Date: January 6, 2012

Title: Olbers’ Paradox

Podcasters: Dr. Kyle E. Kneisl

Description: Olbers’ Paradox is an easily understood realization in astronomy that if the universe were infinitely large, and if it has existed forever, then it should be very, very bright when we look into the night sky. The fact that the sky is mostly dark is one of the strongest pieces of armchair evidence that we have about the finite extent of the universe. This miraculous bit of thinking, given its far-reaching conclusions, is fairly easy to follow. In this podcast, we define Olbers’ Paradox, and show how logical reasoning leads directly to it, without requiring any advanced scientific background.

Bio: Dr. Kneisl grew up in Buffalo, New York, and has had a lifelong interest in astronomy that really began in earnest when he stumbled upon his first book about black holes when he was seven years old. He is still an avid amateur astronomer and sky watcher. He holds a Ph.D. in mathematics from the University of North Carolina and lives in the Washington, DC area, where he hopes to introduce his two young children to astronomy in the coming years.

Transcript:

This is Dr. Kyle Kneisl, and we are talking about Olbers’ Paradox. There is a little more thinking and even some math in this episode, but it is nothing that anyone with an interest in science and astronomy should find at all challenging. And in any case, you need not follow everything precisely to be amazed at the result.

With neither pencil nor paper, we can sometimes come to elegant and powerful conclusions about the nature of the world around us, using nothing more than clearly reasoned thought.

For example, we all know there are infinitely many prime numbers, but HOW did we know that? Suppose there weren’t. Then, take all of the prime numbers that there are (and we can make the complete list, since there are not infinitely many), multiply them together, getting a huge number of some kind, and add 1 to the result. By definition, no prime number will divide this big number we just made (since we added 1, so there is guaranteed to be a remainder of 1 when dividing by any prime number). Thus, this huge number we just made must, itself be prime. But that’s impossible, because then it would have been on the list in the first place! What went wrong? We made no mistake in our logic–the only thing that can be wrong must have been with our assumption, namely that there are NOT infinitely many primes. Cosmology sometimes gives us a chance to use that sort of thinking to arrive at powerful conclusions.

Cosmology is the study of the universe as a whole, particularly: its extent, its past, its future, and the laws that oversee it. Cosmology is, today, a vastly complicated arena of mathematical physics and philosophy; the methods behind the discoveries we are now making about the history and ultimate fate of the universe are largely only comprehensible to professional physicists and astronomers.

Fortunately, the same is not true about many of the historical cosmological breakthroughs on foundational questions; many of these are answered through a straightforward application of logic and reasoning that anyone familiar with scientific inquiry will readily understand using nothing more than mathematics familiar to a high school student.

One of the fundamental questions in cosmology is whether or not the universe has existed forever, whether or not it is infinitely large, and whether or not it is homogeneous (that is to say, with matter distributed in largely the same way throughout, no matter how far away we go). Anyone that has looked in the sky can answer questions like these, with the help of a little high school mathematics. One of the main immediate arguments that the universe has not existed forever and/or is not infinitely large, has become known as Olbers’ Paradox. Any reasonable resolution of Olbers’ Paradox seems to contradict the idea of an infinite, never beginning, never-ending universe. This remarkable bit of very logical mathematical thought is the subject of todayís podcast. In a nutshell, Olbers’ Paradox “proves” that the night sky should be very bright, not very dark, in a static, infinite, eternal universe.

Dr. Heinrich Wilhelm Matthaeus Olbers was a physician who lived from 1758 to 1840 (an extraordinarily long life for that time). At night, he worked as an amateur astronomer on the top storey of his home, which he had converted to an observatory. From here, he made a number of enduring discoveries that assured his legacy, not the least of which was the discovery of Pallas (the third most massive asteroid in the solar system) and the development, for the very first time, of an ingenious method for calculating cometary orbits.

