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为什么你不能走的比光快? – 译学馆
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为什么你不能走的比光快?

Why you can't go faster than light (with equations) - Sixty Symbols

So, we made a few videos about relativity,
我们做过一些关于相对论的视频
and we’ve talked about how distances change in relativity,
讨论了相对论中距离是如何变化的
we’ve talked about how time changes in relativity.
也讨论了相对论中时间是如何变化的
So I thought I’d talk about something,
那么这次我想谈谈
which combines those two things together,
将两者关联起来的
which is how speed, which is distance divided by time, changes in relativity.
那就是速度 距离除以时间 在相对论中是如何变化的
In the previous videos, we made what we called a Gamma Trilogy
在之前的视频中 我们做了个伽玛三部曲
because they all have this gamma factor in it.
因为它们都包含了伽玛因子
Actually one of the interesting things, about the way speed transforms
实际上 还有一点有趣的是 速度的变换方式
is that the gammas all disappear.
就是所有伽马消失的方式
They all cancel out, so there is no gamma in this gamma video.
它们都消除了 因此在这次的伽马视频中没有伽马
So we need to think about how you made your speed,
现在我们需要思考速度是如何产生的
of course, it’s distance divided by time,
当然 是距离除以时间
but particularly how it changes depending on what reference frame you’re in.
但具体来说 它是如何变化的取决于你所在的参考系
So we need to think about
如此我们需要思考一下
how different people measure speed in different reference frames.
不同参考系的不同人如何测量速度
And some of the classic kind of thought experiment you might do
还有一些你可能会做的经典性思维实验是
is you’ve got somebody on a train
你安排了某人在火车上
I’ll call this reference frame S’.
我把这个参考系叫做S’
and they’re gonna roll a bowling ball, along the train
他们将会滚动一个保龄球 让它跟火车一起运动
at some speed or other,
以某一速度运动
and the train itself is moving so this whole reference frame is moving
火车本身就在移动 所以整个参考系也在移动
in this direction at some other speed v.
并以速度v朝这个方向运动
The way you’re on the train you’d measure the speed is that
你在火车上测量速度的方式就是
you’d see how far the bowling ball had gone in your reference frame.
你所看到的 保龄球在你所在参考系中的移动距离
What should we call this? Δx: how far it’s gone,
我们应该称它为什么呢? △x:移动距离
Δx’ because it’s in your reference frame.
Δx’ 因为它是在你所在的参考系
and at what time you’ve measured that at,
以及你什么时候测量的
some time Δt’ after you’d let go of the ball,
Δt’:你放开小球后的一段时间
and then you could actually measure the speed in your reference frame of that bowling ball
然后就可以测量 你所在参考系中保龄球的速度
is just how far it’s gone
就是它移动的距离
divided by the time it took to get there.
除以它到那里所花费的花间
That’s the definition of speed,
这就是速度的定义
it’s how far something’s gone divided by the amount of time it took to get there.
它就是就是某物的移动距离除以它到达目的地所花费的时间
It’s the definition of speed.
也就是速度的定义
So that’s in one reference frame,
这是在一个参考系中
so then the question is what does the person in this reference frame,
那么问题是 对于此参考系中的人员所做的事
the other reference frame see?
其他参考系的人能否看见?
And let’s deal with the non-relativistic case first:
我们先来处理非相对论条件下的情况:
What are they gon na measure?
他们要测量什么?
The distance that they’ll measure that the ball has gone
他们要测量的球移动的距离
is going to be how far the bowling ball has rolled away from you
就是保龄球与滚离你的距离
plus how far you’ve moved down the line.
加上你沿着这条线走的距离
That extra distance, how far the train has moved,
额外的距离 火车移动的距离
depends on basically how fast the train was going.
主要取决于火车的移动速度
You’re sort of adding it all on top of each other?
就是所有的分子都加上它 对吧?
Yeah, exactly! That’s all you’re doing.
是的 完全正确 这就是你要做的
So that distance is vΔt.
所以这段距离就是vΔt
If the train is going at some speed v,
如果火车以速度v行驶
and it has traveled for some time Δt, then
它行驶的一段时间为Δt
the distance it has covered is just the speed times
它所行驶过的距离就是速度乘以
the time it has taken to do it.
