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#### 尺缩与钟慢——狭义相对论第五讲

Length Contraction and Time Dilation | Special Relativity Ch. 5

《分钟物理》
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 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 called“length contraction”and“time dilation”.

And while their names make them sound like they’re two sides of the same coin,

they’re definitely not.

Suppose 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 apart.

Accroding to you, it takes about 2.12 seconds between each tick,

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 worldline 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,

and the expression looks like this.

But 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 our 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 transformation

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

And this is indeed true.

From 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

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 it means to have a slanted worldline,

you’re changing position as time passes, aka moving.

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 call “length 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, 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?

Is “Time contraction” a thing?
“时间收缩”存在吗？
Yes, yes it is.

Although 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 according to the new perspective.

Let’s imagine I’ve put a lightbulb at every point in space

even in between here 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.

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 called “duration contraction”!

So, to summarize, when changing to a moving perspective on universe,

this was time dilation and time duration contraction,

and length of contraction and distance dilation.

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

We simply need to be more careful

with our ideas of distance and time intervals

when applied to extended objects in space time.

Do we mean the time between two exact events (time dilation),

or the time passes 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 which is the length (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
Brilliant.org网站上的狭义相对论问题

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

all of Brilliant’s courses and puzzles,

and lets Brilliant know you came from here.