h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

> it requires that you can go in a straight line in any direction forever, but it doesn't require that the universe be infinite.

Surely the universe only need be inifinite in one direction / dimension to be infinite overall?

> if you go in a certain direction for long enough you get back to where you started.

That's unbounded... but once you've got back to where you started, can't you continue on round again, hence making it infinite?

I am trying but failing to think of a case where something is finite and unbounded... care to help me out?

me[Andy]g

Think of the Earth. If you pick a direction, you can walk in that direction forever. At no point will you walk off the edge of the turtle^H^H^H^H world. But the Earth only has a finite surface area.

I see. Thanks.

I think for the purposes of the entry it doesn't particularly matter whether "unbounded" or "infinite" is used at present, does it? I don't think there's any real evidence of either case being true.

me[Andy]g

Umm, well most scientists seem to think that the universe is unbounded and finite (in space). Some think it's unbounded and infinite (in space). I've not read any scientists claiming that it's bounded in space, but...

A finite and unbounded universe would have to be non-Euclidean. It'd be possible to restrict this entry to Olber's paradox as applied only to Euclidean universes. I don't think that's desirable, myself, because it makes the paradox far less powerful, given that most scientists seem to think that the universe is non-Euclidean anyway. But it's up to the author.

If the entry does restrict itself to Olber's paradox for Euclidean space, then this line: "This means that one of our initial assumptions was wrong - the Universe is not isotropic, homogeneous and infinite in space and time." should be changed to "...the Universe is not EUCLIDEAN, isotropic, homogeneous and..."

> "By simply looking at the night sky and seeing that it is dark, you can work out that the universe cannot be infinite in both space and time, supporting things like Big Bang theory."

Isn't technically true, because the universe might be infinite in space and time, but not isotropic, homogenous, (or Euclidean, if the paradox is restricted to Euclidean spaces). Perhaps you could change "cannot be" to "is probably not"? Or replace "the universe" with "an isotropic, heterogenous universe".

Lucinda!

Where were you *before* this entry was scouted?

I'm not paying as much attention to Peer Review as I did when I was a Scout. Time and tide and all that.

Hi Lucinda

Well, I'm not sure what to do now because the entry's already with a subeditor...

I don't want to move the maths bit; it makes sense to put it there to mathsy people, and I think that the occasional integral filtering into the consciousness of non-mathsy people is healthy and maybe will help them not to be afraid...

Edgar Allen Poe was a poet not a cosmologist last time I checked. Hence the surprisingly. I dunno, I only think it is a little ambiguous; I don't personally think it's worth changing. (Obviously I wouldn't - I wrote it... hmm... *looks at self suspiciously*)

> I'd like to see another Maths bit explaining how an expanding universe fails to account for a dark sky. It should be another integral, right? Since this is apparently the most common mis-conception, I think you should conclusively disprove it.

It is talked about in the entry I referenced. The trouble is, the expansion thing is general relativity, which alas since I have just graduated (I got a 2:i!) I will never study. I don't think that it is within the scope of this entry to include that; I think that anyone sufficiently interested can look it up themselves, after reading the paper I referenced. IMO I reached the 'allowable maths' limit with the integrals I've included...
Having looked up the paper saying why it does, I don't really understand it too well... http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1987ApJ...317..601W&data_type=PDF_HIGH&type=PRINTER&ext=.pdf
I recognise some of the equations (the Friedmann Equation was in my exam), but I'm certainly not prepared to reproduce them here. This paper is referred to in the one that I referenced... and it is from a well respected journal. I know that isn't foolproof but it is a start...

> I do think that under 'explanations that do work' you should at least *mention* the existance of other explanations. Just a single line to say that some people propose other resolutions of this paradox. The way it comes across at the moment is that these two explanations are the only ones that work, which isn't true.

Good point. I shall contact the subeditor with a suitable amendment, when I think of one.

> re: absorbing dust. Is it sensible to mention black holes here? That's one example of material that absorbs energy and doesn't emit it later (Hawking Radiation excepted).

I don't *think* so; one would still be able to detect the Hawking radiaton, as you say, still I will see if I can find anything on that. I'm not hopeful, though...

