infinite space (and infinitely old) means sky too hot?

[halfeye]

I've heard it said that the fact that the sky isn't as hot as a star means that space isn't infinite because if it was every line though it would end on the surface of a star. That's never made sense to me, as stars get further away, the angular width of them gets smaller, so the foregoing assertion just doesn't seem right to me.

I'm not saying I disbelieve in the big bang, I'm saying it looks to me as if someone has got some maths about infinity wrong, it may be that they haven't, but it just doesn't seem right to me.

[warty goblin]

I don't think the math is wrong, as follows.

A star can be abstracted as an angle and an angular size - technically the angular size is determined by both size and distance, but they all have an angular size with some distribution that isn't a point mass at zero. At the very least a visible star is occluding an area proportional to the square of the amplitude of visible light.

Let p be an arbitrary point on the sphere, and let's generate stars uniformly on the surface of the sphere. Because stars have nonzero angular width, every star has a nonzero probability of including p, say pi. Then we can let I(p, n) = 1 if p is covered by star n and 0 otherwise.

But I(p) is just the probability of at least one success in n Bernoulli trials with probability pi, or 1 - P(0 successes). That is simply 1 - (1 - pi)^n, which goes to 1 for all nonzero pi. If pi is exactly zero and stars are perfect points with probability 1, then yes, the heavens would not be occluded by a countably infinite number of stars.

[halfeye]

I don't think the math is wrong, as follows.

A star can be abstracted as an angle and an angular size - technically the angular size is determined by both size and distance, but they all have an angular size with some distribution that isn't a point mass at zero. At the very least a visible star is occluding an area proportional to the square of the amplitude of visible light.

Let p be an arbitrary point on the sphere, and let's generate stars uniformly on the surface of the sphere. Because stars have nonzero angular width, every star has a nonzero probability of including p, say pi. Then we can let I(p, n) = 1 if p is covered by star n and 0 otherwise.

But I(p) is just the probability of at least one success in n Bernoulli trials with probability pi, or 1 - P(0 successes). That is simply 1 - (1 - pi)^n, which goes to 1 for all nonzero pi. If pi is exactly zero and stars are perfect points with probability 1, then yes, the heavens would not be occluded by a countably infinite number of stars.

I don't follow that I'm afraid. The case that I'm sort of thinking of is that the radiation from a single star wouldn't after infinite time paint all the points on a sphere at infinity.

[warty goblin]

I don't follow that I'm afraid. The case that I'm sort of thinking of is that the radiation from a single star wouldn't after infinite time paint all the points on a sphere at infinity.

The short version of my argument is this: a star is not a perfect point and therefore has surface area when painted on the celestial sphere, i.e. everything you could see if you were a giant eyeball floating in space. You are correct that as the distance from a star tends towards infinity, the area of that star tends to zero, but for a visible star it is never zero, since this would be a point. One cannot see a point, because what one sees is light, and because light has wavelike properties, it has an amplitude (which for light is inversely proportional to the wavelength). In short, if you can see it, it has area. This means you don't even need an infinite number of stars to completely block out the sky, a strictly finite number will do. Technically this only shows a finite number of stars in the observable universe, but because information cannot propagate faster than the speed of light, stuff outside the observable universe can't effect us and doesn't matter.

It's important to keep in mind that not all infinities are equal, and none of them are real numbers that can be plugged into existing equations*. I get the sense you're trying to just plug in infinity when you talk about infinitely distant stars painting infinite spheres after infinite time. Unless you can be extremely rigorous in how things are tending to infinity, you won't arrive at any sensible conclusions. And the only way to be rigorous is to work with the limit as things tend towards infinity.

*Sometimes it looks like mathematicians are doing this. Really we're just not writing out the limit because it's boring and everybody reading it can do the limit calculation in their heads without needing a lot of extra notation.

[johannessmid]

I don't follow that I'm afraid. The case that I'm sort of thinking of is that the radiation from a single star wouldn't after infinite time paint all the points on a sphere at infinity.

