Ok, this is a bold question, I know. But, let me explain: After first hearing about Olbers' paradox, I found that something seemed 'off' about it, so I looked into the subject as much as my skills (and time!) would allow. And I came to the conclusion that the paradox was probably kind of like Zeno's paradox: That is, something that 1., would not correspond to physical reality, 2., would not because some concept(s) was/were 'left out' during its formulation, in Zeno's case, the concepts of basic calculus.

Now, of course, I'm well aware of the fact that many of the best ideas in physics are counter-intuitive: General Relativity (and Special Relativity to a lesser degree, in my view), Quantum Physics, The Big Bang, even certain Elements of Thermodynamics, etc... However, Olbers' paradox doesn't seem problematic because of its counter-intuitiveness; but instead because it does not seem to me that it would happen in a steady-state, infinite, infinitely old version of our universe. Here are the reasons why:

1. Logical / Mathematical:

Lets invent the universe, lol. Well, a universe with an observer, and a light $D_1$ distance away such that the light source appears as a Small Disk of light. Now imagine adding another light source at twice $D_1$ away (retaining the same x, and z), $D_2$, and then 3 times as far away (retaining the same x, and z), $D_3$: And, so on, $D_1, D_2, D_3, D_4....$, and so on forever. Given the inverse square law, the observer would not see a field of view filled with light. They'd see the sum of the ever fainter lights, (The answer to which, not the sum itself) would be finite and pretty faint. And this shouldn't change, as far as I can tell, if they waited forever, or if 'god' copied and revolved these light sources around a range of axes, so long as there were a finite (not too big) number of near-by light sources.

2. Empirical:

A lot of the light would become unseen for many reasons everyone here knows.

Alright, Olbers' paradox is unimportant today, as the universe isn't static or infinitely old. However, I think it shouldn't be presented as a sound idea if it is invalid.

So, I'd like for someone to explain to me, 1. Why I'm wrong, or 2. if I'm not wrong, why the paradox hasn't been examined more carefully.

Notes For Answerers:

Please note that people tend to answer questions like this by 'adding' stars of some brightness to the sky until the sky is super bright. This doesn't make sense as a star far enough away would be invisible to the naked eye due to: 1. The brightness of nearer stars. 2. The fact that if far enough away it would simply be impossible to see. In an "Olber Universe,' it seems that the sky 'around' the visible stars would seem dark to the human eye, but with tek you could 'find' a star anywhere you chose to look.

  • $\begingroup$ Olber's paradox fails to take into account many elements of the real universe: the age of the universe, the distribution of stars, the finite lifetime of stars, IGM absorption,... And I have never seen it presented as a sound idea, only as an historical anecdote. $\endgroup$ – usernumber Jan 31 at 9:01
  • $\begingroup$ "A lot of the light would become unseen for many reasons everyone here knows" is too vague to be a meaningful objection, so it's difficult to see how anyone can answer you in that context. You need to be more specific. $\endgroup$ – Peter Erwin Jan 31 at 9:29
  • $\begingroup$ @usernumber the "paradox" can be seen in the light of the xvii xviii centuries. As for the real major cause of a dark sky there are calculations even nowadays. One must to accept the premises (eternal static universe with an infinite number of stars) to appreciate the paradox. Ps I read that E. A. Poe was among the first writing down that perhaps all the light didn't yet had time to reach us, posing a kind of beginning $\endgroup$ – Alchimista Jan 31 at 10:58
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    $\begingroup$ Note that this question was cross-posted to Physics.SE, where it is now closed. $\endgroup$ – PM 2Ring Jan 31 at 13:03
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    $\begingroup$ @Alchimista You're not using standard English. $\endgroup$ – Jinny Ecckle Feb 2 at 8:13

Let me see if I can show why your "inverse-square-law means you can't get light from distant sources" intuition is wrong.

For the sake of argument let's assume stars really are point sources, and look at how much light you would receive from an infinitely old, infinitely large universe uniformly filled with point-source "stars", each with luminosity $L$, at some finite density $\rho_{0}$ (# stars per cubic light year).

Imagine a thin spherical shell located at distance $D$, with thickness $dD$ (small compared to $D$). Each star in that shell will give you a flux of $L / (4 \pi D^{2})$ (there's your inverse-square law). How many stars are in that shell? That's just the volume of the shell -- approximately $4 \pi D^{2} dD$ -- times the density: $4 \pi D^{2} dD \rho_{0}$.