它所花费的时间
So now we can put all this stuff together…
所以现在我们归纳一下
That says that Δx is equal to
就是说:Δx等同于
(how far you see the ball as having moved is)
(你看到的球移动的距离)
how far the train has moved,
火车移动的距离
plus how far the ball has moved relative to the train.
加上小球相对于火车的移动距离
Just the sum of the two.
就是这两者之和
And we can rearrange this in the non-relativistic case
我们可以在非相对论情况下重新排列一下
to say that
就是
Let’s just divide through: Δx divided by Δt
我们这样除一下:Δx/Δt
is equal to v plus Δx’ divided by Δt.
就等同于v+Δx’/Δt
And in this non-relativistic case
在非相对论情况下
we don’t have to worry about different people seeing different times…
我们没有必要担心不同的人看到不同的时间……
So it doesn’t matter whether it’s Δt or Δt’,
所以是Δt还是Δt’都没有关系
we can call it the same thing.
我们可以说这是一样的
It’s all the same thing.
完全等同
Or we can rewrite this again: it just says that
要不我们再写一遍:就是说
the speed you see the balls moving is equal to v.
你看到的球速等同于v
So that’s Δx divided by Δt.
所以就是Δx/Δt
Plus the speed that the person on the train sees the ball moving.
加上火车上的人看到的球速
And that, as you just said,
然后 按刚刚你所说
is you basically just add the speed.
你只需要加上速度就可以了
The speed at which the bowling ball is moving is
保龄球移动的速度就是
the speed that the bowling ball is moving relative to the train
保龄球相对火车的移动速度
plus the speed that the train is moving relative to you.
加上火车相对于你的移动速度
That’s the simple Galilean transformation, no relativity.
这就是简单的伽利略变换 没有相对论
Everything works in the way that we’re used to it working.
一切都按我们习惯的方式运行着
One of the things that get’s people interested in relativity is that people say,
人们对相对论感兴趣的原因之一是 有人说
you can’t ever travel faster than light.
你永远不能超越光速
So one obvious question you could ask is:
所以你可能会问一个显而易见的问题:
why can’t I just travel faster than light by just doing these kinds of additions?
为什么不通过速度相加来超越光速呢?
Supposing the person on the train, instead of rolling a bowling ball,
假设火车上的人 并非滚动的保龄球
was actually shining a torch along the train
而是沿着火车前进方向照亮手电筒
Then the torch beam would be traveling at the speed of light relative to them
那么相对于他们来说 手电筒光束将会以光速运动
and then if you ask
然后你可能会问
how fast the torch beam was travelling relative to you.
相对于你 火光移动的速度是多少
Well, it would be the speed of light, plus
嗯 它会以光速移动
the speed that the train is moving away from you,
加上火车远离你的移动速度
and that’s faster than the speed of light.
它将会比光速还要快
Done it, you’ve hacked the universe.
搞定 宇宙奥秘已被你破解
Unfortunately, relativity takes care of that,
可惜 相对论涵盖了这一点
and it’s not really the way that things work.
事情不是这样的
So we need to start again,
我们需要重新开始
but we need to do the problem properly with relativity.
但我们要运用相对论来解决这个问题
Is this your Minute Physics audition, professor?
教授 这是你的《分钟物理》的试镜吗?
Unfortunately, my drawing is not quite up to the job, I suspect.
可惜的是 我觉得我画的不是很好
And then we’ve got the other chaps in here.
我们在这里再画一个小伙子
Moving along at some speed v.
以速度v移动
Previously the formula we had said that
我们在之前的公式里提到过
the distance that the bowling ball is down the line
保龄球的直线距离
is equal to the distance relative to the train
等同于相对于火车的距离
plus how far the train has moved.
加上火车移动的距离
Now the extra factor that we haven’t put in yet is special relativity.
现在 我们还没考虑的额外因素是狭义相对论
And what we saw before was that what special relativity does is enter these factors of γ.
而正如我们先前所看到的:狭义相对论所做的就是引入γ因子
And so this Galilean transformation that we had before
所以我们之前做过的伽利略变换
is a little bit modified by an extra factor of γ.
就是通过引入因子γ稍做变换
γ is 1 over the square root
γ 就是1除以
of 1 minus v squared over c squared.
1-v²/c²差的平方根
we have a whole series of videos you can go and watch about gamma should you wish to do so.
我们有一系列的视频 如果你想看伽玛的相关视频 也可以去看看
But then the other thing we know about relativity is that
关于相对论 我们要知道的一点是
not only does the distance depend on the reference frame you’re in,
不仅是路程取决于你所在的参考系
but actually the time depends on which reference frame you’re in.
