Tom Levenson at the Inverse Square Blog posts a discussion of Olber’s Paradox. The gist is that if we lived in an infinite, static, homogeneous universe, there would be light everywhere. That empirical falsehood was tough on folks way back when, many of whom believed the universe subscribed neatly to Olber’s little enumeration.

I would rather not repeat what’s already been said on Wikipedia, so I’ll assume that you’re already mildly familiar with the argument. Also, I’m not especially concerned with what real cosmology has to say about things. I want to think about this under the same terms good ol’ Olber could have. Let’s hash out some implications of this static, infinite universe without worrying about all that Stephen Hawking shit.

The first thing to point out is that if the distribution of stars were inhomogeneous, we could avoid the problem. For example, if the density hot star matter went as $\rho \propto r^{n}$, then we would have infinite flux of light received on Earth for $n\geq-1$ and finite flux for $n<-1$. (Technically, it would produce infinite flux even in this second case due to the singularity at $r=0$, but we’ll assume there is some small region near Earth for which the distribution no longer holds). We could even estimate the absolute size of the universe by sampling the density of stars at a few depths to obtain the power law, then finding the size the universe would need to yield the correct average brightness of the night sky.

One problem with this power-law crisis resolution is the creation of a center of the universe – the spot where $r=0$. Historically, once we trashed geocentricism, we pretty much trashed inhomogeneity (on large scales only, since otherwise there would be no point in a PB&J sandwich, which under perfect homogeneity would become an abominable blenderized bastardization) at the same time. Even though this particular solution to Olber’s Paradox does not require the Earth to be at the center of the universe (the flux is finite there, but it is also finite everywhere else), it’s still rather philosophically unattractive. We’ll throw it out.

Instead, focus on the case $n=0$, that is, the homogeneous universe. Many sources claim that in this universe, all points on the sky would be as bright as a star, because wherever you looked, there was sure to be a star in that direction some distance off. (Both Tom’s post and Wikipedia make this claim.)

That claim is wrong. You wouldn’t have every point in the sky as bright as a star. You would have every point in the sky as bright as infinitely many stars. That is, you would get infinite flux density from every single point on the sky. Even if you looked at a patch of sky one arcsecond on a side, you would get infinite light from that patch. Sure, when you look at any direction you’d see a star, but then if you looked further in that direction you’d see another star, and another, and another. The “anothers” never end in an infinite universe.

Let’s say we look at a patch of sky the size of the moon in our toy $r^n$ star density universe. If we only count the stars back to some finite depth $d$, then the total amount of light we receive scales as $d^{n+1}$. The exception is $n=-1$, in which case flux scales as $\ln(n)$, and hence still diverges. (Here I’m referring to the catastrophe cases $n\geq-1$. For $n<-1$ we get some constant flux minus $d^{n+1}$, so the total flux converges towards a constant value.)

That disagreement on the brightness of the sky is crucial. If every point on the sky were as bright as a star, it would get quite toasty around here. In fact, the second law of thermodynamics ensures that the Earth would heat up to the temperature of a star, until it (the Earth) also glowed star-hot, and hence lost heat as quickly as it came in.

This “constant light everywhere” situation is not really so far from the truth, since every direction in the sky does glow the same temperature. It’s just our luck that the temperature of the night sky (more commonly, the Cosmic Microwave Background) is two orders of magnitude colder than the Earth, and that the Earth is about an order of magnitude colder than a star. Nice place to be, thermodynamically, as Sean Carroll pointed out in a public lecture I previously wrote about.

However, if we have the situation predicted by the universe in Olber’s Paradox, we would actually have the Earth getting infinitely hot. For that matter, since the Earth is nowhere special in this model, there would be infinite energy density everywhere. That’s a bit stronger of a quandary than “why is the sky is dark?”.

Where could this infinite energy come from? I have to admit, this “infinite universe” thing is pretty tricky to wrap your mind around. It’s clear that an infinite universe would have infinite total energy. But that doesn’t mean it couldn’t have finite energy density. We concluded, though, that it doesn’t. Something is wrong. Of course, that happens a lot, with infinity.

Energy is conserved, but only in a closed system. Something that’s infinite is not closed. Basically, the infinite universe has infinite energy the same way we Americans figured out how to get infinite money with Social Security – by borrowing it from our infinite future. (That does work, right? My current career plan is to go into stasis for forty years right after I graduate and then start collecting my dues first thing on thawing out.)

