The Twin Paradox (Part ONE)

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The Twin Paradox

a TWIN PAIR OF DUCKS! O_O

No you misheard. I said “twin PARADOX”. As in: a situation where the outcome seems counterintuitive or contradictory. The twin paradox is a famous paradox closely associated with Einstein’s Theory of Special Relativity.

Simply put, you take a set of twins. Born on the same day, on the same year, to the same Mom. We’ll call one of them Liz, and the other Ben. We take Ben and send him on a trip into space on a rocket (or throw him too close to a black hole)… and after a couple years, he returns from his exile and goes to see his twin. It’s their birthday after all (suppose that it’s also their birthday).

But as they put candles on their cake, they realize that Ben is MUCH YOUNGER than Liz! A whole handful of candles! How can this be? Neither twin has access to a TARDIS. Each twin denies undergoing any magical or science-y procedure (aside from the rocket trip, but Ben contends that he mostly just sat around in a room playing nintendo). Each twin recalls each and every day since they parted, years ago, and claims that nothing strange has happened. It’s a mystery! No! It’s more than a mystery: it’s a PARADOX.

Explaining the twin paradox involves a very straightforward, very visual argument. It’s so elegant, you’ll puke! But for you to properly appreciate it, I’m going to have to introduce you to Relativity. No, not Einstein’s Theory of Relativity, but rather  Galilean Relativity.

The Story of Naomi and Quincy (or, The Dalek and the School bus)

Okay. So. The truth of the world is that there are circumstances where two different witnesses will give honest and accurate accounts of the same situation in two very different ways. For instance… … …

Well… lets just cut to the chase. Suppose that we have two siblings: Quincy, who is bald and likes to ride at the back of the bus; and Naomi, who is a Dalek. And suppose that Quincy, while riding his bus down the street, passes his sister. Look at this awesome drawing to see the setup.

You're looking at things from above, which is why this photorealistic drawing contains images which seem unrecognizable.

Okay, now I’m going to draw something called a “reference frame” onto the drawing.

Listen. These drawings took a long time! So print them out and put them on your fridge... and i don't want to hear another damn word about it.

I don’t own a ruler, so the grid is a little ugly, but you get the gist of it. A reference frame is just a grid that you superimpose around yourself. It’s a way of marking the location of everything you see around you. Naomi’s reference frame can be used to tell how far the hedge is away from her, or how far the tree is from her, or how far apart they are from each other.

Now, reference frames are useful for tracking  motion. Because Naomi is a Dalek, and because Daleks are achronal beasties, we can see past images of where the bus was located in the preceding few seconds (in blue). We note that the BUS is moving, and the trees and the hedge and the sidewalk and Naomi are NOT MOVING in Naomi’s reference frame.

You know how, when you were a kid, you'd look out the window of your car, and stick out your hand and imaging smacking all the telephone poles/people that streamed by the window? it's like that!

The main thing about *your* reference frame is that it follows you as you move around. Thus, you’re never really moving in your own reference frame. So Quincy, on the bus, will have his own reference frame. The drawing above shows Quincy’s reference frame, and the location of everything relative to his bus.

Assume that the bus is magic, and so Quincy ALSO has the ability to see the prior locations of objects. We see that now Naomi (the Dalek) and the tree and the hedge are moving! Quincy is sitting still. The bus is sitting still. The world about them is moving.

Two Different Realities

So if I were to take Naomi and ask her to describe the situation, Naomi would say “Quincy was moving”.

If I were to take Quincy aside, he would say “Naomi (the Dalek) was moving”!

Who is RIGHT?  IS ONE LYING?

If you say “Quincy is obviously the one moving, because he was on the bus. busses move, trees don’t… and you’re just playing with semantics.”

Then I will reply “HA! Daleks are science fiction beasties!  How do you know that Naomi, standing atop the HULL of her giant spaceship (by the space-hedge and the space-tree ), did not just FLY PAST Quincy (whose space-bus is sitting at rest)?!?”

And then I will add “Haven’t you ever gotten motion sickness from an IMAX movie? Haven’t you ever panicked when the car parked next to you backed out? Figuring out who is moving and who is still is a tricky business!” (It’s actually a subtle business which lies at the heart of all Einstein’s Relativity).

Can I pay for this tea with puzzles?

Then I will sip some tea like professor Layton.

The truth is that both realities are correct. And they even agree!

Galilean Relativity

So Galileo (I guess? That’s who it’s named after. ask a science historian. shit.) hammered down the BIG TRUTH. Although there are multiple ways to describe the motion of objects in a scenario — who was moving where, and how fast— there is also a formal way to translate one description of reality to another! To translate the description of events given by Naomi’s reference frame; to those described by Quincy’s.

In other words: There is no absolute truth! Everything is Relative! but also: The Laws of physics let us translate between one observer’s perspective and another’s! So… it’s relative, but not arbitrary. :p

The point to today’s entry was to introduce the subject of the twin paradox, and also to introduce the fundamental idea of relativity: that two observers can describe the same scene in DIFFERENT ways, but still be telling the truth. The laws of Galilean relativity allow us to translate between the two stories, to check to see if they agree.

In the context of Galilean relativity, the stories of two different observers usually only differ in the details of who is moving, how fast, and in what direction. Next week I’ll introduce Einstein’s “special relativity”, where the two accounts will differ in a WAY CRAZIER WAY. (hint: they disagree on everything)

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5 Responses to The Twin Paradox (Part ONE)

  1. SeanNo Gravatar 4 June, 2010 at 8:03 am #

    Great introduction into relativity! I also like the fact you were able to work in Professor Layton (best games ever!) and Doctor Who (greatest show ever!). Keep up the good work.

  2. pauloNo Gravatar 4 June, 2010 at 10:46 am #

    Bravo Ben and you managed not to mention the RCMP once.

    • benNo Gravatar 4 June, 2010 at 6:19 pm #

      RCMP?

  3. Acid QueenNo Gravatar 6 June, 2010 at 6:20 pm #

    I prefer Gallifreyan Relativity.

  4. benNo Gravatar 8 June, 2010 at 7:57 pm #

    that’s in part 4.

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