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尺缩与钟慢——狭义相对论第五讲 – 译学馆
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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

第五章 《尺缩效应 时间膨胀 以及更多》
《分钟物理》
In our universe,
在我们的宇宙中
when you change from a non-moving perspective to a moving one
当你由静止视角转化为运动视角
or vice versa,
或者反过来时
that change of perspective is represented by what’s called Lorentz transformation,
这样的视角变化可以通过洛伦兹变换表示
which is a kind of squeeze-stretch rotation of spacetime
而我们会用这个时空盘
that I’ve mechanically implemented with this spacetime globe.
以机械表现出这种时空的挤压 扩张 旋转
Two of the most famous implications of Lorentz transformations
洛伦兹变换揭示的两个最著名的效应
are phenomena called“length contraction”and“time dilation”.
就是“尺缩效应”与“时间膨胀”
And while their names make them sound like they’re two sides of the same coin,
虽然它们的名称听起来像是一枚硬币的两面
they’re definitely not.
但事实并非如此
We’ll start with time dilation which is easier to see.
我们先从更易理解的时间膨胀说起
Suppose I have a clock with me that ticks every two seconds.
假设我有一只钟 它每两秒钟嘀嗒一次
But if you’re moving at a third the speed of light to my left,
但若你以1/3光速向我左侧移动
then from your perspective, the time coordinates at which my clock,
那么在你看来 我的时钟每嘀嗒一次的
now ticks are slightly farther apart.
时间间隔就会变长
Accroding to you, it takes about 2.12 seconds between each tick,
对你来说 我的时钟每嘀嗒一次需要2.12秒
instead of 2 seconds.
而不是2秒
Time literally is running slower for me relative to you,
我的时间对你而言确实变慢了
because Lorentz transformations, which represent how relative motion works in our universe,
因为表示宇宙中相对运动原理的洛伦兹变换
kind of stretch things out a bit.
似乎把东西拉长了一点
Likewise, if you have a clock with you that ticks every 2.83 seconds,
同样 如果你有一个每2.83秒嘀嗒一次的钟
then from my perspective it will tick every 3 seconds.
那么在我看来 它是每3秒嘀嗒一次的
So both of us perceive each other’s perception of time
所以我们各自看来对方的时间
as running slow by the same factor.
都被拉长了相同的因数倍
That is, relative motion causes our perception
也就是说 相对运动导致我们
of the duration of time between events to become longer, or dilated –
感知到的事件持续时间变长了 或者说膨胀了
“time dilation”.
即 “时间膨胀”
If you’re wondering how it can make sense
如果你想知道如何去解释
that we both perceive each other’s time as running slow,
这种两个人都认为对方的时间变慢了的现象
well, I have another whole video on that.
正好 我有另一个视频来说明它
But in short,
但是总而言之
it’s because our respective worldlines which correspond to our own time axes
这是因为分别对应于我们各自时间轴的世界线
are rotated relative to each other,
相对于对方偏离了一定的角度
and so we each only attribute a projection
所以我们只是把对方世界线
of the other person’s worldline length as representing movement through time,
在我们世界线上的投影当成了时间上的运动
and the rest as movement through space.
而将另一边的投影视为空间上的运动
The factor by which intervals are dilated depends on
时间膨胀所对应的洛伦兹因子取决于
how fast we’re moving relative to each other,
我们彼此的相对运动速度
and the expression looks like this.
表达式是这样的
But it’s really just saying
但其实它只表明
how much higher up in time is this point after a Lorentz transformation.
经过洛伦兹变换后某个点在时间轴上上升了多少
And if you plot the expression, you’ll see that for slow speeds
如果作出函数图像 你会发现
relative time intervals are roughly equivalent.
在低速时 相对时间间隔大致相同
But the closer you are to light speed,
但越接近光速
the more our relative perception of times becomes distorted.
在我们相对视角下的时间就膨胀得越多
Length contraction, on the other hand, is a tad more complicated.
另一方面 尺缩效应会更复杂一点
First, we need something with length.
首先 我们需要一件有长度的东西
Let’s suppose we have a cat whose tail is
假设我们有一只猫
at position 0 for all time,
它的尾巴始终处于0点
and whose head is 600 million meters to the right for all time,
而头始终在右方6亿米处
remember each horizontal tick mark here represents 299,792,458m.
记住 这儿横轴一格代表299792458米
So from my perspective,
所以在我看来
the cat is 600 million meters long.
这只猫有6亿米长
However, from your perspective moving at a third the speed of light to the left,
但在你看来就不同了 由于你正以1/3光速向左移动
