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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

and we’ve talked about how distances change in relativity,

we’ve talked about how time changes in relativity.

So I thought I’d talk about something,

which combines those two things together,

which is how speed, which is distance divided by time, changes in relativity.

In the previous videos, we made what we called a Gamma Trilogy

because they all have this gamma factor in it.

Actually one of the interesting things, about the way speed transforms

is that the gammas all disappear.

They all cancel out, so there is no gamma in this gamma video.

of course, it’s distance divided by time,

but particularly how it changes depending on what reference frame you’re in.

So we need to think about

how different people measure speed in different reference frames.

And some of the classic kind of thought experiment you might do

is you’ve got somebody on a train

I’ll call this reference frame S’.

and they’re gonna roll a bowling ball, along the train

at some speed or other,

and the train itself is moving so this whole reference frame is moving

in this direction at some other speed v.

The way you’re on the train you’d measure the speed is that

you’d see how far the bowling ball had gone in your reference frame.

What should we call this? Δx: how far it’s gone,

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,

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

is just how far it’s gone

divided by the time it took to get there.

That’s the definition of speed,

it’s how far something’s gone divided by the amount of time it took to get there.

It’s the definition of speed.

So that’s in one reference frame,

so then the question is what does the person in this reference frame,

the other reference frame see?

And let’s deal with the non-relativistic case first:

What are they gon na measure?

The distance that they’ll measure that the ball has gone

is going to be how far the bowling ball has rolled away from you

plus how far you’ve moved down the line.

That extra distance, how far the train has moved,

depends on basically how fast the train was going.

You’re sort of adding it all on top of each other?

Yeah, exactly! That’s all you’re doing.

So that distance is vΔt.

If the train is going at some speed v,

and it has traveled for some time Δt, then

the distance it has covered is just the speed times

the time it has taken to do it.

So now we can put all this stuff together…

That says that Δx is equal to

(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,

plus how far the ball has moved relative to the train.

Just the sum of the two.

And we can rearrange this in the non-relativistic case

to say that

Let’s just divide through: Δx divided by Δt

is equal to v plus Δx’ divided by Δt.

And in this non-relativistic case

we don’t have to worry about different people seeing different times…

So it doesn’t matter whether it’s Δt or Δt’,

we can call it the same thing.

It’s all the same thing.

Or we can rewrite this again: it just says that

the speed you see the balls moving is equal to v.

So that’s Δx divided by Δt.

Plus the speed that the person on the train sees the ball moving.

And that, as you just said,

is you basically just add the speed.

The speed at which the bowling ball is moving is

the speed that the bowling ball is moving relative to the train

plus the speed that the train is moving relative to you.

That’s the simple Galilean transformation, no relativity.

Everything works in the way that we’re used to it working.

One of the things that get’s people interested in relativity is that people say,

you can’t ever travel faster than light.

So one obvious question you could ask is:

why can’t I just travel faster than light by just doing these kinds of additions?

Supposing the person on the train, instead of rolling a bowling ball,

was actually shining a torch along the train

Then the torch beam would be traveling at the speed of light relative to them

how fast the torch beam was travelling relative to you.

Well, it would be the speed of light, plus

the speed that the train is moving away from you,

and that’s faster than the speed of light.

Done it, you’ve hacked the universe.

Unfortunately, relativity takes care of that,

and it’s not really the way that things work.

So we need to start again,

but we need to do the problem properly with relativity.

Is this your Minute Physics audition, professor?

Unfortunately, my drawing is not quite up to the job, I suspect.

And then we’ve got the other chaps in here.

Moving along at some speed v.

Previously the formula we had said that

the distance that the bowling ball is down the line

is equal to the distance relative to the train

plus how far the train has moved.

Now the extra factor that we haven’t put in yet is special relativity.

And what we saw before was that what special relativity does is enter these factors of γ.

And so this Galilean transformation that we had before

is a little bit modified by an extra factor of γ.

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.

But then the other thing we know about relativity is that

not only does the distance depend on the reference frame you’re in,

but actually the time depends on which reference frame you’re in.

And so we need the equivalent transformation for that.

So there are these things called Lorentz transformations…

That’s the first of them and this is the other transformation that we need.

Turns out, these are the two Lorentz transformations that you need.

Now we can say,

what’s the speed as measured from the person

standing beside the track watching the train go by.

And we just do the same thing that we did before.

And the speed is just the distance divided by the time.

So we can just divide them with Δx divided by Δt.

is equal to… now the γs (gammas) are going to cancel

because if we divide that by the other,

we end up with a γ on the top and a γ on the bottom.

So we can just cancel them out.

I’ll do a little bit of tidying up:

I’ll divide top and bottom through by Δt’,

So I can just write this as, Δx’ over Δt’.

This is kinda the answer we wanted,

because Δx over Δt is the speed of the bowling ball

or whatever it is being thrown along the train

As seen from the perspective of the person standing beside the train track.

And Δx’ over Δt’ is the speed as seen from a person standing on the train.

