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用一刻钟了解相对论

Spacetime Diagrams | Special Relativity Ch. 2

Physics is all about the motion of things,
物理学研究的是事物的运动
how planets and stars move,
行星和恒星的运动方式
how electrons and protons move,
电子和质子的运动方式
how the movement of molecules results in emergent property like temperature and so on.
分子的运动又是如何涌现温度等新质
The role of relativity in physics is to study
在物理学上 相对论研究的是
how that motion looks from different perspectives.
不同观测角度之下的物体运动方式
Here I’m using relativity in a general sense to mean
这里的相对论是广义的 即从诸如匀速 加速运动等
from any different possible perspective moving, accelerating or otherwise.
任何可能的视角研究物体运动的方式
Special relativity in particular is concerned with
而狭义相对论则是
how motion looks just from a limited or special set of perspectives
在一个受限的或特殊的角度下研究物体的运动方式
which will get to later.
我们接下来会讲到的
But either way, relativity special or not
但不论是狭义还是广义
is about how the motions of things look from different perspectives.
相对论研究的都是不同观察角度下物体的运动方式
Like if you were looking at the earth and the moon,
比方说你正在观察地球和月球
depending on where you were and how you were moving,
根据你的位置和运动方式的不同
it might look like the moon is moving around the earth in a giant circle,
月球看起来或是在沿着一个大圆绕地运动
or back and forth on a straight line,
或在一条直线上来回往复
or that the earth and moon together are tracing out a spiralling path through space.
或是地月一起沿着一条路径螺旋着运动
But if the motion of the earth and moon can be described in such different ways.
如果地月运动能用这么多不同的方式来描述
What does any one of these descriptions actually tell us about the earth and moon?
那哪一种描述才是地月运动的真正样子?
Is one of them right and the others wrong?
是否有一种正确 其它的都错了?
Is there some preferred perspective for observing the earth and moon,
是否存在一个观察地月运动的更优视角
that gets closer to the true description of what’s happening?
让能我们更准确地描述发生的事情?
It’s the goal of relativity to answer these kinds of questions.
相对论的目标便是回答这些问题
In fact, relativity can essentially be summed up as two basic ideas:
事实上 相对论的基本任务是:
1. To figure out how objects and their motion look from different perspectives.
1. 从不同的角度了解物体和物体的运动
and two, to notice which properties of objects and motion
2. 研究不同的视角下 物体本身和运动过程
don’t look different from different perspectives.
具有的不变量
We’ve already given an example of number 1,
第一条已经举过例子了
with different ways the motion of the earth and moon can look from different perspectives.
就是从不同视角对地月运动的观察
Number 2, the idea of finding things that don’t look different from these perspectives,
第二条则是要找出不同的视角中不变的量
that’s a little trickier.
这有点棘手
In the earth and moon case, for example, all three perspectives appear quite different.
毕竟在这三个视角下 地月运动的表现是完全不同的
But after a while, you might notice that regardless of the perspective,
但过一会 你也许会注意到不论视角如何
the maximum physical distance between the earth and the moon appeals to be the same
地月之间的最大物理距离是不发生变化的
So you might say: Aha, there’s something that’s independent of perspective,
然后你可能会说 啊 这个量就和视角无关了
maybe it’s a fundamental property of the earth-moon system,
也许这就是地月系统的一个基本性质
and not just an artifact of my particular point of view.
而不是我个人的主观臆造
And this is why relativity is so important in physics.
这也是为什么相对论在物理学中如此重要
By studying what changes and what doesn’t about a physical system
通过研究视角转变过程中
when you change your perspective,
物理系统的变量与定量
you are zeroing in on universal truths.
你正在接近一个普适的真理
Literally, facts that remain true from many perspectives throughout the universe
像地月间的距离大约是384399千米这种
like perhaps that distance between the earth and the moon is 384399 kilometers,
