理解广义相对论：爱因斯坦最美丽的思想

我请回了亚当·布朗。你目前领导谷歌 DeepMind 的 Blue Shift 团队，致力于攻克科学与推理难题。在之前的职业生涯中，亚当是一位多产的物理学家，曾在斯坦福任教，研究范围涵盖宇宙学、弦理论、广义相对论等。据说广义相对论是人类心智所构想或见过的最美之物。我很好奇，像我们这样的普通人，有没有办法理解它到底在讲什么，或者体会到它美在哪里，而不必去上你那 20 讲的博士课程。这就是这次讲座的由来，也感谢你愿意来做这次分享。

I'm back with Adam Brown. You currently lead Blue Shift at Google DeepMind, which is cracking science and reasoning. In a previous life, Adam was a prolific physicist, taught at Stanford, and did research on everything from cosmology to string theory to general relativity. It's said that general relativity is the most beautiful thing the human mind has ever conceived or seen. And I was curious if there's a way that ordinary people like me could understand what is happening or have some advantage on why it's beautiful without taking your 20-lecture graduate course. So, that was the prompt for this lecture, and I appreciate you being willing to do it.

能来这里我非常兴奋，而且，我觉得答案是肯定的。是的，我们可以。所以，我是说，广义相对论，爱因斯坦的引力理论，正如你所说，我认为它是人类单一心智所创造出的最美产物。它是 20 世纪物理学的两大理论之一，另一个是量子力学。与量子力学不同，它基本上就是爱因斯坦一个人的功劳。他得到过一点帮助，但基本上是一个人执着地追求这个想法长达 10 年，然后写下了这个理论，最终描述了太阳系行星的运动，也描述了宇宙的起源和命运。这非常非凡。嗯，它花了爱因斯坦——历史上最著名的头脑之一——大约十年的时间才想明白。但是，你知道，当我教这门课的时候，我会用 10 周的时间，所以在 10 周内，人们对广义相对论的理解会比爱因斯坦在 10 年里真正拥有的更好。这在一定程度上是因为我们拥有爱因斯坦所没有的优势，那就是我们有爱因斯坦和许多像他一样的前人，他们能够把这些极其复杂的想法——在当时被认为任何智商低于爱因斯坦的人都完全无法理解的想法——提炼成精华。嗯，而且，你知道，不会重蹈我们前人的许多错误。我认为，在 10 到 20 分钟里，我无法让你对广义相对论的理解超过爱因斯坦，但我们可以触及核心洞见。也就是爱因斯坦所说的他最美丽的想法。并深入其中，试图理解这个理论的中心思想。

Super exciting to be here, and yes, I think the answer is yes. Yes, we can. Yeah, so, I mean, general relativity, Einstein's theory of gravity, is as I think as you said, like, the most beautiful product of a single mind that we've ever created. It's one of the two great theories of 20th century physics along with quantum mechanics. And unlike quantum mechanics, it was basically Einstein. He had a little help, but basically one person doggedly pursuing this idea for 10 years, and then wrote down this theory that ends up describing the motion of planets in the solar system, and also the origin and fate of the universe. And it's pretty extraordinary. Um, and it took Einstein, one of the most famous minds in history, about a decade to figure it out. But, you know, when I teach it, I'll do a 10-week course, and so in 10 weeks, people will get a better idea of general relativity than Einstein really had in 10 years. And that's kind of because we have an advantage that Einstein didn't have, which is that we have Einstein and many others like him going before us who were able to take these super complicated ideas that were understood at the time as being totally incomprehensible by anybody with a sub-Einstein level of intelligence, and boil them down to their essentials. Um, and, you know, not make many of the same mistakes that were made by our forebears. I think in, you know, 10, 20 minutes, I can't give you a better idea of general relativity than Einstein had, but we can get to the core insight. What Einstein said was his most beautiful idea. And push through it to try and understand what the central idea of this theory is.

好，我们开始吧。

Okay, let's go.

