Astronomical Games: September 2002

The Unwinnable Race

Light from galaxies receding at more than the speed of light can't reach us...or can it?

To dream the impossible dream, to fight the unbeatable foe...

—Don Quixote in Man of La Mancha, by Mitch Leigh and Joe Darion

ONE YEAR, when I was a kid, one of my teachers invited our class over to her house for an end-of-the-year party. On her door was a knocker, and above the knocker was a peephole. Above that, in turn, was a thing I had never seen before.

You may have seen it before: it was an abstract figure of a fish, with letters that looked (to my untrained eye) something like IXOYE. The E, in particular, looked kind of funny, but I recognized that it was kind of abstract—I wasn't that untrained—and abstract letters tend to be a bit unusual sometimes.

In fact, the fish is a well-known Christian symbol, and the letters are in Greek, the language of the New Testament. They are, in order, iota, chi, theta, ypsilon (or upsilon), and sigma. They spell out ichthys, a Greek word meaning "fish." The use of the fish as a symbol of Christianity may have something to do with Christ telling his congregation to "follow me, and I will make you fishers of men." (Matthew 4:19)

But there's more to it than that. Back in early Christian times, when the Roman Empire was still pagan and the Christian movement still weak, the Romans were constantly persecuting and mistreating the Christians for no good reason, even as the Christians were pleading for mercy. It was dangerous and difficult for Christians to congregate.

So they invented a secret sign—the five Greek letters in the fish. They are a Greek acrostic that spells out Iesous Christos Theou Uios Soter, which means, roughly, "Jesus Christ, Son of God, Our Savior." In that way, they could identify themselves to each other in a symbolically meaningful way without being obvious to the persecuting Romans.

(We all know what happened. Within a matter of centuries, the positions were reversed, and the Church became the powerful institution, while the pagans were driven underground. Do you suppose that the Christians showed any of the mercy that they pleaded for themselves? Of course not. They persecuted the pagans with all the vigor with which they had been persecuted not so long before. Memory is short.)

The most popular place for the so-called "Jesus fish," as far as I can tell, is the back of cars. It's so common that it has spawned a number of imitations and parodies. One of the best known is the "Darwin fish," which simply has the six English letters spelling Darwin's name in place of IXOYE, and also has legs growing out of the bottom of the fish. I first started seeing the Darwin fish some 20 years ago.

Now, the Jesus fish is an honest symbol of religious feeling, with historical significance to boot. The Darwin fish is a bit of pictorial wordplay about our being descended from fish (which we are). Both have serious content, although I daresay the Darwin fish is mostly a joke of sorts.

But some Christians, offended by the Darwin fish, have created a third fish, the Truth fish. This fish has teeth and is drawn in the act of eating the Darwin fish. Now what on earth is this supposed to convey? I suppose the idea is that the Truth (as perceived by the Christians) will win out over Darwinian heresy. Yes, but by what means? Predation and consumption? This the best some people can come up with?

It's a sad commentary on life that some people would rather be on the winning side and be done with it, than wonder what might be wrong in a theory they disagree with.

Evolution is only one theory [1] that many religious fundamentalists view with great distrust. The other great scientific folly, as far as they are concerned, is the Big Bang, because it too is at odds with the creationist view—this time about the origin of the universe, not of humans. Apparently, they see it as a concerted attack of science in general, and astronomers in particular, on the fortress of fundamentalism.

The funny thing is that one of the astronomers most deeply involved in discovering evidence for the Big Bang remained unconvinced of it for the rest of his life.

In the early years of the 20th century, the American astronomer Vesto Melvin Slipher (1875–1969) was measuring the radial motions of various spiral nebulae—that is, the part of their motions that brought them closer to us or further away (whatever their side-to-side motions might be). He did this by taking advantage of something called the Doppler-Fizeau effect.

The Doppler effect or shift (as it is usually known) is most familiar to us in the context of the changing pitch of a sound, such as the whistle on a passing train. As the train approaches us, the whistle sounds at one pitch. When it passes us, the pitch then drops suddenly and remains at this lower note as the train recedes.

Many people undoubtedly noticed this before the Austrian mathematician Christian Doppler (1803–1853)—although not with trains, obviously—but since he was the first to give a proper explanation of its cause and to quantify its effects, he is given credit for its discovery. He also tried to extend this analysis to light, since color is to light what pitch is to sound; red light is light of a lower "pitch," and blue light is light of a higher "pitch." Light of a receding object undergoes a "red shift," while light of an approaching object undergoes a "blue shift."

