## Posts Tagged ‘infinity’

December 31, 2008

Tom Levenson at the Inverse Square Blog posts a discussion of Olber’s Paradox. The gist is that if we lived in an infinite, static, homogeneous universe, there would be light everywhere. That empirical falsehood was tough on folks way back when, many of whom believed the universe subscribed neatly to Olber’s little enumeration.

I would rather not repeat what’s already been said on Wikipedia, so I’ll assume that you’re already mildly familiar with the argument. Also, I’m not especially concerned with what real cosmology has to say about things. I want to think about this under the same terms good ol’ Olber could have. Let’s hash out some implications of this static, infinite universe without worrying about all that Stephen Hawking shit.

The first thing to point out is that if the distribution of stars were inhomogeneous, we could avoid the problem. For example, if the density hot star matter went as $\rho \propto r^{n}$, then we would have infinite flux of light received on Earth for $n\geq-1$ and finite flux for $n<-1$. (Technically, it would produce infinite flux even in this second case due to the singularity at $r=0$, but we’ll assume there is some small region near Earth for which the distribution no longer holds). We could even estimate the absolute size of the universe by sampling the density of stars at a few depths to obtain the power law, then finding the size the universe would need to yield the correct average brightness of the night sky.

One problem with this power-law crisis resolution is the creation of a center of the universe – the spot where $r=0$. Historically, once we trashed geocentricism, we pretty much trashed inhomogeneity (on large scales only, since otherwise there would be no point in a PB&J sandwich, which under perfect homogeneity would become an abominable blenderized bastardization) at the same time. Even though this particular solution to Olber’s Paradox does not require the Earth to be at the center of the universe (the flux is finite there, but it is also finite everywhere else), it’s still rather philosophically unattractive. We’ll throw it out.

Instead, focus on the case $n=0$, that is, the homogeneous universe. Many sources claim that in this universe, all points on the sky would be as bright as a star, because wherever you looked, there was sure to be a star in that direction some distance off. (Both Tom’s post and Wikipedia make this claim.)

That claim is wrong. You wouldn’t have every point in the sky as bright as a star. You would have every point in the sky as bright as infinitely many stars. That is, you would get infinite flux density from every single point on the sky. Even if you looked at a patch of sky one arcsecond on a side, you would get infinite light from that patch. Sure, when you look at any direction you’d see a star, but then if you looked further in that direction you’d see another star, and another, and another. The “anothers” never end in an infinite universe.

Let’s say we look at a patch of sky the size of the moon in our toy $r^n$ star density universe. If we only count the stars back to some finite depth $d$, then the total amount of light we receive scales as $d^{n+1}$. The exception is $n=-1$, in which case flux scales as $\ln(n)$, and hence still diverges. (Here I’m referring to the catastrophe cases $n\geq-1$. For $n<-1$ we get some constant flux minus $d^{n+1}$, so the total flux converges towards a constant value.)

That disagreement on the brightness of the sky is crucial. If every point on the sky were as bright as a star, it would get quite toasty around here. In fact, the second law of thermodynamics ensures that the Earth would heat up to the temperature of a star, until it (the Earth) also glowed star-hot, and hence lost heat as quickly as it came in.

This “constant light everywhere” situation is not really so far from the truth, since every direction in the sky does glow the same temperature. It’s just our luck that the temperature of the night sky (more commonly, the Cosmic Microwave Background) is two orders of magnitude colder than the Earth, and that the Earth is about an order of magnitude colder than a star. Nice place to be, thermodynamically, as Sean Carroll pointed out in a public lecture I previously wrote about.

However, if we have the situation predicted by the universe in Olber’s Paradox, we would actually have the Earth getting infinitely hot. For that matter, since the Earth is nowhere special in this model, there would be infinite energy density everywhere. That’s a bit stronger of a quandary than “why is the sky is dark?”.

Where could this infinite energy come from? I have to admit, this “infinite universe” thing is pretty tricky to wrap your mind around. It’s clear that an infinite universe would have infinite total energy. But that doesn’t mean it couldn’t have finite energy density. We concluded, though, that it doesn’t. Something is wrong. Of course, that happens a lot, with infinity.

