Informal physics lesson 1: special relativity

A couple of weeks ago, I directed my web browser to Wikipedia. I was at work, of course--there's no better place to read Wikipedia--and I found myself on the page describing the "Twin Paradox". I'd been there before. I'd bet that once or twice per year for the past five years, I'd been on that page trying to understand what it's all about. I fancy myself a clever guy, and I was miffed that in all those attempts, I hadn't managed to grasp the ideas put forth on what is to me the most easily understandable general knowledge source in the world. Though, to my defense I hadn't yet visited the Simple English Wikipedia article on the topic. Perhaps I should have.

For those of you who are unfamiliar with the Twin Paradox, I'll briefly enlighten you. The Twin Paradox arises when twins who possess an extremely fast spaceship decide that one of them needs to make a quick run to Alpha Centauri to pick up some beer. Usually the slightly larger twin makes the run, since he's a bit of a pushover and the little one gets mad easily. The trip goes smoothly--the larger twin makes it to the best liquor store in the galaxy and back at half the speed of light (sometimes faster, when there are no police around)--until the twins see each other again and realize the little twin aged a couple of extra years. "WTF?!" both twins say in the parlance of their time, "How could this have happened?"

This, my curious friends, is the main question addressed here. I finally understand the basics behind this phenomenon, and I'd like to share them with you. I'm still not sure if I could derive the equations myself, but I'm sure I could give it a damn good try. I'll leave that for another day.

Let me take you back to my sixth grade science class. My science teacher, Mr. M, was a bit weird, though he got his point across. One day, he took our whole class outside into the school parking lot, got into his Ford Rustbucket, and drove away. That wasn't much of a science experiment, I thought. But, alas, Mr. M hadn't left; he had simply turned the car around and was coming back at 35mph, leaning on the car's horn. Several other cars' horns joined in the commotion, but that wasn't the experiment. No, the experiment was that, as the car passed us, the pitch of the horn got lower. At least that is what Mr. M told us afterward, as we student scientists began to theorize about collision avoidance and angry driver provocation.

The Doppler effect was, of course, the lesson of the day. Ask any child what a fast car passing by sounds like, and he can tell you, but he can't always tell you why. The Doppler effect explains how a noisy object coming towards you emits sound waves at regular intervals, and even as it (the car) maintains the same frequency, a stationary observer hears a higher frequency on the approach due to compression, and then a lower frequency due to decompression.

Light works in nearly the same way. Though the speed has to be incredibly fast, an object coming towards you will appear to be more towards the violet (higher frequency) end of the visible spectrum than if the object is stationary. Likewise, the color of the object moving away very quickly will appear to be more red (lower frequency), a phenomenon astronomers have named "red shift" in the context of far-away stars and galaxies moving rapidy away from earth. I should warn you that the color red has very little to do with the actual colors observed; in fact, the only reason the color red is singled out is that it is the lowest frequency light that humans can see with their eyes. Red shifts can and do start and end outside of the visible light spectrum.

Now, imagine we're back in the school parking lot and crazy Mr. M is driving by, honking his horn and screaming out the numbers one through ten. We know that the horn frequency sounds higher when he's approaching and lower when he's driving away, and the same thing happens with the numbers Mr. M is yelling out the window. He seems to us to be counting faster as he approaches and slower as he drives away, but Mr. M thinks he is counting at the same speed the entire time. In some sense, if you think about Mr. M counting along with the seconds on his watch, from our perspective it seems like time is going faster for Mr. M than for us as he approaches ("He's counting the seconds way too fast!") and time is going slower for him than for us as he drives away ("He's counting too slow!").

Here's the funny thing: if I were yelling out the numbers instead of Mr. M as he drives by, Mr. M would hear exactly the same phenomenon. In fact, if two people are travelling quickly towards one another, each one thinks that time is going faster for the other person than for himself. Similarly, two people going away from each other each think that time is going more more slowly for the other person than for himself.

