亚当·布朗——深入浅出的广义相对论入门

如果你把物体降到地球表面，那实际上你从砖块里提取的能量并不多。

If you lower down to the Earth's surface, then the answer is you haven't really extracted that much from the brick.

你在远处做功时，只提取了砖块原始静止质量能量的七乘以十的负十次方。

You you've extracted a fraction seven times 10 to the minus 10 of the original rest mass energy of the brick, doing doing useful work a long way away.

有意思的是，先说第一个观察结果。

Interestingly well, first observation.

这个数值很小。

This is small.

换句话说，在自然单位下，地球表面物体的引力结合能非常微弱。

In other words, the gravitational binding energy of something on the Earth's surface is quite small in natural units.

这就是为什么我们直到做了极其精密的实验，才在地球表面察觉到广义相对论的影响——因为广义相对论在某种意义上是对这个数值的泰勒展开，一阶项就是牛顿力学，更高阶的项才给出广义相对论对牛顿结果的修正。

And that's why we didn't really notice general relativity on the Earth's surface until we did very sensitive experiments, because general relativity is, in some sense, a Taylor expansion in this number, the relativistic effects, where the first order term is just Newtonian, and then the next order terms will give you the GR corrections to the Newtonian answer.

第二个观察点，这有点跑题，但第二个观察是，这个数值恰好非常接近火箭燃料的化学结合能。

Observation number two, and this is something of a digression, but observation number two is that by essentially sheer coincidence, this number here is very close to the chemical binding energy of rocket fuel.

比如，取氧氢混合燃料，你燃烧它推动火箭所释放的化学能，除以氧和氢总质量对应的mc²，结果大约是1.5乘以十的负十次方。

So, if you take a rocket fuel like oxygen hydrogen mix, the chemical energy binding the rocket together, which is the energy that you're going to extract when you burn it to make your rocket go, divided by the mc2 of the oxygen and hydrogen you're going to mix together, is given by 1.5 times 10 to the minus 10.

第一个发现，这两个数值非常接近，尽管它们来自完全不同的计算。

First observation, these two are close to each other, even though they came from completely different calculations.

这是个与地球有关的引力计算。

This was a gravitational calculation that was something to do with the Earth.

而这个则是氢和氧的化学特性。

This is a chemical property of hydrogen and oxygen.

这个数值也很小。

This is also very small.

它这么小的原因是，氢和氧中的绝大部分能量并不储存在它们结合时的化学键能里。

The reason it's very small is that almost all of the energy in hydrogen and oxygen is not stored in the chemical binding energy of these things going together.

绝大部分能量其实藏在质子和中子的静止质量里，而化学燃烧根本碰不到这部分。

The vast majority of it is stored in just the rest mass energy of the protons and the neutrons that chemistry burning doesn't affect at all.

第二大的能量份额来自质子和中子之间的核结合能，涉及强相互作用和弱相互作用，化学反应同样完全影响不到这些。

The second largest amount is stored in the nuclear binding energy of the protons and the neutrons to each other, giving away the strong force and the weak force, which, again, chemical reactions don't don't touch at all.

这个数值很小，是因为化学键的能量相比我们所讨论的物质的静止质量来说，实在太弱了。

This is a small number because chemistry is chemical bonds are very weak compared to the rest mass of the things we're considering.

这两个很小的数值几乎完全相等，所以我们能用化学火箭进入太空，但过程非常艰难。

These two small numbers are almost exactly equal to each other, which is why we can use chemical rockets to get to space, but it's hard.

特别是，这个数值比那个大几倍，这意味着用化学火箭送入太空时，有效载荷的比例极小——因为大部分燃料根本无法进入轨道。

In particular, this number is a few times bigger than this number, which means that almost all of the vast majority or payload fraction is quite small when trying to use chemical rockets to get to space because most of your your fuel cannot get to orbit.

你得承受火箭的效率损耗，也就是说，发射台上大部分的重量都得在抵达太空前烧掉，才能把一小部分火箭送上去。

You have to pay a rocket factor that's going tell you that most of your payload that's sitting there on the launch pad is going to have to be burnt up before you get to space in order to get a small fraction of the the the rocket up to space.

换句话说，我们能用化学火箭进入太空，但如果从太阳表面出发，这根本不可能，但即便如此，它依然很难。

In other words, we can use chemical rockets to get to space in a way that would be totally impossible if we try to do it from the surface of the sun, but it's hard.

明白了。

Okay.

这是在地球上的比例，但这个公式说明，如果天体质量更大、更致密，把物体下放到表面时能释放的能量比例也会更大。

That's the fraction on on the Earth, but this formula tells you that if you have an object that's that's heavier, or or more compact, the fraction of of energy that you extract by lowering the object down to the surface is going to be larger.

比如，如果把物体下放到太阳表面，而不是地球表面，这个比例就会大得多。

So, for example, if you lower it not down to the Earth's surface, but down to the sun's surface, this would be larger.

差不多大一百万倍，因为太阳的质量是地球的几百万倍，不过太阳体积也大，这又抵消了一部分。

Know, a million times larger because the sun is, you know, a few million times the mass of the Earth, but then it's also bigger, so that that takes it away a little bit.

最后你得到的是两乘以十的负六次方，这是太阳表面著名的红移值。

And you end up with two times 10 to the minus six, famous redshift from the sun's surface.

从这里开始，你可以进一步推演。

And you can escalate from there.

你可以想象把一个太阳质量的质量压缩到地球大小的半径里，这样公式的结果会更大。

You know, you can imagine cramming a sun like mass into a Earth like radius, to make this formula even bigger.

事实上，这差不多就是白矮星，比如天狼星B的真实情况——太阳质量，地球半径。

In fact, that's that's pretty much exactly what what, so, you know, sun mass, earth radius, which is about what happens in a white dwarf like like Sirius b.

这种情况下，数值还会更大。

This will get even bigger.

你从物体表面下落时能提取出的质量比例也会更高。

Again, a larger fraction of the mass of the object you'd be you'd be extracting by lowering it down to the surface.

但很明显，当物体质量太大、密度太高时，某种东西必须得崩溃才行。

But it really feels like something has to give before we make a object that is too mass, too massive, and too compact.

特别是，如果你看这个公式，当半径小于或等于引力常数乘质量除以光速平方时，会发生什么？

In particular, if you look at this formula, you know, what happens for r less than or equal to gm over c squared.

如果这个物体如此致密、质量如此之大，以至于它的半径小于质量除以光速的平方，那看起来你就能获得超过100%的能量，也就是提取的比例大于1。

If this object here was so compact and so heavy that it had a radius less than the mass of the object divided by c squared, it sure looks like you could get more than 100%, the fraction with bigger than one.

把这块砖降到这个物体表面时，你能回收的质能甚至超过它本身原有的全部质量。

You can get more than 100% of the mass of your brick back by lowering it down to the surface of this object.

这感觉不对劲。

And that feels wrong.

事实上，这比刚才说的情况还要离谱。

That feels, in fact, more wrong than what's going on here.

因为你现在在远处积累了大量能量，甚至可以用它再造出一块全新的砖。

Because now you've got all this energy a long way away, and you could perhaps use it to make a whole new brick.

你手上还有超过mc²的多余能量，再把这块砖降下去，这就像是凭空造出了大量能量，而之前根本不存在这些能量。

And you've got all this more than MC2 out there, lower that one down, and it feels like we've figured out a way to make a huge amount of energy where there was no energy before.

这个推理强烈暗示，当物体半径小到这个程度时，一定有什么地方出错了。

This argument is pretty suggestive that something has to go wrong by the time you get down to that radius.

事实上，当你用完整的广义相对论来计算时，你会发现，没错，确实出了问题。

And, indeed, when you do the calculation, you know, this is a Newtonian calculation, so it's only suggestive, but when you do the calculation in full general relativity, indeed something does go wrong.

出问题的地方在于，你会形成一个黑洞。

And the thing that goes wrong is that you form a black hole.

你可以想象两种方式来避免这个结论。

You could imagine two ways that you could avoid this conclusion.

一种是，当你靠近大质量物体时，引力会变得非常弱，弱于牛顿定律的预测。

One would be somehow that gravity becomes very weak when you get close to a massive object, weaker than the Newtonian, law would predict.

这有点像在电磁学里重复这个操作时的解法——把一个电荷靠近另一个电荷，试图提取它们之间的静电能量。

That's sort of what saves you if you try and repeat this same trick in electromagnetism, lowering a charge down towards another charge and trying to extract the electrostatic energy between them.

实际上，由于量子效应，当两个电荷靠得太近时，它们的边界会变得模糊，导致能量随距离倒数变化的规律被平滑了，它们不再那么强烈地相互吸引，也就无法提取更多能量。

What happens is essentially due to quantum effects, when one gets too close to the other, they all start to fuzz out, and this the inverse the energy going like inverse r gets softened, and you can't extract more energy because they they stop attracting each other so hard.

所以，这是一种可能：当你靠近另一个物体时，作用力会比牛顿定律预测的更弱。

So that's one possibility, is that the force gets weaker than Newtonian law would predict as you approach the other object.

但实际情况恰恰相反，广义相对论的解决方式是让力变得更强。

That's actually the opposite of how general relativity resolves this.

广义相对论通过让引力比牛顿定律预测的更强，来化解这个悖论。

General relativity resolves this paradox by the force getting stronger than Newtonian law would predict.