从多个视角去观察宇宙后始终正确的事实
are literally more universal than a fact that only holds true at a single place and time,
确实要比地月之间的夹角是150°
like that the angle between the moon and earth is 150 degrees.
这类只在特定时间地点适用的事实更普适
Relativity is a way of thinking
相对论是一种帮助你评估
that helps you to evaluate how universal a given truth is.
给定事实普遍性的思考方式
OK, enough philosophizing.
好了哲学解释到此为止
To make all this tangible,
为了讲得更清楚点
we need a rigorous way of describing moving things
我们要用更严谨的方式来描述物体的运动
and of describing changes to how that motion looks when you change your perspective.
与视角转换时运动在视觉上发生的变化
We’re ultimately going to build up to special relativity, which has to do with motion over time,
我们最后学到的狭义相对论是脱不开运动的
but we’ll start with non-moving things,
但为了对相对论有一个直观的感觉
just to get a sense of how intuitive relativity can be.
我们会从静止的物体入手
You’re probably familiar with specifying the position of a cat on a plane using xy coordinates.
你或许习惯用xy坐标来对平面上的猫定位
This cat is three tick marks to the right of our point of reference, and two tick marks up.
这只猫在原点的右侧三个刻度 上方两个刻度的地方
So we say it’s at position x=3 and y=2,
所以我们说它的位置在x=3 y=2
which typically gets written as just a pair of numbers like (3,2).
通常用(3,2)这样的数字对来表示
However, (3,2) is not a universal truth.
但是(3,2)并不是普适的事实
I mean, it’s just based on where I’m standing, and how I’m oriented.
我的位置和我的朝向都会影响这个坐标
But over here, where you are, maybe you’re rotated by 30 degrees,
然而你如果把视角旋转30°
and you made the tick marks close together,
把坐标标得紧密一些
and suddenly the cat is at a different position, x=9, y=9,
虽然这只猫没有动 但坐标突然就变了
even though the cat hasn’t moved.
也就是x=9 y=9
In fact, it’s possible to specify the cat’s position,
事实上 只要你想的话
using any x and y values we want.
随便找个xy值都能用来表示猫的位置
Depending on our point of reference,
这取决于我们选取的参考点
which corresponds mathematically to
从数学上讲就是
where we put our axes and how we orient and scale them.
改变我们坐标轴的方向和刻度
So, clearly, specifying the position of something is not a universal truth.
因此 一个物体的位置并不是普适的事实
Or, in relativity parlance, “position is relative.”
或者用相对论的说法 “位置是相对的”
A more universal or absolute truth can be found
要是你有两只猫的话
if you have two cats.
你会发现一个更普适 更绝对的事实
Let’s say they’re at x=0, y=0, and x=5, y=0.
假设它们坐标分别是(0,0)和(5,0)
I’m not gonna draw a person at the origin of the axes from now on,
从现在开始 我不会在坐标轴上画小人了
but you should remember that the axes we use
但你得记住 我们用的坐标系
represent a particular perspective and orientation from which we measure things.
代表了我们测量事物的特定视角和方向
The distance between these two cats is clearly 5,
两猫之间的距离显然是5
they’re at the same y value and their x values differ by 5.
它们y坐标相同 但x坐标差了5个刻度
If we move and rotate our point of reference now,
如果我们现在移动并旋转我们的参考点
the cats are at positions, x=1, y=1 and x=5, y=4.
两只猫的位置就成了(1,1)和(5,4)
So they differ by 4 in the new x direction
这样它们在新的x方向上差4个刻度
and 3 in the new y direction.
新的y方向上相差3个刻度
But the overall distance between the cats, which we can find using the pythagorean theorem,
我们能用勾股定理算出俩猫间直线距离
is the square root of 4 squared plus 3 squared,
是4和3的平方和的平方根
which is the square root of 25 which is 5,
也就是25的平方根 还是5
Which is the same distance we calculated with the original axes!
和我们用最开始的坐标算出的距离一样
This turns out to be a general truth.
这证明了一个普适的事实
On a plane, the distance between two things doesn’t change
在任一平面上 通过移动或旋转参考点
if you change the perspective just by shifting your point of reference or your orientation.
来改变你的视角 那两个物体间的距离不会改变
I like to think about this as similar to how if I take a piece of paper,
