The paradox. Suppose the universe is endless, has been there for ever, and is filled evenly with stars. Look in any direction. Your line of sight must eventually end on the surface of a star, just as in a very big forest you eventually see a tree trunk whichever way you look. Then the whole sky would be as bright as the surface of a star (a white-hot glare), and night would not be dark. But the night sky is dark. This is Olbers’ paradox (Heinrich Olbers, 1823, though others had noticed it before).

Why distant stars do not get weaker in total. A star twice as far away appears four times fainter (the inverse square law). But in a shell twice as far away there are four times as many stars. So every shell of stars of the same thickness, near or far, adds the same amount of light. The total light grows steadily as more shells are added, with no limit. (That is the diagram at the bottom right.) In practice stars in front hide stars behind them, so the sky cannot get brighter than the surface of a star.

How far must we look? If stars have radius r and there are n of them in each cubic light-year, a line of sight has travelled on average a distance λ = 1/(nπr²) before it hits a star. About 10²³ stars in a universe of radius 46.5 billion light-years gives λ ≈ 2.5 × 10²³ light-years. A fraction 1 − e^−D/λ of the lines of sight hit a star within a distance D, and this is also the brightness of the sky compared with a star’s surface:

B = 1 − e^−D/λ ≈ D/λ when D is much less than λ

So the sky is dark when D is much smaller than λ and white when D is much bigger. With today’s numbers the sky would be about 10⁻¹³ as bright as a star: black, and 10¹³ times too dark to be white. The universe would need to be about 10¹² times larger and older than it is.

The resolution. Light travels at a finite speed, and the universe has a finite age (13.8 billion years). The light from stars farther than cT has not had time to arrive, so we only see a tiny part of the universe. Making the universe bigger alone does not help: the sky stays dark unless it is also very old. Making it older alone does not help either, unless it is also very big. Both must be enormous. Stars also do not shine for ever: there is not enough hydrogen to keep that many stars burning for 10²³ years. Olbers’ paradox is one of the clues that the universe began a finite time ago.

Things that are not the answer. Dust could not hide the light, because it would soak up the energy, get hot and glow itself. Expansion (redshift) does dim the light from the most distant galaxies, but only by a modest factor, and the finite age matters far more.

How the picture works. Each little square of sky is a line of sight. It is white if a star surface lies along it within the distance we can see: every square has a fixed random “distance to its first star”, so squares that are white at a smaller D stay white at a larger D. The row of lines in the lower picture is the same idea. Because the sky stays black until D comes within about a hundredth of λ, the slider seems to do nothing for a long time, and then the sky turns white within the last factor of 100 or so. That is real: we are twelve orders of magnitude short.