实际上时间也取决于你所在的参考系
And so we need the equivalent transformation for that.
因此我们需要对它进行等价变换
So there are these things called Lorentz transformations…
就是所谓的洛伦兹变换……
That’s the first of them and this is the other transformation that we need.
那是第一个 这是我们需要的另外一个变换
Turns out, these are the two Lorentz transformations that you need.
做完后 就得到你所需要的两个洛伦兹变换
Now we can say,
现在我们可以说
what’s the speed as measured from the person
速度是从此人的角度来衡量的
standing beside the track watching the train go by.
也就是铁轨旁看火车驶过的那个人
And we just do the same thing that we did before.
我们只是重复做了之前的事情
And the speed is just the distance divided by the time.
那么速度就是距离除以时间
So we can just divide them with Δx divided by Δt.
所以我们可以两者相除 即Δx除以Δt
is equal to… now the γs (gammas) are going to cancel
就等同于……现在要把γ消除
because if we divide that by the other,
因为倘若我们用另外一个来除
we end up with a γ on the top and a γ on the bottom.
最终我们得到一个γ在分子上 一个γ在分母上
So we can just cancel them out.
所以我们刚好消除它们
I’ll do a little bit of tidying up:
我来稍微整理一下:
I’ll divide top and bottom through by Δt’,
分子分母同时除以Δt’
So I can just write this as, Δx’ over Δt’.
所以我可以这样写:Δx’/Δt’
This is kinda the answer we wanted,
这种答案就是我们想要的
because Δx over Δt is the speed of the bowling ball
因为Δx除以Δt就是保龄球的速度
or whatever it is being thrown along the train
或者扔在火车上的任何东西的速度
As seen from the perspective of the person standing beside the train track.
这个速度是从站在铁轨旁边人的角度来看的
And Δx’ over Δt’ is the speed as seen from a person standing on the train.
对于站在火车上的人来说 Δx/Δt’就是速度
So I can write that in terms of the”u”s and”u'”s we have before
我可以把它们跟之前那个变换一样 写作u和u’
This just says that
这就是说
u is equal to u’ plus v divided by
u等于u’加上v的和除以
1 plus, u’ v divided by c squared.
1加上u’v/c²的和
Which is the final answer in the non-relativistic limit.
这就是狭义相对论极限下的最终答案
So what we were looking at before, the Galilean transformation,
所以我们之前所看到的 伽利略变换
both u’ and v are small compared to the speed of light.
u’和v相对于光速来说不值一提
That means this term is small. That’s just 1 basically…
也就是说这项非常小 分母就相当于1……
And that just says that the speed you see is just
这就说明 你看到的速度就是
the sum of the speed of the ball relative to the train and the train relative to you.
球相对于火车的速度与火车相对于你的速度 这两者之和
Which is the answer we had before.
这就是之前我们得到的答案
So that’s right, it all kind of comes out right
所以事实如此 即使物体不尽相同
if things aren’t closest be alike
结果仍是如此
Now let’s do the more interesting case of where the relativity actually matters.
现在我们再举一个更有趣的能真正体现相对论发挥作用的例子
And in particular let’s go to the really extreme case
我们来看一个极端的例子
of instead of rolling a bowling ball along the train,
这次不再是火车上滚动的保龄球
let’s shine a light beam along the train.
而是我们在火车上照出一束光
So in that case, u’,
这种情况下 u’
the speed of whatever it is is equal to the speed of light.
即任何物体的速度 都等同于光速
And let’s see what happens when we put that in there.
当把它放进去时 我们看看会发生什么
So in that case we end up with,
这种情况下 我们得到
if u’ is equal to the speed of light…
假设u’=光速
Then u is equal to c (the speed of light) plus v…
那么u就等同于c(光速)加上v的和……
divided by 1 plus (u’ is the speed of light) cv over c squared. Now,
再除以 1加上cv/c²的和(u’=光速)
let me know just do a little playing around with this.
我对这个方程做一下改变
So I will rewrite the top here as…
那么我就要把上面重写一下:
c into 1 (so, I will pull that factor out) plus v over c.