我更喜欢把这想象成 我拿了一张纸
and slide it around and rotate it,
然后移动并旋转它
I haven’t actually changed anything on the piece of paper.
但实际上我没有改变纸上的任何事物
Or, in relativity parlance: Distances are ABSOLUTE.
或者用相对论的说法 距离是绝对的
The geometric intuition for this is that you can move your axes around,
从几何直观来看 你可以四处移动坐标系
slide them up and down, and rigidly rotate them,
把它们上下滑动或刚性转动
without affecting your description of the distance between two things.
最终也不会影响你对两事物间距离的描述
If you like, we can make this mathematically precise by calling the original coordinates x and y,
如果你想 我们能用原始xy坐标
and the new coordinates x new and y new.
和新xy坐标来做更精确的数学表述
Then when we’ve slide the x axis an amount ΔX,
那么当我们沿x轴平移ΔX的距离
technically called a “translation by x”,
专业的说法是进行x转换
we say that Xnew=X-ΔX,
我们就有了 Xnew=X-ΔX
and when we slide the y axis by an amount Delta y,
而当我们沿y轴平移Δy的距离
technically called a “translation by Delta y”,
专业的说法是y转换
we say that y new=y-Delta y.
我们有了 ynew=y-Δy
The minus sign is here because if you slide your origin point closer to something
若你把原点移向某物 新的xy坐标就变小
its new x and y coordinates will be smaller.
这也是在这里用减号的原因
Changes of orientation are a little fancier,
要进行坐标系方向的变换需要一点想象力
but it’s really just some geometry:
但实际上也就是一些几何知识
if you reorient the x and y axes counterclockwise by an angle theta,
如果你把x和y轴逆时针旋转一个θ角
the new coordinates look like x new=x cosθ-y sinθ
所产生的新坐标可以用xnew=xcosθ-ysinθ
and y new = y cosθ+ x sinθ.
和ynew=ycosθ+xsinθ来表示
And if you want a fun algebra exercise,
若你想做个有趣的代数练习
you can use these equations or even their 3D counterparts
你可以用这些等式甚至它们的三维形式
to check that indeed that the distance between two points doesn’t change
来核算一下坐标系的移动和旋转
when you slide or rotate your axes.
的确不会改变两点间距离
But the messiness of all the details here
但这些细节造成的混乱
really clouds the simplicity of what’s going on.
掩盖了其朴素的事实
The important geometric idea I want you to remember
我希望你记住一个重要的几何概念
is that rotating and sliding axes
那就是坐标系的旋转和平移
doesn’t change the distance between two points.
并不会改变两点间的距离
However, the distance between two points does change
可是如果允许我们改变刻度线的间距
if we’re allowed to change the spacing of the tick marks.
两点间的距离就会发生变化
If when we change our axes,
如果在改变坐标系时
we also double the tick marks,
把刻度的数量变成两倍
then the distance between the cats becomes 10, not 5.
那么两猫间的距离不再是5 而是10了
Turns out, distance measured in numbers,
这就证明了 用数字来衡量距离
is not so universal.
没有那么通用
But there is a more universal truth.
但确实存在一个更普适的事实
Suppose we have a stick that’s 1 tick mark long,
假设我们有一根长一刻度的棍子
according to the original axes,
该刻度为原始坐标系刻度
conventionally this thing might be called a meter stick.
一般来说这根棍子可能被称为米尺
And now we can say that the two cats are five sticks apart.
那么现在我们便能称两猫距离为5棍
When we again move and rotate our axes
当我们再次移动和旋转坐标系
and change the spacing of the tick marks,
并改变刻度的间隔大小时
the cats are again 10 tick marks apart,
两猫间的距离又变成了10个刻度
but the stick is also now 2 tick marks long,
但棍子现在也变成2刻度长了
so the distance between the cats is still 5 sticks.
所以两猫之间的距离仍是5棍
This is an example of an even more general physical truth:
这个例子反映了一个更普适的物理事实
The distance between two things, measured in terms of another physical thing,
根据另一实际的事物测量出的两物间距离
doesn’t change when you change your perspective by shifting your point of reference
不会因你移动或旋转参考点亦或改变刻度
or orientation, or the spacing of your tick marks.
而发生的视角变化而改变
In relativity parlance, we’d say that
用相对论的话来描述就是
the ratio of two distances is absolute.
两个物体间距离的比值是绝对的