在广义相对论之前，有狭义相对论。所谓“狭义”，意思是它并不适用于所有情况。那也是在 10 年前，即 1905 年，由爱因斯坦在他的奇迹年中发明的。你知道，如果你想用一句话概括狭义相对论，你会从“没有任何东西能比光速更快”这个观察或假设开始。狭义相对论把这个观察提升为一个原理，极其认真地对待这个原理，把它作为我们理解时空的核心观察，于是你就得到了狭义相对论。狭义相对论适用于电磁学。它也适用于强核力和弱核力，适用于我们所知道的其他基本力——尽管爱因斯坦当时甚至不知道这些力。它显然不适用于引力。所以，10 年后，爱因斯坦在他的广义相对论中纠正了这一点，这个理论更“广义”，因为它包含了引力，补全了基本力的集合。同样，这是爱因斯坦在 1915 年经过 10 年执着追求后发明的。如果你想用一句话概括广义相对论，你可能会说“连引力也不例外”。没有什么能比光速更快，连引力也不例外。它的内涵远不止于此，但它将完成这个“没有任何东西能超过光速”的核心弧线。

Before general relativity, there was special relativity. So, special meaning it doesn't apply everywhere. That was also invented by Einstein 10 years earlier, in 1905, during his annus mirabilis. And you know, if you want to sloganize special relativity, you would start with the observation or the hypothesis that nothing can go faster than light. Special relativity takes that observation, promotes it to a principle, takes that principle extremely seriously as the sort of central observation of our understanding of space-time, and you arrive at special relativity. Special relativity applies to electromagnetism. It applies, though Einstein didn't even know about these at the time, it applies straightforwardly to the strong and weak nuclear forces, to the other fundamental forces that we know about. It does not, obviously, apply to gravity. And so, that was corrected 10 years later by Einstein in his general theory of relativity, a theory more general because it includes gravity, completes the set of fundamental forces. Again, invented by Einstein after 10 years of dogged pursuit in 1915. If you wanted to sloganize general relativity, you might say not even gravity. Nothing can go faster than light, not even gravity. There's much more to it than that, but it's going to complete this arc of the centrality of nothing being able to go faster than the speed of light.

好的，为了了解一些背景，我们得往回倒。我们得一直倒回到爱因斯坦之前存在的引力理论。爱因斯坦时代的引力主流理论可以一直追溯到 17 世纪末的牛顿。所以我们来谈谈这个。牛顿在 1687 年的《自然哲学的数学原理》中提出了牛顿定律。牛顿定律，嗯，他有好几条。你知道，我们今天最可能讨论的是其中两条，一条是他著名的第二定律，它说力引起的加速度 A 由公式 MA 等于 F 给出。如果你有一个力 F，它会对物体产生一个加速度 A，其中质量告诉你物体抵抗加速的程度。质量越大，产生给定加速度所需的力就越大。这条定律，他的第二定律，在我们进入广义相对论后仍然成立。我们将不得不对力和加速度的含义有更复杂的理解，但这条定律会被广义相对论保留。第二定律的一个特例是牛顿第一定律。牛顿第一定律说，如果力为零，那么加速度为零。如果力为零，那么物体将一直沿直线运动。这在广义相对论中也将继续成立，即如果不受到外力，物体沿直线运动。然而，我们将不得不升级我们对“力”和“直线”的理解。好的，这将继续成立。不再成立的是牛顿的万有引力定律。所以，牛顿引力。这告诉你加速度如何响应力，但你需要知道力是什么才能用它做任何事。牛顿的万有引力定律说，两个物体引力相互作用产生的力是，嗯，所谓的牛顿常数，只是某个自然常数，乘以一个物体的质量乘以另一个物体的质量，太阳的质量乘以地球的质量，除以它们之间距离的平方。这就是著名的平方反比定律。它指向一个矢量，指向分离的方向，并且是吸引的。所以，那里有一个负号。这在广义相对论中不成立。事实上，你立刻就能看到这个引力定律与“没有任何东西能比光速更快”的主张之间存在紧张关系。如果这字面上是真的，那么通过晃动太阳，对这个定律的直接解释会说地球上的力立即变化。我改变了地球和太阳的距离，所以我在地球上能立即检测到，不是 8 分钟后，而是立即。所以，那将意味着你可以发送一个比光速更快的影响。牛顿的力定律与这个原理是不一致的。

