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狭义相对论#3:洛伦兹变换

Lorentz Transformations | Special Relativity Ch. 3

第三章:洛伦兹变换
相对论的目的是
The goal of relativity is to explain and
通过不同的角度 尤其是不同的运动角度
understand how motion looks from different perspectives,
来解释和理解运动
and in particular, from different moving perspectives.
描述运动是相当简单的
It’s easy enough to describe motion itself –
如果某物相对我移动
if something is moving relative to me,
这表明它随着时间变换位置
that means it has different positions at different times,
我能把它绘制在时空图上
which I can plot on a spacetime diagram.
这条直线对应理想的匀速运动
This straight line corresponds to motion at a constant velocity
即每秒向右
of say, v units to the
移动v单位
right every second.
我们感兴趣的问题是
And the question we’re interested
从运动的视角看东西是什么样的
in is what do things look like from the moving perspective?
当然 该问题的答案是物理层面的
Of course, the answer to this question is a physical one,
并且是由实际运动
and is determined by experimental
测量得到的实验数据所确定的
evidence gathered by actually moving.
而这些数据都将发挥作用
And that evidence will come into play,
但我们首先要明白
but first we need to understand what it means,
它在时空图上的意义才能从运动视角观察某物
in terms of spacetime diagrams, to view something from a moving perspective.
我们将从时空图的一个主要特性开始当某人从他的视角
We’ll start with a key property of spacetimediagrams: when someone draws a spacetime diagram
绘制一张时空图时 在任何时候 他们
from their own perspective, on that diagram they’re always,
都在图中X等于0的位置
for all time, located at position x=0,
因为他们与他们的位置
since they’re always a distance
之间的距离总是0
of 0 away from where they are.
换句话说
Or in other words:
像这样的时空图仅当
a spacetime diagram like this represents your perspective only if your
你的世界线是竖直线X=0时才代表你的视角
worldline is a straight vertical line thatpasses through x=0. If,
如果在时空图上
on a spacetime diagram,
描述你移动的世界线远离X=0 并且向
the worldline describing your motion leaves x=0 and goes
其他地方偏移 这意味着
anywhere else, that means you’re moving
你正相对于该时空图的视角进行移动
relative to the perspective of that particular diagram,
因此这不是你的视角
and thus it’s not your perspective.
了解了这一点
With this in mind,
为了从移动物体—比如这只猫—的视角
to describe how things look from the perspective of a moving object,
来描述事物
like this cat,
我们需要使用一些方法来将时空图中
we simply need some way to transform spacetime diagrams that makes the
猫的世界线变成一条垂直地穿过X=0的线
worldline of the cat into a straight vertical line through x=0;
换句话说 我们想要
or in other words, we want
制作一张时空图
to make the spacetime diagram where the cat is moving
其中猫的世界线
into one where the cat’s worldline
与时间轴重合
coincides with the time axis.
这不是我们仅仅通过
That’s not something we can do just
像我们在不同位置看而获得不同视角那样
by sliding the whole plot left or right or up or down,
向左 向右 向上 向下滑动整张图就能完成的 不
like we’ve done for perspectives from differentlocations. No,
速度的改变需要由某种
changes of velocity require some sort
旋转物去改变
of rotationy thing to change the angle of
世界线的角度 重要的是
the worldline, and importantly,
由于这只猫正好行进的特定
whatever this rotationy thing does should be generalizable
速度并不特殊
to a world line at pretty much any angle,
所以无论这个旋转物怎么做
since there was nothing special about the
它都应该能在几乎任何角度被概括为一条世界线
particular speed the cat happened to be going.
我们还需要考虑两个重要的
There are also two important pieces of experimental evidence
实验证据:首先
that we’ll need to take into account: first,
如果我测得猫以速度v远离我
if I measure the cat as moving at a speed v away
那么猫测得
from me, then the cat will measure
我也是以同样的速度v远离它
me as moving at that same speed v away from it,
如果我们相向移动也是同理
and likewise if we’re moving towards each other.
这意味着我们对时空图的变换
Which means we not only want to transform the spacetime diagram
不仅想使猫的
in a way that the cat’s
倾斜世界线变垂直
angled line becomes vertical,
我们还希望在经过这个变换之后
but we also want the angle between our two lines to stay
这两条线之间的角度是不变的 这意味着
the same after the transformation – that is,
在猫的视角中 我应该也在动