Or basically, if you want to actually describe a distance,
简单来说 你要是想真实地描述距离
you can’t just specify a number,
就不能只指定一个数字
like I’m five away from you.
比如说用5来表示我们之间的距离就不太合适
You have to say what you’re measuring distance in terms of,
你必须说出你测量的基准物
and what number of those things your distance is equal to.
并且你测量的距离等于几个基准物的长度
This is kind of a subtle point and is very important
若你对研究度量和单位的度量衡学感兴趣
if you’re interested in metrology, the study of measurement and units.
这点细节很重要
But because it doesn’t really play a major role in special relativity,
但它在狭义相对论中没那么重要
from now on I’m going to be a bit sloppy and just assume that
从现在开始 我会默认并假定
whenever we’re talking about distances, we’re talking about distances not as numbers
我们谈及距离的时候 说的都不是数字上的距离
but in terms of some reference distance, like meters, or cats, or whatever.
而是根据米或猫或者别的事物得来的参照距离
And the same will apply to times.
这同样也适用于时间
when we talk about a time interval, we’ll assume it’s a time interval
时间间隔让我们能衡量事物随时间的变化
in terms of some reference time, like the second,
所以我们谈到它的时候
which brings us to the motion of objects over time.
会假定存在一个像秒一样的基准时间间隔
To describe a moving object,
我们要描述一个移动的物体时
it’s customary to use a horizontal coordinate axis for the left-right x position.
通常会用水平坐标轴来描述水平方向的x坐标
But instead of using the vertical axis to represent height y, we use it to represent time t.
而此时纵轴不再代表高度y 而是时间t
So for something not moving,
因此对于一个静止的事物
something that stays at the same position x at time t=0, t=1, t=2 and so on,
即在t=0 t=1 t=2等时刻处于同一x坐标的事物
we draw a straight vertical line through x.
我们可以画一根过x轴的垂线来表示它
For something moving one meter per second to the right,
对于每秒向右移动一米的物体
we draw a line that goes one meter to the right
我们则可以画一条每一秒钟
for every second that transpires vertically.
向右移动一米的直线
It’s important to note that we’re not saying
重要的是 我们并不是说
that the object is moving through 2D space along a 45 degree line.
这个物体在二维空间里沿着一条45度的线在移动
The object is moving purely one-dimensionally along the x axis,
它实际上只是单纯地沿着x轴做一维运动
and we’re just showing those different one-dimensional positions as time passes.
我们只是展示一维空间内不同物体的位置随时间的变化
This whole “time on the vertical axis” thing
但大多数情况下
can also be a bit weird at first since in most other situations
你碰到的时间轴都是横轴
you’ve probably encountered time plotted on a horizontal axis.
所以一开始你可能会觉得时间是纵轴有点怪
But vertical time has its merits, and more importantly,
不过这么做也有好处 而且更重要的是
it’s convention at this point.
这是一个约定俗成的表示方法
So it’s worth getting used to.
所以还是需要习惯
I like to think of each horizontal line
我喜欢把每根横线
as representing a different snapshot of a scene.
看成是同一场景在不同时间拍下的快照
We could show the snapshots one after another as time actually passes, of course,
我们自然可以把这些快照拿起来一张张地看
but it’s useful to be able to see all of the snapshots at once.
但是一次性同时看到这些快照会相当有用
So if we display each snapshot at a consecutive vertical position,
如果我们把这些快照在垂直方向上连续展示出来
we get a nice representation in a single static image
就能很好地在一张静态图片中呈现出
of motion that normally takes place over time.
一般情况下运动随时间的变化
This geometric way of representing motion over time
这种表示运动随时间变化的几何形式
is called a “space-time” diagram.
叫做 “时空图”
And it’s so central to intuitively understanding relativity
这对于直观地了解相对论来说十分重要
that it’s worth doing a few more examples.
所以也值得我们多举些例子
Say we have a cat attached to a spring,
假设我们把一只猫系在了弹簧上
bouncing back and forth, left and right.
让它左右往复地弹动
If we plot this motion on a spacetime diagram,
如果我们把这个运动在时空图上画出来
as time passes we see the cat move left and right,
我们可以看到 随着时间的流动