特别是，当你试图靠近这个半径时，引力会变得极其强大，根本不可能慢慢把砖块放下去。

In particular, the force gets so strong when you try and get within this radius that in fact you cannot slowly lower the brick down towards the surface.

因为你已经形成了黑洞，引力会在某个有限距离上变得无穷大，而不是在 r 等于零的地方，而是在某个具体的有限半径处——砖块会直接从你手里被扯走，你再也无法从中提取任何能量。

Because you've formed a black hole, the gravitational force becomes infinite a finite distance distant way, not at r equals zero, but at some some finite value of r, and the brick simply gets ripped out of your hand, and you're unable to extract the extract any more energy out of it.

这就是广义相对论对这个悖论的解释。

And that's the resolution that general relativity provides to this paradox.

特别是你会发现，你已经形成了一个黑洞。

And in particular, you will find that you've formed black hole.

Crusoe 让我们提前试用了他们的无服务器微调产品，让你在不操心基础设施和资源调配的情况下微调开源模型。

Crusoe gave us early access to their serverless fine tuning product, which lets you fine tune open models without having to deal with infra or provisioning.

我觉得用我过去访谈的录音来微调一个提问生成器挺有意思的。

I thought it'd be cool to fine tuning a question generator using the transcripts of my old interviews.

这个模型现在已经厉害到，如果它掌握了我所有的研究资料，又能实时分析对话内容，它提出的下一个问题甚至比我本人还要精准。

The model's gotten so good that if they had all my research in prep and they could look at a conversation so far, they could ask a next question better than I would.

Crusoe 的实现过程特别简单直接。

Crusoe made the implementation super straightforward.

我刚上传了数据，选了一个开源模型，然后就开始运行了。

I just uploaded the data, picked an open model, and started the run.

我根本没碰任何超参数。

I didn't have to touch any of the hyperparameters.

Crusoe 的 AI 团队为每个模型都准备好了最优配置，所以我直接全设成自动模式就行了。

Crusoe's applied AI team maintains optimal recipes for each model, so I just set everything on auto.

运行结束后，我把它部署成一个自助式接口，还给团队做了一个评估工具。

When the run finished, I deployed it as a self serve endpoint and built an eval for my team.

我让他们从三个匿名选项里挑出最好的下一个问题。

I had them choose the best next question out of three anonymized choices.

一个是基础模型生成的，一个是微调后模型生成的，还有一个是我自己实际问的。

One that was produced by the base model, one that was produced by the fine tuned model, and one that I actually asked.

幸运的是，团队有三分之二的时间更喜欢我亲自问的问题。

Fortunately, my team preferred my actual questions about two thirds of the time.

希望这个基准测试别很快达到上限。

Hopefully, benchmark doesn't saturate.

在剩下的情况里，他们几乎总是更喜欢经过微调的模型，而不是基础模型。

And in the remaining cases, they almost always preferred the fine tuned model over the base model.

无服务器推理现在已经上线，无服务器微调功能下周正式推出。

Serverless inference is live now, and serverless fine tuning goes live next week.

了解更多内容，请访问 crusoe.aidwarkesh。

Learn more at crusoe.aidwarkesh.

到目前为止，我们在黑板上写下的所有内容都是牛顿力学的范畴。

So far, everything we've written down on the board is is Newtonian.

这纯粹是牛顿力学，只要你一加上光速，就会开始感到困惑。

It's just Newtonian, and you just stop plugging in the speed of light, you start getting confused.

要真正回答我们提出的这些问题，你得转向广义相对论——这个理论正确地把光速和引力统一了起来。

To actually answer some of these questions that we're asking, you need to go to general relativity, the the theory that correctly unifies the speed of light with gravity.

最早在黑洞的背景下提出这一点的是施瓦西，他推导出了施瓦西度规，描述了中心质量周围的引力场，包括黑洞周围的场。

And this was first done in the context of black holes by Schwarzschild, who wrote down the Schwarzschild metric that describes the gravitational field around a central mass, including potentially around a black hole.

让我写下一些由此得出的公式。

And let me just write down some of the formulas that emerge.

实际上，我想写下三个公式，它们都是施瓦西度规的直接推论，能帮我们直观理解黑洞内外的状况。

In fact, I think I'm going to write down three formulas, the three direct consequences of short shelled metric that are going to give us an intuition for what it's like outside and indeed inside a black hole.

我先写第一个公式，它描述的是，如果你试图在中心质量外部保持静止，会感受到多强的引力。

And so the first formula I'm going to write down is the formula for the gravitational field that you would experience if you were trying to remain static outside a central mass.

我们先讨论静止观察者的情况，之后再扩展到那些在移动的观察者。

And so, you know, let's just talk about for static observers so can discuss how these will get upgraded for observers who are moving around.

但现在，我假设你正停留在某个固定半径 r 的位置，远离黑洞。

But for now, I'm just going to imagine that you're trying to sit here at some radius r, some fixed radius r away from the black hole.

你之所以没掉下去，是因为你保持静止——也许我用滑轮把你放下来，你现在正抓着绳子悬在那里。问题是，你需要多大的力才能阻止自己坠入黑洞？

The reason you don't fall the reason you're static, the reason you don't fall into a black hole, maybe, you know, I've lowered you down on a pulley, and you're just sitting here, holding the pulley, and the question is how strong a force do you need to stop you falling down?

你的身体正缓慢地往下沉。

Your ab's sailing down very slowly.

你保持静止不动。

You you're you're static.

那你此刻感受到的局部引力有多大？

What is the local force of of gravity that you experience?

或者你可以想象，你坐在那儿，之所以能保持静止，是因为你开着一台强大的火箭推进器，那你 locally 感受到的加速度有多大呢？

Or you can imagine that you're sitting here, the reason you're static is you're finding a rocket very hard, and how much acceleration do you locally feel?

不管用什么方法，只要你保持静止，你感受到的局部引力是多少？

So by whatever mechanism, you're remaining static, what is the local force of gravity that you feel?

你感受到的局部引力，在牛顿物理里，答案其实很清楚。

The local force of gravity that you feel, well, in Newtonian physics, you know what the answer to that question would be.

引力就是 G M 除以 r 的平方，也就是牛顿著名的平方反比定律。

The force of gravity is g m over r squared, which is Newton's famous inverse square law.

但在广义相对论里，这个公式要加一个修正项。

But this gets a correction from general relativity.

修正项是 1 减去 2 G M 除以 c 的平方再乘以 r。

And the correction is one minus two g m over c squared times r.

这个 2 G M 除以 c 的平方，我们之前已经多次见到过了。

So this this this same two gm over c squared that we that we find all over the place.

这意味着，如果你离黑洞非常非常远，这个修正项就几乎等于 1。

And what this tells you, well, first of all, if you're a very long way away from the black hole, this here is essentially one.

当r非常大时，你就又回到了牛顿的引力定律。

R very big, and you get Newton's force law back again.

对于地球来说，这个修正值非常小。

And for the Earth, this this this is very small.

正如我们之前说的，它大约小了十的负十次方左右。

This you know, as we discussed, it's down by a factor of of 10 to the minus 10 or so.

然后你再开平方根。

And then you take the square root.

你平时根本察觉不到，但在r很大的时候做展开，就会发现有修正项。

So you don't you don't really notice it, but you can tailor expand this at at large r, and you find out that you get corrections.

会得到一个平方反比定律，加上一个立方反比的修正，再加一个四次方反比的修正，结果发现，在短距离上，引力比牛顿理论预测的要强。

You get an inverse square law plus an inverse cube law correction plus an inverse, fourth law correction, and you find that gravity at short distances is stronger than it would have been in Newtonian physics.

这就是广义相对论的修正，它让引力场变得更强了。

This is the general relativity correction, and it's making the gravitational field stronger.

为了不掉进黑洞，你得用更大的加速度来抵抗。

You have to accelerate harder to not fall into the black hole.

特别是当 r 等于 2GM 除以 c 平方时，也就是所谓的史瓦西半径，你需要无限大的加速度。

And in particular, once r is equal to two g m over c squared, this, what's called the Schwarzschild radius, you have to accelerate infinitely.

引力造成的加速度——也就是为了保持 r 值不变所需的固有加速度——会变得无穷大。

The gravity the acceleration, the proper acceleration required to not, not move in r goes to infinity.

所以，如果我们把这个应用到黑洞上，这个 2GM 除以 c 平方的半径就非常关键了。

So in fact, if we now convert this one to the black hole, this is a very significant radius over here, two g m over c squared.

它被称为事件视界。

It's, it's called the event horizon.

之所以叫事件视界，是因为只要你还在视界之外、离得更远的地方，想保持静止，只需要有限的加速度就够了。

It's called the event horizon because if you want to remain static outside the event horizon, further away from the event horizon, you just need to accelerate with some some finite velocity in order to remain static.

你需要一个有限的引力场，但当你靠近事件视界时，引力场会趋向无穷大。

You need to have a finite gravitational field, but the gravitational field as you approach the event horizon becomes infinite.

所以，一旦你到达或越过事件视界，就不可能再保持静止了。

So once you're at or beyond the event horizon, it is impossible to remain static.

不管你把火箭推到多强，你都注定会被吸入黑洞。

You will inevitably get sucked into the black hole no matter how hard you fire your rocket.

这只是静态情况下的公式。

Now, this is just the static formula.