So I can write that in terms of the”u”s and”u'”s we have before

This just says that

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.

So what we were looking at before, the Galilean transformation,

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…

And that just says that the speed you see is just

the sum of the speed of the ball relative to the train and the train relative to you.

So that’s right, it all kind of comes out right

if things aren’t closest be alike

Now let’s do the more interesting case of where the relativity actually matters.

And in particular let’s go to the really extreme case

of instead of rolling a bowling ball along the train,

let’s shine a light beam along the train.

So in that case, u’,

the speed of whatever it is is equal to the speed of light.

And let’s see what happens when we put that in there.

So in that case we end up with,

if u’ is equal to the speed of light…

Then u is equal to c (the speed of light) plus v…

divided by 1 plus (u’ is the speed of light) cv over c squared. Now,

let me know just do a little playing around with this.

So I will rewrite the top here as…

c into 1 (so, I will pull that factor out) plus v over c.

Because if I multiply this out,

I just end up with c plus cv over c which is just v

so c plus v, divided by

And then all I’m going to do is cancel the c over c squared

so I end up with 1 plus v over c.

Which you know is just the same as the term there.

So this is the same as that

which just basically means that this whole thing is equal to the speed of light still.

So there’s the bizarre thing

That we’ve taken. Remember, what we’ve done here is said,

Okay, So, there’s a light beam

that’s moving relative to the train at the speed of light.

Now if you’re watching from beside the train,

what speed do you see that light beam going at?

The answer is still the speed of light.

You haven’t actually added to its speed at all.

And I guess the physics behind it is that

you have to worry about both space and time

being changed by what reference frame you’re in.

Which means that not surprisingly,

the speed that comes out is going to be change

in a slightly strange way as well.

And it turns out that

relativity takes care of this invariance in the speed of light.

That no matter what reference frame you’re watching the light beam travel from,

you’ll always see it travel at the speed of light.

Is the universe wanting to keep things at the speed of light

and everything changes to cater for that?

Or does everything change all the time

and the speed of light just falls out of that?

Is it chicken or egg?

It’s a very good question.

And I actually really like your first explanation that

actually the universe arranges in things in such a way

that the speed of light always comes out as the speed of light.

no matter how much you mess around with things

by trying to run away from the light beam or run towards it…

When you come to measure it’s speed,

you’ll always find that it’s speed comes out at the speed of light.

and everything else kind of adjusts the distances and the times

adjust in just such a way,

so that when you come to combine those distances and times

to measure a speed of a light beam,

It will always come out the speed of light.

Is there something that is important about the speed of light being constant and unchanging?

So what motivated Einstein to come up with all this in the first place is…

the laws of physics should the same

whatever reference frame you’re in.

So that if you’re in sealed box,

there should be no experiment you can do that will tell you

if that sealed box is moving at a constant speed or is stationary.

And in fact, in his view of things,

it is a kind of meaningless question.

And what he also knew is that

the speed of light comes out from the laws of electromagnetism.

And so

then the question is,

in what reference frame do those laws of physics work

And his argument is that

those laws of physics should work in whatever reference frame you’re in.

which means that actually the speed of light has to be invariant

if you believe the laws of physics are the same

in all the different reference frames.

What if it hadn’t been that way?

What if the universe said,”nah, I can be different for different reference frames”?

Like would you and I be ripped to pieces in some cosmic rip hole?

would the universe just be a bit different and quirky?

Like does the universe benefit from the fact that this is what happens?

It’s very hard to construct a universe in which this wasn’t true.

(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,

you know, all the other things that come from relativity flow from that.

So think that famous things like E=mc squared

all flow from the invariance of the speed of light.

And that means that

the equivalence of mass to energy

is a natural consequence of relativity

and so, for example, what powers the sun (fusion),

converting mass into energy

wouldn’t work?

And if we weren’t living in a relativistic universe,

Now you could imagine that

the laws of physics in such a universe would come up with some other way

of generating energy from fusion.

But the picture we currently have of it really would be completely different.

And the quantity of the speed of light (the speed that it actually is)

Does that matter?

Would the universe be the same

if the speed of light had been halved, or it was quiet,

you know, if it was something close to the speed we walk at,

does it need to be as fast as it is to our human brains?

So it comes out of the laws of electromagnetism

it’s to do with how strong magnetic fields are,

how strong electric fields are.

which are really just arbitrary constants of nature as far as we know.

which I guess means

that actually the speed of light which is some combination

of these things is also an arbitrary law of nature.

The universe would be a very strange place

if the speed of light were very much slower.

Because of course all of these relativistic effects

that lead to all the weird things

that come out of relativity

we don’t have to worry about them in every day life.

But if it turned out

the speed of light was walking pace,

then all sorts of strange relativistic effects would happen

every time you walked to the post box.

all the relativistic time dilation and length contraction effects.

So it would be like, if I go to post this letter

am I going to die before I get there?

oh, yeah, would all your relatives have died of old age before you get home again.

So, yes indeed, it would be a very very strange universe.