Doppler therefore predicted that stars should appear bluer when they approach us, and redder when they recede. He even suggested that certain blue or red stars were that color because of their motions toward us or away from us.

Actually, the terms "red shift" and "blue shift" are somewhat misleading, because only visible light, between red and blue, shifts toward the red in a "red shift" and toward the blue in a "blue shift." Violet light, for example, does not shift toward the blue in a blue shift, because violet light is even higher in "pitch"—that is, of a shorter wavelength—than blue light. Instead, violet light shifts to ultra-violet light, which is of shorter wavelength still. Ultra-violet light is invisible to our eyes, but that is a limitation of our eyes, not any qualitative difference between violet and ultra-violet.

At the other end of the spectrum, beyond red, is the infra-red. Like ultra-violet light, infra-red light is also invisible to our eyes, and again, this is merely a feature of our eyes. In a red shift, infra-red light does not shift to the red, since infra-red light is of even lower "pitch"—that is, of a longer wavelength—than red light. Instead, it shifts deeper into the infra-red.

Doppler was on the right track generally, but mistaken in his explanation of the color of the stars. The problem is that stars emit light of all frequencies, not just visible ones. When a star recedes from us, true, the visible frequencies shift to the red (and the red light shifts to the invisible infra-red frequencies), but previously invisible light of higher "pitch," the ultra-violet, shifts down to the visible blue and violet, and the star on the whole remains about the same color. A similar thing happens in the opposite direction when the star approaches us.

The actual effects of radial motion on light were analyzed by the French physicist and astronomer Armand-Hippolyte-Louis Fizeau (1819–1896). The light of distant stars and galaxies only contains light of most of the colors of the spectrum. Here and there are tiny gaps called Fraunhofer lines (see "How to Cook a Star"), caused by the absorption of those missing colors by certain chemicals in the stellar atmospheres.

These gaps can be used not only to identify the chemical composition of those stars, but also as "notches" on the spectrum. When a star recedes from us, the whole spectrum of the light emitted by the star, both visible and invisible, shifts to lower frequencies, but so do the Fraunhofer lines. Since the Fraunhofer lines have a recognizable pattern, by detecting how far the lines have moved toward the red or the blue, the radial motion of the star, or a galaxy of stars, can be determined.

By 1914, Slipher had recorded the spectra of 15 galaxies. He found that only two of them, the Andromeda Galaxy and one other, emitted light that was blue-shifted—therefore, they were approaching us. The other 13 galaxies were all red-shifted and were therefore receding from us.

As it so happens, the galaxies Slipher recorded were not evenly distributed across the sky. Overall, the galaxies approaching us were on one side of the sky, and those receding from us were on the other. Slipher concluded, quite reasonably, that our galaxy must be drifting among the galaxies—away from the 13 galaxies and toward the two. (Much earlier, the German-English astronomer William Herschel had come to a similar conclusion about our motion amongst the nearby stars, and he had been perfectly right, so Slipher had some precedent for his guess.)

However, as Slipher and others imaged more and more spectra, they found that almost all of the galaxies were receding, and in all parts of the sky. The only ones that were approaching us seemed to be relatively close to us. All the distant ones were flying away from us, and at enormous velocities, some as high as several percent of the speed of light.

Enter the American astronomers Edwin Hubble (1889–1953) and Milton Humason (1891–1972). They decided to investigate a possible relationship between distance and recession velocity. They weren't the first to do so—they may have gotten the idea from a preliminary study done by the German astronomer Carl Wilhelm Wirtz (1876–1939)—but they were the first to do it so extensively and systematically. They divided their attentions: Humason would use his extraordinary ability at the telescope to record the spectra of ever more distant galaxies to determine their radial velocities, and Hubble would devise ways to figure out how far they were.

Unfortunately, there's no simple way to reliably determine the distance to faraway galaxies, because you can't quickly tell the difference between a dim nearby galaxy and a bright but distant one. Parallax, the only definitive method for determining astronomical distances, was only good out to a few dozen light-years in Hubble and Humason's time—about a million times too close to do any direct good. Even with today's sophisticated instruments, the best we can do is a thousand light-years or so, still too short to measure the distances to the galaxies. Instead, astronomers must find some way to use the parallax distances as a foundation to define new yardsticks good to, say, a few thousand light-years. Those distances are not as reliable as the parallax distances, but they are the best we can do, and they in turn are used as the foundation for still more new yardsticks out to millions of light-years, and so on. These steps constitute a "distance ladder" that astronomers are constantly trying to strengthen and diversify.