It's because everything in this question scales as r^2. Consider a small bit of solid angle looking out. Look out 1 ly - you see X stars, with a certain average luminosity coming at you. Now look out 2 ly. The energy from a standard star is ~1/r^2, so you'll see 1/4 the luminosity from a star at that distance. But now that solid angle covers a physical distance ~r^2, or 4 times bigger, so you should see 4 times as many stars! The luminosity in that solid angle should have the same contribution from stars 1 ly away as 2 ly away.

So as you look farther away, yes, you are getting less energy from each star - but there are more and more stars, so you get equal energy from every distance.

[Fat Rooster]

Another way of thinking about it is to forget stars; the question is why are we not surrounded by 'big bang'? It happened everywhere in the universe simultaneously, and was extremely hot. Every straight line will eventually see the big bang, and stars would be cold spots against that. Every point should be always surrounded by big bang, and so remain at big bang temperatures. The question can then be phrased as "how did the universe cool down"?

The answer is that we actually do see light in every direction, but the expansion of the universe since it was created has doppler shifted it down to the microwave range (the CMB). Basically the distance between the peaks of the waves that were really close together (low wavelength means high energy) when they started is now orders of magnitude bigger because the space between them expanded! The waves are suddenly much lower energy.

From there you just reintroduce time, and you see that even if every line did eventually hit a star, the light from that star would not have reached us yet (and the expansion of the universe actually means it never will), and we see the big bang* occurring at a point between us and the star. If you pick a 'dark' direction, the light coming from that direction is the light that was there when universe first became transparent. It formed then, and has been travelling ever since, getting colder all that time.

*Actually we see a moment a bit after the big bang, because things were so hot that the universe was opaque. What we see is the moment it suddenly became transparent.

[Anymage]

The argument was against the idea of the universe just having existed forever. Like Johannesmid noted, if the light from all the stars did have time to reach us (whether due to a universe that has been around for an infinite amount of time or due to an infinite speed of light), the dimming of stars at a given range would be exactly balanced by the fact that there are more stars at a farther away range, and every sight line would end in a star. Otherwise would imply absolutely nothing in that direction out to infinity, which would be a very interesting observation.

(A finite universe that had either been around forever or has an infinite speed of light would also be interesting, but neither of those are really relevant to our case.)

As Fat Rooster noted, all this really means is that the universe is not infinitely old. Plenty of other observations have allowed us to put together the big bang theory and ballpark just how old the universe is. So long as there's a limit to how far we can see, even going with the simplified case where all we have to think about is stuff being so far away that its light hasn't had time to reach us yet, the problem solves itself.

[Radar]

It's because everything in this question scales as r^2. Consider a small bit of solid angle looking out. Look out 1 ly - you see X stars, with a certain average luminosity coming at you. Now look out 2 ly. The energy from a standard star is ~1/r^2, so you'll see 1/4 the luminosity from a star at that distance. But now that solid angle covers a physical distance ~r^2, or 4 times bigger, so you should see 4 times as many stars! The luminosity in that solid angle should have the same contribution from stars 1 ly away as 2 ly away.

So as you look farther away, yes, you are getting less energy from each star - but there are more and more stars, so you get equal energy from every distance.

This is still not quite right as the further away the source is, the more likely the light is to be absorbed by something on the way. So the estimation of infinite energy passing through any given point in the universe cannot be correct. Since absorption is a nicely random process, the flux of energy from any given source is additionally mitigated by an exponential factor e^(-a r), where a is some absorption coefficient. Obviously whatever absorbed the light between the star and our observation point will also emit the energy back, but we already count that source.

At the end of the day the conclusion is that an infinite, not expanding and infinitely old universe has to be in thermal equilibrium but it can still exist with finite temperature as one of the unchanging parameters.

edit: actually there is another thing as all the above considerations assume flat spacetime. As far as i can tell, it all turns out the same for hyperbolic and spherical geometry but I am eyeballing it.

