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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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長度收縮和時間擴張|狹義相對論 第五章 (Length Contraction and Time Dilation | Special Relativity Ch. 5)

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    Summer 發佈於 2021 年 01 月 14 日
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