把c变成1(所以 我要把这个因子提取出来)加上v/c
Because if I multiply this out,
因为倘若我把这个乘出来
I just end up with c plus cv over c which is just v
最终得到c加上cv/c也就是v
so c plus v, divided by
所以就是c+v 除以
And then all I’m going to do is cancel the c over c squared
接下来我要做的就是消去c
so I end up with 1 plus v over c.
所以我最终得到1+v/c
Which you know is just the same as the term there.
你知道的 它就和这项一样了
So this is the same as that
所以它们是一致的
which just basically means that this whole thing is equal to the speed of light still.
大致来说就是这个整体的速度还是等于光速
So there’s the bizarre thing
所以奇怪的是:
That we’ve taken. Remember, what we’ve done here is said,
我们已经这样做了 记住 要言行一致
Okay, So, there’s a light beam
好了 现在 这里有一个光束
that’s moving relative to the train at the speed of light.
相对于火车 它是以光速运动
Now if you’re watching from beside the train,
如果你是从火车旁边看
what speed do you see that light beam going at?
你看到光束的速度是多少?
The answer is still the speed of light.
答案依旧是光速
You haven’t actually added to its speed at all.
实际上你根本没有增加它的速度
And I guess the physics behind it is that
我猜着它背后的物理原理是这样的
you have to worry about both space and time
你必须同时考虑空间和时间两个因素
being changed by what reference frame you’re in.
它们会随着你所在的参考系而改变
Which means that not surprisingly,
这样说并不奇怪
the speed that comes out is going to be change
最终的速度也会
in a slightly strange way as well.
以一种有点奇怪的方式发生变化
And it turns out that
事实证明
relativity takes care of this invariance in the speed of light.
相对论兼顾了光速的恒定性
That no matter what reference frame you’re watching the light beam travel from,
无论你看到的光束来自哪个参考系
you’ll always see it travel at the speed of light.
你总能看到它在以光速运动
Is the universe wanting to keep things at the speed of light
是因为宇宙想要保持光速运行
and everything changes to cater for that?
一切变化都要符合这个规律吗?
Or does everything change all the time
还是一切都在变化着
and the speed of light just falls out of that?
而光速就此产生了?
Is it chicken or egg?
到底是哪一种?
It’s a very good question.
这是一个值得思考的问题
And I actually really like your first explanation that
实际上我非常喜欢你的第一个解释
actually the universe arranges in things in such a way
实际上宇宙就是以这样的方式运行着
that the speed of light always comes out as the speed of light.
光速还是光速
no matter how much you mess around with things
不管你把事情弄得有多乱
by trying to run away from the light beam or run towards it…
意图远离光束抑或靠近光束……
When you come to measure it’s speed,
当你要测量它的速度时
you’ll always find that it’s speed comes out at the speed of light.
你总会发现它的速度就是光速
and everything else kind of adjusts the distances and the times
调整距离和时间相关的其他所有一切因素
adjust in just such a way,
也还是这样
so that when you come to combine those distances and times
因此当你把这些距离和时间结合起来
to measure a speed of a light beam,
测量光束速度的时候
It will always come out the speed of light.
光束的速度还是光速
Is there something that is important about the speed of light being constant and unchanging?
光速保持恒定不变 是有什么至关重要的因素嘛?
So what motivated Einstein to come up with all this in the first place is…
所以爱因斯坦最初提出这些的动机是……
he had this idea that
他有这种想法
the laws of physics should the same
物理定律是一致的
whatever reference frame you’re in.
无论你在什么样的参考系
So that if you’re in sealed box,
那么如果你深处一个封闭空间
there should be no experiment you can do that will tell you
那没有任何理论试验能告诉你
if that sealed box is moving at a constant speed or is stationary.
那个封闭空间是以恒定的速度移动的 还是静止的
And in fact, in his view of things,
实际上 在他看来
it is a kind of meaningless question.
这是毫无意义的问题
And what he also knew is that
他也同样知道
the speed of light comes out from the laws of electromagnetism.
光速来自于电磁定律
And so
所以
then the question is,
问题是
in what reference frame do those laws of physics work
在什么样的参考系里 这些物理定律是成立的
And his argument is that
他的论点是
those laws of physics should work in whatever reference frame you’re in.
无论是在哪个参考系 这些物理定律都是成立的
which means that actually the speed of light has to be invariant
这就意味着光速必须是不变的
if you believe the laws of physics are the same
倘若你相信 物理定律
in all the different reference frames.
在所有的不同的参考系都是一致的
What if it hadn’t been that way?