[Fat Rooster]

This is still not quite right as the further away the source is, the more likely the light is to be absorbed by something on the way. So the estimation of infinite energy passing through any given point in the universe cannot be correct. Since absorption is a nicely random process, the flux of energy from any given source is additionally mitigated by an exponential factor e^(-a r), where a is some absorption coefficient. Obviously whatever absorbed the light between the star and our observation point will also emit the energy back, but we already count that source.

At the end of the day the conclusion is that an infinite, not expanding and infinitely old universe has to be in thermal equilibrium but it can still exist with finite temperature as one of the unchanging parameters.

edit: actually there is another thing as all the above considerations assume flat spacetime. As far as i can tell, it all turns out the same for hyperbolic and spherical geometry but I am eyeballing it.

I can't say exactly where your mistake is, but you have definitely made one, because there are no energy sinks in that description. Stars definitely turn matter into different matter and release energy. That is a source. Without some sink to reverse the process it cannot be in equilibrium. Every point in space would be being heated by the local stars to the point where it becomes the same temperature. It cannot dump energy into other points in space because they will all be experiencing the same thing, so be at the same temperature. Every direction is either the same temperature, or hotter. There will be no colder regions of the sky to radiate more energy into than is received. I guess you could put enough black holes into the system that only ~1 in a million straight lines hits a star before a black hole. Even they fail eventually though, because they would need to be continually growing. A shrinkage term could be balanced against the extra material required for new stars though I guess.

It does not turn out the same for hyperbolic geometry, because in order for space to be hyperbolic; space-time must be hyperbolic, or you break the relativity principle (observers see the same rules no matter their relative velocities). To see this consider the force on a bar holding two masses. If the bar is moving sideways, each of the two masses tries to follow a straight line. In hyperbolic space that would mean the distance between them will try to increase over time (as straight lines diverge), resulting in a tensile force on the bar that is not seen on a stationary bar. That is not allowed. The only way to resolve this is that even 'stationary' points will diverge over time, meaning everything experiences the same force. In other words, the universe is expanding, with the implied doppler effects.

An expanding universe that is continually forming material is actually a solution to it (though breaks our understanding of entropy, but infinite time requires that), because the expansion does provide the required cooling term. Even though we would see a star in every direction, the light would have been doppler shifted to the point of being cold. It is only ruled out by actually looking, and noting that what we actually see is this 'thing' at ~14 billion years ago that happened everywhere and was absurdly hot. We can see the big bang's afterglow (though thankfully expansion has tamed it to not fry us). Closer than that we see structures that are noticeably different than what we see today (far less metals).

Edit: As for whether every line hits a star, the relevant concept is mean free path. Assuming our local space is representative we can define a mean free path to hitting a star. We can then determine the distance (d) where the probability we will hit a star if we draw a random straight line of that distance is 50% (or any other percentage). For every direction we can ask "what is the probability we will not hit a star in 1x that distance. That is basically the same as asking whether there is a star in the random straight line of length d: 50%. What about 2d? That is the chance you don't hit a star in the first section, multiplied by the chance you don't hit one in the second section, or 25%. At n x d distance, the probability there is no star on that straight line is 0.5ⁿ. If n is infinity then the chance we do not see a star is 0, so the chance we do see a star is 1.

[Radar]

I can't say exactly where your mistake is, but you have definitely made one, because there are no energy sinks in that description. Stars definitely turn matter into different matter and release energy. That is a source. Without some sink to reverse the process it cannot be in equilibrium. Every point in space would be being heated by the local stars to the point where it becomes the same temperature. It cannot dump energy into other points in space because they will all be experiencing the same thing, so be at the same temperature. Every direction is either the same temperature, or hotter. There will be no colder regions of the sky to radiate more energy into than is received. I guess you could put enough black holes into the system that only ~1 in a million straight lines hits a star before a black hole. Even they fail eventually though, because they would need to be continually growing. A shrinkage term could be balanced against the extra material required for new stars though I guess.