It is hard to imagine living in an infinite universe, especially a static one. If all times and all places are the same, then how did humans come to choose this time and this place to exist? In an infinite universe, wouldn’t it be true that anything that can happen already has happened, infinitely many times? Wouldn’t someone exactly like me have written this exact blog post over and over endlessly back into eternity? Far out, dude.

I simply cannot imagine an infinite universe. The finite speed of light effectively allows us to borrow energy from the past. But it’s an infinitely-large, infinitely-long past, and consequently we would have infinite energy. There’s no energy conservation paradox, because the universe never transitions from a starting point with finite energy density to an ending point with infinite energy. A infinite universe simply does not have that starting point to begin with. It’s all way too insane.

You could postulate an infinitely-large, infinitely-old universe with finite energy density everywhere, but you’d have to kill off this light-travel mechanism for borrowing energy from the past (which is also a mechanism from broadcasting energy into the future.) You’d have to keep things where they are. Which, with a universe like this, is in your imagination.

### 9 Responses to “Olber’s Paradox”

1. Tommi Says:

In an infinite universe, wouldn’t it be true that anything that can happen already has happened, infinitely many times?

No, not really.

I can, for example, take an infinite number of infinite sequences of natural numbers. The numbers sequences are (1, 0, 0, 0, …), (0, 1, 0, 0, …) and so on, with always a single 1 at nth place in the sequence. Although there is an infinite number of such sequences, it does not contain every possible sequence of natural numbers.

So in your infinite universe there always could be different sequences in this very blog post, with everything else remaining constant.

No argument with regards to the main point of the article.

2. sophismata Says:

“Sure, when you look at any direction you’d see a star, but then if you looked further in that direction you’d see another star, and another”

But once you see a star you can’t see further, because the light of further stars would be blocked by the closest one. At least that’s the argument.

3. meichenl Says:

Sophismata,

That is a common argument, but it is incorrect. The one star doesn’t block the stars behind it and just block them. If it blocks the star behind it, it absorbs light from that star and heats up. When it heats up, it glows more brightly. That’s true for every single star out there – there’s nothing special about those few that happen to be in the line of sight of Earth. So all stars must get hotter. But then if they all get hotter, the radiation they absorb from each other increases, and they get hotter again, to infinity.

4. sophismata Says:

Tommi, your “infinite sequences” example is flawed. You say “it does not contain every possible sequence of natural numbers”, but that’s because it can’t by definition. If your sequences are made as “always a single 1 at nth place in the sequence”, then of course it can’t include the sequence (0,2,0,0,0…)

5. meichenl Says:

Tommi – good point!

I was thinking along the lines of Poincare Recurrence Theorem.

6. sophismata Says:

“If it blocks the star behind it, it absorbs light from that star and heats up”

Yes I know, but did Olber know it? We are trying to think about this under the same terms good ol’ Olber could have…

7. Tommi Says:

Sophismata;
My example is not flawed. It contains an infinite number of sequences (construction a bijection between the sequences and natural number is trivial), each of which only contains natural number and each of which is infinite. This is what was required.

Of course I defined it in such a way that not all sequences of natural numbers are there. If it did contain all the natural numbers, then it would not be a counterexample and would be flawed indeed.

Counterexample for the notion that if (1) there is an infinite number of something and (2) each case is different (alternatively, there are infinitely many different cases), then it follows that all possible cases happen.

Think of it in this way: There is a space of infinite sequences of natural numbers. I showed one way of selecting an infinite number of them such that not all sequences are selected.

meichenl;
I’m just saying that something is logically possible, not that it would be actually possible. The recurrence theorem is interesting (I don’t study physics), but it seems to fail patently given infinite space.

8. sophismata Says:

Tommi,

You provided a counterexample to something else. The main point was “anything that can happen already has happened”, but your counterexample purposefully creates a scenario where X cannot happen, breaking the premise.

9. Tommi Says:

sophismata;

I’ll try to be as clear as possible. Where, exactly, do you disagree?

What are all the possibilities, or, what can happen: Infinite amount of (countably) infinite sequences of natural numbers.

What must happen, or, what are the necessary conditions: An infinite number of infinite sequences of natural numbers.

Some examples that fill the requirement of “must”, but still do not contain all possible infinite sequences of natural numbers:
{(0, 0, 0, …), (1, 0, 0, …), (0, 1, 0, 0, …), …}
{(0, 0, 0, 0, …), (1, 0, 0, …), (1, 2, 0, 0, …), …, (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, ….), ….}.

So: If only an infinite amount of infinite sequences of natural numbers is required, it is not necessary that all infinite sequences of natural numbers must be included, as the two examples show.