made by Rindler is that no matter how fast the acceleration period is, when it is over Tom has gone halfway to his destination [198, p. 31]. This is a result of the length contraction effect: from Tom’s point of views when the acceleration ends, the Sun is moving away from him at 86% of the speed of light and Alpha Centauri is moving toward him at that speed. Because of relativistic length contraction, the distance between them has shrunk by 50%. To Tom, his clock is normal, but the distance is contracted by 50%. So it all works out.4
12.4 CONSTANT ACCELERATION IN RELATIVITY
There is no such thing as constant acceleration in special relativity, for a simple reason: under constant linear acceleration (with initial condition v = 0 at time t = 0), the velocity after time t is v = at. Because of this, after a sufficiently long time (about one year if a = g), the spacecraft is traveling faster than the speed of light. We therefore must be careful about how we define acceleration.
An astronaut on an accelerating spacecraft will feel the sensation of weight, so this is what we will adopt as our definition of acceleration.
1. Weigh the astronaut on Earth (W0).
2. Then, once on the spacecraft, put a scale under her and measure her effective weight (W).
The “proper” acceleration of the spacecraft, a, is then
Since g ∼ 10 m/s2, if the weight aboard the craft is only 10% of the weight on Earth, the acceleration of the spacecraft is about 1 m/s2.
It turns out that we can make life very simple for ourselves in talking about acceleration if we adopt a system of units in which distances are measured in light-years and time is measured in years. That is, the speed of light is c = 1 LY/yr. In this system of units, g works out almost exactly to 1 LY/yr2, which makes our calculations very easy. Let’s assume the following:
• The ship starts from rest and travels out with a constant acceleration of g along a straight line.
• x is the distance the ship travels.
• t is the time from the beginning of the trip as measured on clocks on Earth.
• v is the speed the ship gets to after time t.
• τ is the time that has passed on the ship.
Then the motion of the ship follows the equation
This is often referred to as “hyperbolic motion” in relativity because if x is plotted against t, the figure is that of a hyperbola.
We can express everything in terms of the time on board the ship:
For those unfamiliar with the terms, “cosh,” “sinh,” and “tanh” are the hyperbolic cosine, sine, and tangents, respectively:
Table 12.2
Time and Space under Relativistic Acceleration
and
12.4.1 A Trip to the End of the Universe
Let’s plan a trip to the great beyond. After all, a trip of a billion light-years begins with a single step, right? The nice thing about relativistic acceleration is that although nothing can go faster than the speed of light. It will take at least a billion years to go a distance of a billion light-years, but time dilation makes it seem much, much shorter for the voyagers. Here’s the idea: we’ll take our ship and accelerate for half the time (i.e., half the distance) of the trip (as measured by shipboard clocks), turn the ship around, and decelerate the other half. How far do we get in this time? How much time has elapsed on Earth? How fast were we going at midpoint? In table 12.2, xend is the distance the ship travels by the end of the voyage, tend and τend are the times as measured on Earth and on the ship’s clock, respectively, and vmid is the speed at midpoint.
The results are clear: given a ship capable of accelerating at 1 g continuously, one could reach the nearest stars in a few years, the center of the galaxy in under 25 years, other galaxies in 40 years, and the edge of the universe in 50 years. So the universe is within our grasp! Unfortunately, the energy requirements for a trip like this mount up considerably, but I’ll leave this as an exercise for my readers to work through.
NOTES
1. One point concerning this novel is that much of the time dilation experienced by Corbell is the result of diving near the event horizon of the supermassive black hole at the center of the galaxy. Gravitational time dilation is from the general theory of relativity, which isn’t covered in this chapter.
2. Indeed, the website Crank Dot Net, which specializes in providing links (I quote) to “cranks, crackpots, and loons on the net,” lists no less than 48 sites dedicated to showing that the theory of relativity (really, the two theories of relativity, special and general) are wrong [3].
3. I am being careful to avoid saying “it appears that clocks run slow” because this implies that the rate changes are somehow an illusion. This is wrong. By any test you can make, a clock moving in relation to you runs slow.
4. The readings on Pat’s clocks are harder to explain, but one can show that there are more ticks on Pat’s clock than on Tom’s using a combination of time dilation and what is called the Doppler effect from Tom’s point of view.