你可能会想，虽然在那个半径以内没法保持静止，但也许我可以绕着黑洞高速公转，这样就不会掉进去。

You might imagine, okay, it's impossible to remain static outside the black hole closer than that, but maybe I could not fall into the black hole by orbiting, really, really fast.

如果我转得够快，离心力就会把我往外推，就能避开黑洞了。

And if I orbit really, really fast, I have a huge centrifugal force that pushes me away from the the black hole, and I can stay out of the black hole that way.

但实际上这行不通，而且这个原因对理解广义相对论中的引力机制很有启发。

That actually doesn't work, and the reason it doesn't work is somewhat instructive, for the way gravitational attraction happens in general relativity.

举个例子，国际空间站为什么不会掉向地球？

Of course, if you think about the International Space Station, why doesn't it fall towards the Earth?

正是因为它们在绕行，这种运动产生的离心力正好抵消了地球引力，所以宇航员才会感觉失重。

It is precisely the fact that it's that it's orbiting, and the fact that it's orbiting gives it a centrifugal force that shoots the astronauts away from the Earth and precisely balances the the gravitational field of the astronauts, which is why they they feel weightless there.

在远离黑洞的地方，轨道角动量确实能帮你维持距离，防止被吸进去。

So orbital angular momentum, if you're a long way away from the black hole, helps you escape from stay stay away from the black hole, stops you falling in.

但有一种常见的科幻误解，以为黑洞会把周围的一切都吸进去。

There is this kind of sci fi notion that black holes just suck in everything around them.

不对。

Not true.

如果你离黑洞足够远，完全可以围绕它运行，就像围绕任何中心质量公转一样。

You are perfectly able to orbit around a black hole, if you're a long way away from it, just like you would orbit around any central mass.

你并不会不可避免地掉进黑洞。

You are not inevitably falling into the black hole.

你完全可以稳定地绕行。

You can you can orbit just fine.

但当你离黑洞太近时，绕行就不再有帮助了。

But orbiting stops helping when you get too close to the black hole.

我们之前说过，事件视界的半径是 2GM 除以 c 的平方。

We said that the event horizon is is two g m over of c squared.

实际上，一旦你进入 3GM 除以 c 平方的范围内，绕行反而会适得其反，让你更难远离黑洞。

In fact, already once you're within three g m over c squared, orbiting is counterproductive if you're trying to stay away from the black hole.

这是因为绕行会产生两种效应。

And that's because there are two effects of orbiting.

轨道运动的一个作用是帮你远离黑洞。

One effect of orbiting helps you stay away from the black hole.

这就是离心效应。

That's the centrifugal effect.

由于离心效应，轨道角动量会把你推离黑洞。

Orbital angular momentum pushes you away from the black hole due to the centrifugal effect.

如果我们写出这个公式，其实并不难。

And if we wrote the it's not too hard to write down.

如果你写出考虑了角动量的版本，就会发现它确实把你往外推。

If you wrote the version of this formula that applies when you have angular momentum, you would see that pushing you away from the black hole.

但还有另一个效应，会把你拉向黑洞。

But there's another effect which drags you towards the black hole.

那就是广义相对论中的一个事实：所有形式的能量都会产生引力。

And that is the fact that in general relativity, all energy gravitates.

不只是静止质量会产生引力，动能也会产生引力。

Not just rest mass energy gravitates, kinetic energy also gravitates.

所以，绕行的效果是，由于黑洞质量与你轨道角能量之间的引力耦合，你会额外受到一股朝向黑洞的拉力。

And so the effect of orbiting is that you have an additional pull down towards the black hole from the coupling between the mass of the black hole and the gravitational attraction between the mass of the black hole and your orbital angular energy.

当你离黑洞较远时，离心力是主导因素。

And when you're far away from the black hole, the centrifugal force is the more important term.

但当你靠近黑洞时，这种耦合效应就变得更为关键。

When you're close to the black hole, that coupling is the more important term.

事实上，一旦你进入三倍史瓦西半径以内，轨道角动量就不再帮你远离，反而开始把你拉向黑洞。

And in fact, once you get within three gs m, orbital angular momentum stops helping and starts hurting.

在三倍史瓦西半径以内的区域，没有任何弹道轨道能让你进入后再逃出来。

There are no ballistic orbits that go within three gs m and that manage to escape again.

明白了。

Okay.

这就是第一个公式。

So that's formula number one.

它告诉你，在距离黑洞 r 的位置，引力场是什么样子。

It tells you that what the gravitational field is, a distance r away from a black hole.

特别是，它表明一旦你到达这个临界半径，引力场就会变得无限强，只要你跨过这个界限，无论你把火箭推到多强，都注定会坠向黑洞中心。

And in particular, it shows you that once you get to this critical radius, the gravitational field becomes infinite, and you must if you cross that, you must proceed to the black to the center of the black hole no matter how hard you fire a rocket.

这被称为事件视界。

That's called the event horizon.

在事件视界上，你还没死。

At the event horizon, you are not yet dead.

但你已经无法逃脱了。

You are, however, doomed.

只要你跨过事件视界，就再也没法逃出去了。

If you cross the event horizon, you will never be able to escape.

就算你把自己变成光，拼命往外射，也逃不掉。

Not if you convert yourself to light and try and shoot yourself out.

就算你把火箭推到极限，用尽所有动力，也无济于事。

Not if you fire your rocket infinitely hard.

另一个地方当然是 r 等于零，那里才是你真正丧命的地方，也就是奇点。

The other place, of course, is r equals zero, which is that's where you actually die, and that's at the singularity.

我们一会儿再详细讲一讲这个。

And we'll describe that a little bit in a moment.

在牛顿力学里，引力只有在这里才会变得无穷大。

In Newtonian physics, the gravitational force only becomes infinite here.

而在广义相对论中，只要你试图抵抗引力，引力在事件视界处就已经无穷大了。

In general relativity, it becomes infinite already at the event it raised if you try and resist the force of gravity.

好了，这是第一个公式。

Okay, that's formula number one.

接下来我们看第二个公式。

Now let's do formula number two.

第二个公式，还有接下来的两个公式，其实都彼此紧密相关。

Formula number two, and all three formulas I'm going to write down are just going to be you know, heavily related to each other.

它们本质上都是彼此的另一种表达方式。

They're really going to be reformulations of each other.

讲的是引力时间膨胀。

Asks about gravitational time dilation.

我们再来想象一下，你就坐在这里。

So let's again imagine that you're sitting here.

你看，这是Dwarkesh，坐在离黑洞某个半径R的地方。

You know, this is Dwarkesh sitting here some radius R away from the black hole.

而我呢，坐在远处，远远地在无穷远处，就看着你。

And I'm sitting, you know, out here, way off at infinity, just watching you.

我们俩相对于彼此是静止的。

And we're static relative to each other.

没有任何相对运动。

There's no there's no relative motion.

你只是靠着你的滑轮系统悬停在这里。

You're just suspended here by your by your pulley system.

问题是，你的手表相对于我的手表走得有多快？

And the question is, how fast does your watch go relative to mine?

当然，对你来说，你的手表就是一秒一秒地正常走着。

Of course, as far as you're concerned, your your watch is is ticking at one second per second.

至于我这边，我的手表也是每秒滴答一次。

As far as I'm concerned, my watch is ticking at one second per second.

但如果我看着你，观察你的话，我会觉得你的手表走得慢。

But if I watch you, if I look at you, I see your watch as running slow.

而如果你看我，你会觉得我的手表走得快。

If you look at me, you see my watch is running fast.

所以第二个公式就把这个差异量化了。

And so the second formula makes that quantitative.

你离黑洞这么近，你的手表比我这个远处的手表慢了多少呢？

How much slower you, who is close to the black hole, how much slower your wristwatch runs than mine?

公式说，你手表测出的时间间隔，等于我这个远处手表测出的时间间隔，乘上这个一直出现的平方根因子，再乘以根号下1减去2GM除以Rc平方。

And it says that the time interval as measured by your wristwatch is given by the time interval as measured by my wristwatch a long way away times this exact same square root factor that's showing up all over the place, times the square root of one minus two g m over r, c squared.

所以这个因子是小于1的。

And so this factor here is less than one.

如果我觉得过了一秒，你那边会觉得还没到一秒。

So if I think one second has passed, you think less than one second has passed.

换句话说，如果我慢慢把你降到黑洞附近，你在离黑洞一定距离的地方待上一整年，感觉就像过了一年，等我再把你带回远离黑洞的地方时，你会发现外面的世界已经老了更多。

In other words, if I slowly lower you down towards the black hole, you hang out some finite distance away from the black hole for what feels to you like a year, and then raise you back up a long way away from the black hole, you will return to a world that has aged a lot more than you have.

这个公式精确地描述了这一现象。

And this formula makes that makes that precise.

我看到你的手表走得比我的慢。

I observe your wristwatch to be running slow.

你看到我的手表走得比你的快。

You observe my wristwatch to be running fast.

在这里，时间过得比上面慢。

Time passes slower down here than it does up here.

这个现象早在20世纪50年代，就已被哈佛大学物理系通过实验反复验证。

This is a fact that has by now been extremely well observed experimentally in the 1950s in the Harvard Physics Department.

他们把两个原子钟放在大楼的不同高度，发现位置较高的那个走得比位置较低的快。

They put two atomic clocks at two diff two different heights in in the building and noticed the one that the one that was higher, was running faster than the one that was slower.

