The Twin Paradox (part TWO)

virginia lottery

Preamble

Welcome to the Twin Paradox part 2. In this section, i will explain the concept of time dilation, which lies at the heart of the twin paradox.

i was once asked "what is time?" and i replied "time is the thing that lets me eat more tomorrow"...

Remember Relativity?

Hi so last time I introduced you to Galilean Relativity. The main idea was that we want to discuss how people, when they look out the windows of their moving bus, observe a different reality than the people standing on the sidewalk watching the bus go by.

Today i’m going to introduce you to a part of Einstein’s Special Relativity called time dilation. (Spoiler: If you were to carefully watch the wristwatches of the people on the bus passing you, you would notice that time is moving SLOWER for the people on the bus. Than for the people surrounding you on the sidewalk.)

What is Time?

What an ambitious header for a section. How am I going to answer it?

A cornered Physicist won’t give you a straight answer to this question. And the reason is simple: Physics is the study of motion or change, and implicit to both of these concepts is the concept of time. So any attempt to describe time using the ordinary physics toolkit will be full of circular reasoning.

Thus,  I’ve cut all the physical/philosophical jargon, and instead i’m going to present you with the physicist’s WORKING definition for Time:

Time is what a clock measures.

Now. I realize that this is a circular definition. Who cares? (here’s a riddle: “what is a waste of itself when you try to define it”) Perhaps you’d like to hear people speculate about the  problem of the arrow of time. (Sean Carroll says that time only exists because we had a big bang. )

In any case, the working definition is all we need. That said, we still need toanswer the question:

What’s A Clock?

Okay. So a clock is a system which changes in a predictable and regular way in ONE and ONLY ONE direction.

  • An hour glass is a good example. An hour has gone by when all the sand has gone from the top chamber to the bottom, and the sand never flows back up. it’s never ambiguous when an hour has passed.
  • The moon in the sky is also a good example. It always goes east-to-west, moving about 15 degrees an hour.
  • The hands on a wrist-watch work well as a clock, and the mechanism INSIDE the wristwatch ensures that the hands only move clockwise and do so with regularity.
  • Plutonium’s Radioactivity makes an okay clock. Plutonium has a radioactive halflife of 24000 years. If I measured the radioactivity of a lump of plutonium, and then if I came back later and found that it was only half as radioactive, I’d know that 24000 years had passed.

So clocks are all around us, from the seasons to the tides to the wrinkles on our faces. Some clocks have very consistent rates of change, some even let us reliably measure very small intervals of time.

Lets assume, from here on in, that I have access to some VERY AWESOME CLOCKS.

this "clock" is ineffective. because it's timeless!

Speed of Light Is Constant

So imagine that you’re on a bus and you’re driving down the highway and you see a bird flying alongside the bus. It’s going pretty slow, right? If you were standing on the roadside you’re probably disagree and say that the bird was flying pretty friggin’ fast.

In general, two people who are moving at two different speeds (well, velocities actually but…) will disagree about what velocity they see a third person/object/bird moving at.

Albert Einstein was a genius, and he was one of the first people to come to terms with the idea that LIGHT (an electromagnetic wave) will move at exactly the same speed, regardless of how fast you are moving.

When I say “the speed of light is constant” maybe you’d like a scenario to illustrate exactly how crazy (counterintuitive) this proposition is. Lets call it “Photon Phil, the space cowboy”.

yeee hawww ben's gone off on a tangent!

Imagine that, in the distant year 2000, when everyone owns a personal atom powered rocket; the earth is passed by a GREAT HERD of Photon buffalo. A photon buffalo is like a regular buffalo (well, bison technically), but it’s made out of photons, it lives in space, and it runs through the universe at the speed of light. the herd is hundreds of lightyears long, and it’s fantastic. (no i’m not going to waste another hour looking for a good cheesy internet image to steal. just imagine it.)

Photon Phil, the Space cowboy, decides to make it his business follow the herd of photon buffalo. So he gets into his rocket ship, and tries to catch up to the herd. when he first approaches the galloping herd, his ships’ sensors tell him that each bison is passing him at a speed of 300,000,000 meters per second. Okay, so he accelerates along with the herd. after a good day or two of accelerating, he asks his ships’ sensors if he’s caught up with the herd, but the sensors tell him that each bison is passing him at a speed of 300,000,000 m/s.  So he really puts on the juice, and puts the pedal to the metal, and accelerates hard for a year… but each bison still passes him at a speed of 300,000,000 m/s.

