What's the issue with Olbers' paradox?

Question

I'm not quite grasping the reasoning behind Olbers' Paradox, or why an eternal, non-expanding and spatially infinite universe would be incompatible with a dark sky. For simplicity, let's suppose that all the stellar matter in the universe is transparent, such that to look at any direction in the night sky is equivalent to looking at an infinite number of stars or light sources. Yet I still don't see why it must follow that every point in the night sky will be infinitely bright (or at least sufficiently bright), provided that the sum of the apparent luminosity for all the stars at a given point converges to some finite value. For example, assume we drew a line to the infinite horizon and the first star intersecting that line had an apparent luminosity of 1 lumen, the second star .1 lumen, the third star .01 lumen and so on and so forth...

Furthermore, if this finite convergent value should happen to be lower than the detectable optical threshold of the human eye, that point of the night sky would still appear to be dark. Of course, this assumes that all the stars are transparent, and that they do not occlude any light. In the real world this effect will be lessened due to stellar opacity, but even this hypothetical scenario involving complete transparency doesn't seem to necessarily invoke a paradox, provided that the stellar density is such that the apparent brightness falls off at a level fast enough.

To put it another way, in an eternal, non-expanding, and spatially infinite universe, every point in the night sky will indeed intersect with some distant star, but there is no reason to think that the received luminosity of that star will be bright enough to be noticed. So, what I am I missing? Why is it commonly assumed that Olbers' paradox precludes an infinite static universe?

Answer 1

Olber's paradox is - as you state - the phenomenon, that the night sky is dark, but would have to be bright as the sun if the universe was infinite and infinitely old. The unspoken assumption here is also that the universe is at least on larger scales isotropic, that is about the same everywhere.

So if the universe is the same everywhere (and with everywhere we can be generous and consider distances between our galaxy and its neighbours still small), then it means that we will see some star or galaxy whereever we look. Galaxies are made of stars, so we will see stars everywhere - and every star is approximately as bright as the Sun - thus we would have a sky as bright everywhere as the disk of our sun. Maybe put the argument differently: consider the sky covered by suns at different distances. But in every direction you see a sun. So in essence it means that every direction you look at a Sun's surface. The contribution of every single Sun may be tiny (depending on how far it is) - but there are infinitely many Suns and thus the surface brightness is the same everywhere. The stars are only so faint because they are so far. Mathematically: you double the distance, you get a quarter of the intensity from every single star (inverse square law). But doubling the distance also means we have four times as many stars (the volume of a spherical shell is $\propto d^2$. Thus both effects taken together, effectively every distance doubled adds the same brightness as the previous one as the reduction in the brightness of each single source is exactly countered by the amount of sources we look at. Now, in an infinite, static universe there are infinitely many such shells, and we add infinitely many constant brightness values, leading to an infinite brightness (or at least the same brightness if we (wrongly) assume that the foreground stars hide the stars behind them.

Now, the argument may be "there is matter between the distant stars and us which absorbs the light". Yes. But we are talking of a static, infinitely old universe. The matter will be heated by the incoming light - and eventually it will become similarily hot as the source(s) which are heating it, thus also reach the temperature of stellar surfaces and emit light at the same intensity. Stellar opacity is not an argument here either - it also just is matter and energy does not get lost and will simply heat up all the same like all other non-stellar matter up to the same temperature as it is heated by: the radiation from the stellar surfaces.

Answer 2

A couple of good answers already, but this might be an easier way of thinking about it.

The light you see from a star comes only from the photons emitted at the tiny specific angle just right enough to enter your eye. That angle is very close to zero, but not zero. As a star moves further away, that angle gets smaller and smaller, so more photons miss your eye, the result is the star appears dimmer because a tiny portion of the sky that was sending photons to your eye is now dark sky. Those photons reaching your eye are not less intense, there are just fewer of them. If, instead of dark sky, you fill that darkness with another star (or stars) of equal brightness. So, your eye will interpret that region just as bright as it was before. Extrapolate that across the whole sky, and there is no region not sending photons directly towards your eye at full intensity.

Answer 3

It's an easy integral assuming four things:

• The universe is infinite in extent,
• Stars are uniformly distributed in this infinite universe,
• The universe is infinitely old,
• Newtonian mechanics applies, implying that light from remote stars is inversely proportional to the distance squared.

Given these assumptions, we not only would not have non-dark skies, we wouldn't even exist because the sky would be infinitely bright. As I mentioned at the top, it's an easy integral with these four assumptions: the brightness is $\int_0^\infty \text{some constant}\, dr$. In other words, infinitely bright. This is Olber's Paradox.

General relativity combined with big bang theory offers a solution: The last two assumptions are false. The universe is not infinitely old, and light from ever more remote stars is diminished not only by distance but also by redshift due to expansion.

Answer 4

Another version of Olber's Paradox: consider a cube with side length, say, 1kLY (1000 light years). This cube will have stars inside of it, and the light from those stars will be exiting the cube through its sides. But we can make a symmetry argument that there should also be light coming into the cube through the sides, and for the average cube, the light coming in will equal the light going out; there's no reason for one quantity to be larger than the other. So while there will be individual photons leaving the cube, there won't be any net light leaving the cube, and so we might as well imagine the cube being surrounded by mirrors, reflecting all the light back into the cube. If the stars are infinitely old, and have been producing light infinitely long, and none of that light has left the cube, then there is an infinite amount of light in the cube.

