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• In our universe, when you change from a non-moving perspective to a moving one, or vice versa,

• that change of perspective is represented by a what's called Lorentz transformation,

• which is a kind of squeeze-stretch rotation of spacetime that I've mechanically implemented

• with this spacetime globe.

• Two of the most famous implications of Lorentz transformations are phenomena calledlength

• contractionandtime dilation” – and while their names make them sound like they're

• two sides of the same coin, they're definitely not.

• We'll start with time dilation, which is easier to seesuppose I have a clock with me

• that ticks every two seconds.

• But if you're moving at a third the speed of light to my left, then from your perspective,

• the time coordinates at which my clock now ticks are slightly farther apartaccording

• to you it takes about 2.12 seconds between each tick, instead of 2 seconds.

• Time literally is running slower for me relative to you, because Lorentz transformations, which

• represent how relative motion works in our universe, kind of stretch things out a bit.

• Likewise, if you have a clock with you that ticks every...

• 2.83 seconds, then from my perspective it will tick every 3 seconds.

• So both of us perceive each other's perception of time as running slow by the same factor

• that is, relative motion causes our perception of the duration of time between events to

• become longer, or dilated - “time dilation”.

• If you're wondering how it can make sense that we both perceive each other's time as

• running slow, well, I have another whole video on that, but in short, it's because our respective

• worldlines (which correspond to our own time axes) are rotated relative to each other,

• and so we each only attribute a projection of the other person's worldlines' length as

• representing movement through time, and the rest as movement through space.

• The factor by which intervals are dilated depends on how fast we're moving relative

• to each other, v, and the expression looks like thisbut it's really just saying

• how much higher up in time is this point after a Lorentz transformation?”

• And if you plot the expression, you'll see that for slow speeds, relative time intervals

• are roughly equivalent, but the closer you are to light speed, the more relative perception

• of times becomes distorted.

• Length contraction, on the other hand, is a tad more complicated.

• First, we need something with length.

• Let's suppose we have a cat whose tail is at position 0 for all time, and whose head

• is 600 million meters to the right for all time (remember, each horizontal tick mark

• here represents 299,792,458m).

• So from my perspective, the cat is 600 million meters long.

• However, from your perspective moving at a third the speed of light to the left, the

• ends of the cat get stretched out from each other by the Lorentz transformationwhich

• at first might seem like dilation of distance, not contraction.

• And this is indeed truefrom your perspective, the distance between the cat's tail at my

• time t=6 and the cat's head at my time t=6 is indeed longer: it's now 636 million meters

• (dilated by exact same factor as in time dilation).

• However, this dilated distance doesn't represent the length of the cat from your blue perspective,

• because these measurements of the positions of its head and tail no longer happen at the

• same time, and the cat moves in between when the measurements are taken - that's what having

• a slanted world line means - changing position as time passes, aka movement.

• And if something moves while you're measuring it, that measurement doesn't represent its

• length.

• So to correctly measure the length of the cat from your perspective, we need to measure

• the positions of its front and back at the same time according to your perspective.

• Which is this distance here, which is clearly shorter - 566 million meters.

• In fact, it turns out it's exactly the inverse factor from the other distance - instead of

• multiplying 600 million by 1.06, it's divided by 1.06.

• The same thing happens the other way, too: if you have a cat that's stationary in your

• perspective, then when I view it from my perspective, I'll measure its length (by measuring the

• head and tail at the same time, according to me), as being shorter.

• This is the phenomenon we calllength contraction” – the measured lengths of moving things

• are shorter than when those things are viewed as not moving.

• The precise factor by which lengths are contracted again depends on how fast we're moving relative

• to each other, v, and looks like this.

• And similar to the case of time dilation, the closer you are to light speed, the more

• relative perception of lengths becomes distorted.

• So let's recap: time dilation of moving objects is simply the direct effect of Lorentz transformations

• stretching consecutive time coordinates apart in time, while length contraction of moving

• objects is a combination of the stretching effect of lorentz transformations on spatial

• distances (which is kind of like a “distance dilation”) PLUS then changing the times

• at which we're comparing things because they were no longer simultaneous.

• This is what I meant when I said earlier that time dilation and length contraction aren't

• two sides of the same coin: time dilation compares the times of the same events in the

• new perspective, and it pairs with distance dilation, which compares the positions of

• the same events in the new perspective.

• Length contraction, in contrast, compares positions at the same time according to the

• new perspective.

• So you might be wondering, is there a time version of length contraction, then?

• IsTime contraction” a thing?

• Yes, yes it is (though people almost never talk about it and it doesn't have an official

• name, but I think it's nice to complete the full picture.

• the one missing piece is to compare times at the same position): Let's imagine I've

• put a lightbulb at every point in space (even in between where I can attach them to the

• time globe) and I turn them all on simultaneously at one time, and then turn them off simultaneously

• a little bit later.

• From your moving perspective, any particular one of my lights will have its on-off time

• interval dilated, of course, but at any particular location in space (like, where you are), the

• duration of time between when the lights go on and when the lights go off will actually

• be shorter.

• Maybe it should be calledduration contraction”!

• So, to summarize, when changing to a moving perspective in our universe, there's both

• time dilation and length contraction, but there's also distance dilation and duraction

• contraction.

• These four ideas said aloud as words certainly sound super contradictory and impossible (I

• mean, how can the time be both shorter and longer?!), but if you have a spacetime globe

• it's easy to understand there are no paradoxes or contradictionswe simply need to be

• more careful with our ideas of distance and time intervals when applied to extended objects

• in spacetime: do we mean the time between two exact events (time dilation), or the time

• between the versions of those events that happen at the same place (duration contraction)?

• Do we mean the distance between two events regardless of when they happen (distance dilation),

• or the distance between the versions of those events that happen at the same time (length

• contraction)?

• This is subtle stuff, and words and equations by themselves make these concepts really really

• hard to keep straight; but a spacetime diagram doesn't lie.

• To get more experience with time dilation and length contraction yourself, I highly

• recommend Brilliant.org's course on special relativity.

• There, you can do problems that build off what you learned in this video and explore

• real world scenarios where it's important to take time dilation and length contraction

• into account, like the famous Michelson Morley experiment.

• The special relativity questions on Brilliant.org are specifically designed to help you go deeper

• on the topics I'm including in this series, and you can get 20% off of a Brilliant subscription

• by going to Brilliant.org/minutephysics.

• Again, that's Brilliant.org/minutephysics which gets you 20% off premium access to all

• of Brilliant's courses and puzzles, and lets Brilliant know you came from here.

In our universe, when you change from a non-moving perspective to a moving one, or vice versa,

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長さの収縮と時間の拡張｜特殊相対性理論 第5章 (Length Contraction and Time Dilation | Special Relativity Ch. 5)

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Summer に公開 2021 年 01 月 14 日