Nowadays, there is an entire alphabet soup of methods that we can use, but in the 1920s, the distance ladder was in its infancy, and Hubble and Humason had to define their own methods. One way that they tried to estimate distances to galaxies was to select galaxies grouped into little clusters. Our own galaxy is a member of the Local Group, a small cluster of a couple of dozen galaxies of varying size. Hubble and Humason's reasoning was that galaxies everywhere have roughly the same intrinsic brightnesses. If we knew how bright a galaxy really was, and we could see how bright or dim it appeared in the sky, then we could figure out how far it must be to appear as bright or dim as it does.

Any individual galaxy might be of any brightness, so the fact that it was dim would tell us little. A cluster of galaxies, however would give us some context. Hubble realized that any cluster of galaxies might have one or two galaxies that were especially bright, intrinsically. If he were to use those as a measure of how far the cluster was, he might underestimate its distance, just as a very big building might appear closer than it really is, when it's very dark outside. Clusters were very unlikely, however, to have five especially bright galaxies, so Hubble's clever trick was to use the fifth brightest galaxy in the cluster as the standard by which to measure the cluster's distance from us.

When Hubble and Humason combined their results, they found something extraordinary. The more distant a galaxy was, the faster it seemed to be receding from us. Moreover, the recession velocity was directly tied to its distance from us: if one galaxy was twice as far away from us as another, it was moving away at twice the speed; if it was five times as far away, it moved away at five times the speed. This simple relationship between distance and velocity has come to be known as Hubble's Law.

When Hubble and Humason first published their results, it seemed that for every million parsecs (a distance approximately equal to 3.26 million light-years) that a galaxy was away from us, it receded at 500 km/s. That is, if a galaxy was 6 million parsecs (about 20 million light-years) away, it was moving away from us at 6 times 500 km/s, or 3,000 km/s. (This is about 1 percent of the speed of light.) We can therefore write Hubble's Law as

v = H0 d

where v is the galaxy's recession velocity, d is its distance from us, and H0 is known as Hubble's constant, which Hubble determined to be 500 km/s per million parsecs (written 500 km/s/Mpc).

Later studies showed that although Humason had properly determined the recession velocities of the galaxies—it was based on the more reliable Doppler-Fizeau effect—Hubble had systematically underestimated the distance to the galaxies, so that although the general form of the law was correct, the value for Hubble's constant had to be much smaller than 500 km/s/Mpc. Much of 20th century cosmology focused on finding a better value for H0, and it turned out to be no mean feat. At times, estimates of its value varied by as much as a factor of 5 or so, but more recently, better methods have reduced the uncertainty, so that Hubble's constant is generally understood to be somewhere between 60 and 75 km/s/Mpc.

What does this mean? One possible interpretation is that our galaxy is sitting still in space, and all the other galaxies (except a few of the closest) are running away from us at enormous speeds. But the mere fact that Hubble's Law is a direct proportion suggests a different interpretation. If a galaxy 10 Mpc away is moving away from us at 750 km/s, and a galaxy 10 Mpc behind that one is moving away at 1,500 km/s, then the first galaxy would see the second galaxy moving away at only 750 km/s, and would also see us moving away at 750 km/s, in the opposite direction. In other words, all galaxies see all other galaxies moving away (except, again, the few that are closest to them), all according to Hubble's Law. That law would apply not just to us, but to the entire universe.

If we run the galaxies backward in time, we see that there must have been a time, then, when all the galaxies were in the same place, or very nearly so. This is not an ironclad conclusion, but it is the simplest one, given the evidence of the Doppler shifts. Many billions of years ago, the galaxies must have all begun speeding away from each other. They can have done this in one of two ways: either by moving through space, or by being carried along by expanding space. Since the former would violate the special theory of relativity (it would require some galaxies to move faster than the speed of light), it must be the latter. This event—the beginning of the expansion of the universe—is called the Big Bang.

Ironically, although he helped to establish it as a theory of the origin of the universe, Hubble was never satisfactorily convinced that the Big Bang actually happened. He only maintained that the law of proportionality that carried his name was a real, physical property of the cosmos. Since his name carries so much weight, his reluctance has led some people to conclude that the Big Bang is not so viable a theory after all.

But science by authority is a dangerous game to play. Hubble did have trouble with the Big Bang theory, but his reasons made sense at the time. For various reasons, a Hubble constant as large as 500 km/s/Mpc means that the universe could not be much older than about 2 billion years, and it was already known that the solar system had to be older than that—perhaps more than twice as old. (The Sun and the Earth and the rest of the solar system are, in fact, about 4.6 billion years old.) Hubble felt that this precluded the Big Bang explanation of the receding galaxies—he even felt that it might not permit the interpretation of the red shifts as Doppler shifts. Perhaps light shifted to lower frequencies for a reason completely unrelated to their radial motions. A lower Hubble constant means an older universe, however, and today, there is no such problem with the Big Bang cosmology.