Uh oh, photon phil. you’re about to waste your life!

you won't catch the devil's herd because they're going at the speed of... what? these are steer and aren't buffalo? whatever. it's late.

As a quick aside, we call it the speed of light because it’s the speed that light travels at in a vacuum. Specifically, it’s like the speed a wave travels at if there’s no internal fiddley mechanism that take time to transmit the wave. augh. this explanation is taking too long. Okay, so Imagine you are trying to knit a scarf. and you are as agile as you can be, but your fingers are too wide and keep getting in the way. so (use your magic to…) make your fingers narrower. now you can knit faster. but the needles are too wide and keep getting in the way and slowing you down. so you get narrower needles. and you can knit faster. but now the yarn is too wide. get narrower yarn. now your fingers are too wide…. etc.  So you keep getting narrower, and you keep knitting faster. What speed would you be able to knit if your fingers, needles and yarn all had zero width? you’d knit at the speed of light. In any case other things travel at the “speed of light” that aren’t light. like gravity!

In which Cary Grant wishes that he had the nimble fingers of a child.

Take a Second to Think about Lazy Snakes (this will be useful later)

The blue snake is SHORTER than the brown snake. If i opened their mouths, and used my magic flashlight to shine a photon down the length of their hollow cores, it would take MORE TIME for the photon to go down the length of the brown snake. For instance, if it took 1 second for a photon to go from tail to tip of the blue snake, it would take Longer than 1 second for  a photon to go from tail to tip of the brown snake.

Photon Ping-Pong

Alright. So lets analyze a rad scenario. Suppose that Cam and Rod are twins and they like playing Ping-pong. Only Suppose that it’s the future and that Cam and Rod are playing SUPER PHOTON PONG, where the ping-pong ball is made out of photons. It’ll travel across the table at the speed of light. Instead of ping-pong paddles, they use mirrors.

because it's the future cam and rod look like ZARDOZ or something.

Assume that because Cam and Rod are twins (and thus, not particularly competitive) (also, assume that they’re twins) they’ve set up their paddles to reflect the ping pong ball straight back along it’s initial path. It takes 1 second for the photon to travel from cam’s mirror, to rod’s and then back. It’s not clear whether the clock at Cam’s back has the visage of The King on it, but we can assume that the clock is TOTALLY AWESOME and accurate, and that it shows 1 second ticking past each time Cam touches the photon ping-pong ball.

This Just In: different people disagree on how long it takes to do something.

Lets talk about how this system would look If it were moving. Imagine that Cam and Rod are playing photon ping pong across a table, inside of a big space-ship  with a big glass top, so that we can look down on their game. In fact, Lisa (their sibling, but not a twin) is standing on a meteor and watches them play together as they pass.

I'm just going to admit it now: I don't know how to use a computer to make pictures that don't look crummy

Now, to Cam and Rod inside their ship, they can sense no motion. everything around them is still.

But to Lisa, sitting out beyond the moving ship, everything in Cam and Rod’s ship is moving and It’s all moving together, horizontally across the sky.

The blue line (a) is the stationary distance across the Ping-Pong Table, the Red Line (b) is the trajectory of the pingpong ball, as seen by Lisa

Lets discuss the Photon ping-pong ball specifically. It goes up and down between Cam and Rod’s mirror’s, but it also tracks horizontally along with the rest of the ship. Lisa Sees the photon ball take a path that looks like path b in the previous image.

So Lisa has a wristwatch on, and she knows that it takes a photon ball 1 second to move up and down the length of path “a”. It’s clear (from the snake explanation earlier) that  path “b” is Longer than Path “a”… therefore it will take Longer than 1 second (by Lisa’s accounting), for the pingpong ball to return to Cam.

So here’s the thing. According to this reasoning, as Lisa watches the Clock behind Cam’s head tic-tock; she’ll notice that it seems to be running SLOW. It takes MORE THAN 1 SECOND of Lisa’s time, for 1 SECOND of Cam’s Time to pass.

Holy Hell!

We just showed that Time passes SLOWER for people we see moving. We call this effect Time Dilation. It only really kicks in when you move really close to the speed of light (or if you have a very very accurate clock); but it’s still a very real effect.

next time:

There’s a problem with what i’ve just explained. If you can figure out the problem, email me at ben@sciencesortof.com and if you get it right, i’ll give you a prize…. that is to say, the first person who gets it right who is not a physicist will get a prize. If you ARE a physicist (or at least, if you *should* know the answer) and you get it WRONG, then you’ll have to send me a prize. next week is interesting people!

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