如今，这种效应在GPS等技术的精度范围内已经完全可以测量。

This is an effect that is now, considerably within the precision of, for example, GPS.

系统必须减去这个效应，不然所有数据都会乱套。

It just has to subtract that effect, otherwise everything would drift all over the place.

地面上的GPS时钟比轨道上发送信号的原子钟走得慢，必须把这个差异计算进去并修正，才能得到准确的读数。

GPS clocks that are sitting on the Earth's surface are running slow compared to the atomic clocks that are in orbit sending out the signal, and you have to account for that difference and subtract it off in order to get an accurate read.

这被称为引力时间膨胀。

This is known as gravitational time dilation.

注意，这和狭义相对论里由于两个物体相对运动产生的时间膨胀完全不同。

And notice it's quite different from the relativistic time dilation you see in special relativity, which is caused by two objects being in motion relative to each other.

在这里，我们之间并没有相对运动。

Here, we're not in motion relative to each other.

我们俩都是静止的。

We're both static.

我们都是固定的。

We're fixed.

这源于我们处在引力势的不同位置——你比我更深地陷在引力势阱里。

This is caused by us being at a different place in the gravitational potential, you deeper in the gravitational potential than me.

所以这两种时间膨胀效应是相互叠加的。

So those are two different sources of time dilation, and they stack.

假设你现在不是静止在这里，而是在绕行。

So let's say instead of being static here, you're orbit.

你离得足够远，可以围绕黑洞运行。

You're far enough away that you can orbit the black hole.

那我问你，我看到你移动得有多慢？

And I ask, how slow do I see you as moving?

现在有两种因素，都会让你在我眼里显得更慢。

There are now two contributions, both of which make you look slow relative to me.

其中一个因素是引力时间膨胀，由这个公式决定。

The one contribution is the gravitational time dilation, given by this formula.

另一个因素是狭义相对论里的经典效应——运动的观察者看起来时间会变慢。

A second contribution is the good old special relativity correction where moving observers look like they're going slow.

这两种效应都会同时存在。

And we'll have both of those effects.

所以当你围绕黑洞运行时，你看起来会比平时更慢。

So you'll look like you're going even slower than you would have done otherwise as as you go around the black hole.

这和狭义相对论的一个不同之处在于，这里没有对称性。在狭义相对论里，两个观察者都会觉得对方过得更慢，因为彼此相对运动的速度一样，没有哪个是真正惯性参考系。

So so one thing that seems different between this and special relativity is that there's no symmetry, where in special relativity, both observers will feel that the other one is aging slower than they are because they're both moving relative to each other at the same rate, and there's no true inertial path.

但在这里，确实能感觉到有一个全局意义上的参考系更准确、更相关。

But here, it actually does seem like there's a global sense in which one is a more correct the the the more relevant inertial frame than the other one.

你说得完全对。

You're exactly right.

没错。

Yeah.

在狭义相对论中，如果你我彼此相对运动，我会觉得你的表走得慢。

So in special relativity, if you and I are moving relative to each other, I think your watch is moving slow.

而你觉得我的表走得慢。

You think my watch is moving slow.

我们俩谁都没错。

Neither of us is more correct than the other.

相对性原理告诉我们，我们的两种视角同样有效。

The principle of relativity tells you that both of our perspectives is equally valid.

但在这里，我们的两种视角并不等价，因为这里没有狭义相对论中的那种对称性。

Here, both of our perspectives are not equally valid because there is is not the symmetry that there was in special relativity.

特别是，这种对称性被黑洞打破了。

In particular, the symmetry is broken by the black hole.

我们都同意，你处于比我还深的引力势阱中。

You really we both agree that you are deeper in the gravitational well than I am.

这一点我们都认可，而且你的钟比我的走得慢。

We both agree on that, and your clock runs slower than my does than mine does.

你并不会反过来觉得我的钟变慢了。

You do not see my clock reciprocally running slow.

事实上，你看到的是我生活得更快。

You, in fact, see me living sped up.

如果你在观察我，你会看到我像快进一样过着日子。

If you if you are observing me, you see me living my my life in fast forward.

有意思。

Interesting.

这是第二个公式。

So this is the second formula.

它描述了我们的手表彼此之间走时快慢的差异。

It says how fast our wrist wristwatches move relative to each other.

现在想象你在这里，戴着走得很慢的手表，然后朝我发射一束光。

Now let's imagine that you're here with your slow moving wristwatch, and you shine a light towards me.

假设这束光有特定的频率。

And let's say the light has a particular frequency.

比如，你是用钠原子跃迁产生的光，频率是固定的。

You made it with with a sodium transition, for example, a particular frequency of light.

当这束光向上传播到我这里时，我会觉得它的频率比你发射时低。

As that light travels upwards, by the time it reaches me, I'm going to think that it is lower frequency than it was when than you thought it was when you sent it.

为什么？

Why?

因为频率反映的是振荡的快慢，而在我看来，你所有的一切都进行得比较慢。

Because frequency is about how rapidly it oscillates, And I just think that everything you do is moving slow relative to me.

你觉得它的振荡变慢了。

You think it's oscillating slower.

频率变低了，也就是说，光向光谱的红端偏移了。

It has lower frequency, which means it gets shifted towards the red part of the spectrum.

我们管这个叫红移，引力红移。

The word the word that we use is redshift, gravitational redshift.

光发生了红移，频率变低，能量也就跟着降低了。

It gets redshifted, lower frequency, and therefore less energy.

如果你向上发射一个光子，它的能量就取决于频率。

If you send one photon up, the energy of the photon is given by the by the frequency.

它到达我这里时，红移更明显，能量也比离开你那里时更低。

It'll arrive at me more more redshifted and with lower energy than it had when it left you.

反过来，如果我在这上面向你发射一个由钠跃迁产生的光子，对你来说，当光到达你时，我这边的一切就像快进一样。

Conversely, if I am up here and I send you a photon generated by the sodium transition, as observed by you, by the time the light reaches you, you see me moving in a fast forward.

所以你觉得，它离开你的时候频率比现在低。

So you think that it has higher frequency than it had when left you.

它的光谱向蓝色端偏移了。

It's moved towards the blue part of the spectrum.

我们称之为蓝移。

We say that it is blueshifted.

这个思想实验告诉你，时间在不同高度流逝的速率，直接决定了能量在不同高度的价值换算关系。

And so this thought experiment tells you that knowing the exchange rate for how, time passes at different altitudes directly gives you the exchange rate for how much energy is worth at different altitudes.

如果你试图把能量传给我，等它到达我这里时，对我而言的价值，已经比你当初认为的低了。

If you try and send me some energy, by the time it reaches me, it's worth less to me than you perceived it as being worth to you.

而价值降低的幅度，正好由那个控制一切的平方根公式精确决定。

And the amount it's less is going to be precisely given by the same square root formula that's controlling everything else.

这就给我们带来了第三个公式。

And so that gives us our third equation.

第三个公式是这样的：假设你，Dwarkesh，有一个质量为 m c 平方的物体，正停在你下方的固定半径处。

And so the third formula says, suppose that you, Dwarkesh, have an object of mass m c squared sitting with you at this fixed radius down there.

从我这个远离黑洞的地方来看，这个物体到底值多少能量？

How much energy, as measured by me a long way away from the black hole, how much energy is that worth to me?

当然，如果它在我这儿，它的能量就是 m c 平方，但它并不在我这儿。

Of course, if I had it with me, it would be worth m c squared worth of energy, but I don't have it with me.

它不幸地正处在你那边很深的引力势阱里。

It's unfortunately sitting with you deep in a gravitational potential.

所以对我而言，它的价值低于 m c 平方。

So I have it's worth less than m c squared to me.

事实上，用的正是同一个公式。

In fact, it's just the exact same formula.

当它传到我这儿时，我所接收到的能量值是 G M 除以 R C 平方。

The amount of energy that it's worth to me by the time it reaches me is g m over r c squared.

这个结论有几种理解方式。

And there are a couple of ways to see that.

其中一种，就是我们刚才讲过的那种。

One is the way that we just that we just set.

假设你拿一个质量为 m 的物体。

Suppose you take your object of mass m.

这其实就是阿伏伽德罗常数个碳原子。

It's just Avogadro's numbers of carbon atoms.

或者说，它是一半阿伏伽德罗常数的碳原子，加上一半阿伏伽德罗常数的反碳原子。

Or let's say it's half an Avogadro's numbers of carbon atoms and half a Avogadro's numbers of anti carbon atoms.

你可以通过让它们相互碰撞，引发剧烈爆炸，来把能量传给我。

And one way you could send me the energy is by smashing them together, violent explosion.

你把所有这些能量转化成光，然后试图把光能传送到我这里。

You convert all of that energy to light, and you try and beam that light energy up to me.

但正由于引力时间膨胀效应，当光到达我这里时，我得到的能量已经不到 mc² 了。

But what you find precisely because of this gravitational time dilation is by the time it reaches me, I'm not getting MC squared worth out.

根据我们刚才的推导，我得到的能量确实少于 mc²。

I'm getting, by the argument we just gave, less than MC squared worth out.

我实际得到的是 1 减去 2gm 除以 r c 平方。

I'm getting one minus two gm of r c squared out.

这里的质量在上升过程中会发生红移，等它到达无穷远时，能量已经比最初少了。

Mass down here suffers this red shifting as it goes up and has less energy by the time it reaches infinity than it did to begin with.

还有一种方式可以把能量传给我。

There is another way that you could have got the energy to me.