Now here's an interesting question. Suppose, for the sake of argument, that Hubble's constant is 75 km/s/Mpc. That means that a galaxy 4,000 Mpc distant (about 13 billion light-years) should be receding from us at 300,000 km/s, which is just about the speed of light.

This doesn't violate the special theory of relativity because the galaxy isn't travelling through space at the speed of light. The space between us is growing at 300,000 km every second, and space isn't subject to the speed of light restriction. The galaxy mostly just goes along for the ride.

Still, we might well wonder: How is it that light from such a galaxy ever gets to us, when the distance between us and it is growing at the speed of light? Worse yet, there may be galaxies even further away than 13 billion light-years, and the space between us and them is growing even faster than the speed of light. By all rights, the light from these distant objects should be running an unwinnable race against the expansion of the universe, and we shouldn't be able to see them at all.

And yet, astronomers a couple of years ago reported the detection of one object that was 26 billion light-years away, and another that was even more distant, estimated at 78 billion light-years distant. How could those astronomers make such a basic mistake? Can't they see that there's a serious contradiction here?

When faced with a contradiction between what we know and what others tell us, the human thing to do is to vote on the side of what we know and impugn what others tell us. This gives us artifacts like the Truth fish. However, it's hard to learn anything new that way, so let's see if we can't figure out a way that those astronomers might be right after all.

Part of the problem arises from thinking about light as though it were an ordinary object. If a car is speeding away from us at 30 m/s (about 44 miles per hour), and someone in the back seat rolls a tennis ball back in our direction at 30 m/s, relative to the car, then the two motions cancel each other out and the ball remains stationary on the ground. By that interpretation, it sure looks like light shouldn't be able to reach us, from a galaxy that's receding from us at the speed of light.

But the situation with the galaxy and its light doesn't work like that, and it's not just because light travels about 10 million times faster than the tennis ball. First of all, you don't need to "throw" light. You can just "drop" a photon, and it instantaneously speeds off at 300,000 km/s; in fact, you can't get it to go any speed other than 300,000 km/s. In particular, you can't generate a stationary photon by moving at the speed of light in one direction and ejecting the photon in the other. It stubbornly insists on moving at the speed of light no matter what you do.

Secondly, as we noted above, the galaxy doesn't travel through space; the expanding space carries it along. It's as though the car weren't running at all, but instead, the road between us and the car was expanding at a rate of 30 m/s.

So let's rethink our car and tennis ball model in light of those observations. Instead of the car driving away from us, we'll imagine that the car and we are both resting on a huge and infinitely elastic rubber band that is initially, say, 3 km (that is, 3,000 meters) long. If this rubber band were totally stationary, then the tennis ball (representing a single photon of light), travelling at 30 m/s, should take 100 seconds to reach us.

If, on the other hand, we start the rubber band stretching at 30 m/s at the same moment the tennis ball comes rolling out of the rear of the car, also at 30 m/s, then the behavior is more complex. To simplify it, let's have the rubber band and the tennis ball take turns every second. At each turn, we'll first stretch the rubber band 30 meters, and then we'll move the ball along the rubber band by 30 meters, and we'll see what happens.

The first second, the rubber band stretches 30 meters, to 3,030 meters. The light travels 30 meters back toward us along this slightly stretched out rubber band, and in doing so, is back to 3,000 meters away, and it would seem that no progress is being made.

But that's a somewhat misleading way to look at it. Instead of giving the linear distance between us and the tennis ball, let's instead figure out the progress the tennis ball has made as a fraction of the total distance between us and the car. At the beginning, the tennis ball is still at the rear of the car, so its progress, as a fraction of the 3,000 meter distance between us and the car, is 0/3,000, or just 0.

At the end of 1 second, however, the tennis ball is now 30 meters along, and the rubber band is 3,030 meters long. The fractional progress of the tennis ball is thus 30/3,030, or 1/101. It's not much, but it's a start! And if this fractional progress hits 1, that's when the tennis ball reaches us, at the other end of the rubber band.

Let's see if that happens in fact. In the second second (ahem), the rubber band has stretched another 30 meters, to 3,060 meters. In so doing, the tennis ball is carried back almost, but not quite, 30 meters. Because it's 1/101 of the way back to us, it is 100/101 of the way from us to the car, and it is thus carried back a distance equal to 100/101 of 30 meters, or about 29.7 meters. Then the light travels 30 meters along the rubber band. Its net gain is 30 meters minus 29.7 meters, so it's now 2,999.7 meters away from us.