from the cat’s perspective, I should be moving.
这是第一个实验证据
So that’s the first piece of experimental evidence.
第二个证据我们将会在之后讲到
The second piece of evidence we’ll come tolater.
让我们只关注猫的世界线中的这个部分
Let’s focus just on the section of the cat’s worldline from time t=0,
从时间为0 位移为0的点
where it’s at x=0,
到时间为4 位移为2的点
to t=4, where it’s at x=2.
这两点间部分是一条直线段
This section is a straight line between those two points,
我们想让它变成一条
and we want it to end up as a
垂直线 所以我们可以使时间为0
straight vertical line, so we can simply leave the t=0,
位移为0的点保持不变
x=0 point unchanged while moving the
而将t=4 x=2的点移动到时间轴(位移为0处)
t=4,x=2 point onto the time axis (where x=0).
其实要做到这样通常只有三种可能:要么让这一点
And there are really only three general possibilitiesfor how to do this: either this point gets
移动到时间轴上 同时使它
moved onto the time axis while keeping it
时间保持为4 或者让它
at the same point in time, t=4, or it gets
更早地移到时间轴上(比如时间为3) 或者更晚地移到(比如时间为5)
moved onto the time axis at an earlier time(say, t=3), or a later time (like t=5).
有一个非常棒的几何学方法能画出这些可能
There’s a very nice geometric way to picturethese possibilities.
如果我们把时空图中的运动视为
If we think again of motion on a spacetime diagram
一系列快照 比如
as a series of snapshots, like, at
当时间为0时猫在位移为0的位置
time t=0 the cat is at position 0,
时间为1时猫在位移为0.5的位置 时间为2时
at time t=1 the cat is at position 0.5, at time t=2
猫在位移为1的位置等等
the cat is at position 1, etc,
所有点保持时间不变地移到时间轴的这种变换
then the transformation where points move to the time axis and keep
看起来就像将每一张快照滑动相应的距离
the same time just looks like sliding each snapshot over a corresponding amount;
而所有点时间增加地移动到时间轴的变换
the possibility where points move to the time axis
看起来有点像是
at a later time looks kind of like some
绕着原点做某种旋转
sort of rotation around the origin;
而所有点时间减少地移动到
and the possibility where points move to the time
时间轴的变换看起来有点像是某种压缩了的旋转
axis at an earlier time looks kind of like some sort of squeezy rotation.
后两种变换
The reason these last two involve
选择旋转快照而不是滑动它
rotating the snapshots rather than just sliding is
是为使猫和我的世界线
to make sure that the angle
所夹角度在转变前后
between the cat’s worldline and my worldline stays the same
保持不变
before and after the transformation –
要理解其原因是个有趣的几何学问题
it’s a fun little geometry puzzle to understand why.
现在 在这三种可能性中
Now, among these three,
我们根据自己日常生活中对时间流逝体验的经验
the option that makes the most intuitive sense based on our everyday
所做出的最直观的选择是
experiences of the passage of time,
一个给出的点所在时间
is that a given point in time should stay at the same
应该保持不变 即它们应该滑到时间轴上
point in time, and just slide over to thetime axis.
我的意思是指 当我们搭乘火车
I mean, we don’t noticeably experience
自行车或者飞机时 我们都不能明显地感受到
time travel every time we hop on a train or bike
时间的流逝
or plane.
并且这样的滑动在数学上是成立的 例如如果我们
And this sliding does mathematically work – if we move things
将时间t等于1上的东西向左移
at time t=1 a half meter
半米 将时间t等于2上的东西都
to the left, and things at time
向左移1米
t=2 one meter to the left, and
后面同理 然后我们将获得
so on, then we’ll have a
一个从猫的视角的运动描述 这只猫没有移动
description from the cat’s perspective – the cat’s not moving,
而我在每秒半米地
and I’m moving to the left
向左移
half a meter every second.
这对其他速度的情况也成立
It works for other speeds, too.
如果我们想要得到一个以每秒1米
If we want the perspective of somebody who’s going a meter
相对猫向右移动的人的
per second to the right relative
视角 我们可以将这些快照滑得更远
to the cat, we can slide the snapshots over even farther,
这样的话这只猫
and now the cat’s going a meter
每秒向左运动1米
per second to the left,
而我每秒向左移动1.5米
and I’m going a meter and a half per second to the left.
当然我们也可以滑回我自己的视角
And of course we can slide back to my perspective
在这个视角下这个新来者每秒
from which the newcomer is going a meter and
向右移动1.5米
a half per second to the right.
这种通过滑动实现视角转换的方式
This kind of sliding change
通常被叫做“剪切变换”
of perspective is normally called a “ shear transformation, ”
但这是在两个维度都是在空间维度的情况下 由于我们的维度中有一个是时间维度
