# Mathematical proof of Olber's paradox

Is there a mathematical proof regarding Olber's Paradox?

For me this feels a false paradox as I think that it IS possible to have an infinite (in space and in time) Universe with an infinite number of stars, homogeneously spaced but where there are many rays in the sky that never intersect a star. i.e. the sky will be mostly dark.

Mathematically, consider $$\mathbb{R}^3$$ as the space, and an infinite (but countable) number of unit length balls $$B_i = \{x\in\mathbb{R}^3: \|x-c_i\| \leq 1\}$$ for specified centers. Can we distribute the infinite balls homogenesouly throughout the infinite space such that we will have at least one ray (a half line starting from the origin) that never intersects any such balls?

My intuition (which is wrong more than half the time) tells me that we can! But many sources online seem to claim that we cannot and use this fact to prove that either the universe in finite in space or in time (or in both)

Is there a mathematical proof for this mathematically simple formulation of Olber's paradox?

## 1 Answer

If the balls are distributed randomly with statistical uniformity in space, and we examine some spherical shell of some width, we can say that on average the balls will occupy some fraction $$F$$ of the shell's cross-sectional area (as seen from Earth). The probability of a ray making it through the shell without intersecting a ball will then be $$1-F$$. But now the ray has to make it through a second shell, and that probability will also be $$1-F$$, and so on. So the probability of a ray making it through $$N$$ shells is $$(1-F)^N$$. Since $$1-F<1$$, this probability goes to zero as N goes to infinity. It would seem then, that given an infinite, homogeneous, statistically random distribution of stars, almost all rays will intersect a star at some point. There may be some set of rays that will make it "all the way to inifinity" but that set will surely be of measure zero.

Now, as a pure math problem, there might be many ways to arrange stars homogeneously so that there are big patches of dark in the night sky. Maybe we could arrange the stars in some crystal or quasi-crystal lattice, for instance. But, physically, no such arrangement would be stable; such an arrangement would last for only a short time, giving way to some random distribution very quickly. And there is indeed no evidence that stars, galaxies, what have you, were ever arranged in such a fashion. So to discuss Olber's paradox as a true physical problem, we have to deal with a random distribution. And I think this eliminates pretty much any arrangement any determined mathematician might come up with to evade the paradox.