He described the paradox named after him in 1823, though research seems to reveal that it had been known in similar form for hundreds of years before 1823. Furthermore, Dr. Olbers’ discussion of the paradox was not particularly important and so it is quite a coincidence that the paradox seems permanently named for him. The paradox, in a nutshell, points out that in an infinitely large universe, existing eternally, that has a homogeneous distribution of matter, every point in the sky should have the brightness of a star’s surface. Now, I don’t know about you, but when I look up into the night sky, I see almost complete blackness, except for little points of light from stars and other objects; far from what Olbers’ Paradox tells us to expect. If the reasoning behind the Paradox is correct, this would seem to be proof that the universe cannot be simultaneously eternal, infinite, and homogeneous, and that, in fact, one or more of these properties fall apart.
Let’s see if we can come to the same conclusion as Olbers’ Paradox?

Most physical effects that we see or feel become four times weaker when we are twice as far away (two “squared” is four). Nine times weaker when we are three times as far away (three “squared” is nine), and so on, always squaring. This is true for gravity, for the brightness of stars, for the heat from a fire, for the force of electromagnetic attraction, for the loudness of sounds, and for almost everything that radiates in all directions equally.

So let’s think about this. The average star that’s a million light years away contributes four times as much light as the average star that’s TWO million light years away. The average star that’s three million light years away contributes only one-ninth as much light. It seems that the amount of light contributed is going down far, far faster than the distance is going up. And, it is! In fact, mathematicians can show if you keep adding up the fractions with squared denominators, 1 + 1/4 + 1/9 + 1/16 + 1/25 and so on, forever, the answer stops at about 1.645 (try it on your calculator). If that’s all there were to it, it would be no surprise at all that the sky is dark.

But we are forgetting something. It *is* true that a star at twice the distance has one quarter the brightness. But, at twice the distance (as you can see by carefully looking at surface area and/or volume equations for spheres), you will have four times as many stars, because the surface area available to put stars on has been multiplied by four! The loss of brightness is exactly balanced by the increased number of stars that we will find at that distance! So instead of having 1 + 1/4 + 1/9 + 1/16 and so on in our total light calculation, we actually have 1 + 4/4 + 9/9 + 16/16, in other words, 1 + 1 + 1 + 1 + 1, forever, which is, of course, infinity. Thus, the total light the universe should be showering down upon us from stars is infinity! Hence, the paradox, since there is, in fact, nothing at all wrong with this logic, unless there is something wrong with the underlying assumptions.

In fact, the reason Olbers’ Paradox occurs, according to the consensus of modern science today, is that the universe is believed to not be infinite in space, time, and may not be homogeneous and isotropic. For example, few mainstream cosmologists believe the universe to have extended back in time forever; that is to say, the universe is almost unanimously believed to have had a distinct beginning, about 14 billion years ago. Thus, in our above calculations, the assumption that we can continue adding terms from farther, and farther, and farther stars, forever, appears to be incorrect since there will be distant points whose lights has not yet (even after 14 billion years) reached us.

All kinds of explanations exist for what might be going wrong in Olbers’ Paradox. For example, maybe there are, at some point in space, large non-transparent clouds that block the light of the distant stars. Maybe the stars are distributed in some complicated fractal distribution. There are ways to explain why the conclusion of Olbers’ Paradox might not obtain. The simplest, though, is simply that the universe is finite in time and space, and the darkness of the night sky is mostly due to that.

It is remarkable that something so fundamental as whether or not the universe is infinite in space and time can be profitably pondered from the comforts of our armchairs using nothing more than our facility for logical thought, and that truly scientific statements can be made.

Do you want to know more about Olbers’ Paradox? A quick search on any search engine will turn out hundreds of results, but be careful: there are several controversial and even incorrect discussions of the Paradox, especially from those for whom the conclusions of Olbers’ Paradox is troublesome for religious or philosophical reasons. The book “Mathematical Fallacies and Paradoxes” by Bryan Bunch, published by Dover, is inexpensive and provides an excellent description of several paradoxes and the ways they are, or are not, resolved.

This is Dr. Kyle E. Kneisl wishing you a happy 2012.

End of podcast:

365 Days of Astronomy
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