I am not talking about energy sinks (and black holes are very much not energy sinks - nothing truly is): I am talking that the calculations counted the same energy many times as the light will be absorber and reemitted on the way to to observer by the stars closer to the observer than some distant sources. This is a minuscule difference as the angular cross-sections are small, but it is crucial in properly estimating the equilibrium. It is also important to notice that stars are not exactly sources of energy - they are just energy converters. So in an infinite universe they could not possibly burn bright forever.

And yes, at the end of the day, everything will have the same temperature - I said just as much. It is simply that the temperature will not be infinite.

As to the rest of your post, I might be wrong about hyperbolic geometries as I assumed they could be stable, but indeed they actually force expansion, so no static universe.

[halfeye]

Edit: As for whether every line hits a star, the relevant concept is mean free path. Assuming our local space is representative we can define a mean free path to hitting a star. We can then determine the distance (d) where the probability we will hit a star if we draw a random straight line of that distance is 50% (or any other percentage). For every direction we can ask "what is the probability we will not hit a star in 1x that distance. That is basically the same as asking whether there is a star in the random straight line of length d: 50%. What about 2d? That is the chance you don't hit a star in the first section, multiplied by the chance you don't hit one in the second section, or 25%. At n x d distance, the probability there is no star on that straight line is 0.5ⁿ. If n is infinity then the chance we do not see a star is 0, so the chance we do see a star is 1.

My point about the star not hitting every point on a sphere at infinity is that just because a star appears at the end of every line, it does not follow that the star sends a photon along that line, there are an infinity of lines from a star, and a star in its lifetime emits a huge but finite number of photons. So a star at a huge enough distance might not send any photons to a particular point.

[johannessmid]

And yes, at the end of the day, everything will have the same temperature - I said just as much. It is simply that the temperature will not be infinite.

I don't think anyone is arguing that the sky's temperature would be infinite. The discussion is whether every line of sight will see a star. The energy coming in from any solid angle would be equivalent to seeing a star.

[Fat Rooster]

My point about the star not hitting every point on a sphere at infinity is that just because a star appears at the end of every line, it does not follow that the star sends a photon along that line, there are an infinity of lines from a star, and a star in its lifetime emits a huge but finite number of photons. So a star at a huge enough distance might not send any photons to a particular point.

Uh, it doesn't work like that. To see why you can sort of apply the same logic to anything, the sun for example. If we break up the sun into 1mm² squares, what are the chances we are going to receive a photon from any particular square in our piddly lens? Extremely small. Does that mean we can't see the sun? absolutely not. The fact that a square twice as far away is 1/4 times as likely to send a photon your way is precisely accounted for by the fact that your pixel that covers that arc is receiving light from 4x the surface. Eventually you will be so far away that the probability of any particular star sending even one photon your way becomes vanishingly small, but the area of the sky it covers is so tiny that it doesn't matter. We would expect the same number of photons from anything covering that area of the sky at that temperature, such as a particular area on the sun.

If it helps, think of sections in a cone that a single pixel looks at instead of lines. You can tile the sky with finite cones, where as lines don't actually fill it. Consider a section 1d away with thickness w. Lets say there are 1 billion stars in this section. Each of those stars sends x photons (w small compared to d means they all send roughly the same number. We receive x billion photons from these stars. Now lets consider a section 2d away with thickness w. Each of those stars is 2x further away, so sends x/4 photons, but the volume is 4x bigger, so there are 4x as many stars. We receive just as much light in our detector from these stars as the nearer section. If we consider a cone that only covers a star so far away that we don't expect a photon from it, and consider how many photons we should receive if we pointed that cone straight at the sun, the answer would be as many as we got from the original star; unlikely to get any.

Light does not include depth information, so a black body that covers x% of the sky it will give just as much light whether it is very big and far away, or very small but very near. By saying that all lines end in a star we say that 100% of the sky is star. That will give the same result as if we were surrounded by a small amount of star surface much nearer, instead of a vast amount of star surface very far away.

[Radar]

I don't think anyone is arguing that the sky's temperature would be infinite. The discussion is whether every line of sight will see a star. The energy coming in from any solid angle would be equivalent to seeing a star.