Okay, so to see some background here, we're going to have to rewind. And we're going to have to rewind all the way back to the theory of gravity that existed before Einstein. The reigning theory of gravity at the time of Einstein stretches all the way back to Newton in the late 17th century. So let's talk about that. So Newton's laws in his Principia in 1687. Newton's laws. Well, he had a few. And you know, maybe the one that we could most talk about today is two of them, which is his famous second law that says the acceleration A caused by a force is given by the formula MA equals F. That if you have a force F, it'll cause an acceleration on an object given by A where the mass tells you how much an object resists being accelerated. The bigger the mass, the bigger the force you need to cause a given acceleration. And this law, his second law, will turn out to be still true once we come to general relativity. We'll have to have a more sophisticated understanding of what we mean by force and acceleration, but this will be preserved by general relativity. A special case of the second law is Newton's first law. Newton's first law says that if the force is zero then the acceleration is zero. If the force is zero, then objects continue to move on a straight line at all times. And that will also continue to be true in general relativity, that if not subject to an external force, objects move along straight lines. However, we'll have to upgrade our understanding of what we mean by force and what we mean indeed by straight line. Okay, that's going to keep being true. The one that's not going to keep being true is Newton's law of gravity. So, Newtonian gravity. This tells you what the acceleration is in response to a force, but you need to know what the force is to be able to do anything with that. And Newton's law of gravity says that the force caused by the gravitational interaction of two bodies is, well, what's called Newton's constant, just some constant of nature times the mass of one body times the mass of the other body, the mass of the Sun times the mass of the Earth divided by the distance between them squared. It's famous inverse square law. And it points to a vector that points in the direction of separation and it's attractive. So, there's a minus sign there. This will not be true in general relativity. And in fact, you immediately see that there's a tension between this gravitational force law and the claim that nothing can go faster than the speed of light. If this was literally true then by jiggling the Sun, a straightforward interpretation of this law would just say that the force at the Earth varies immediately. I've changed the distance of the Earth and the Sun and so, I can immediately detect it at the Earth, not eight minutes later, but just immediately. So, that would imply that you could send an influence faster than the speed of light. It is inconsistent, Newton's force law with this principle.

当然，一种可能是，这对非引力作用力成立，但一旦涉及引力就不成立了。事实上，利用引力，你或许能造出一种超光速电话。这是一种可能性，但爱因斯坦并不想接受这种可能性。他花了很多年去排除任何超光速或超光速影响的可能。所以爱因斯坦，以及当时很多人，都认为这个（光速不变原理）必须让步，而事实上，这确实就是最终的结果。

One option, of course, could be that this is true for non-gravitational forces, but not once you have gravity. And indeed, using gravity, you could perhaps build a faster-than-light telephone using gravitational effects. That's a possibility, but not a possibility that Einstein really wanted to embrace. He'd spent many years chasing out any possibility of going faster than light or any superluminal influences. So Einstein, and in fact many people at the time, thought that this is the one that has to give, and indeed that is in fact what's going to turn out to be true.

那么，我们进展到哪了？实际上，这里有一个先例，即平方反比定律被修正，最终与狭义相对论相容。这个先例就是另一种自然力——电力。所以，还有静电力的定律，不是牛顿写下的，而是大约一个世纪后写下的，它说的是，由两个带电物体的静电相互作用（而非引力相互作用）产生的力，与引力有非常相似的形式。它告诉你，力等于某个常数乘以一个物体的电荷再乘以另一个物体的电荷，方向也指向两物体之间的连线，除以距离的平方——又是一个平方反比定律。

So, where are we? There's actually a precedent here for an inverse square law getting modified in such a way that it ends up being consistent with special relativity. And that precedent is the other force of nature, the electric force. So, there's also the electrostatic force law, not written down by Newton, but written down a century or so later, which says that the force caused not by the gravitational interaction of two objects, but by the electrostatic interaction of two charged objects, has a very similar form to the gravitational force. It tells you that the force is equal to some constant times the charge of one object times the charge of the other object, pointing also in the direction of separation between the two objects, divided by the distance squared—another inverse square law.

而且，出于完全相同的原因，静电学看起来与狭义相对论不一致。但最终并非如此，或者说，这并非故事的全貌。静电学只是电磁学真正理论（即麦克斯韦定律）的一个极限，它不仅包含电力，还包含磁力。而电力只有在一切静止时才看起来完全如此。当物体开始运动时，会出现额外的修正，所有这些修正共同作用，使静电力定律与狭义相对论完全相容。

And again, for exactly the same reason, electrostatics looks to be inconsistent with special relativity. But ultimately it's not, or ultimately this is not the full story. Electrostatics is just one limit of the true theory of electromagnetism, which is Maxwell's laws, that has not just electric forces, it also has magnetic forces. And the electric forces only look exactly like this when nothing is moving. When things do start to move, there are additional corrections to this, all of which conspire to make the electrostatic force law fully consistent with special relativity.