Thermodynamically, if heat is being produced, but the region producing the heat is at finite temperature, it must be shedding that heat to somewhere. And that somewhere must be cooler than the region. But if the universe is at large scale uniform, then each region is on average the same temperature as everywhere else, so there's nowhere for the heat to go.

Answer 5

It seems there are generally some misunderstandings regarding the nature of Olber's paradox: it is usually claimed that the paradox arises from the assumption of a universe infinite in space and time. This is not at all the case. The crucial point of Olber's paradox is that it assumes permanent production of energy in any given volume element but no losses of energy. We can write the local change of energy as

$$ \frac{dE}{dt} = P_L - L_L + P_T -L_T $$

where $P_L$ is the local production rate of energy in the volume element (assumed sufficiently large), $L_L$ the local loss rate, and $P_T$ and $L_T$ the 'transport' production and loss rates associated with energy entering and leaving the volume from/to neighbouring volumes. Hovever, for a homogeneous universe, we have obviously $P_T$=$L_T$, so we are just left with the local terms

$$ \frac{dE}{dt} = P_L - L_L $$

and for $L_L=0$ (as assumed in the formulation of the Olber's paradox) we would then have $dE/dt = P_L$ , that is the energy per volume element would be increasing with time, so no steady state would be possible. For the latter we require $P_L=L_L$ so that $dE/dt=0$.

This shows that Olber's paradox is not directly related to the question whether the universe is finite or infinite but merely whether the local energy loss rate balances the local production rate. Olber's paradox inconsistently implies that it does not, as it assumes that a given volume element has sources of radiation continuously producing energy but no sinks removing energ. This is inconsistent because in a closed system (and even an infinite universe is closed as no energy can enter or leave it) the energy conservation law would be violated as the total energy would not be constant. And this should not be interpreted as an equality of energies mysteriously entering and leaving the system (which would still mean the system is not closed despite energy conservation being preserved) but as the existence of some cycle that changes the different forms of energy into each other within the closed system.

Answer 6

Olbers's Paradox actually fails to demonstrate a finite or nonuniform universe and was supposed to be a paradox because its model for the night sky was missing crucial aspects that have since been filled in, at least in part. It ignores the existence of non-luminous interstellar matter, which we have since learned about (and which modern astronomical theories explicitly rely on). Such matter is known to attenuate light according to an exponential decay law (see the Beer-Lambert attenuation law). Olbers himself disregarded this law as it was not documented with respect to starlight until 1930, more than a century after he proposed this supposed paradox as a thought experiment. Those arguing against the existence and prevalence of such matter are asking us to presuppose the non-existence of planets, moons, asteroids, dust and gas (from which the stars themselves are said to form), nebulae, dark matter, etc.

I have written a basic simulation to validate the effects of non-luminous matter quantitatively and qualitatively. Code here. It directly applies the exponential decay law by assuming a uniform random density of stars and a uniform density of non-luminous matter throughout space. The simulation even cheats massively in favor of Olbers's Paradox by modeling the infinite backdrop of stars at varying distances as a purely luminous wall of light at a fixed, finite and even shortest possible distance beyond the foreground model of direct sampling. This means we have a strict theoretical upper bound on the total light emanating from distant stars. The density of non-luminous matter is controllable as a parameter p. The simulation shows that for even very small values of p, a purely luminous wall would be practically invisible, or else be indistinguishable from cosmic microwave background radiation (interstellar attenuation not only decreases apparent intensity, it also induces reddening).

In other words, Olbers's thought experiment would fail to falsify even the hypothetical existence of a contiguous sphere of light as luminous as the surface of the brightest star enveloping the entire visible portion of the universe. It can say nothing about what is beyond the photons that reach us--nor does it tell us anything about the photons that didn't reach us.

If one sets p=0, the entire field of view is indeed saturated by pure, contiguous starlight under this simulation. But for any non-infinitesimal density p of attenuating matter, the visualization -- and the mathematical solution for expected light intensity over all points in space--converges sharply towards a model that perfectly resembles what we see in our night sky, regardless of all other parameter settings.

In the following two images, the absorption value is increased only slightly, starting as close as the simulation allows to zero at first:

In both images, there is a perfectly bright wall of simulated stars a short, fixed distance away from the observer. Radiation from that wall leaks and is visible as a uniform grey background when the simulated wall is very close and the density of attenuating matter approaches zero, but if the wall of light is slightly more distant or the density of attenuating matter is only slightly increased, the background approaches pitch black, despite there still being an infinite number of perfectly bright stars intersecting every possible line of sight.

You can bump up the stellar density, radii and luminosity as high as you want, and even a tiny percentage of non-luminous matter still overwhelms the resultant apparent brightness calculations with the exponential decay induced by material attenuation.

Of course arguments about thermodynamics are just begging the question. How nonluminous matter is able to be there and coexist with stars is a separate matter of discussion. (How stars got there is too, come to think of it). The fact that it is there and that we know it is there, even at great distances, disproves the idea that Olbers's paradox is a paradox at all, as it completely fails to falsify an infinite universe given what we know.

In case someone thinks this idea is flamingly absurd, that even a small amount of matter between stars can block nearly all of the light from them, remember that despite how brightly the Sun shines at noonday and how close it is to the Earth, you can block almost all of that light using a thin cardboard cereal box or a parasol consisting of cloth a couple millimeters thick. This gives you an idea of how effective exponential attenuation is. Can there not be the equivalent matter of one millimeter of cardboard between you and a star 3 billion light years away? If not then what are stars made of?