Now consider a shell at twice the distance ($2 D$). Each star in that shell contributes 1/4 the flux of a star in the first shell, since they're twice as far away. However, the volume of the thin shell is four times larger than the volume of the first shell ($4 \pi (2 D)^{2} dD = 16 \pi D^{2} dD$, and so the number of stars in that shell is four times larger, which perfectly cancels out the decrease in flux. So the light coming from stars in that shell is the same as the light coming from stars in the first shell. Since there is an infinitude of such shells in the default infinite universe of the Paradox, and each shell contributes the same flux, the total flux you get from all the stars will be infinite.

Fortunately for the hypothetical inhabitants of that universe, the fact that stars are really finite in size prevents this from happening, since the finite disk of a star blocks light from all the stars directly behind it. But that means that wherever you look, your line of sight intersects with the surface of a star, which is the same as looking within the disk of the Sun.

  • $\begingroup$ Thank you for the post! I'm very grateful that you took the time to go over this interesting topic with me. Unfortunately, the shells and integral and everything else are just a formalized version of the paradox. $\endgroup$ – Jinny Ecckle Feb 1 at 20:22
  • $\begingroup$ The crux of the issue for me is that there are physical arrangements of stars with (instant light propagation) that won't yield the paradox. This means that the amount of EM energy (per unit v) in space would be totally governed by how the stars were arranged; which seems like a ridiculous physical interpretation. This, to me, suggests that something is wrong here. The way to address this is: Messing with the geometry and noting other factors like the first law of thermodynamics. If you're interested, there's a chat on stack physics where we'll be working on this. Thank you! $\endgroup$ – Jinny Ecckle Feb 1 at 20:22
  • $\begingroup$ BTW, I never said anything like ""inverse-square-law means you can't get light from distant sources,"" I said the amount you'd get is the finite answer to a particular infinite sum.... $\endgroup$ – Jinny Ecckle Feb 1 at 20:30
  • $\begingroup$ Would you mind taking that out of the quotation marks? $\endgroup$ – Jinny Ecckle Feb 1 at 20:33

The error I think you are making in your "logical" argument is "A light [ie a star] at a distance D1 would a appear as a point of light". If D1 is a star, it would appear as a disc of light. A very small disc, but a disc with a finite and nonzero size.

We then add stars at distance D2, D3 etc. Each star covers more of the sky. This is because each star appears as a disc and not a point. Add infinitely many discs of light and you cover the whole sky.

If in every direction that you look you see a disc of a star, then every direction is as bright as a star. This is the crux of the paradox.

The brightness of the star (as a point source) is irrelevant, so the argument from an "inverse square law" is a red herring.

  • $\begingroup$ Thank you for the answer. But, unfortunately, you made a few errors, 1. $D_n$ are distances not a light. 2. The shape disk, sphere, or point makes no difference; the issue at hand is the inverse square law, not the shape (there are no points in reality other than maybe black holes, anyway lol). $\endgroup$ – Jinny Ecckle Jan 30 at 21:19
  • $\begingroup$ The significant issue here is that you just reaffirmed the paradox without really thinking about my Q. The thought experiment gives a finite sum, at least I think so. Really imagine spacing the lights like that, it should sum finite and small. Further, if you could really create this universe, as far as I can tell, it would act as I say. If not, please explain why? $\endgroup$ – Jinny Ecckle Jan 30 at 21:24
  • $\begingroup$ "If the universe were infinitely old, infinite in space and homogenous then every line of sight would, end on a star, and the night sky would be a uniformly as bright as the sun." This is the issue, every line of sight would end on a star, of course, the issue is that I can't see any reason why the sky would be as bright 'as the sun.' $\endgroup$ – Jinny Ecckle Jan 30 at 21:28
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    $\begingroup$ Why is the inverse square law a red herring? In Olber's model all stars are the same size & absolute brightness, i.e., we have a homogeneous isotropic eternal universe populated by average stars. A star at unit distance occupies an angular area of $a$ with apparent brightness $L$, so its brightness per unit area is $L/a$. A star at distance $d$ occupies an area $a/d^2$ with brightness $L/d^2$ due to the inverse square law, so its brightness per unit area is once again $L/a$. $\endgroup$ – PM 2Ring Jan 30 at 22:35
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    $\begingroup$ "1. The brightness of nearer stars. 2. The fact that if far enough away it would simply be impossible to see." This is what's wrong. $\endgroup$ – Rob Jeffries Jan 30 at 23:05

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