> Doesn't the paradox require that the universe is *unbounded*, not necessarily infinite? IE, it requires that you can go in a straight line in any direction forever, but it doesn't require that the universe be infinite. EG, the paradox still holds if the universe is finite, and if you go in a certain direction for long enough you get back to where you started.

I don't know. I think it needs infinity for an infinitely bright sky. I don't know how one would work the mathmatics in that circular kind of case - I mean, you could integrate round and round a circle infinitely many times, but light that's going to hit your eye or your detector will the first time round, I think? Still, there's no evidence for unboundedness but not infinity, as far as I know.

> It'd be possible to restrict this entry to Olber's paradox as applied only to Euclidean universes. I don't think that's desirable, myself, because it makes the paradox far less powerful, given that most scientists seem to think that the universe is non-Euclidean anyway. But it's up to the author.

Technically, I believe that 'Olbers' Paradox' only applies in a Euclidean universe, because I don't believe that Olbers considered non-Euclidean ones in his paper... ...but not being able to read German or having a copy to hand, I wouldn't know. There are non-Euclidean Universes and non-Euclidean Universes, right? What I mean is, some it would and some it would not work in? I can't prove in in a non-Euclidean universe (I don't know the math) but I don't see why it shouldn't work in at least some non-Euclidean universes...
I think I'm going to make the change you suggest, as I will with your last point...

Wot? Down a deep well the sky looks dark and you can see the stars?! How deep is this well and whare can I find one?

I think there is another way of deriving the paradox, which might answer some questions. I'm not sure if this derivation is right or not though, I'd be interested in hearing what others have to say.

If the distribution of stars is homogeneous (in space and time) and isotropic (in space) then no position or direction is special, so the energy per unit volume must be everywhere constant (just by symmetry). Since no energy is ever lost (an additional assumption we have to make?) and there is always more energy pouring in the energy per unit volume would be everywhere infinite. (Energy has to be constantly added to the system if we're to demand homogeneity in time, since stars have a finite lifetime.)

I think this analysis would also show that the paradox works in a non-Euclidean universe, provided the geometry of that universe was itself homogeneous and isotropic (i.e. the geometry is everywhere locally the same [maths note: this condition is actually called local homogeneity, homogeneity is stronger, but they are the same if the universe is simply connected I think], and isotropic). Euclidean space satisfies this condition, so we could see it as a special case.

Finally, this derivation also shows that there is no reason why we need the universe to be infinite. (Although it would have to be unbounded because a universe with boundary wouldn't have a locally homogeneous geometry.) So, for example, it would work for a 3-spherical (finite, constant positive curvature), 3-toroidal (finite, zero curvature) or hyperbolic universe (infinite, constant negative curvature).

If there are black holes, and there are, then the universe is certainly not going to be homogeneous, although it might be locally homogeneous which I suspect is not enough to make the argument work. (As Lucinda said, they could, if correctly placed, soak up all the energy.)

> "but light that's going to hit your eye or your detector will the first time round, I think?"

Only if your eye never moves and has been there forever.

Good point, Lucinda, & thanks Dogster...

Right. I think I'm going to restrict this for the case of a Euclidean universe. I searched for papers on the paradox in non-euclidean spaces and drew a blank. I searched for black holes and also drew a blank...
I'm going to contact the subeditor to let him know that there's amendments to come. If anyone else is going to have a brainwave, please do so now because I don't want to unreccomend this entry!

I'm really beginning to resent Olbers...

But how big is that well? I've started digging and it still looks pretty light up there. OK so I've only dug down to my knees but I could do with some advice here. I may even be wasting my time!

It would be quicker to find an old mine shaft, but be carefull you might get trapped if it caves in.

Alji

And while you're down there, you might want to set up a neutrino detector!

Sounds cool, J'au-æmne - let us know when you've done

Massive swimming pool will do!

Alji

There, I've updated the entry (it took so long as I had a cold & was to ill to think about physics)

Is it better now?

How do we know, your the one with the cold

I think it stil is good.

Only one more thing. In 'explanations that work' it might be better to say 'AT LEAST one of the assumptions is wrong' (in normal letters).

And that is the whole of the paradox: we call them assumptions that may be wrong, they thought them to be the absolute unchangable reality.