不是用光束把能量传上来，而是直接把你的质量物体挂到滑轮上，让我把它拉出来。

Not by beaming it up as light, but by just taking your mass object, attaching it to the pulley, and having having me pull the object out.

等我把它拉出来之后，我就在这儿，离黑洞很远的地方，得到了完整的 m c 平方能量。

By the time I pulled it out, I've now got m c squared sitting out here, a long way away from the black hole.

所以，我确实得到了完整的 m c 平方能量。

So I do have m c squared, the full m c squared worth of energy.

但要得到它，我得付出代价。

But to get it, I needed to pay.

我付出的代价，就是克服引力势能把它拉出来。

And what I needed to pay was precisely, pulling it out of the gravitational potential.

从这个角度看，我就明白了为什么最后剩下的能量不到完整的 m c 平方——因为要把质量拉出来，我得先为克服引力势能买单。

So that from that way of thinking about it, that's why I have less than m c squared worth of energy left, because I had to pay the pulling it out for the potential in order to accrue that mass.

所以这个公式告诉你，如果我手里有一块质量为 m c 平方的砖，放在距离黑洞 r 的位置，而我本人离黑洞很远，我能从这块砖里提取多少能量？

So this formula tells you, if I have a brick of mass m c squared sitting some radius r away from the black hole, how much energy can I extract from that brick if I'm if I start if I'm a long way from the black hole?

如果我们知道了这个公式，就能精确算出这个值。

And so if we know that formula, then we can in fact calculate exactly this formula.

当我把砖块慢慢降到半径 r 的位置时，到底提取了多少能量？

How much energy have I extracted from the brick by lowering it down to a radius r?

这个问题的答案我们其实已经知道了。

Well, we know the answer to that question.

它一开始的能量是 m c 平方。

The energy it started with is m c squared.

现在它剩下的能量就是这个值。

The energy it now has is this.

所以，通过滑轮系统慢慢把砖块降下来的过程中，我提取的能量，就是初始能量 m c 平方，减去它现在剩下的能量。

So the energy I've extracted from the brick while slowly lowering it down using my pulley system must be the energy I started with, m c squared, minus the energy it now has.

换句话说，我通过把砖块降到半径 r 处所提取的能量占比，就是 m c 平方减去这个值，再除以 m c 平方，也就是 1 减去根号下 1 减去 2 G M 除以 c 平方 r。

Or in other words, the fraction of the energy that I've extracted by lowering it down to a radius r is, m c squared minus this, all divided by m c squared, one minus root one minus two g m over c squared r.

这个公式准确给出了提取能量的比例。

And this is the exactly correct answer for the fraction of the energy extracted.

它看起来和这个不太一样，因为那个只在牛顿力学的近似下成立。

It doesn't look exactly like this because this is only correct in the Newtonian limit.

我们是用牛顿物理推导出那个公式的。

We derive this using Newtonian physics.

而这个公式不仅在牛顿近似下成立，即使在广义相对论效应显著的地方也完全准确。

This is exactly correct not just in the Newtonian limit, but all the way to where the effects of general relativity are important.

如果你离黑洞非常非常远，r 远远大于 2G M 除以 c 平方，那就可以对这个公式做近似展开。

Now, if you're a very, very long way away from the black hole, r is much, much bigger than two g m over c squared, then you can tailor expand this formula.

而一阶近似项，就是原来的牛顿公式。

And the first order term is just the old Newtonian formula.

这本来就应该如此。

It better be.

广义相对论在远距离下的结果，必须还原成我们最初发现的牛顿物理。

It better be that the long distance limit of general relativity recovers the Newtonian physics that we we originally discovered.

但当你越来越靠近黑洞时，这个结果就开始偏离牛顿力学的预测，而这种偏离恰恰能解决我们最初那个关于把砖块往黑洞下方放的思维实验。

But as you get closer and closer to the black hole, this starts to deviate from the Newtonian answer and deviate in the Newtonian answer that exactly is gonna end up resolving our original thought experiment to do with lowering a brick down towards a black hole.

那么，根据这个公式，当我把砖块慢慢往黑洞靠近时，到底提取了多少能量呢？

So how much, then, looking at this formula, have I extracted from the brick as I lower it down towards the black hole?

如果 r 是无穷大，也就是说砖块还离黑洞很远，那我提取的能量就是一减一等于零。

If r equals infinity, if I haven't if the brick's still a long way from the way from the black hole, then I've extracted one minus one equals zero.

我从黑洞里一点能量都没提取出来。

I haven't extracted any energy from the black hole.

当我把砖块慢慢往黑洞靠近时，一开始得到的还是牛顿力学的公式。

As I lower it closer and closer to the black hole, well, initially, I just get the Newtonian formula.

实际上，这些公式在广义相对论里也基本靠谱，因为只有当这个项接近1的时候，修正项才会变得显著，而目前它还很小。

So in fact, these are pretty close to correct in general relativity as well because the corrections are only going to start getting large when this term becomes order one, and it's still very small here.

所以这些公式在现阶段都基本是正确的。

So these are all essentially correct.

但一旦我靠得更近，它们就不再准确了。

But once I get closer and closer to the black hole, they they stop being correct.

我发现，当 r 趋近于黑洞的事件视界时，这个公式趋近于零，意味着我已经把砖块的全部能量都提取出来了。

And and what I see is that as r approaches the black hole event horizon, as this formula goes to zero, I have extracted exactly all of the energy from the brick.

我一开始把砖块放在离黑洞非常远的地方，用一根绳子系住，慢慢把它往事件视界方向放下去。

So I start off with a brick a very, very long way from a black hole, attach it to a rope, slowly lower the brick down towards the event horizon.

当然，我不能把它放到视界以下，否则就失去控制了；但我会把它放到能放的最低点——就在视界正上方，然后轻轻松手，让它从静止状态自由下落。

Of course, I can't lower it past the event horizon, otherwise, I'll lose control of the brick, but I'd lower it as you know, right above the event horizon, the the last possible place I can lower it to, and then just let go of it with zero velocity.

砖块掉进黑洞后，我通过外面的滑轮系统，把原本属于砖块的全部质能 E=mc² 都提取出来了。

The brick falls into the black hole, and I have extracted the entire MC2 that used to be in the brick in my pulley system out there.

这样一来，我们之前的问题就完全得到了解答。

And so it exactly resolves this question we had.

有可能从砖块中提取超过 mc² 的能量吗？

Is it possible to extract more than MC2 from the brick?

不可能。

No.

能用黑洞把砖块的全部 mc² 能量都提取出来吗？

Is it possible to extract the full m c squared from the brick using a black hole?

是的，可以做到。

Yes, it is.

这其实很巧妙，也正是人们考虑把黑洞当作发电站的原因。

And that's actually pretty neat and why people talk about using black holes as power plants.

如今大多数发电站都是通过燃烧化学能来运作的。

Most power plants today operate by burning chemical energy.

这种做法效率并不高。

That is not very efficient.

你需要付出十的负十次方的代价，因为化学键相比物体的静止质量来说太弱了，你实际上只提取了燃料静止质量中极小的一部分。

You have pay a factor of 10 to the minus 10, because chemical bonds are super weak compared to the rest masses of objects, and you're really only extracting a tiny fraction of the rest mass of the fuel that you're you're considering.

你可以进一步提升效率，转向核能——不再依赖原子间微弱的电磁力，而是利用原子核内质子与中子之间的核力。

You can level up from there by going to by going to nuclear energy, which, instead of it dealing with the feeble electromagnetic bonds between atoms, starts to concern itself with the nuclear forces between the the protons and the neutrons within the within the nucleus.

这样，能量提取效率就能从大约十的负十次方提升到裂变的十的负三次方，或者聚变的十的负二次方。

And so you can go up from about 10 to the minus 10 to about 10 to the minus three for fission or 10 to the minus two for fusion.

但即便在裂变和聚变中，这已经是极限了。

But that's about as good as you can go even with fission and fusion.

尽管你可以从强核力中提取能量，比如裂变和聚变，但这些过程都不会改变质子和中子的总数。

Because even though you can extract energy from the strong nuclear force, fission and fusion, neither of them change the total number of protons plus neutrons in your process.

大部分能量——99%的能量——其实都储存在质子和中子的静止质量里，而不是电磁相互作用或者核力中，这一点化学反应和核反应都碰不到。

And the bulk of the energy, 99% of the energy, is stored not in the electromagnetic interaction, not in the stronger direction, but in the rest mass energy of the protons and neutrons, something that neither chemical reactions nor nuclear reactions can touch.

但引力可以做到。

But gravity can touch them.

如果我从一个质量为 m 的物体开始，理论上我能提取出接近全部的能量，几乎能100%利用它原本的静止质量能。

If I start off with a mass object of m, I can extract up to quantum fractions essentially, I can extract essentially 100% of the rest mass energy that I've gone in.

这可能是效率最高的发电方式，因为只要设计出这样的装置，原则上就能把初始物质的全部能量都提取出来。

It is the most efficient possible power plant because by building an apparatus like this, in principle, I could extract a 100% of the energy of whatever I started with.

我大概能理解能量和质量是等价的，化学键断裂时会释放能量，物体的质量也会因此减少。

I intuitively get how energy equals mass, and then there's, like, these chemical bonds, those get dissolved, they release energy, the thing weighs less if the those bonds are released.

