## A Piece on a Paradox

Posted in The Universe and Stuff with tags , , , , , , , , , on March 7, 2012 by telescoper

Not long ago I posted a short piece about the history of cosmology which got some interesting comments, so I thought I’d try again with a little article I wrote a while ago on the subject of Olbers’ Paradox. This is discussed in almost every astronomy or cosmology textbook, but the resolution isn’t always made as clear as it might be. The wikipedia page on this topic is unusually poor by the standards of wikipedia, and appears to have suffered a severe attack of the fractals.

I’d be interested in any comments on the following attempt.

One of the most basic astronomical observations one can make, without even requiring a telescope, is that the night sky is dark. This fact is so familiar to us that we don’t imagine that it is difficult to explain, or that anything important can be deduced from it. But quite the reverse is true. The observed darkness of the sky at night was regarded for centuries by many outstanding intellects as a paradox that defied explanation: the so-called Olbers’ Paradox.

The starting point from which this paradox is developed is the assumption that the Universe is static, infinite, homogeneous, and Euclidean. Prior to twentieth century developments in observation (Hubble’s Law) and theory  (Cosmological Models based on General Relativity), all these assumptions would have appeared quite reasonable to most scientists. In such a Universe, the intensity of light received by an observer from a source falls off as the inverse square of the distance between the two. Consequently, more distant stars or galaxies appear fainter than nearby ones. A star infinitely far away would appear infinitely faint, which suggests that Olbers’ Paradox is avoided by the fact that distant stars (or galaxies) are simply too faint to be seen. But one has to be more careful than this.

Imagine, for simplicity, that all stars shine with the same brightness. Now divide the Universe into a series of narrow concentric spherical shells, in the manner of an onion. The light from each source within a shell of radius $r$  falls off as $r^{-2}$, but the number of sources increases in the same manner. Each shell therefore produces the same amount of light at the observer, regardless of the value of $r$.  Adding up the total light received from all the shells, therefore, produces an infinite answer.

In mathematical form, this is

$I = \int_{0}^{\infty} I(r) n dV = \int_{0}^{\infty} \frac{L}{4\pi r^2} 4\pi r^{2} n dr \rightarrow \infty$

where $L$ is the luminosity of a source, $n$ is the number density of sources and $I(r)$ is the intensity of radiation received from a source at distance $r$.

In fact the answer is not going to be infinite in practice because nearby stars will block out some of the light from stars behind them. But in any case the sky should be as bright as the surface of a star like the Sun, as each line of sight will eventually end on a star. This is emphatically not what is observed.

It might help to think of this in another way, by imagining yourself in a very large forest. You may be able to see some way through the gaps in the nearby trees, but if the forest is infinite every possible line of sight will end with a tree.

As is the case with many other famous names, this puzzle was not actually first discussed by Olbers. His discussion was published relatively recently, in 1826. In fact, Thomas Digges struggled with this problem as early as 1576. At that time, however, the mathematical technique of adding up the light from an infinite set of narrow shells, which relies on the differential calculus, was not known. Digges therefore simply concluded that distant sources must just be too faint to be seen and did not worry about the problem of the number of sources. Johannes Kepler was also interested in this problem, and in 1610 he suggested that the Universe must be finite in spatial extent. Edmund Halley (of cometary fame) also addressed the  issue about a century later, in 1720, but did not make significant progress. The first discussion which would nowadays be regarded as a  correct formulation of the problem was published in 1744, by Loys de Chéseaux. Unfortunately, his resolution was not correct either: he imagined that intervening space somehow absorbed the energy carried by light on its path from source to observer. Olbers himself came to a similar conclusion in the piece that forever associated his name with this cosmological conundrum.

Later students of this puzzle included Lord Kelvin, who speculated that the extra light may be absorbed by dust. This is no solution to the problem either because, while dust may initially simply absorb optical light, it would soon heat up and re-radiate the energy at infra-red wavelengths. There would still be a problem with the total amount of electromagnetic radiation reaching an observer. To be fair to Kelvin, however, at the time of his writing it was not known that heat and light were both forms of the same kind of energy and it was not obvious that they could be transformed into each other in this way.

To show how widely Olbers’ paradox was known in the nineteenth Century, it is worth also mentioning that Friedrich Engels, Manchester factory owner and co-author with Karl Marx of the Communist Manifesto also considered it in his book The Dialectics of Nature. In this discussion he singles out Kelvin for particular criticism, mainly for the reason that Kelvin was a member of the aristocracy.

In fact, probably the first inklings of a correct resolution of the Olbers’ Paradox were contained not in a dry scientific paper, but in a prose poem entitled Eureka published in 1848 by Edgar Allan Poe. Poe’s astonishingly prescient argument is based on the realization that light travels with a finite speed. This in itself was not a new idea, as it was certainly known to Newton almost two centuries earlier. But Poe did understand its relevance to Olbers’ Paradox.  Light just arriving from distant sources must have set out a very long time ago; in order to receive light from them now, therefore, they had to be burning in the distant past. If the Universe has only lasted for a finite time then one can’t add shells out to infinite distances, but only as far as the distance given by the speed of light multiplied by the age of the Universe. In the days before scientific cosmology, many believed that the Universe had to be very young: the biblical account of the creation made it only a few thousand years old, so the problem was definitely avoided.

Of course, we are now familiar with the ideas that the Universe is expanding (and that light is consequently redshifted), that it may not be infinite, and that space may not be Euclidean. All these factors have to be taken into account when one calculates the brightness of the sky in different cosmological models. But the fundamental reason why the paradox is not a paradox does boil down to the finite lifetime, not necessarily of the Universe, but of the individual structures that can produce light. According to the theory Special Relativity, mass and energy are equivalent. If the density of matter is finite, so therefore is the amount of energy it can produce by nuclear reactions. Any object that burns matter to produce light can therefore only burn for a finite time before it fizzles out.

Imagine that the Universe really is infinite. For all the light from all the sources to arrive at an observer at the same time (i.e now) they would have to have been switched on at different times – those furthest away sending their light towards us long before those nearby had switched on. To make this work we would have to be in the centre of a carefully orchestrated series of luminous shells switching on an off in sequence in such a way that their light all reached us at the same time. This would not only put us  in a very special place in the Universe but also require the whole complicated scheme to be contrived to make our past light cone behave in this peculiar way.

With the advent of the Big Bang theory, cosmologists got used to the idea that all of matter was created at a finite time in the past anyway, so  Olber’s Paradox receives a decisive knockout blow, but it was already on the ropes long before the Big Bang came on the scene.

As a final remark, it is worth mentioning that although Olbers’ Paradox no longer stands as a paradox, the ideas behind it still form the basis of important cosmological tests. The brightness of the night sky may no longer be feared infinite, but there is still expected to be a measurable glow of background light produced by distant sources too faint to be seen individually. In principle,  in a given cosmological model and for given assumptions about how structure formation proceeded, one can calculate the integrated flux of light from all the sources that can be observed at the present time, taking into account the effects of redshift, spatial geometry and the formation history of sources. Once this is done, one can compare predicted light levels with observational limits on the background glow in certain wavebands which are now quite strict .