Thus, the twins I mentioned above will both watch each other age very slowly as the big twin makes the beer run to Alpha Centauri, and they will see each other age very quickly as the big twin makes the trip back. In the past, always thought, If that happens, the twins should still be the same age when they chug their first beer together, but I was wrong. Here's why:

First of all, we still havent discussed exactly how fast each twin sees the other one aging. Now in the examaple of Mr. M, if we both count as he approaches, we both hear the other counting too fast, but the speed increase is not equal. In general, I would hear him counting faster than he hears me, due to the fact that sound travels through air. Think of it this way: if Mr. M broke the sound barrier (though I don't think his car could even break the speed limit), his one... two... three... would get to me after he did, and I might actually hear him counting in reverse. Mr. M would never hear me counting in reverse (not if I can help it!) since I am stationary in the parking lot and not moving with respect to the air around us. Therefore, it matters a lot if you're the one moving or the one standing still, with respect to an atmosphere of air.

But, that brilliant guy Einstein said that, in general (time and space), no one is moving and no one is standing still, absolutely; you can be moving only relative to one another. In fact, this follows directly from the assumption that the speed of light in a vacuum is constant no matter where you are and how fast you're going. In other words, chasing a beam of light is like having a stick attached to your head with a carrot dangling from the end of it; no matter how fast you run towards it: you can't get nearer, and it even speeds up whenever you do.

To the point: with sound and air, who you are matters, but not with light, time, and space. So, when the big twin is on his way to pick up the beer, he sees the little twin aging exactly as slowly as the little twin sees him aging. The same thing happens on the return trip. But, there is one little difference: the turnaround.

Like I said before, if the twins see each other aging slowly and then quickly, and they see the same rates of change, they would be the same age when they shared a beer. That is true if they see each other aging slowly for the same amount of time as they see each other aging quickly. However, if for some reason the little twin is sitting at home watching TV and doesn't realize that the big twin is already on his way back, he might think that the big twin is still aging slowly. This is more true than it sounds.

Let's look from the perspective of each twin. The big twin hops in the spaceship and puts on ZZ Top as he cruises to Alpha Centauri. Every time he looks back, his twin is aging slowly. Big twin gets the beer, switches to Aerosmith on the CD changer, and watches his brother age quickly for the entire trip back. Now, from the other end, little twin curses his brother's music tastes as the spaceship takes off, and he watches his twin age slowly while listening to Lady Gaga. When the big twin finally gets to the AC Carry Out, the little twin doesn't see it immediately. Light goes only so fast, and the little twin can't possibly witness the turnaround (possibly through the larger of two giant telescopes in the twins' house) until light has time to get from Alpha Centauri back to earth. That means that the big twin has already been on his way back for a while when the little twin finally sees him buy the beer and turn back.

Thus, while the big twin sees equal amounts of his brother's slow and fast aging (the big twin watches himself turn around at exactly the halfway point), the little twin sees more of his brother's slow aging than fast aging (the little twin watches the big twin turn around only after the light has time to reach him). And that is why the big twin is now the younger twin when they share the beers, well earned from this experiment.

The moral of the story is: if you are the one out there doing stuff, you have the advantage since you are there when you do it; it takes other people a while to realize what you did, and by the time they do, they're already behind. Or ahead. Something like that.

"Next time," says the little twin over his second beer, "I'll get the beer."


The good guys, and using definitions

Continuing along from my last post--where I typographically realized that most of my ideas are in fact not mine, but belong to dudes that are bigger, badder, and more thorough than myself--I'd like to draw your attention to an organization that magically appeared out of nowhere the instant I publicly called for a movement based on rational empiricism with a bit of humanism thrown in. Ladies and gentlemen, meet The Center for Inquiry and its affiliates. They even have representation at the U.N. here in Vienna, as well as Geneva and New York.

The Center for Inquiry defines itself based on--or in contrast to--religion, as do all of the major "freethinking" organizations I've come across, but I think I can forgive them for that if they do a real good job with the whole rational empiricist part.

In other news, there was an unassailably brilliant post at Rationally Speaking a few days ago. In it, Julia Galef concisely attacks the tendency to let definitions drift when under fire and directs some of her comments at her fellow blogger Massimo Pigliucci, who responded on the following day with a post that is equally unassailable, though less satisfying. If you fancy yourself a good arguer, reading these two short essays are well worth the time. In the end, you'll see that it's not only the definition that is important, but also how you use it.