but that’s when both dimensions are spacedimensions: since one of our dimensions is time,
一个剪切变换就代表了
a shear transformation represents a change
一个东西速度的改变 所以
in the velocities of things, so in
在物理学上它被叫做推进
physics it’s called a “boost.”
就像火箭推进器将你推进到一个更高的速度 然而
As in, rocket boosters boosting you to a higherspeed. However,
事实是
it turns out
物理领域的这种推进实际上并不能用剪切变换
that boosts in the physical universe are not actually described by shear
来描述
transformations.
第二个最有名的实验证据就是
This is where the second and most famous pieceof experimental evidence comes in: the speed
光的速度
of light.
你可能早就听过了 从19世纪末起
As you’ve probably heard, starting in the late 1800s,
物理学家就建立了大量的
physicists built up mountains
实验和理论证据来论证
of experimental and theoretical evidence that the speed
得到了即使通过不同视角来测量
of light in a vacuum is always the same,
同个空间里的光速仍然不变的这个结果
even if you measure it from a movingperspective.
这个当然
This is, of course,
以我们日常生活中对速度理解的经验来看是非常不直观的
entirely unintuitive from our everyday experiences with velocities,
当你静止的时候扔下一个球 然后
where if you throw a ball from a standstill and then
在一辆移动的车里扔下一个球 这个
from a moving vehicle, the ball thrown
从汽车中扔下的球相对地面的移动速度更快
from the vehicle will be moving faster relativeto the ground.
但是 实验结果显示光不像平常的实物那样
And yet, experimental results show that lightdoes not behave like everyday objects: shine
静止时投下的光束 或移动汽车中投下的光束
light from a standstill, or from a moving vehicle,
测得的相对地面的移动速度
and its measured speed relative to
是相同的
the ground will be the same.
剪切变换不能接受光的这种特性 它们通过
Shear transformations simply can’t accomodatethis feature of light’s behavior: they change
滑动每个快照到其时间与原来成一定比例的位置来改变所有东西的速度
all velocities equally by sliding each snapshot an amount proportional to its time.
没有东西的速度是可以保持不变的 如果你画出光线的
No velocity remains unchanged – if you draw the worldline
世界线 然后使用
of a light ray and then change
剪切变换来变换到一个移动的视角
to a moving perspective using a shear transformation,
光线的速度将会改变 这是
the speed of that light ray will change, which
错误的 幸运的是
is wrong. Luckily,
另外两个为了推进到移动视角
one of the other two options
的选择中的一个能够与
for boosting to a moving perspective can accomodate
恒定的光速契合
a constant speed of light:
记得有种变换是快照做一种压缩旋转
remember the transformation where the snapshots do a kind of squeeze rotation,
即点到达时间轴的时间更短了
and points move to the time axis at earliertimes?
即使是在其他的速度都改变的情况下
This kind of transformation can amazingly leave one speed unchanged,
这种转变仍令人惊奇地
even while it changes
可以保持一种东西的速度不变
all other speeds.
更令人惊奇的是 这个速度还能保持在各个方向上的不变
More amazingly, the unchanged speed is leftunchanged in all directions.
让我们建立一个模型
Let’s do an example.
这里是从我视角获得的一系列关于
Here’s a set of snapshots from my perspective
一只慢速移动的羊和一只快速移动的猫的快照
with a slow-moving sheep and two fast-moving cats,
让我们假设有实验证据证明
and let’s suppose that we have experimental evidence
这只猫在所有视角中的
that cats always move at the same
速度都是不变的
speed regardless of perspective.
如果我们想从羊的视角来描述
If we want to describe this situation from the perspective
这种情况 我们就
of the sheep, we can’t simply
不能只是简单地滑动
slide the snapshots
这些快照使这只羊静止且使它的世界线和
over so the sheep isn’t moving and its worldline coincides with the
时间轴保持一致 因为这会改变猫的速度 但是
time axis, since that would change the speedof the cats. But,
如果像这样我们滑动旋转并且延伸
if we slide and rotate and stretch the snapshots
这些快照 然后观察:我们将
like this, then look – we’ve transformed
这张表转变到了羊的视角
the diagram to both describe things
并且猫的速度
from the sheep’s perspective and keep the cats moving
没有发生变化
at the same speed they were before.
你可能注意到每只猫好像
You might note that the various cats appear
沿着它们的世界线被分隔开
to be spaced out differently along their worldlines,
但这只代表着这些在我视角看来时间不变的
but that just means that the constant-time snapshots
快照在羊的视角中
from my perspective aren’t constant-time
时间是变化了的
snapshots from the sheep’s perspective.
重要的是