CHAPTER THIRTEEN
FASTER-THAN-LIGHT TRAVEL AND TIME TRAVEL
Regular space is deep and wide,
Hyperspace is just outside.
—THE SPACE CHILD’S MOTHER GOOSE
13.1 THE REALISTIC ANSWER
Faster-than-light travel and time travel are both impossible.
13.2 THE UNREALISTIC ANSWER
I’m in a somewhat difficult position writing this chapter, for several reasons:
• I don’t believe that the laws of physics allow faster-than-light travel or time travel into the past.
• However, both of these ideas are strong elements in many science fiction stories.
• Compounding the problem, there is a plethora of recent books on the subject, and almost anything I can say about the subject has already been said by those authors, and better.
Faster-than-light (FTL) travel is a key component of many science fiction stories, for obvious reasons: traveling to the stars is an undertaking that requires enormous resources and takes a very long time even if we push light speed in doing so. The d
istances between the stars are just not commensurate with human length scales or time scales. There are a number of different approaches to getting around this problem:
• Ignore relativity entirely. If we assume that relativity is wrong, then we can go as fast as we like. No one likes this approach much today, as there is overwhelming evidence that relativity works. The original Star Trek series seemed to ignore relativity, although I realize that the mere act of writing this down will have some fan quoting chapter and verse to contradict me. The surrealistic TV series Space:1999 certainly ignored it, although this wasn’t its biggest break with reality or common sense by a long shot. E. E. “Doc” Smith in the Lensman series had an “inertialess” drive that somehow worked around relativity to allow ships to be accelerated to near-infinite speeds [221]. This isn’t so much ignoring relativity as bypassing it, but it’s still impossible.
• Use tachyons. Tachyons are completely hypothetical particles that have never, ever been seen and that in principle travel faster than the speed of light [80]. They almost certainly don’t exist. Robert Erlich is one physicist who thinks they do. He thinks that neutrinos might be tachyons. He’s probably wrong, but his ideas are interesting [79].
• Use some idea concerning “quantum entanglement.” This is probably what Ursula K. Le Guin based her idea of the “ansible communicator” on [145]. It stems from the fact that in quantum mechanics, the “collapse” of a wave function for two “entangled” particles happens instantaneously—at least in some reference frame. This doesn’t violate the theory of relativity, however, because (at least according to the Copenhagen interpretation of quantum mechanics) wave function collapse cannot transmit information or energy.1 So much for the ansible.
• Use hyperspace. Hyperspace is some “multidimensional” analog to real space; how it works is a little vague. I think the idea is that the speed of light is effectively infinite in hyperspace and that there is a point-to-point correspondence between hyperspace and regular space, so that by putting a ship there and moving around a bit, you emerge back in regular space light-years away. Books using this idea in various forms include The Mote in God’s Eye and its sequels, Larry Niven’s Known Space stories, the TV show Babylon 5, and possibly the reimagined Battlestar Galactica as well.
• Use folded space-time and wormholes. In the general theory of relativity, space behaves in some ways (in higher dimensions) like a rubber sheet distorted by the masses placed in it. There is nothing to prevent the rubber sheet from being bent so that two points, although far away on the sheet, are close in a direction perpendicular to the sheet (as seen from a higher-dimensional perspective). Some sort of bridge between those points would allow a ship to travel from one to the other in a much shorter time than it would take to move along the sheet. The first place I read about this was in Madeleine L’Engle’s A Wrinkle in Time, in which the author incorrectly described this means of transportation as a “tesseract” [152]2. This is just about the only idea that has any scientific merit; the relativist Kip Thorne investigated whether this would work for FTL travel at the request of his friend Carl Sagan for use in the novel Contact [209]. Wormholes are the practical way to exploit folded space, as they form the aforementioned bridges.
Kip Thorne has discussed using wormholes for effectively FTL travel extensively, so I will just summarize here. I do need to discuss some ideas from Einstein’s theory of relativity to introduce the subject.
13.3 WHY FTL MEANS TIME TRAVEL
It is a fairly common statement that FTL travel implies time travel into the past. However, the reasons for this aren’t often discussed, as it takes a fairly subtle understanding of the theory of relativity to appreciate why this should be so. What is also true, at least as far as the special theory of relativity is concerned, is that allowing matter or energy to travel faster than light results in some bad paradoxes—real ones, not seeming paradoxes like the twin paradox discussed in the last chapter but ones that cannot be resolved without abandoning one of the two principles of relativity. It isn’t clear whether the general theory of relativity resolves these paradoxes and allows FTL travel and time travel. For now, let’s discuss things in the context of special relativity.