Yet, the calculations presented by you result exactly in an infinite energy flux as each 1 ly shell gives the same contribution regardless of the distance, so once you integrate over distance you get infinity as a result. I simply wanted to add a proper correction as the infinite temperature came from accounting for the same light multiple times.

[Palanan]

Originally Posted by halfeye
I've heard it said that the fact that the sky isn't as hot as a star means that space isn't infinite….

This is Olbers’ paradox.

[johannessmid]

Yet, the calculations presented by you result exactly in an infinite energy flux as each 1 ly shell gives the same contribution regardless of the distance, so once you integrate over distance you get infinity as a result. I simply wanted to add a proper correction as the infinite temperature came from accounting for the same light multiple times.

Nope, what you get is that everything in the sky looks like a star. Not infinite.

[Radar]

Nope, what you get is that everything in the sky looks like a star. Not infinite.

The way you wrote it says otherwise:

It's because everything in this question scales as r^2. Consider a small bit of solid angle looking out. Look out 1 ly - you see X stars, with a certain average luminosity coming at you. Now look out 2 ly. The energy from a standard star is ~1/r^2, so you'll see 1/4 the luminosity from a star at that distance. But now that solid angle covers a physical distance ~r^2, or 4 times bigger, so you should see 4 times as many stars! The luminosity in that solid angle should have the same contribution from stars 1 ly away as 2 ly away.

So as you look farther away, yes, you are getting less energy from each star - but there are more and more stars, so you get equal energy from every distance.

Equal energy from each distance means infinite energy once you integrate over infinite distance. It might not have been what you meant, but it is exactly what is written above.

Stars are not points, so they also obscure everything behind them in a narrow angle. So the further away the source is, the more likely the light is to be absorbed by something on the way. Hence the exponential correction that solves the emerging infinity problem. Whatever the stars absorb is obviously emitted again, but we count that light already, so without the correction one would count the same energy multiple times, which is the source of the error.

[Fat Rooster]

The way you wrote it says otherwise:

Equal energy from each distance means infinite energy once you integrate over infinite distance. It might not have been what you meant, but it is exactly what is written above.

Stars are not points, so they also obscure everything behind them in a narrow angle. So the further away the source is, the more likely the light is to be absorbed by something on the way. Hence the exponential correction that solves the emerging infinity problem. Whatever the stars absorb is obviously emitted again, but we count that light already, so without the correction one would count the same energy multiple times, which is the source of the error.

That looks like a typo. Equal from each direction, rather than distance.

A bigger problem is that stars would experience this too, and would continually heat up. Regions of higher density may still form stars, but most of the stellar nurseries would literally boil away, which would prevent new stars forming.

The basic problem is that you are trying to talk about the equilibrium conditions of a system that is not in equilibrium. Stars will be adding to the energy available by fusion and no energy sink has been proposed. It is like asking for the equilibrium point reached by x_n+1=x_n+1. It sort of has a limit at infinity, but you are better saying it doesn't have one. It usually means your model is wrong. The hypothetical doesn't reach one until it gets hot enough for stars to 'fail' (probably just star formation rather than the stars themselves, the timescales involved for this to happen are stupidly long, and the temperatures required to prevent stars forming are vastly lower than those required to boil a star). At that point not all lines end in a star, because there are no stars!

That could be in equilibrium. Uniform warm gas doing nothing. Bathed in radiation so it cannot cool down. Profoundly dull. Slightly more interesting is the push pull between neutron stars and black holes, but that also does not have an equilibrium. Even the profoundly unlikely random collision between multiples is going to happen in 'infinite' time, The density of black holes is going to change over time, never reaching a limit. Even once all the matter is in black holes, and they have brought the temperature down to almost absolute zero, they will exchange mass by hawking radiation. Those below a certain size will be 'warmer' than their surroundings, and shrink, while those largest will grow. Then the temperature will drop further and larger black holes will start to evaporate. The timescales for this would be absolutely stupid, but infinity doesn't care. If one day is not quite equal to the next you are not at equilibrium. It would only end when the universe consists of a sole survivor, except that an infinite universe would always have a bigger one somewhere.