Is the universe infinite?

There's always been an argument that has irritated me about why the universe cannot be infinite. It goes like this:

If there were an infinite number of stars, every line of sight would end on a star, hence the night sky would be white. The night sky is not white, hence the number of stars is finite.

I don't see how that holds water at all personally, yet I still find it mentioned in major books such as a Brief History of Time.

It's well known that if you add up an infinite number of positive numbers, you don't necessarily get infinity.

For example, 1 + 1/2 + 1/4 + 1/8 + 1/16 etcetera = 2; not, as you might be inclined to think, an infinite value, nor even a remotely large value.

So here's the deal with the whole infinite stars thing: as a star gets further away from us, it takes up less space in the sky. As a very crude way of expressing my argument (yet nonetheless valid, I think), let's say there is a star 1 lightyear away, which takes up 1 unit of area in the sky. The next star in this particular universe is 2 lightyears away in another direction and takes up 1/2 a unit of sky. The next star is 4 lightyears away, and takes up 1/4 units of area; the next 8 lightyears and 1/8 units, 16 and 1/16, and so on.

Surely if this universe existed, although there would be an infinite number of stars in the sky, the stars would only take up a total of 2 units of area.

Hence, an infinite number of stars does not necessarily imply a white sky; and a sky which is not white does not necessarily imply a finite number of stars.

This is Olbers' Paradox.

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Originally Posted by Wikipedia

If the universe were homogeneous and infinite, then at a distance, r, away from the earth, the shell of the sphere would have a certain flux (viewed from earth) due to the individual flux of the stars on the shell (brightness) and also the number of stars in the shell (cumulative flux). When an observer from earth looks to a farther distance to another shell, r + x, the number of stars increases by the square of the distance, while the flux decreases by the inverse squared. Comparing the total brightness of the first shell to the second shell, one notices that both shells have equal flux, since the flux of each individual star decreases due to distance but is equally made up for by the number of stars. This means that no matter how far away an observer on earth views the sky, the brightness of each consecutive shell would not diminish, rather they would be equal.

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There are many possible explanations here.

That argument that you hate is incredibly dumb. :eek: Two thoughts come to my mind. The first is, an infinite number of stars would not equal a white sky because not all of the stars' light may have reached us yet. Secondly, I don't see why an infinite universe implies that there's an infinite number of stars. It could be a finite cluster for all anyone knows.

Usually, an "infinite universe" refers to our space-time continuum not having any physical boundaries. Basically, you could point a ship in one direction and go on forever without ever reaching the "edge" of the universe. It doesn't emply that there is an infinite amount of matter in the universe, which is technically impossible, since all empty space would be occupied by matter.

No that's not true. If the universe is infinitely large and only a certain fraction of space is occupied by matter, then there is still infinite matter, as any positive number, be it fraction or not, multiplied by infinity, is infinity.

Basically, no, infinite matter does not mean no space between it; in an infinite universe there could be infinite matter and infinite empty space, and infinity plus infinity gives infinite total space; they're just in a fixed proportion.

And yeah, just to make that clear, I'm not talking about infinite space as such, I'm talking about infinite space with an infinite number of stars dispersed through it.

Gnome; hm, actually that article seems to present the statement 'every line of sight would end on a star', as an axiom. I think I just showed that that is not necessarily true.

It's a bit like how [whathisname] showed that there are miniscule 'gaps' in the real number number line; there are many more numbers which are not real in these little gaps. In the same way there are many miniscule gaps in the dome of the sky which do not land on a star.

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Originally Posted by Xei

Gnome; hm, actually that article seems to present the statement 'every line of sight would end on a star', as an axiom. I think I just showed that that is not necessarily true.

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The paradox assumes an infinite universe with randomly dispersed stars. In a universe like this, there would be a certain non-zero probability of a star occupying any particlar point along a line of sight. With an infinite amount of points along any line of sight, the probability would amount to one.

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Originally Posted by Xei

No that's not true. If the universe is infinitely large and only a certain fraction of space is occupied by matter, then there is still infinite matter, as any positive number, be it fraction or not, multiplied by infinity, is infinity.

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But in the case of our universe, there is a finite distance to how far matter has spread since the Big Bang (the observable universe). So the space that matter holds is a known constant, rather than a fraction of infinity (if the space-time continuum is infinite). Since it's physically impossible for matter to have gone further than a certain point, you end up with infinite mass in a fixed volume, thus zero free volume (by the way, how can you fill up all empty space with round particles :P?).

Anyways, that's arguing semantics, since my point was that the universe doesn't (seem) to contain an infinite quantity of matter. The debate should be whether the space-time continuum is infinite or not.

There are stars passed where the human eye can see. Astronomers have also seen passed the limit of where their are stars. So there isn't an infinite number of stars which reinforces the big bang "theory" and there is not an infinite amount of stars.

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But in the case of our universe, there is a finite distance to how far matter has spread since the Big Bang (the observable universe). So the space that matter holds is a known constant, rather than a fraction of infinity (if the space-time continuum is infinite). Since it's physically impossible for matter to have gone further than a certain point, you end up with infinite mass in a fixed volume, thus zero free volume (by the way, how can you fill up all empty space with round particles :P?).