我也明白，当质子和中子之间的键被打破时，也会释放能量，导致质量减少。

I even get that if the bonds between the protons and the neutrons are broken, that releases energy and makes the thing have less mass.

但如果一个包含质子和中子的物体刚好在事件视界上方，是不是意味着这些质子和中子就在那一刻彻底消失了？

But if something with protons and neutrons is just slightly above the event horizon, is the interpretation that those protons and neutrons stop existing right at that point?

那它们只剩下原始质量的1%、2%或5%，这到底意味着什么？

What does it mean for them to be have 1% or 2% or 5% of their original mass?

是的。

Yeah.

这是个非常好的问题，真正要讲清楚就得涉及量子力学了，不过那已经超出今天讨论的范围了。

This is that's a great question and really becomes relevant once you turn on quantum mechanics, which is beyond the scope of today's discussion.

但从经典物理的角度来看，黑洞就一直待在那里，所以你完全可以问：那些质子和中子去哪儿了？

But classically, the black hole just sits there forever, and so you can just say, well, what happened to the protons and neutrons?

答案就是，它们现在被禁锢在黑洞里面了。

You say, well, they now live inside the the black hole.

而宇宙中的质子和中子总数，仍然是守恒的。

And the number of protons plus neutrons is still conserved out there in the universe.

你只需要给黑洞本身赋予一个‘重子数’就可以了。

It's just you need to assign a called a nucleon number to the black hole itself.

从经典物理的角度看，这样解释是没问题的。

That's fine as far as it goes classically.

从量子力学的角度来看，这远远超出了今天讲座的范围，霍金和贝肯斯坦发现，黑洞其实会辐射能量。

Quantum mechanically, way beyond the scope of today's lecture, Hawking, and Beckenstein discovered that black holes, radiate away energy.

最终，黑洞会完全消失，所有能量——如果你仔细计算的话——都会变成引力子、光子，也许还有一些中微子。

Eventually, the black hole will be gone, and all of the energy, if you calculate it, ends up in gravitons and photons and perhaps some neutrinos.

其中几乎没有一点会变成质子或中子。

None of it or almost none of it ends up in protons and neutrons.

一旦引入量子引力，就会发现一个非常有趣的现象：黑洞会吞噬重子数。

It is a very interesting fact, once you turn on quantum gravity, that black holes eat nuclear number.

原本看起来在电磁力和核力作用下都守恒的量，最终却被引力给吞掉了。

This thing that seems like it's conserved, at least perturbatively, both both by electromagnetism and by the nuclear forces, ends up being eaten by by gravity.

对的。

Mhmm.

很多人喜欢把这一点推广成一个更普遍的原理：量子引力不尊重任何全局对称性。

And people like to promote this we're talking about quantum gravity now to a general principle that quantum gravity doesn't respect any global symmetries.

它不尊重重子数对称性。

It doesn't respect nuclear number symmetry.

它也不尊重这些对称性。

It doesn't respect any of these symmetries.

这可是另一个大话题了，咱们改天再细聊吧。

And that's a whole other can of worms that we can open some other day.

我最近写了一篇博客，推测过去几年里，训练时的样本效率其实并没有太大提升，真正大幅改善的是数据分布的广度和多样性。

I recently wrote this blog post where I speculated that sample efficiency during training actually hasn't improved that much over the last few years, and rather we've just dramatically improved and widened the data distribution.

最近我和朋友吃饭时，突然想到一个办法，可以实际验证这个问题。

And I was having dinner with friends recently and then I had this idea of how you could get some empirical information on this question.

现在有个叫 NanoGPT 的速度挑战赛，参赛者们比谁能在固定损失值的前提下，用最少的计算资源训练出 Karpathy 的 GPT-2 基线模型。

There's this NanoGPT speedrun where people compete to train Karpathy's GPT-two baseline to a fixed loss with less and less compute.

训练数据是固定的，所以我想到，如果分析每个参赛记录的损失曲线随时间的变化，大概就能看出样本效率提升的速度。

The training data is frozen, so I wondered if the loss curves over time of each record could tell you roughly how fast sample efficiency is improving.

于是我掏出手机，把这想法录进 Cursor 应用的语音备忘录里，然后继续吃饭。

So I pulled up my phone, I dumped this idea into a voice note in the Cursor app, and I went back to dinner.

大概十五分钟后，我收到一条通知——Cursor 的智能代理已经克隆了修改版的 NanoGPT 仓库，分析了所有参赛记录的损失曲线，估算出样本效率每年大约提升两到五倍。

And then I got a notification about fifteen minutes later, the Cursor agent had cloned the modded NanoGPT repo, it had analyzed all the loss curves for all the records, and it estimated that sample efficiency had been improving about two to five x every single year.

当然，这只是一个很初步、很零散的线索，但它让我萌生了和朋友一起写一篇完整文章的想法，用多种方法来深入探讨这个问题。

Of course, is very naive and circumstantial evidence, but it inspired me to start writing a full post with a friend where we investigate this question using many different methods.

而这里的便捷性真的非常关键。

And the friction really mattered here.

如果不能立刻用 Cursor 应用启动这个调查，这个想法早就溜走了。

The idea would have just floated away if I wasn't able to just kick off the investigation right then and there with the Cursor app.

如果你想试试 Cursor 的 iOS 应用，可以去 cursor.com/dwarkesh。

If you wanna try Cursor's iOS app, go to cursor.com/dwarkesh.

好的，亚当。

Okay, Adam.

我希望这个播客不仅能传递理论知识，也能带来实际的洞见。

I'd like to think of this podcast being part not only theoretical or practical knowledge as well.

假设有人把所有这些公式都背得滚瓜烂熟，结果一不小心掉进了黑洞，他会看到什么？

So suppose, one learns all these equations, but then finds himself in the unfortunate position of falling into a black hole, what would they see?

好问题。

Great question.

实际上，你可以从两个不同的角度来看这个问题。

So there's actually two different perspectives you could take.

一个是我在外面看着你掉进黑洞的情形。

One is the perspective of me watching you falling into the black hole.

另一个是你自己掉进黑洞时的体验。

The other is the perspective of you falling into the black hole.

这两种视角彼此并不矛盾，但却有很有趣的差异。

And those two perspectives are consistent with each other, but interestingly different.

所以我最好把这两种情况都解释一下。

So maybe I should describe them both.

首先，我们来想想，当你掉进黑洞的时候，我能看到什么？

So first, let's ask the question, what do I see as you fall into the black hole?

我看到的是，你本来需要多大的推力、多拼命地点火才能避免被黑洞吸进去，但你根本不会这么做。

What I see this is, of course, how you how fast you need to how hard you need to fire your rocket towards not falling to the black hole, but you're not gonna do this.

你就只是远远地、远远地待在黑洞外面。

You're just gonna sit here a long, long way away from the black hole.

你会关掉火箭，顺其自然。

You're gonna turn off your rocket and accept what comes.

接下来，你会慢慢加速朝黑洞坠落，起初遵循牛顿力学的公式，但靠近黑洞时，就要开始加入广义相对论的修正，调整牛顿的平方反比定律。

And what comes is you'll slowly accelerate towards the black hole with a rate first given by the Newtonian formula, and then when you get close to the black black hole, start picking up corrections, start picking up general relativity corrections to the Newtonian inverse square law.

当我看着你坠向黑洞时，一开始你会越掉越快，因为你在黑洞的引力势中不断下落。

And so what I will see as I watch you fall towards the black hole is first, you'll go faster and faster and faster towards the black hole as you fall down the gravitational potential of the black hole.

速度越来越快，但接着奇怪的事发生了——你不再加速，反而开始变慢。

And faster and faster and faster, but then something strange will happen, which is you'll stop going faster and you'll start going slower.

你变慢的原因是，随着你下落，引力时间膨胀效应开始显现，我看到你的动作变迟缓了，你的时钟也走得越来越慢。

And the reason you're going slower is that as I watch you, you start to get gravitational time dilation as you fall down here, and I start to see you running slow, your clock running slow.

这个公式适用于你静止不动的情况。

This is the formula for if you were static.

但你并不是静止的。

You're not static.

你正在运动。

You're moving.

所以还有另一个公式。

So there's some other formula.

但这个公式的效果是一样的：当你越来越接近黑洞时，你的手表会越来越慢。

But the formula has the same effect, which is that as you get closer and closer towards the black hole, your wristwatch starts running slower and slower and slower.

事实上，如果你做正确的积分计算，我永远看不到你越过事件视界。

And in fact, if you do the appropriate integral, I never see you cross the event horizon.

我只看到你越来越靠近事件视界，但同时速度越来越慢，越来越慢。

I just see you getting closer and closer to the event horizon, but slowing and slowing and slowing as you get closer and closer to the event horizon.

当我观察你的时候，我大概是用光来观察的，而这些光会变得越来越红移。

And as I watch you, I'm presumably using light to watch you, that light gets more and more red shifted.

光的波长会变得越来越长。

The wavelength gets longer and longer.

光的波长越长，就越难真正看清你。

The longer the wavelength of light, the harder it is even to really see you.

我开始因为光的波长而无法准确定位你。

I start getting you start getting delocalized by the wavelength of the light.

最后，我就完全看不到你了。

And eventually, I just stop seeing you entirely.

我最后看到的，是你发出的那一个光子，然后你就渐渐变暗，从红到黑。

There's a final photon that I see that that you emit, and then you just fade to black, fade through red.