事实上，历史的理解方向是相反的。首先，麦克斯韦在 19 世纪中叶写下了麦克斯韦方程组。直到后来人们才注意到：“嘿，麦克斯韦方程组实际上完全符合‘没有任何东西能超过光速’这一点。”而这种一致性体现在一种对称性上，即麦克斯韦场方程的洛伦兹对称性，这是在它们被写下之后才被注意到的，最终引导爱因斯坦提出了他的狭义相对论。

In fact, the historical direction of understanding ran the opposite way. First of all, you have Maxwell in the middle of the 19th century writing down Maxwell's equations. And only later do people notice, "Hey, Maxwell's equations actually are sort of fully consistent with nothing can go any faster than the speed of light." And that consistency is reflected in a symmetry called the Lorentz symmetry of the Maxwell field equations, only noticed later after they were written down, that eventually led Einstein to formulate his special theory of relativity.

所以，我们有一个先例，从一个平方反比定律开始，然后把它包装成一个完整的相对论不变理论。因此，你可能会说：“好吧，我们就把引力拿来，对引力做和静电学完全一样的事情，以构建某种引力磁理论，使牛顿第二定律成为一个最终与狭义相对论相容的近似。”从某种宏观意义上说，这正是我们最终要做的。这正是爱因斯坦最终要做的。但这将比麦克斯韦对静电学的推广更为激进。

So, we have a precedent for starting with an inverse square law, and then dressing it up in a full relativistically invariant theory. And so, you might say, "Well, let's just take gravity and do exactly the same thing to gravity that we did to electrostatics in order to make some gravitomagnetic theory that makes Newton's second law an approximation that's ultimately consistent with special relativity." And in some grand sense, that is what we're going to end up doing. That is what Einstein's going to end up doing. But it's going to be a much more radical departure than the Maxwell generalization of electrostatics.

而且这里有两个提示，都体现在这个公式中，表明我们将不得不做一些与静电学略有不同的事情。首先要注意的是，静电定律和牛顿引力定律之间的第一个区别是符号差异。有一个很大的区别，即这里是减号，这里是加号。这反映在，如果你有两个正质量，比如地球和太阳，它们会相互吸引。相反，如果你有两个同种电荷，它们会静电排斥。这就是为什么那里是减号，这里是加号。

And there are really two hints, both of which are visible in this formula, that we're going to have to do something slightly different than we did for electrostatics. The first thing to notice, the first difference between the electrostatic law and Newton's law of gravity, is this sign difference. There is a big difference, which is that here it is a minus sign, and here it is a plus sign. That is reflected in the fact that if you have two positive masses, you know, the Earth and the Sun, they gravitationally attract each other. Conversely, if you have two like charges, they electrostatically repel each other. Which is why that's a minus sign and that's a plus sign.

这意味着你不能对引力做和电磁学完全相同的事情，因为否则，如果你在数学上使用同样的技巧，你会得到数学上相同的结果，即你会发现同号质量会排斥而不是吸引。不过我们不要超前，但最终，这是因为静电学由自旋为 1 的粒子（光子）介导，而引力将由自旋为 2 的粒子介导。这就是导致符号变化的原因。

That means that you cannot do literally the same thing for gravity that you did for electromagnetism, because otherwise, if you did mathematically the same trick, you'd end up with mathematically the same result, which is that you would find that like masses would repel rather than attract. Not to get ahead of ourselves, but ultimately, that's because electrostatics is mediated by a spin-1 particle, the photon, and gravity is going to be mediated by a spin-2 particle. And that's responsible for the change in that sign.

好了，这就是为什么你不能完全照搬静电学的方法。所以，爱因斯坦必须寻找其他方法。他必须寻找其他途径，试图将这一理论提升为相对论不变的理论。在这样做时，他有一个线索。你知道，有很多事情在发生。爱因斯坦的核心天才之一，就是专注于这个线索，将其视为他应该寻找方向的高度重要线索。有时这被描述为他最美好的思想。他就是这样描述的。

Okay, so that's why you can't do exactly the same thing as electrostatics. And so, Einstein had to look for something else. He had to look for some other way to try and lift this to a relativistically invariant theory. And in doing that, he had one clue. And you know, there's lots of stuff going on. It's part of Einstein's central genius to focus on this as a highly significant clue of where he should look. Sometimes described as his most beautiful thought. That is how he would describe it.