Energy is conserved, but only in a closed system. Something that’s infinite is not closed. Basically, the infinite universe has infinite energy the same way we Americans figured out how to get infinite money with Social Security – by borrowing it from our infinite future. (That does work, right? My current career plan is to go into stasis for forty years right after I graduate and then start collecting my dues first thing on thawing out.)

It is hard to imagine living in an infinite universe, especially a static one. If all times and all places are the same, then how did humans come to choose this time and this place to exist? In an infinite universe, wouldn’t it be true that anything that can happen already has happened, infinitely many times? Wouldn’t someone exactly like me have written this exact blog post over and over endlessly back into eternity? Far out, dude.

I simply cannot imagine an infinite universe. The finite speed of light effectively allows us to borrow energy from the past. But it’s an infinitely-large, infinitely-long past, and consequently we would have infinite energy. There’s no energy conservation paradox, because the universe never transitions from a starting point with finite energy density to an ending point with infinite energy. A infinite universe simply does not have that starting point to begin with. It’s all way too insane.

You could postulate an infinitely-large, infinitely-old universe with finite energy density everywhere, but you’d have to kill off this light-travel mechanism for borrowing energy from the past (which is also a mechanism from broadcasting energy into the future.) You’d have to keep things where they are. Which, with a universe like this, is in your imagination.

### Let’s Read the Internet! Week 2

October 19, 2008

Davisson-Germer Experiment Chad Orzel at Uncertain Principles
The first observation of the wave properties of electrons came by accident. Just like you.

A Beautiful New Theory of Everything Garrett Lisi on TED.com
In case you were wondering how everything works…

Didn’t quite catch that? Don’t worry. You can always read the paper.

Infinity is NOT a Number Mark Chu-Carroll at Good Math, Bad Math
More comprehensible than the previous post, if less profound. The fundamental problem with making infinity a number seems to be that it lets you prove all manner of foolishness, such as 1=2.

The Universal Declaration of Human Rights

It’s awfully pretentious to claim your document to be “universal”. Who has the authority? Further, what does it mean for everyone to be equal in rights? Clearly, we are not equal in many senses. Separating out “these things are rights” and “these things are what you have to deal with because of the circumstances of your life” is a tough task. For example, according to this declaration, everyone has a right to marry. But marriage is simply not a universal concept among humans. It’s perfectly conceivable to have viable, righteous societies with absolutely no concept of marriage. The concepts of privacy and property ownership could be sacrificed in righteous societies, under the right circumstances. Creating a list of rights that’s simultaneously universal and specific seems nearly impossible. But the visualization is nice.

Dead Waters Romain Vasseur et. al
Boats that get stuck in plain water. I don’t understand why this works, but the video is really cool.

Chimpanzees Make Spears to Hunt Bushbabies Not Exactly Rocket Science
Like it says, chimps make weapons and kill shit with them. In case you were wondering where we get it from.

Late Bloomers Malcolm Gladwell in The New Yorker
Just because you’re old doesn’t mean you’re useless. Therefore, you might as well slack off for another year or two before beginning that “great life’s work” stuff.

Where’s the Algebra? Michael Alison Chandler on X = Why?
Some chick with a seriously ugly smile asks whether algebra is important. But her “education” from her brother sadly misses the point. She asks, “what good are equations?”, and he replies “We have to learn equations to install lights.” But the entire article is written with the attitude that these equations are magical things that pop out of nowhere to describe lighting systems, their goal being to confuse blue-collar workers to the greatest extent possible. I don’t think there’s any understanding here the equations actually come from somewhere. Someone used a more basic set of principles to derive the equations, or else conducted experiments and then found equations to describe the results. Applying equations to describe real situations is not supposed to be a matter of plugging numbers into formulas.

The Cartoon-Off Farley Katz at The Cartoon Lounge
Normally, I wouldn’t bother linking to something that’s already been Slashdotted, but I bookmarked this page for “Let’s Read the Internet on Wednesday, and then the Slashdot post comes up just hours before I compile my links for the week. I guess the fact that the entire geek culture already knows about doesn’t really impact how funny it is.

The Sun
The web page that makes you go blind if you stare directly at it.

Fabry, Perot, and Their Wonderful Interferometer Skulls In The Stars
The author consistently produces wonderful posts explaining concepts in optics from a historical point of view. I actually used a Fabry-Perot interferometer in physics lab once. What I learned there is that they make surprisingly bad hammers.