倘若不是这样的怎么办?
What if the universe said,”nah, I can be different for different reference frames”?
倘若宇宙说:“天啊 不同参考系我是不一样的?”
Like would you and I be ripped to pieces in some cosmic rip hole?
比如你和我在宇宙缝隙中会被撕成碎片嘛?
would the universe just be a bit different and quirky?
或者宇宙刚好有点不同 有点奇特呢?
Like does the universe benefit from the fact that this is what happens?
还是宇宙是从这样的事实中获益呢?
It’s very hard to construct a universe in which this wasn’t true.
要构建这样一个不真实的宇宙是非常困难的
(In retrospect it’s hard to construct a universe in which this isn’t true.)
(追溯历史 构建一个这样不真实的宇宙是非常困难的)
Bear in mind, for example,
你要记着 比如说
you know, all the other things that come from relativity flow from that.
源自相对论的其他所有东西都是由此演化的
So think that famous things like E=mc squared
动下脑子想想 比如有名的E=mc²之类的
all flow from the invariance of the speed of light.
它们都是基于光速是不变的这一原理
And that means that
这就意味着
the equivalence of mass to energy
质能等价
is a natural consequence of relativity
是相对论的自然结果
and so, for example, what powers the sun (fusion),
所以 比如说 什么给太阳提供动力(核聚变)
converting mass into energy
把质量转换成能量
wouldn’t work?
不成立嘛?
And if we weren’t living in a relativistic universe,
如果我们不是生活在相对论的宇宙
Now you could imagine that
现在你可以想象一下
the laws of physics in such a universe would come up with some other way
物理定律在这样的宇宙中以另外一些
of generating energy from fusion.
把核聚变变成能量的方法出现
But the picture we currently have of it really would be completely different.
但我们现在所看到的的情况将会完全不同
And the quantity of the speed of light (the speed that it actually is)
光速的数值(实际上就是是真实速度)
Does that matter?
有关系吗?
Would the universe be the same
如果光速减半 或者静止的话
if the speed of light had been halved, or it was quiet,
宇宙还会是一样的吗
you know, if it was something close to the speed we walk at,
你知道的 如果它和我们走路的速度接近
does it need to be as fast as it is to our human brains?
那么它要不要和我们大脑保持同步?
So it comes out of the laws of electromagnetism
于是电磁学定律应运而生
it’s to do with how strong magnetic fields are,
这和磁场的强弱有关
how strong electric fields are.
和电场的强弱有关
which are really just arbitrary constants of nature as far as we know.
据我们所知 什么才是真正的自然任意常数
which I guess means
我觉得这就说明
that actually the speed of light which is some combination
实际上是光速把这些东西集合在了一起
of these things is also an arbitrary law of nature.
而所有的这些都只是自然规律的任意变化
The universe would be a very strange place
宇宙将会是一个非常奇怪的地方
if the speed of light were very much slower.
如果光速比之前慢很多
Because of course all of these relativistic effects
这当然是所有的这些相对论效应
that lead to all the weird things
产生了所有的稀奇古怪的事情
that come out of relativity
而这些都来自于相对论
we don’t have to worry about them in every day life.
我们不必每天都为此焦虑
But if it turned out
但如果事实证明
the speed of light was walking pace,
光速是步行的速度
then all sorts of strange relativistic effects would happen
那么各种稀奇古怪的相对论效应都会发生
every time you walked to the post box.
每当你都走过邮局的时候
You’d have to worry about
你不得不担心
all the relativistic time dilation and length contraction effects.
所有的这些相对论时间膨胀和长度收缩效应
So it would be like, if I go to post this letter
所以就像这样 如果我去寄信
am I going to die before I get there?
我在到那里之前就会死嘛?
oh, yeah, would all your relatives have died of old age before you get home again.
是啊 你所有的亲戚都将会在你到家之前老死
So, yes indeed, it would be a very very strange universe.
所以没错 这会是一个非常非常奇怪的宇宙

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视频概述

知道为什么你不能走的比光快吗?宇宙之大,有没有什么的速度比光速还要快呢?本视频中诺丁汉大学的迈克·梅里菲尔德教授将会用爱因斯坦相对论,通过简单通俗易懂的实验方程式来告诉你答案,学过伽马三部曲的也不要错过喔~

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视频来源

https://www.youtube.com/watch?v=DGpwkWhnWAI

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