线索是这样的。引力定律和静电学之间还有另一个区别。那就是这个扮演电荷角色的对象，你知道，在静电学和引力中类似于电荷的东西。事实是，这里坐着的就是质量。这就像是牛顿物理学中的一个奇怪巧合。所以，在静电学中，质量和力、加速度的关系中，质量只扮演一个角色。它就在这里。它是物体的惯性，是抵抗加速度的东西。这有时被称为惯性质量，就在这里。

And the clue is this. There is another difference between the gravitational force law and the electrostatics. And that is this object that plays the role of the charge, you know, the analog of the charge in electrostatics and gravity. And it's the fact that it's the mass sitting here. And that's like a strange coincidence from Newtonian physics. So, mass in electrostatics, forces and accelerations, plays exactly one role. It's sitting here. It's the inertia of the object and it's what is resisting being accelerated. This is sometimes called the inertial mass that's sitting here.

而电荷则完全不同，与质量无关。你可以有重而无电荷的物体，比如中子。你可以有轻而高电荷的物体，比如电子。粒子的电荷与其质量之间没有必然联系。它们只是两个完全独立的东西。但在引力中并非如此。在引力中，牛顿第二定律中的这个质量，即抵抗力的惯性质量，恰好等于牛顿引力定律中的这个质量，它告诉你你被拉动的程度。这是同一个质量。

And then the charge is completely different and unrelated to the mass. You can have heavy objects that have no charge like the neutron. You can have light objects like the electron that have high charge. There is no necessary relation between the charge of a particle and its mass. They're just two entirely separate things. Not true in gravity. In gravity, this mass that's sitting here in Newton's second law, the inertial mass that's resisting the force, is exactly equal to the mass that's sitting here in Newton's gravitational law that's telling you how much you're pulled along. It's the same mass.

这有时被称为引力质量。这有时被称为惯性质量。所以，与静电学不同，这个公式中出现的引力质量等于有时被称为惯性质量的东西，它就在这个公式中。这个等式在牛顿物理学中已经成立。事实上，牛顿注意到了这一点，并做了许多实验来确认这一点，精确到大约千分之一。到爱因斯坦时代，我们知道它精确到十亿分之一。而现在我们知道它精确到 10 的 15 次方分之一。

This is sometimes called the gravitational mass. This is sometimes called the inertial mass. And so, unlike in electrostatics, the gravitational mass that appears in this formula is equal to what's sometimes called the inertial mass that sits in this formula. This equation is already true in Newtonian physics. Newton noticed it in fact and did a number of experiments to confirm that this was true to you know one part in a thousand or so. By the time of Einstein, we knew it was true to one part in a billion. And now we know it's true to one part in 10 to the 15.

你知道，这两者在牛顿物理学中本质上完全是巧合，但它们被观测到是完全相同的。爱因斯坦正是抓住了这一点，这是他接下来行动的核心线索。这有时被称为等效原理，它解释了为什么如果你在真空室里拿一根羽毛和一块砖头同时放下，它们会同时落地。它们会同时落地，因为尽管砖头受到的力比羽毛大得多（因为它更重），但这恰好抵消了砖头对力的抵抗比羽毛对加速度的抵抗更大的事实，所以它们下落的速度完全相同。因此，这两者的相等性导致了那种精确的相等。所以，爱因斯坦的天才之处在于把这一点作为他最终取代牛顿定律的核心线索。而它之所以是核心线索，是因为实际上还有另一类力，不是像电磁力或引力那样的基本力，而是一组涌现的力，它们天生就具有这种性质——在这些理论中，这是必然的。为了解释这一点，我们现在要进入这次讨论的实验部分。

You know, it's striking that these two, that in Newtonian physics it's just a complete coincidence essentially that those two are the same thing. Nevertheless, they were observed to be exactly the same thing. And this was Einstein's—he honed in on this fact and it was his central clue for what to do next. This is sometimes called the equivalence principle, and it's responsible for the fact that if you take a feather and a brick in a vacuum chamber and drop them both, they will both fall and hit the ground at the same time. They'll fall and hit the ground at the same time because even though the force on the brick is much stronger than the force on the feather because it's heavier, that exactly cancels out the fact that the resistance to force on the brick is larger than the resistance to acceleration of the feather, and they exactly fall at the same rate. So the equality of those two is responsible for that exact equality. So Einstein's genius was to hone in on this as a central clue for how he is going to end up replacing Newton's law. And the reason it's a central clue is because there is in fact another class of forces, not fundamental forces like electromagnetism or gravity, but a set of emergent forces that exactly have this property baked into them—that they must, that's guaranteed in those theories. And to explain that, we're now going to move over to the experimental section of this discussion.