The important thing is
代表着猫的速度的
that the angle of the cats’ worldlines – which represents their
世界线的角度是保持不变的
speed – has remained unchanged.
这对我来说太神奇了
It’s kind of amazing to me that this works at all;
在数学和物理上
that it’s mathematically and physically
在所有速度中只有一个速度改变都是可能的
possible for all speeds except one to change!
对于这种压缩旋转来说这是有可能的
But it is possible with these squeeze rotationy things,
它们就是怎样通过移动的视角
and they’re the answer to our question
来描述运动的问题的答案
of how to describe motion from a moving perspective. Well,
不过不是保持猫的速度不变
not by keeping the speed of cats constant,
而是保持光的速度不变
but by keeping the speed of light constant:
通过进行压缩旋转
by doing squeeze rotations so
使得能够在不改变光速的前提下
that a moving perspective’s angled worldline becomes vertical
将有角度的移动视角下的世界线变垂直
without changing the speed of light – that is,
而且不改变世界线
without changing the slope of the worldlines
相对光线的斜率
for light rays.
这种压缩旋转被称之为洛伦兹变换
These squeeze rotationy things are called Lorentz Transformations,
是以第一个
named after one of
推导出它们的正确数学表达式的人
the first people to derive the correct mathematical expression
命名的 看起来有点像
for them – it looks kind of like
我们上个视频中讲到的旋转的表达式
the equation for rotations that we saw in the last video,
我将附上一个视频
and I’ll post a followup video
来演示怎样通过几个简单的假设和实验证明来得出它的
showing how to derive this using just a few simple assumptions and experimental facts.
洛伦兹变换是狭义相对论的
Lorentz Transformations are at the heart
核心 它是被
of special relativity – they’re the thing that
洛伦兹 爱因斯坦 闵可夫斯基
Lorentz and Einstein and Minkowski and
以及其他专家学者认定为能够正确描述
others figured out was the correct description of
在我们宇宙中移动视角下的运动的
how motion looks from moving perspectives in our universe,
而它也将是
and they’ll be the foundation
这个系列余下视频的基础 现在
of the rest of this series, too. Now,
物理学家像这样来画他们的时空图刻度:如果一个垂直
Normally, physicists draw their spacetimediagram tickmarks such that if every vertical
刻度代表一秒 那一个水平刻度就代表299792458米
tickmark represents one second, a horizontaltickmark represents 299,792,458 meters, which
这代表着光的速度
means that the speed of light,
即299792458米每秒 它通常被画成一根45度的
which is 299,792,458 meters per second, is drawn as a 45° line
向右移动的光向右45度
– to the right for right-moving light,
向左移动的光向左45度
and to the left for left-moving light.
通过这种比例
With this scaling,
一个光速不变的洛伦兹变换
a Lorentz Transformation that leaves the speed of light constant simply
包括将45度线上的所有东西压缩
consists of squeezing everything
和将其余的东西以特定比例
along one 45° line and stretching along the other in
拉伸
a particular, proportional way.
你立即就可以发现其他这些世界线的角度
You can see immediately how this changes the angles
是怎样改变的 也就是
of all of the other worldlines, that is,
我们对他们速度的感受的改变
changes how we perceive their speeds,
同时光速还是不变的
and yet doesn’t change any of the light rays.
这样的结果是
And it turns out
建立一个相对你做洛伦兹变换的机械装置
that it’s possible to actually build a mechanical device that does Lorentz
是可行的 这个就是
Transformations for you: here it is!
就如同地球仪以一种基本的方式
Just like how a globe has the structure
构建旋转结构一样
of rotations built into it in a fundamental way,
你可以通过轻轻地转动这个地球仪来观察旋转
and you can simply turn the globe to see how rotations work,
而不用经过大量
rather than doing a lot of
复杂的数学计算 这个时空地球仪里面也构建了洛伦兹变换模型
complicated math, this spacetime globe hasLorentz Transformations built in: it does
它为我们演示了狭义相对论
the math of special relativity for you,
还能够让我们通过不同的视角来
allowing you to focus on understanding the physics
理解运动的物理现象
of motion from different perspectives!
这里有一个简易模型
Here’s a quick example:
在我的视角中 随着时间的流逝我的位置是保持不变的
from my perspective, I’m always at the same position as time passes,
而一只猫正以三分之一光速远离我
while the cat is moving away from me to the right
向右移动 而
at a third the speed of light, and the
我灯泡的光线在向左向右发散
light rays from my lightbulb are moving out to the right and left.
使用这个时间球仪
Using the time globe,
我能做一个洛伦兹变换来推进到猫的视角