彻底变黑，我永远也看不到你越过事件视界。

To black, I never see you cross the event horizon.

在广义相对论刚出现的时候，人们就注意到了这一点，当时特别困惑，还以为你掉进事件视界的时候会感受到什么奇怪的现象。

This was noticed by by people in the early days of of general relativity and and greatly confused them, and they started to think that something you would experience something funny yourself as you fell across the event horizon.

但其实并不是这样。

That is that is not true.

事实上，如果我换到你的视角来看，你的时钟并没有变慢。

In fact, if I instead adopt the perspective of you, from your point of view, your clock isn't running slow.

它还是每秒走一秒，正常得很。

It's running at one second per second.

如果你回头看我，那边倒是有点奇怪，可能我这边的时间过得特别快。

If you look back at me, there's some some funny stuff going on to do with me me running fast, perhaps.

但对你来说，一切都很正常。

But as far as you're concerned, everything's totally normal.

你朝着黑洞加速飞去。

You accelerate towards the black hole.

越靠近黑洞，你就越快，最后平平稳稳地穿过事件视界，跟平常一样。

You're getting faster and faster and faster as you approach the black hole, and you just sail across the event horizon totally as as normal.

对你而言，事件视界并不是什么危险的地方。

The event horizon is not a particularly violent place for you.

你可以算一下穿过事件视界时，或者接近并穿过时的潮汐力，其实并不算大。

You can calculate the tidal forces as you cross the event horizon or as you approach and then across the event horizon, and they're not particularly big.

准确说，对于质量大的黑洞，潮汐力并不算强。

Or rather, for large black holes, they're not particularly big.

如果是太阳质量那么大的黑洞，潮汐力就会很强，会让你非常痛苦。

For a solar mass black hole, they they would be pretty big and would be pretty painful for you.

你会感觉到脚部受到的引力远比头部强，因为脚离黑洞更近，结果整个人被拉长了。

You'd find that the your feet are being attracted to the black hole much more vigorously than your head is because they're closer, and you end up getting stretched.

但如果我选一个足够大的黑洞，你根本不会察觉到任何异常。

But if I take a big enough black hole, you wouldn't notice anything funny happening whatsoever.

黑洞越大，潮汐效应就越小。

The bigger the black hole, the smaller the tidal effects.

如果我选一个质量相当于整个星系的黑洞，你穿过事件视界时几乎不会有任何感觉。

And if I took a black hole the mass of the galaxy, you'd be basically fine as you cross the event horizon.

如果再选一个更大的黑洞，你甚至能在穿过事件视界后，活完整个人生，直到撞上奇点——那才是致命的。

If I took an even bigger black hole than that, you could live out your entire life having crossed the event horizon with a black hole before you hit the singularity, which is fatal.

一旦你穿过事件视界，你就注定无法回头了。

When you cross the event horizon, you are doomed.

你注定无法回头，因为穿过事件视界后，你只能朝着奇点前进。

You are doomed because once you cross the event horizon, you must proceed to the singularity.

不管你用火箭怎么推进，都无法阻止自己撞上奇点。

There's no way you can fire a rocket to stop yourself, hitting the singularity.

你注定难逃一劫，但还活着。

You are doomed, but you are not dead.

你只有在撞上奇点、被潮汐力拉成意大利面、彻底碾碎的时候，才算真正死了。

You are only for sure dead once you hit the singularity and get spaghettified, get mangled by the by the tidal forces.

但对于足够大的黑洞，你即使已经注定灭亡，也可能完全察觉不到。

But for a large enough black hole, you can be doomed and not even know it.

事件视界其实不是一个局部能测量出来的概念。

The event horizon is is really a not locally measurable quantity.

它是一个具有目的性的事实。

It is a teleological fact.

它意味着一旦你跨过了事件视界，就只能朝奇点去，不过对于足够大的黑洞，这段路可能要花上很久。

It says that once you have crossed the event horizon, you must proceed to the singularity, but it can take a long time to get there for a large enough black hole.

理论上，如果你面对的是一个横跨数百光年的黑洞，你完全可以寿终正寝。

You could, in principle, for a large enough black hole, live out your entire life if you had a black hole that was many light centuries across.

你的一生完全可以正常度过。

You could live out your life.

你甚至能有后代，而且他们一生都生活在黑洞内部。

You could have descendants, all of whom live in inside the black hole.

只有当你真正接近奇点时，潮汐力才会变得足够强大，把你撕碎。

And only once you really approach the singularity do the tidal forces get strong and kill you.

正如你所说，广义相对论解释或预测了很多现象，其中一些我们认为是正确的，还有一些我们还不确定。

As you were saying, GR explains or predicts a lot of phenomenon, and some we think are correct, some we don't know are correct.

那为什么我们觉得黑洞是正确的，却不相信虫洞呢？

Why why do we think black holes are correct, but not wormholes?

是的。

Yeah.

这是个很好的问题，当初根本没人相信这回事。

That's a great question, and people did not believe it to begin with.

施瓦西在爱因斯坦提出场方程后不久，就推导出了他的解。

Schwarzschild wrote down his solution almost immediately after Einstein wrote his field equations.

当时人们觉得这个方程有点不对劲。

People thought that that equation was sick in some way.

他们认为这只是一种概率为零的极端情况，根本不可能发生。

It was a measure zero thing that would never happen.

那只是个数学上的怪胎，人们觉得在现实宇宙中不可能自然形成黑洞。

It was some mathematical monstrosity, but it was impossible to make black holes naturally in the real universe.

但他们错了，因为黑洞确实存在。

And they were wrong because black holes do exist.

我们现在非常确信。

We're extremely confident now.

既有理论上的突破，也有实验证据证明黑洞的存在。

There was theoretical developments, there was experimental evidence that black holes exist.

最大的理论突破来自彭罗斯，后来霍金和彭罗斯进一步发展了这一理论，彭罗斯也因此获得了诺贝尔奖，他们从理论上证明了黑洞的形成是广义相对论的普遍特征。

The biggest theoretical development was Penrose, and then later Hawking and Penrose, for which which he won the Nobel Prize, who showed theoretically that the formation of black holes is a generic feature of general relativity.

这并不是只有在精确调整初始条件时才会发生的异常现象，只要初始条件是普通的，黑洞的形成就是必然结果。

It's not just some sick thing that happens if you fine tune the initial conditions, that if you just start off with generic initial conditions, the development of black holes is a generic feature of that.

这可是个重大的突破。

That was a huge development.

然后还有实验方面的证据。

And then there was the experimental side.

我们现在拥有的关于黑洞的实验证据已经非常多了。

And the pieces of experimental evidence we have for black holes is now huge.

爱因斯坦那个年代根本没有任何这类证据，甚至在他之后的五十年里，人们还对黑洞充满困惑，觉得它们根本不存在。

It did not exist in Einstein's day, and for fifty years after Einstein, people were extremely confused about black holes and thought they didn't exist.

但现在已经有很多确凿的证据了。

But there's numerous pieces of evidence.

我觉得最直观的证据，就是观测我们银河系的中心。

I think the most visually appealing piece of evidence is just observing the center of our galaxy.

如果你看银河系中心——剧透一下，那里确实有个黑洞。

So if you look at the center of the galaxy, is I mean, spoiler alert, there is a black hole there.

我们把它叫做人马座A*。

We call it Sagittarius a star.

那是一个质量高达太阳好几百万倍的超大黑洞。

A huge black hole weighs many millions of times the mass of the sun.

你没法直接看到黑洞，毕竟它是黑的嘛。

You can't see the black hole, or at least you can't see it, you know, directly, because it's black.

你能看到的是围绕它的那些恒星。

What you can see is the stars around it.

如果你观察这些恒星几十年，而我们现在确实已经有了几十年的观测数据，就会发现它们的运动轨迹并不是直线，而是沿着漂亮的椭圆轨道运行，甚至有些是进动的椭圆。

If you watch these stars over the course of decades, and we now have a number of decades of observations of them, you will see the stars not moving along what we would call straight lines, instead moving in in nice little ellipses or or precessing ellipses.

这些椭圆轨道看起来像是在围绕某个东西旋转。

And those ellipses look like they are orbiting something.

你看不到那个东西本身，但你能看到围绕它运转的恒星。

You cannot see the something, but you can see the stars that are orbiting that something.

你可以计算出它的大小和质量，结果发现它质量极大，体积却极小。

And you can calculate how big is it, how massive is it, and what you find is that it's very massive indeed, and it's also very small indeed.

你知道它体积小，是因为恒星能靠得非常近，却似乎从未与它相撞。

You know it's small because the stars get super close to it but don't seem to collide with it.

通过追踪这些轨道，你可以推断出银河系中心存在一个质量极大、极其黑暗、又极度致密的天体。

And so by tracing these orbits, you can tell that there is something super heavy, super dark, and super compact at the center of that galaxy.

那就是人马座A*，我们银河系中心的黑洞。

That is Sagittarius A star, the black hole in the center of our galaxy.

这是其中一个非常有力的证据。

That's one compelling piece of evidence.

大约十年前，我们又有了另一个有力的证据——我们不仅看到了黑洞，还感受到了它们。

Another piece of compelling evidence was about a decade ago, we not only saw black holes, we felt them.

LIGO是一个巨大的激光干涉仪，我们在多个地点建造了它，对时空本身的微小震动极其敏感。

So LIGO is this huge interferometer, laser interferometer, that we built at a number of different sites that is super attuned to vibrations in spacetime itself.