If you're like me, you'll also find yourself with a renewed appreciation for philosophy. Until now, I thought that descriptive, linguistic definitions (as Pigliucci mentions) were the only definitions worth using. Now I realize that prescriptive, conceptual definitions can also be worthwhile, and they are in a sense the basis of many necessary branches of philosophy. My delay in coming to this realization is likely a byproduct of my mathematics education; mathematicians don't do prescriptive. Thus, I'm finding myself without the tools I need to intellectually pursue moral, ethical, and humanistic goals. I'm slowly getting there, though.


I've got a bad habit

Often, I state my ideas as if I'd invented them, and only afterwards do I realize that I'm neither the first nor the most eloquent person to arrive at the same conclusions. My lack of prior research (call me a shoot-first academic outlaw if you like) for my previous post on this blog once again became painfully evident as only days later I came across a couple of articles--by writers more educated than myself--discussing the exact same points, among others.

I've been reading whatever internet "literature" I could find on topics relating to my recent spate of posts on the politics of atheism, and I've come across a few worthwhile blogs. One of the first blogs that caught my eye was Atheism Analyzed, which is written by a self-proclaimed former atheist. The statements of purpose found in the blog header and in the margins seem to align nicely with my own, but in the first few articles I read I found more antagonistic and rhetorical jargon than interesting thought. In fact, one article attempts to dismantle the arguments presented in a blog post by Massimo Pigliucci but fails miserably. Between overgeneralizations and false accusations, the anonymous author of Atheism Analyzed ("Stan") misreads quotes taken from Pigliucci's post and incorrectly describes Pigliucci's conclusions in his own words. This was obvious even before I read the original article. Given that Atheism Analyzed--which reads more like anti-Big Science propaganda than anything else--took Pigliucci's words out of context, I find it amusing that the author didn't manage to make me doubt Pigliucci even for a second. I'd never heard of Pigliucci before, but even his out-of-context statements convinced me that he had a valid point. Before I left the blog, I made a short comment on the Atheism Analyzed post that refuted the statement that Pigliucci's "strawman" was somehow a fallacy (I'd been in the same situation before on my blog). I didn't understand the response I received, either because I'm not educated enough to understand the ill-defined jargon or because the argument wasn't coherent. Time might tell.

So, I went over to Pigliucci's blog, Rationally Speaking--which he shares with two others--and found something worth reading. I'm very impressed with the articles I've read, even though the informal setting leaves plenty of room for poorly defined terminology and colloquialisms. Despite this, the central message of the post that Atheism Analyzed so desperately wanted to refute comes through loud and clear. I might be biased, of course, but Pigliucci is absolutely correct in saying

Conceptions of gods are infinitely... flexible (or vacuous, if you prefer)... and they are thus simply not falsifiable. This is often (naively) mistaken to imply that no specific claim made by these theories can be rejected on empirical grounds. That’s as manifestly not true as it is besides the point: of course modern science can firmly reject the empirical claim that the earth is a few thousand years old; but since “the god hypothesis” doesn’t behave as a hypothesis at all from the epistemological standpoint, it doesn’t matter. In the cases we are discussing there is no science-like connection between theoretical constructs and empirically verifiable facts, so to “falsify” the latter is equivalent to shooting into a cloud of gas. It unnecessarily flatters and elevates religious belief to treat it as science.

I don't like how he judges religious belief to be figuratively below science, but other than that Pigliucci is spot on. He goes even further with an idea I hadn't thought of yet:

...even science itself is far from being an activity rooted in reason alone. A standard distinction in philosophy of science is made between the context of discovery and the context of justification. The first one deals with how scientists come up with new theories or ideas, the second one on how they proceed to empirically test or establish them. The notion is that the context of justification is where science works in a rational way, by logically connecting hypotheses and empirical facts. But discoveries are often haphazard and non-rational in nature, with scientists themselves being unable to account for how exactly they came up with a particular idea...

I think that's brilliant. If I didn't hate the phrase "thinking out of the box" so much, I'd probably use it right now to explain how unjustifiable (read: not empirical) thinking is beneficial--if not necessary--as an antecedent to scientific justification.

I'll be reading more from Pigliucci and his gang, and next time I have a brilliant idea, I'll make sure to check his old posts to see if he's already written about it.