你最好懂点物理，亚当。否则我就把演播室砸了。

You better know your physics, Adam. Otherwise I'll destroy the studio.

呃，没有花招。你愿意把手指伸进去确认它是湿的吗？好的。你看，这是水桶。在底部，水为什么不掉出来毫无悬念。它不掉出来是因为我们理解的引力是指向桶底的。但现在我们要做的是稍微加速，然后做圆周运动。现在有一个你可能觉得表面上看很令人惊讶的现象，那就是即使桶倒过来，水也不会掉出来。有两种理解方式。一种就是直接的方式，你会说，你知道，当水在弧线顶端时，它想从桶里掉出来。但当它准备好加速到足以从桶里掉出来的时候，桶已经移动了，现在在下面了，它根本没有时间掉出来，你知道，或者你可以说这和宇航员不会掉到地球上是同一个原因。还有第二种视角。第二种视角同样有效，就是想象你和水一起在桶里运动。从这个角度看，对于水为什么不会掉出来，有另一种解释。那就是离心力，从随桶一起运动的人的角度看，有一个力把他们推向桶底。这个力就是所谓的离心力。它被称为假想力或惯性力。离心力只是说，你知道，由于处于旋转参考系中，会产生一个力，大小等于你的速度除以你绕圈半径，方向指向外，正向外。所以这就是离心力，它把你压在桶底，或者当你转弯时把你压在汽车外侧。好的，我们注意到什么？我们注意到，你受到离心力的作用，如果你愿意这么说的话，你感受到离心力的强度，再次像引力一样，但不同于静电学，是由你的质量决定的。你知道，告诉你获得多少离心力的那个质量，是由你的惯性质量决定的。但当然，这里完全没有神秘之处，为什么右边的这个质量是由你的惯性质量决定的。它由你的惯性质量决定，恰恰是因为这种倾向——你感受到这个力的原因恰恰是质量倾向于沿直线运动的倾向，而你没有沿直线运动，你在做圆周运动，正是这种惯性倾向最初导致了质量。换句话说，任何时候你遇到这种仅仅由惯性引起的惯性力，可以保证这个力的电荷是由惯性质量决定的。所以，惯性力的电荷总是由惯性质量决定。引力的电荷——引力的电荷是由惯性质量决定的。所以，爱因斯坦跳了起来，会不会是这样，这是他的核心思想，会不会引力本身就是一种惯性力？这是允许的，因为引力质量等于惯性质量。对于像电磁学这样的东西，这完全不可能，因为它要求电磁电荷等于惯性质量，而这对于电磁学来说显然是错误的。爱因斯坦问，会不会引力就是如此？这个事实允许了这一点，它也会把这个事实解释为不再是牛顿定律中的偶然真理，而是关于世界的必然事实。所以，这是爱因斯坦在 1907 年的核心思想，他最美丽的思想。但这听起来完全疯狂。它听起来完全疯狂，因为它要求我们对直线的理解是错误的。这是一个极其激进的主张，原因我现在就来描述。像离心力或科里奥利力或我们熟悉的任何其他惯性力，惯性力是当你没有沿直线运动时感受到的力。当你沿直线运动时，你不会感受到任何力。你不会感受到惯性力。所以，为了使这成立，我们必须说自由漂浮和自由下落的宇航员是在沿直线运动。我们必须说，你，就坐在那里，看起来没有动，却感受到引力把你压进椅子里，我们必须说你没有沿直线运动。所以，我们必须对谁在沿直线运动、谁没有沿直线运动有相当大的误解。