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That would be true, but the point is that one of the major reasons that the Big Bang is a widely accepted theory is due to accepting that there are a finite number of stars in the first place, so it's circular logic really (infinite stars could exist forever because they do not all gravitate towards a central point, as there is no central point). There are other factors, yes (red shift, for example), but what I'm really talking about here is simply the validity of the mathematical argument about a white sky.

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The paradox assumes an infinite universe with randomly dispersed stars. In a universe like this, there would be a certain non-zero probability of a star occupying any particlar point along a line of sight. With an infinite amount of points along any line of sight, the probability would amount to one.

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As I said, that may not be entirely true...

Do you happen to know the summation expression for the area in the 'dome of the sky' that the stars would take up and the proof that it iterates to infinity? Or just how the area of the perspective parallel cross sectional area of an object (so for a sphere like a star, the circle which it appears to us as) is related to distance from an observer?

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Originally Posted by Xei

That would be true, but the point is that one of the major reasons that the Big Bang is a widely accepted theory is due to accepting that there are a finite number of stars in the first place, so it's circular logic really (infinite stars could exist forever because they do not all gravitate towards a central point, as there is no central point). There are other factors, yes (red shift, for example), but what I'm really talking about here is simply the validity of the mathematical argument about a white sky.

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Ah, got ya.

Actually, the theory is that the universe is growing. But the thing is, light is not instantaneous. But if you follow that theory, then eventually, in n years, the night sky would be white. But light also doesn't travel in a straight line (as a whole). It spreads out, getting dimmer. Most of the stars in the distance, by the time they get here, will have spread out, being dimmer. It then goes up exponentially from there. Just like the "1 + 1/2 + 1/4 + 1/8 + 1/16" thing.

Another thing to take in account is the theory itself. This assumes that the universe is an infinite size, with infinite stars. Having two infinites would mean that the stars take up all space. This creates another paradox. There isn't infinite stars for any amount of space, but for infinite space there is infinite stars.

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Originally Posted by Venomblood

Actually, the theory is that the universe is growing. But the thing is, light is not instantaneous. But if you follow that theory, then eventually, in n years, the night sky would be white. But light also doesn't travel in a straight line (as a whole). It spreads out, getting dimmer. Most of the stars in the distance, by the time they get here, will have spread out, being dimmer. It then goes up exponentially from there. Just like the "1 + 1/2 + 1/4 + 1/8 + 1/16" thing.

Another thing to take in account is the theory itself. This assumes that the universe is an infinite size, with infinite stars. Having two infinites would mean that the stars take up all space. This creates another paradox. There isn't infinite stars for any amount of space, but for infinite space there is infinite stars.

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Don't forget that light may also be bent by gravity. It has been observed curving around other stars to offset its actual position relative to its actual position. Another example is a blackhole in which the light is so bent out of its path that it is never viewed.

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Originally Posted by Spartiate

But in the case of our universe, there is a finite distance to how far matter has spread since the Big Bang (the observable universe). So the space that matter holds is a known constant, rather than a fraction of infinity (if the space-time continuum is infinite). Since it's physically impossible for matter to have gone further than a certain point, you end up with infinite mass in a fixed volume, thus zero free volume (by the way, how can you fill up all empty space with round particles :P?).

Anyways, that's arguing semantics, since my point was that the universe doesn't (seem) to contain an infinite quantity of matter. The debate should be whether the space-time continuum is infinite or not.

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In the Big Bang theory, there is not a fixed volume. The fabric of spacetime is actually what is expanding.

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Originally Posted by Xaqaria

In the Big Bang theory, there is not a fixed volume. The fabric of spacetime is actually what is expanding.

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Why aren't we stretching along with the fabric? Wouldn't the distances stay the same?

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Originally Posted by Xaqaria

In the Big Bang theory, there is not a fixed volume. The fabric of spacetime is actually what is expanding.

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Space-time expands faster than c though. If we assume that normal matter is bound by c as its maximum velocity, the observable universe has a calculatable volume. There isn't a physical boundary, it's just impossible for particles to have travelled further by now.

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Originally Posted by Bonsay

Why aren't we stretching along with the fabric? Wouldn't the distances stay the same?

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Well according to relativity, we are not 'woven' into the fabric, we sit on top of it and it stretches beneath us. This is why when calculated, it appears as though we are in fact at the center of the universe, since everything seems to be expanding and moving away from us. The explanation for this is that no matter is actually moving outward from a center point, the space between is simply expanding.

Cool.
Can't the universe still be infinite in a way? Like a 2D being walking on a sphere. It could go on forever seemingly into infinity. That way we stretch the ball, make it bigger, or make it grow while it would still be "infinite" for the 2D beings.

So you're suggesting Spacetime could be a hypershpere? :)

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Originally Posted by Bonsay

Cool.
Can't the universe still be infinite in a way? Like a 2D being walking on a sphere. It could go on forever seemingly into infinity. That way we stretch the ball, make it bigger, or make it grow while it would still be "infinite" for the 2D beings.

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The shape of the universe is still a subject of debate.