Olbers's Paradox

Alan Cai

Decemer 8, 2023

Consider the following: assuming the universe contains an infinite amount of stars spaced approximately evenly with respect to Earth, why is the entire night sky not entirely covered in light as every single ray from Earth will eventually end in a star?

Olbers’s paradox, named after German astronomer Heinrich Olbers, states that in such a static universe, one which is infinite, one would expect that any celestial surface would be covered in light due to the necessity of the existence of a star at the end of any ray. However, prudent observation is not necessary to quickly determine that such is not the case.

Wikipedia provides another way to think of it mathematically. Imagine spherical shells of thickness one light year radiating from the earth. For extremely large radii, the volumes of these shells will approximately equal the surface area. Denote the radius of a particular large shell as N. The shell of radius 2N will have area 4N. However, note that light coming from twice the distance away will be one-quarter as bright. This is because light and distance follow an inverse square relationship as further light must shine over a larger area, a quantity that increases quadratically. Assuming that the shell with thickness 2N has the same star density as the shell with thickness N and the stars are spaced approximately evenly apart(given the large number of stars in the universe and its expansion, this is reasonable to conclude), then the total amount of light coming from the larger shell and the total amount of light coming from the smaller shell are approximately equal. Thus, if there are infinitely many shells arranged with radii increasing geometrically by a factor of 2, then the positive amount of light emitted by the shell with radius N will be multiplied by infinity, resulting in our planet, and any point in the universe, to be lit up by light in any direction.

The reason why our planet’s surface is not the brightness of a star is because although the universe may be infinite, the observable universe is finite. In other words, despite the infinite amount of stars, only a finite amount of them could be seen. From the mathematical perspective, shell multiplication must eventually stop because they will approach the edge of the visible or observable universe. Technically, the visible and observable universe are distinct terms as the latter includes light emitted from a slightly more distant time, for the purposes of this article, we will consider the two to be the same.

Einstein and other theoretical physicists postulated that light travels at a fixed finite speed in a vacuum and that no object can travel faster than the speed of light. Therefore, assuming the universe is around 14 billion years old, the maximum distance light reaching the earth could have traveled is 14 billion light years. However, this is not the maximum extent of the visible universe because although the light that reaches our eyes may have traveled up to 14 billion years give or take, the universe has expanded since then and the distance between the earth and the light source has increased manyfold. Therefore, the visible universe is around 100 billion light years across. This celestial sphere is what we call the observable universe. Nevertheless, it is important to note that the universe is much larger as universe expansion has caused objects to move from the earth faster than the speed of light, making their visibility impossible.