Uh, no tricks. Will you put your finger in that and confirm that it's wet? Okay. So you know, here is the bucket. At the bottom, no mystery why the water is not falling out of the bucket. It's not falling out because the point force of gravity as we would understand it is pointing down to the bottom of the bucket. But now what we're going to do is go a little bit faster and loop-the-loop. And now there is what you might find superficially surprising, which is that the water doesn't fall out of the bucket even when the bucket is upside down. And there are two ways to understand that. One way is just the straightforward way, which is that you would say, you know, the water wants to fall out of the bucket when it's at the top of its arc. But by the time it's got itself together to accelerate enough to fall out of the bucket, the bucket's moved on and is now below, and it just didn't have time to fall out of the bucket, you know, or you might say the same reason that astronauts don't end up falling to Earth. There is a second perspective. The second perspective, which is an equally valid perspective, is imagining that you're riding along with the water in the bucket. And from that point of view, there's another explanation for why the water doesn't fall out of the bucket. And that is the centrifugal force that from the perspective of somebody moving along with the bucket, there is a force pushing them towards the bottom of the bucket. And that force is known as the centrifugal force. And it's what's known as a fictitious or inertial force. And the centrifugal force just says that you know, there is a force caused by being in a rotating reference frame given by your speed divided by the radius of the circle you're going round in, pointing outwards, positively outwards. And so this is the centrifugal force that pins you to the bottom of the bucket or pins you to the outside of the car as you go round a bend. And okay, what do we notice? What we notice is that you're charged under the centrifugal force, if you will, how intensely you feel a centrifugal force is once again, just like with gravity, but unlike with electrostatics, given by your mass. You know, the mass of centrifugal force that tells you how much centrifugal force you get is given by your inertial mass. But of course here is absolutely no mystery whatsoever why the mass that's sitting here on the right-hand side is given by your inertial mass. It is given by your inertial mass precisely because it is the tendency—the reason you're experiencing this force is precisely the tendency of masses to wish to move along straight lines, and the fact you're not moving along a straight line, you're moving in a circle, it is precisely that inertial tendency that causes the mass to begin with. Another way to say it is for any time you have one of these inertial forces caused just by your inertia, it is guaranteed to be the case that the charge under that force is given by the inertial mass. So, inertial forces always have a charge given by the inertial mass. Gravity has a charge—the charge of gravity is given by the inertial mass. So, Einstein leapt, could it be the case, and this was his central idea, could it be the case that gravity itself is an inertial force? That's permitted because the gravitational mass is equal to the inertial mass. It would be totally impossible straightforwardly for something like electromagnetism because it would require that the electromagnetic charge was equal to the inertial mass, which is just simply false for electromagnetism. Could it be the case, Einstein asked, that it's true for gravity? It's permitted by this fact, it would also explain this fact as now not an accidental truth like in Newtonian laws, but a necessary fact about the world. So, this was Einstein's central idea in 1907, his most beautiful thought. But it sounds totally crazy. And it sounds totally crazy because it requires us to be wrong about what straight lines are. It is an extremely radical proposition for the reason that I will describe right now. Inertial forces like the centrifugal force or like the Coriolis force or any of these other ones that we're familiar with, inertial forces are forces you experience when you are not moving on a straight line. When you are moving on a straight line, you don't experience any forces. You don't experience inertial forces. So, in order for this to be true, we'd have to say that astronauts who are free floating and free falling are moving along a straight line. We'd have to say that you, who's just sitting there not seemingly not moving, who is experiencing the force of gravity pushing you into your chair, we'd have to say that you're not moving along a straight line. So, we'd have to be pretty wrong about who's moving along a straight line and who's not moving along a straight line.

随着我逐渐了解 Jane Street 的人，我注意到他们中很多人有物理背景。我最近有机会和 Jed Thompson 聊了聊，他在成为交易员之前是粒子物理学家，我们谈到了他的物理训练如何帮助他在 Jane Street 的工作。

As I've gotten to know the folks at Jane Street, I've noticed that a lot of them have physics backgrounds. I recently got a chance to talk to Jed Thompson, who was a particle physicist before he was a trader, about how his physics training helps him with his work at Jane Street.

我认为 Jane Street 的交易员或研究员中，很少有具备金融背景或交易背景的。当我还从事物理学时，我常说的一句话是：我几乎从不会在还没有对答案有个不错猜测的情况下就去做计算。在交易中，我认为同样如此。这些本质上都是关于世界如何运转的模型。你可以通过反复观察模式来建立良好的直觉，最终达到一种状态：从一开始就大多能提出正确的问题，从而省去大量工作。

I think very few Jane Street traders or researchers come in with any finance background or any trading background. When I used to be in physics, something that I would say is I almost never do a calculation without already having a pretty good guess at the answer. In trading, I think the same is true. These things are fundamentally models for how the world is behaving. You can build good intuition by seeing patterns over and over again and come to a point where you're mostly asking the right question from the beginning, which short circuits a lot of the work.

所以，即使你没有金融背景，甚至没有物理背景，也应该考虑申请。访问 janestreet.com/torkash 了解更多。

So, even if you don't have a finance background, or for that matter a physics background, you should still consider applying. Go to janestreet.com/torkash to learn more.