First some definitions from the theory of relativity. An event is anything that happens, plus the time and place where and when it happens. Generally speaking, we reserve the term for things that are of short duration so that we can define the time with reasonable precision. Let’s take two events; with complete lack of imagination I’ll call the first one A and the second one B. Let’s say A is lighting a fuse and B is a firecracker going off. Did A cause B?
This is an interesting question. Let’s put a long fuse on the firecracker, say five feet long, which burns at a rate of 1 ft/s. If the firecracker explodes six seconds after the fuse is lit, then lighting the fuse may have caused the firecracker to explode. If it explodes four seconds after, then A couldn’t have caused B because there wouldn’t have been enough time for the fuse to burn down. However, we can (in principle) make a fuse that burns faster.
Let’s make a really long fuse: five light-seconds long, or roughly 1.5 million km. We’ll light the firecracker using light—imagine that we have some sort of optical fiber, a laser to send the signal, and a detector on the firecracker. The signal travels at a speed of one light-second every second, so it takes five seconds for the light to reach the firecracker. Again, if the cracker explodes in four seconds, then sending the signal could not have caused the explosion; if it explodes after six seconds, then it could have. The point here is that according to all we know about physics, we cannot increase the signal speed beyond the speed of light. If the firecracker explodes four seconds after the light signal was sent, there is no possible way that A could have caused B.
What is interesting is that in that case it doesn’t really matter that A occurred before B. Philosophically speaking, if A cannot possibly influence B, it doesn’t matter when they happen relative to each other. Amazingly, this philosophy is embedded in the mathematics of the theory of relativity! As we saw in an earlier chapter, time flows at different rates for people moving relative to one another. If A cannot possibly cause B, then in some reference frames A will precede B, in others B will precede A, and in some both A and B will happen at the same time. This is known as the “relativity of simultaneity,” and is one of the most subtle and interesting things about the special theory.
Now lets modify things so that the fuse burns with a speed faster than light—say, twice the speed. Then the firecracker will explode 2.5 seconds after the fuse is lit. The problem is, in which reference frame? In one reference frame the firecracker explodes after the fuse is lit, in another before the fuse is lit, and in a third at exactly the same time. What this means is that if we accept that the special theory of relativity works, we cannot assign a consistent time ordering to these events. For those interested in reading about this in more detail, I recommend the two books It’s about Time by N. David Mermin and Spacetime Physics by Edwin F. Taylor and John Archibald Wheeler, in particular the section on Lorentz transformations and the parable of the “great betrayal” [165][234].
The special theory does work in regions of space that are free of large masses, such as the vast gulfs between the stars. There is overwhelming proof that we can use the postulates and the results of the special theory; trillions of particles generated in accelerators all over the world can’t be wrong, can they? The conclusion has to be that a spaceship simply can’t travel faster than the speed of light through normal space.
Science fiction writers who include things that travel faster than light in their stories generally do so by the implicit assumption of a preferred reference frame from which times and distances are measured. This is usually time and space as measured by clocks on Earth. It is the assumption that Robert Heinlein makes Time for the Stars, when Tom begins to sound slow to Pat and Pat fast to him as his ship approaches the speed of light [113]. It may also be true in a more subtle wa
y for the Alderson drive used in The Mote in God’s Eye. The following quotation is from the essay “Building the Mote in God’s Eye,” in which Niven and Pournelle describe how one uses the drive in the novel:
There are severe conditions to entering and leaving the continuum universe [i.e., hyperspace]. To emerge from the continuum universe you must exit with precisely the same potential energy … as you entered. You must also have zero kinetic energy relative to a complex set of coordinates that we won’t discuss here [186].
I remember reading this essay in Galaxy magazine when I was about ten years old and wanting to know more details of how it worked. It seems that the Alderson drive implicitly violates the special theory by imposing a privileged reference frame from which times, distances, and energies are measured. Again, the ships disappear in one place and reappear in another, light-years away, “instantly,” but again, there are reference frames in which the ship will reappear before it disappears.
Physicists like causality: the idea that if event A causes event B, A must unambiguously happen before B does. Therefore, directly traveling faster than the speed of light is out. Is there any other way?