As far as its fractional progress is concerned, however, the stretching of the rubber band to 3,060 meters doesn't affect it at all. It stays at 1/101 until the tennis ball moves its 30 meters. This increment of fractional progress is equal to 30/3,060, or 1/102, so that the cumulative progress of the ball is 1/101 + 1/102.

In the third second, the rubber band stretches another 30 meters. This doesn't affect the fractional progress of the tennis ball, but the rubber band is now 3,090 meters long. When the tennis ball moves 30 meters, therefore, its incremental progress is 1/103, so now the cumulative progress of the ball is 1/101 + 1/102 + 1/103.

By now, it should be clear that we can figure out where the ball is after any time t seconds. The rubber band should be 3,000 plus 30t meters long, and the fractional progress of the ball should be

1/101 + 1/102 + 1/103 + · · · + 1/(100+t)

The interesting question is whether this series ever adds up to 1 or more for any finite t. If so, then the tennis ball will reach us within a finite interval; if not, it never does. If the rubber band never stretched, then all the terms would be equal to 1/100, and it would take 100 terms to sum up to 1. That just tells us that the tennis ball takes 100 seconds to reach us when the rubber band isn't stretching, as we saw before.

However, the rubber band is stretching, and the terms are not all equal to 1/100, but are instead constantly decreasing. With that in mind, you might well wonder if it's not possible that the series will eventually stall and never reach 1.

It's not! The terms do decrease, but they don't decrease nearly quickly enough for the series to stall, and the sum does reach 1 after a finite time t. In fact, it exceeds 1 for the first time when t = 173 seconds, meaning that it will take nearly 3 minutes for the tennis ball to reach us. That's not quite twice as long as it would take if the rubber band didn't stretch, and it's certainly not infinite.

You might think that if the rubber band were stretching at 60 m/s—that is, twice the speed of the tennis ball—then the tennis ball would never reach us. But that's not so. Since the road is stretching twice as fast as before, the series in question would be

1/102 + 1/104 + 1/106 + · · · + 1/(100+2t)

and although that series would take longer to reach 1, it would still reach it: it exceeds unity for the first time at t = 323 seconds. And, if the rubber band started out not at 3 km, but at 30 km, then the series would be

1/1002 + 1/1004 + 1/1006 + · · · + 1/(1000+2t)

which reaches 1 first when t = 3,198 seconds. Amazing as it might seem, it is the nature of such series that the tennis ball always reaches us in a finite period of time. Such a bold statement can be rigorously proven, but in case you find such mathematical arguments unconvincing (see "Escape!"), you are perfectly welcome to search for a counterexample.

And there's no fundamental difference between the way the tennis ball can reach us from a car receding faster than the speed of the ball, and the way that photons reach us from a galaxy receding faster than the speed of light. As a result, we can see those galaxies; they are not hidden from us. Light wins the (apparently) unwinnable race.

But—you may have seen this coming—there's a fly in the ointment.

Hubble had to use a number of different tricks to measure the distance between the Earth and the faraway galaxies. The speed with which those galaxies receded from us was, however, quite easy to measure, with the red shift. Any red shift should be simple to convert into a recession velocity. For example, a recession velocity of three-fifths the speed of light corresponds to a red shift of 1. A recession velocity of four-fifths the speed of light corresponds to a red shift of 2.

And a recession velocity of five-fifths the speed of light—that is, the speed of light itself—what kind of red shift does that give?

The answer from theory is unambiguous: the red shift, for an object receding at the speed of light, is infinite. All photons of light have their wavelengths stretched infinitely long, to the point where their energies drop to zero. The photons would reach us, but be petering out to nothingness just as they arrive.

The situation is even worse for photons emitted by galaxies receding from us at more than the speed of light. To use the formulas in a straightforward manner is to derive a negative red shift, and that has no physical interpretation we know of. So perhaps, despite our best efforts, we can make no sense out of the claims of the astronomers after all.

And yet, there is one more wrinkle in the maddening field of cosmology that we haven't taken into account—but, we'll get to that next month.

[1] Non-scientists are often swayed by the wishy-washy sound of the word "theory" into thinking that evolution and the Big Bang are flimsier propositions than, say, Newton's Three Laws. In point of fact, Newton's Three Laws are no more cast in stone than those theories, but the mere use of the word "law" convinces people to consider them sacrosanct.

Copyright (c) 2002 Brian Tung