leaving behind a trace in the shape of a sine wave.
猫左右移动留下了一道正弦波
On the other hand,
换句话说
if we’re given a spacetime diagram and want to recover the motion of the cat
如果我们想要还原某张时空图上的猫的运动
we simply slide the diagram downwards at a constant rate
我们只需要把图向下匀速滑动
and move the cat left and right so that it follows along the traced-out path.
让猫沿描绘出的痕迹左右移动就好
This is important: a traced-out path in a spacetime diagram
重点来了 时空图中描绘出的痕迹
is a faithful recording of an object’s motion.
是对物体运动准确且可靠的记录
And these paths are called “world-lines”, presumably because they show
大概是因为这些这些这些记录揭示了事物最终去向
where in the world the object has gone
所以它被叫做世界线
though by “world” we often mean
不过我们所说的“世界”
“room” or “solar system”, or “universe”.
一般指的是 空间 太阳系或宇宙
Any particular point on a worldline has coordinates t and x,
世界线上任一点都有其特定的t和x坐标
which we write as a pair telling us
我们常写作数对来进行表示
for time t what position x the object was located.
它能告诉我们时间为t时 物体的位置x
So far we’ve just been representing one-dimensional motion on our spacetime diagrams,
到目前为止 我们在时空图上看到的还只是以为运动
just one spatial direction the object is moving in,
即物体的运动仅在一个空间方向上进行
and then time as the vertical axis.
且时间为纵轴
If we want to use a spacetime diagram to represent motion in two dimensions,
如果我们想用时空图描绘二维运动
like the moon orbiting the earth,
比如地月轨道运动
we actually need three dimensions to do so.
我们实际上就需要用三个维度
The two horizontal directions for the moon and earth to move in,
两个水平的方向用来表示地月相对运动和自转
and the vertical direction to trace out the snapshots as time passes.
垂直方向则是按时间展示出那些快照
It’s pretty cool,
这挺酷炫的
but if you have multiple particles moving complicatedly,
可如果有多个粒子进行复杂运动
this can get really messy on a 2D screen.
在2d屏幕上看起来就十分混乱
And it’s physically impossible to make a full spacetime diagram
而要用时空图完全地描绘三维运动
for three dimensional motion,
在现实中是不可能的
because you would need four spatial dimensions to do so.
因为你需要用到四个空间维度
Three horizontal directions for the motion
用其中三个水平的方向表示运动
and a vertical direction for time,
然后用垂直方向代表时间
which of course is impossible in our universe
这在我们仅有三个空间维度的宇宙中
with its measly three spatial dimensions.
自然是无法实现的
So physicists normally restrict their spacetime diagrams
因此物理学家一般把时空图
to just one or two spatial dimensions,
限定成一到两个空间维度
and time going vertically.
然后用垂直方向表示时间
So how does relativity work with spacetime diagrams?
但相对论对时空图有什么影响呢?
That is, now that we know
换言之 既然我们知道了
how to describe motion geometrically,
如何用几何方法描述运动
how do changes in perspective affect that description?
那么视角的改变又有什么影响呢?
Let’s take as an example me staying put right at x=0,
举个例子 如果我在x=0处不动
and a cat moving one meter per second to the right away from me
一只猫从t=0时开始以1m/s的速度
starting at time t=0.
向右远离我
It may not surprise you to notice
毫无意外 你会注意到
that when you slide the x axis to the left or right,
当你左右移动x轴的时候
the particular x positions of the cat and I have
我和猫在某一时刻t的
at any particular time change,
坐标x会发生变化
but the distance between us doesn’t change.
但我们之间的距离保持不变
At time t=0 we’re still 0 meters apart,
在t=0时 无论你怎样移动x轴
no matter how much use like the x axis back and forth,
我和猫的距离始终是0米
at time t=2 we’re 2 meters apart, and so on.
在t=2时也是如此 我们始终会相距2米
So you can slide the x axis back and forth however you like
你可以随你喜欢把x轴来回移动
and spacial distances don’t change.
空间距离始终会保持不变
Similarly, if you slide the time axis up and down,
同样的 如果你上下滑动时间轴
the absolute time like when the cat starts moving away from me changed.
像猫远离我的时刻这种绝对时间会发生改变
But time intervals don’t change:
但时间间隔却仍然不变