the ends of the cat get stretched out from each other by the Lorentz transformation
根据洛伦兹变换 这只猫的首尾会彼此远离
which at first might seem like dilation of distance, not contraction.
看上去应该是距离膨胀 而非缩短
And this is indeed true.
但事实确实如此
From your perspective, the distance between the cat’s tail at my time t=6
以你的视角来看 猫尾(在我t=6的时刻)
and the cat’s head at my time t=6 is indeed longer,
与猫头(在我t=6的时刻)之间的距离确实更长了
it’s now 636 million meters,
现在是6.36亿米
dilated by exact same factor as in time dilation.
延展倍数与时间膨胀中的洛伦兹因子相同
However, this dilated distance doesn’t represent
然而 这一延展的距离并不代表
the length of your cat from your prespective,
从你的视角所看到的猫的长度
because these measurements of the positions of its head and tail
因为猫的头尾位置
no longer happen at the same time
并不是同时测量的
and the cat moves in between when the measurements are taken –
并且测量时猫在这之间是运动的
that’s what it means to have a slanted worldline,
这就是倾斜的世界线带来的效果
you’re changing position as time passes, aka moving.
你随着时间流逝而改变位置 即运动
And if something moves while you’re measuring it,
而如果你在测量一个运动物体的长度
that measurement doesn’t represent its length.
但测得的距离并非它的长度
So to correctly measure the length of the cat from your perspective,
所以为了从你的角度准确测量猫的长度
we need to measure the positions of its front and back
我们需要在你的视角下
at the same time according to your perspective.
同时测量猫的首尾位置
Which is this distance here, which is clearly shorter – 566 million meters.
这里测得的距离只有5.66亿米 显然更短了
In fact, it turns out it’s exactly the inverse factor from the other distance
事实上 结果表明它正好是距离的反向因子
instead of multiplying 600 million by 1.06, it’s divided by 1.06.
用6亿除以1.06 而不是乘1.06
The same thing happens the other way, too:
反过来也一样
if you have a cat that’s stationary in your perspective,
假设你有一只相对你静止的猫
then when I view it from my perspective,
当我从我的角度观察它时
I’ll measure its length
我可以同时测量
by measuring the head and tail at the same time according to me,
猫的首尾位置以测得它的长度
as being shorter.
同样也变短了
This is the phenomenon we call “length contraction”
这个现象就被称为“尺缩效应”
The measured lengths of moving things are shorter
测得的运动物体的长度
than when those things are viewed as not moving.
总比它静止状态下的长度更短
The precise factor by which lengths are contracted again depends on
长度收缩所对应的洛伦兹因子同样取决于
how fast we’re moving relative to each other, and looks like this.
我们相对运动的速度 就像图上这样
And similar to the case of time dilation,
类似于时间膨胀
the closer you are to light speed,
你越接近光速
the more relative perception of lengths becomes distorted.
相对视角下的长度就收缩越多
So let’s recap:
让我们回顾一下
time dilation of moving objects is
运动物体的时间膨胀
simply the direct effect of Lorentz transformations,
是洛伦兹变换的直接影响
stretching consecutive time coordinates apart in time,
导致时间坐标被拉伸
while length contraction of moving objects is
而运动物体的长度收缩则是两种效果的叠加
a combination of the stretching effect of Lorentz transformations on spatial distances
一是洛伦兹变换在空间距离上的拉伸效果
which is kind of like a “distance dilation”
这有点像“距离膨胀”
Plus then changing the times at which we’re comparing things
一是我们测量物体两端时的时间改变
because they were no longer simultaneous.
因为变换后的测量时间已经不是同时了
This is what I meant when I said earlier
这就是为什么我之前说
that time dilation and length contraction aren’t
时间膨胀和尺缩效应
two sides of the same coin:
并非一枚硬币的两面
time dilation compares the times of the same events in the new perspective,
时间膨胀是比较同一事件在不同视角下的时间
and it pairs with distance dilation,
它与距离延长相对应
which compares the positions of the same events in the new perspective.
距离延长是比较同一事件在不同视角下发生的位置
Length contraction, in contrast,
相比之下 尺缩效应
compares positions at the same time according to the new perspective.
是比较在不同视角下物体两端在同一时间的位置
So you might be wondering,
那么你可能会想
is there a time version of length contraction, then?
那有没有一种时间版的尺缩效应呢?
Is “Time contraction” a thing?
“时间收缩”存在吗?
Yes, yes it is.
是 它存在的
Although people almost never talk about it
尽管人们不怎么谈论