I can do a Lorentz transformation to boost into the cat’s perspective.
通过这只猫的视角
And from the cat’s perspective,
这只猫 当然 随着时间的流逝位置保持不变
the cat – naturally – stays at the same position as time passes,
但是在猫看来
while the cat views me as moving away from it
我以三分之一光速向左远离
at a third the speed of light to the left,
而我灯泡光线的速度一直不变
and the speeds of the light rays from my lightbulb are still the same,
一直是45度角的
still at 45° angles.
我喜欢这种切实而实践的
I just love how tangible and hands-on this
方式 通常当人们第一次听说
is – normally when people are first introduced
狭义相对论和移动视角下运动
to special relativity and how motion looks
是怎样的时 听到的
from different perspectives, it’s done with
都将是一连串混乱的不完整的
a bunch of messy, incomplete,
代数方程 但其是你并不需要方程来理解
algebraic equations – but you don’t need the equations to understand
狭义相对论以及移动视角下运动看起来是怎样的
the ideas of special relativity and how motion looks from different perspectives.
你只需要对这个时空图和时间地球仪的理解
You just need an understanding of spacetime diagrams, and a time globe.
所以这个系列的余下视频里
And so in the rest of this series,
我将使用这个时间地球仪来
I’m going to be using the time globe extensively to
深入解释当你们在听到狭义相对论时
dive into all of the normally confusing things you’ve heard
通常会产生的疑问
about in Special relativity:
钟慢 尺缩 双胞胎悖论
time dilation, length contraction, the twins paradox,
同时性的相对性 为什么人
relativity of simultaneity, why you
不能突破光速等等
can’t break the speed of light, and so on.
我非常感谢
I have to say a huge thank you
我的朋友麦克•罗贝尔 是他的帮助让做出时间地球仪模型
to my friend Mark Rober for helping actually make the time
成了可能(你可能熟悉
globe a reality ( you may be familiar
他的YouTube频道 在那里他做出了
with his youtube channel where he does incredible
令人难以置信的设计
feats of engineering,
比如通过移动来让人能永远射中靶心的靶子)
like this dartboard that moves so you always hit the bullseye ).
他花费了大量的时间 精力
He devoted a huge amount of time, effort,
和技术知识来让我的疯狂的
and engineering expertise to turn my crazy
想法变成这个漂亮的 精密的
idea into this beautiful, precision,
手动的狭义相对论模型 我
hands-on representation of special relativity and I’m
对他感激不尽 如果不是他 我这个系列都是没有办法做成的
supremely indebted to him – this serieswouldn’t be possible otherwise.
如果你想了解更多细节
And if you’re eager for more details,
我正在计划做另一个完整的介绍这个
I’m planning another whole video about the time
时间球仪的视频
globe itself.
与此同时 想要获得更多的手工
In the meanwhile, to get more hands-on
狭义相对论数学模型 经济学知识
with the math of special relativity, or economics,
或者机械知识的话 我强烈推荐这个视频的赞助者Brilliant.org
or machine learning, I highly recommend Brilliant.org,this video’s sponsor.
结合我的视频系列
In conjunction with my video series,
Brilliant通过他们独特的情景演示来展开
Brilliant has rolled out their own course on special
狭义相对论的课程
relativity with their own unique illustrative scenarios –
如通过相对论的激光标签
like relativistic laser tag
来理解洛伦兹变换
to understand lorentz transformations.
如果你想更深入地理解数学模型
If you want to understand a mathematical topic deeply,
没有比
there’s really nothing better than
认真地思考并且自己解决它更好的办法
thinking through the ideas and solving problemsyourself.
而Brilliant就是帮助你做这些的
And Brilliant helps you do just that.
一旦你准备好更深入地
Once you’re ready to go in-depth
了解狭义相对论(线性代数或者群论
into special relativity ( or perhaps linear algebra or group theory,
这两个与相对论都相关的理论)
which are both also relevant for relativity ),
访问Brilliant.org/minutephysics你将获得20%的折扣 还有
you can get 20 % off by going to Brilliant.org/minutephysics. Again,
再次重审 Brilliant.org/minutephysics会再给你20%的折扣
that ’ s Brilliant.org/minutephysics which gets you 20 %
抵额外的看所有的
off premium access to all
Brilliant课程和问题费用
of Brilliant’s courses and puzzles,
让Brilliant知道你是从这里来的
and lets Brilliant know you came from here.

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视频概述

洛伦兹变换

听录译者

收集自网络

翻译译者

ふうらいぼう

审核员

审核员 EM

视频来源

https://www.youtube.com/watch?v=Rh0pYtQG5wI

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