the cat still takes 2 seconds to get 2 meters away from me.
猫还是要花2秒才能离我2米远
So you can slide the t axis up and down,
因此你把t轴上下滑动
and distances in time are left unchanged.
某一时间点上我和猫的距离并不会改变
If we have 2-dimensional motion,
如果我们讨论的是二维运动
then changes in the orientation of the two spatial axes
那么改变两个空间轴的方向
also don’t change the distances between objects at any particular time.
也不会改变某一时间点上 两物之间的距离
Essentially, you can re-orient the xy axes however you like.
实际上 你想怎么调整xy轴的方向都行
So the relativity we applied to purely spatial diagrams
我们用在单纯的空间图上的相对论
applies pretty well to space-time diagrams, too.
在时空图上也相当适用
To summarize the major takeaways:
要点总结:
relativity is about understanding
相对论要弄明白的是
how changes in perspective do or don’t affect motion
观察视角的不同对于物体运动的影响
We can describe motion over time geometrically
我们可以用时空图 以几何的形式来
using spacetime diagrams.
描述运动随时间的变化
And spacetime diagrams can help us see
而且时空图能让我们看到
how changes of perspective affect how the motion of objects looks.
视角的变化是怎样影响了我们对于事物运动的观察
Like how changing your position
比如你可以通过滑动坐标系
and orientation correspond to sliding the axes around
和绕时间轴的刚性转动改变空间轴方向
and rigidly reorientating the spatial axes,
来改变你的位置和方向
without changing the spatial distance between two points at the same time
但这不会改变同一时间两点之间的空间距离
and without changing the temporal distance between two points at the same location in space.
也不会改变同一空间位置 两点之间的时间距离
However, all of this relativity is static.
不过前面讨论的都还只是静态的相对论
and by that I mean that we haven’t yet talked about how motion looks
我的意思是我们没有在一个移动的视角下
from a perspective that is itself moving.
对于物体的运动情况进行讨论
That’s ultimately the key to special relativity,
这是狭义相对论的关键
and will be the subject of the next video.
也会是下一期视频的主要内容
If you’d like to play around with some spacetime diagrams yourself,
如果你想自己画些时空图玩玩
I highly recommend checking out the “propagation of light” interactive quiz
我强烈推荐你到本视频赞助商的网站Brilliant.org
on Brilliant.org, this video’s sponsor.
做一下“光传播”的交互式测试
This quiz is seriously cool-
这个测试真的很厉害
it uses spacetime diagrams to guide you through
它会通过时空图引导你去了解
how astronomer Ole Rømer deduced that the speed of light must be finite
天文学家Ole Rømer仅通过观察木卫IO的轨道
just by observing the orbit of Jupiter’s moon Io.
就推理出“光速是定值”的全过程
It’s a super clever idea,
他所用的办法特别聪明
and the quiz does a great job
而且这个测试
of using spacetime diagrams to help visualize the situation
用时空图很好地把整个过程可视化了
and guide you through calculating the speed of light yourself.
还会引导你自己一步步算出光速来
In fact, this quiz is part of a whole course on Special Relativity
实际上这个测试只是Brilliant网站上
that Brilliant has available at brilliant.org/StaathofMinutePhysicsSpecialRelativity.
一整套在售的狭义相对论课程的一部分
And doing problems like theirs after watching videos like mine
在看过这一类视频之后再去做这样的测试
is a great way to practice
是个不错的练习方式
and really internalize the ideas of special relativity.
而且还能真正地消化狭义相对论的观点
If you decide to sign up for premium
如果你决定注册高级会员
access to all of their courses and quizzes,
来获得所有的课程和测试
you can get 20% off by going to Brilliant.org/minutephysics,
通过(1)网站可以给你打八折
or even better, go to brilliant.org/MinutePhysicsSpecialRelativity,
或者进入(2)网站
which lets Brilliant know you came from here
让Brilliant知道你是本视频观众
and takes you straight to their relativity course.
然后就可以直接跳转到相对论课程的页面

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视频概述

用几何图示的方法让普通人也能了解相对论

听录译者

收集自网络

翻译译者

风荷一一

审核员

审核员 EM

视频来源

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

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