and it doesn’t have an official name,
它也没有一个正式的名字
but I think it’s nice to complete the full picture.
但我觉得把表格填完比较好
the one missing piece is to compare times
缺失的一格是比较同一位置
at the same position according to the new perspective.
在不同视角下的时间
Let’s imagine I’ve put a lightbulb at every point in space
想象一下我在空间中每一点都放一个灯泡
even in between here where I can attach them to the time globe
包括时空盘两点之间的任意位置
and I turn them all on simultaneously at one time,
然后在某一时刻我把它们同时点亮
and then turn them off simultaneously a little bit later.
一会儿后把它们同时熄灭
From your moving perspective,
当然 在运动的你看来
any particular one of my lights will have its on-off time interval dilated, of course,
任何一个灯泡开和关的时间间隔都变大了
but at any particular location in space (like, where you are),
但是在空间任意位置(比如你的位置)
the duration of time between when the lights go on
灯泡变亮和变暗的
and when the lights go off will actually be shorter.
时间间隔(即事件持续时间)会缩短
Maybe it should be called “duration contraction”!
也许这应该称为“持续时间收缩”
So, to summarize, when changing to a moving perspective on universe,
简而言之 当我们在宇宙中切换到运动视角时
this was time dilation and time duration contraction,
会出现时间膨胀和持续时间收缩
and length of contraction and distance dilation.
以及尺缩效应和距离延长
These four ideas said aloud as words
这四个概念光看字面意思
certainly sound super contradictory and impossible.
好像极度矛盾而又毫不现实
I mean how can the time be both shorter and longer?
我是说 时间怎么能又变快又变慢?
but if you have a spacetime globe
但如果你有一个时空盘
it’s easy to understand there are no paradoxes or contradictions.
就很容易理解 这并不相悖或自相矛盾
We simply need to be more careful
我们只需要
with our ideas of distance and time intervals
在将洛伦兹变换应用于时空中延伸的物体时
when applied to extended objects in space time.
谨慎考虑时空间隔的概念
Do we mean the time between two exact events (time dilation),
我们研究的是两个事件之间的时间间隔 (时间膨胀)?
or the time passes between the versions of those events
还是在同一位置不同视角下
that happen at the same place (duration contraction)?
事件发生的持续时间(持续时间收缩)?
Do we mean the distance between two events
我们研究的仅仅是两个事件的间距
regardless of when they happen (distance dilation),
而不管它们是何时发生的(距离延长)?
or the distance between the versions of those events
还是在同一时刻不同视角下
that happen at the same time which is the length (length contraction)?
事件之间的距离或长度(尺缩效应)?
This is subtle stuff,
这个东西很微妙
and words and equations by themselves
光看它的名称和方程
make these concepts really really hard to keep straight;
你真的很难搞懂这些概念
but a spacetime diagram doesn’t lie.
但是有了时空盘问题就简单很多了
To get more experience with time dilation and length contraction yourself,
要想进一步了解时间膨胀和尺缩效应
I highly recommend Brilliant.org’s course on special relativity.
我强烈推荐Brilliant.org的狭义相对论课程
There you can do problems that build off what you learned in this video
您可以在那儿深入了解在本视频中学到的东西
and explore real world scenarios
并且需要重视尺缩效应
where it’s important to take time dilation and length contraction into account,
和时间膨胀的在探索现实情境中的应用
like the famous Michelson Morley experiment.
就像著名的Michelson Morley实验一样
The special relativity questions on Brilliant.org
Brilliant.org网站上的狭义相对论问题
are specifically designed to help you go deeper
是为帮助您深入理解我的系列视频
on the topics I’m including in this series,
而特别设计的
and you can get 20% off of a Brilliant subscription
您可以打开Brilliant.org/minutephysics
by going to Brilliant.org/minutephysics.
获取Brilliant的八折订阅优惠
Again, that’s Brilliant.org/minutephysics
再次强调Brilliant.org/minutephysics
which gets you 20 % off premium access to
可以让您享受八折优惠
all of Brilliant’s courses and puzzles,
该优惠适用于所有课程和题目
and lets Brilliant know you came from here.
并会让Brilliant知道您是从这里了解到的哦

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视频概述

这个视频通过结合使用方程和时空盘,以浅显的语言介绍狭义相对论的尺缩效应和时间膨胀,以及在不同观察方式下它们的另外两种衍生效应,“距离延长”和“”持续时间收缩”。

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视频来源

https://www.youtube.com/watch?v=-NN_m2yKAAk

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