就在我们2015年底刚启动它后不久，就发生了一件著名的事：我们感受到了时空的震动。

And there's a famous event pretty much immediately after we turned it on in late twenty fifteen where we felt spacetime shaking.

你知道这确实是时空在震动，而不仅仅是地球在晃动，因为我们当时已经有多个探测器，先是两个，现在有四个，分布在地球的不同位置，它们的反应完全一致。

And you knew it was spacetime shaking, not just the Earth shaking, because we had a bunch of these detectors, then two, now now four, at different points on the Earth, and they all shaped in exactly the same way.

这不可能是某处有卡车经过或发生地震，而其他地方却没有影响造成的。

So it couldn't just be explained by a truck passing one and and not the other or a seismic event on one and not the other.

所有探测器的震动模式都完全吻合，我们反向推算出，引起这种震动的是两个超大质量黑洞的碰撞——两个黑洞，每个质量都约为太阳的30倍，位于宇宙另一端，距离我们约16亿光年。

They all shook in exactly the same way, and we were able to back calculate that the thing causing them to shake was the collision of two ginormous black holes, two black holes, both of which made weight about 30 times as much as the sun on the other side of the universe, about 1,600,000,000 light years away.

这场碰撞发生在大约16亿年前，却刚好在我们启动LIGO探测器后的几周内，信号传到了地球。

That collision happened about one point six billion years ago and just happened to reach the Earth within weeks of us turning on the LEGO detectors.

我们现在不仅感受到了，更准确地说，已经观测到成千上万次这样的时空扰动，每一次都对应着一次黑洞合并。

We've now seen we've now felt more accurately thousands of such shakings corresponding to thousands of black hole mergers.

还有更多的证据。

And then there's more evidence.

后来，我们建成了事件视界望远镜，这是由遍布全球的众多射电望远镜组成的庞大网络，能够极其清晰地观测银河系中心的人马座A*黑洞，以及邻近星系中心那个更大的黑洞，捕捉到物质坠入黑洞时发出的微弱射电信号——这些物质在坠入过程中会剧烈发光，而我们正是通过这些射电信号看到了这一切。

There is a, later, we had what's called the Event Horizon Telescope, which is a ginormous conglomeration of radio telescopes all over the Earth that were able to look very closely at the black hole at the center of our galaxy, Sagittarius a star, and the black the even bigger black hole at the center of the our neighboring galaxy and see, very, very faintly, the radio emissions of matter falling into these black holes that shine super brightly as it does so, and we were able to see that in the radio emissions of these things.

所以我们感受到了它们。

So we felt them.

我们也亲眼看到了它们。

We've seen them.

我们还观测到了黑洞对周围恒星轨道产生的引力影响。

We've seen their gravitational effects on orbiting stars.

到目前为止，我们对黑洞的存在已经非常确信了。

We're extremely confident at this stage that black holes exist.

这太美妙了，不仅有人能提出这样的理论，而且这个理论的解释力如此广泛。

It's so beautiful that not only can a single mind come up with this theory, but the theory has so much reach.

而且我们还能设计出各种复杂的工具，去检验、推演并理解它在这么多不同领域中的深远影响。

And that we can come up with the machinery to evaluate and perturb and understand its implications in so many different wild ways.

从自由度的数量，到涵盖的量级范围，

It's crazy the number of degrees of freedom, the number of orders of magnitude that

简直太广了。

it covers.

你最初思考它的时候，可能只是从电梯里上下跳动的思想实验开始，接着它延伸到解释水星的轨道，探测太阳系内天体运动的微小扰动，再到光线的弯曲。

It covers you you first start thinking about it by doing thought experiments to do with jumping up and down in elevators, and then it reaches out to describe the orbit of Mercury and detectable perturbations of orbital dynamics within the solar system and then the bending of light.

然后它又解释了星系的旋转，再进一步描述了整个宇宙的膨胀以及可能的结局。

And then it describes the rotation of the galaxy, and then it describes the expansion and potential fate of the entire universe.

这确实跨越了多个数量级，而这一切竟然几乎出自一个人的头脑，实在令人惊叹。

That's many orders of magnitude indeed, and it's pretty impressive that it was the work of almost a single mind.

坦白说，我们的宇宙真该为能被如此优美的理论所描述而感到荣幸。

Frankly, our universe should be honored to be described by such a beautiful theory.

你能讲讲广义相对论是怎么从爱因斯坦的一个理论，变成全世界都接受的真理的吗？

Can you tell the story of how GR went from a theory that Einstein had to something that the world came to believe is true?

哦，对，那就是光线的弯曲。

Oh, yeah, that would be the bending of light.

其实，在那之前，牛顿物理就已经存在一些无法解释的异常现象。

So, there were known anomalies with Newton's physics beforehand.

我们一直没法精确算出水星的轨道。

We couldn't get the orbit of Mercury exactly right.

而广义相对论早期一个很漂亮的验证，就是它准确算出了水星的轨道。

And one of the very nice early tests of general relativity is that it did get the orbit of Mercury exactly right.

这算是一个相当有力的证实。

So that was a pretty good confirmation.

但同时，这也没那么让人震撼，因为你先有了一个理论，再去解释一个已经知道的数据。

But at the same time, that's not quite so satisfying because it was a number that's already known that you invent a theory, and then it correctly predicts.

真正更了不起的是，在还不知道答案的情况下，你的理论却预测对了。

It's considered more impressive if you get the right answer without knowing what the right answer is in advance.

所以，光线弯曲才是更令人信服的例子。

And so that would be the bending light.

从历史上看，这无疑是最具影响力的。

Certainly, historically, that was the most influential.

根据广义相对论，所有形式的能量都会产生引力，也会受到引力的影响。

So according to general relativity, all energy gravitates, and all energy is affected by gravity.

所以当光经过太阳这样的大质量天体时，会朝太阳的方向发生偏折。

And so light, as it's passing a massive object like the sun, will will get bent in the direction of the Sun.

其实，牛顿力学里也会出现同样的情况。

Actually, the same will happen in Newtonian physics.

如果你假设有一个粒子飞过太阳，它的偏折程度取决于它与太阳的接近距离，还有它的速度。

If you just say, suppose you have a particle going along, and you know how much it gets bent as it passes the Sun, depending on its impact parameter, but also its velocity.

速度越快，偏折就越小。

And the faster it's going, the less it gets bent.

你只要把牛顿的公式拿过来，把速度设成光速，算一下结果就行。

So you just take that Newtonian formula and you plug in velocity equal to speed of light and see what the answer you get at.

用牛顿力学算出来的偏折角度是某个特定值。

You get a certain amount of bending through Newtonian physics.

而根据广义相对论，做同样的计算，结果会是牛顿力学预测的两倍。

And general relativity, you can do the same calculation, and you actually get double the Newtonian answer.

所以这个历史过程其实有点特别。

So this was a slightly strange history of it.

在爱因斯坦还没完全写完广义相对论的时候，他就已经根据等效原理，结合自己对它的理解，提前预测了这个数值。

Before he had finished writing down general relativity, Einstein had a prediction based on what he thought was the equivalence principle, based on his understanding of the equivalence principle for what this answer should be.

于是他写下了这个预测值，随后因为他的观点引发了不少人的兴趣，陆续有团队组织远征，试图实际测量这一现象。

And so he wrote down the answer, and then in response to him and a number of other people being interested in this, there were people sending out expeditions to go and try and measure it.

其实，我印象中牛顿最早做的第一件事，就是打电话给天文台，问：你们能不能观测太阳背后遥远的恒星，测量光线经过太阳时的偏折？

Actually, I think the very first thing that Newton did is he phoned up the observatory and said, can you measure look at distant stars behind the Sun and measure how light bends as it passes the Sun?

这完全是理论物理学家的典型做法——我觉得威尔逊山天文台的台长当时直接回绝了：不可能。

And this was, you know, this was the true theorist move because the, I think the director of the Mount Wilson Observatory said, like, absolutely.

我们做不到。

We no.

我们根本办不到。

We cannot do that.

如果你把望远镜对准太阳，你就会失明。

If you point a telescope at the sun, you'll go blind.

如果你把望远镜对准太阳旁边，会被日冕的强光淹没，什么都看不到。

If you point it just next to the sun, you'll just get washed out by the corona of the sun, you won't see anything.

但有一个时候不会被太阳光淹没，那就是日全食的时候，月亮挡住了太阳，你就能看到紧贴太阳边缘的恒星，测量它们背后的光线弯曲情况。

Except there's one time when you won't get washed out by the sun, and that's during a total solar eclipse, when the moon blocks the sun and you're able to see stars very close to the sun and measure the bending of the light behind them.

所以在20世纪10年代，人们派出了很多考察队去测量光线偏折。

So during the 1910s, there was a whole bunch of expeditions sent to measure the deflection of light.

他们跑到日全食的路径上，用望远镜观测太阳旁边的恒星，看它们在天空中的位置有没有移动，以及移动了多少。

They'd go park out in the path of totality and look through telescopes at the stars right next to the sun and see if they moved in the sky, and if they moved, how much they moved.

我觉得第一次尝试是在1911年。

And I think the first one that they did was in 1911.

他们去了阿根廷观测日食，所有设备都准备好了。

They went to, Argentina for an eclipse, and then everything's set up.

这事儿的问题就在这儿。

I mean, this is the problem with this thing.