所以，这是一个激进的想法，因为它要求我们对什么是直线产生错误认识。特别是，你坐在这里，就坐在椅子上，这是你相对于地心的高度随时间的变化。而这是 Torkash，就坐在这里，高度不变。因为你正在经历重力，如果重力是一种惯性力，因为你感受到向下的力，那就意味着你必然在沿着一条非直线运动。所以，那就是你。

So, this is a radical idea because it requires us to be wrong about what a straight line is. In particular, you sitting here, just sitting in your chair, here is your height above the center of the Earth as a function of time. And here is Torkash, just sitting here at constant height. Because you are experiencing the force of gravity, if gravity is an inertial force, because you are experiencing a force down, that means that you have to be moving along a not straight line. So, that's you.

相比之下，这支粉笔，当它上下运动时，其轨迹近似于一条抛物线。粉笔上下运动。粉笔在我接住它之前处于自由落体状态，这意味着如果重力是惯性力，那么这条轨迹必须是直线。当然，按照我画的方式，这条看起来是直的，而那条不是。所以，如果重力要成为惯性力，我们就必须对什么是直线、什么不是直线感到困惑、产生错误。

By contrast, this piece of chalk, as it goes up and down, executes something that's well approximated by a parabola. There is the chalk up and down. The chalk is in free fall until I catch it, which means that if gravity is an inertial force, this has to be straight. Now, certainly the way I've plotted it, this looks straight and that one does not. So, if gravity is to be an inertial force, we have to be confused, wrong, about what is a straight line and what is not a straight line.

然而，如果你曾坐在飞机座位上看着面前的屏幕，你应该对这种情况很熟悉。想象一下你在飞机上看到的地图，想象你正从旧金山飞往伦敦。虽然我不擅长画地球，但这是我的版本。我们在这里，旧金山。这里是格陵兰。从北极方向过来。这里是英格兰。这里是伦敦。

However, this is actually a situation to which you should be familiar if you've sat in an airplane seat and looked at the screen in front of you. If you imagine the map that you see on an airplane, imagine you are flying from San Francisco to London. Now, I'm not good at drawing the Earth, but here is my version of it. Here we are in San Francisco. Here is Greenland. Coming in from the North Pole. And here is England. Here's London.

我可以告诉你我是怎么想明白的，因为大陆的形状非常理想化。

I can tell you how I figured this out because of the very idealized forms of the continents.

而且你知道，有时坐在飞机后排座位上会让人很沮丧，因为你显然知道飞机应该这样飞，沿着两地之间的最短距离飞行，但相反，他们却绕了一个大弯，擦过格陵兰然后南下。你知道实际上并非如此。你知道实际上，尽管在图上看起来是这样，但这并不是直线。这条有时被称为等角航线的线并不是直的，当然也不是从旧金山到伦敦的最短路径。而这条实际上，近似来说，是一条直线。所以事实上，从旧金山到伦敦的直线确实经过格陵兰，我现在就来演示。

And you know, sometimes it can be quite frustrating sitting there in the back seat of the airplane because you know, obviously the plane should be flying like this moving along the shortest distance from one place to another, but instead they take this massive detour that clips Greenland and heads on down. You know that in fact that's not what's going on. You know that in fact, despite what it looks like on the graph, this is not a straight line. This rum line that's sometimes called is not straight and would certainly not be the shortest path from San Francisco to London. And this is in fact, to good approximation, a straight line. So in fact, the straight line from San Francisco to London does indeed go over Greenland as I will now demonstrate.

所以，这里是旧金山。这里是伦敦。你可以看到，从一地到另一地的直线会沿着这个方向经过格陵兰，然后到达伦敦。在这张地图上很明显，因为这张地图显然反映了地球的曲率。这张地图之所以让人困惑，是因为它试图假装地球是平的。它试图忽略地球的曲率。而且因为它试图将球形的地球映射到平面面板上，必然会产生畸变。每当你试图把弯曲的东西假装成不弯曲时，你最终必然会对什么是直线、什么不是直线产生错误认识。

So, here is San Francisco. Here is London. And you can see that the straight line that goes straight from one to the other would go in this direction over Greenland and hit London. That's obvious on this map because this map obviously reflects the curvature of the Earth. This map is getting confused and it's getting confused because it's trying to pretend that the Earth is flat. It is trying to ignore the curvature of the Earth. And because it's trying to map a round Earth onto a flat panel, there has to be distortions. And whenever you try and take something that is curved and pretend it's not curved, you will inevitably end up being wrong about what is and is not a straight line.

你在地球上看到这一点，这条先上升后下降的线，实际上是直线。你在时空中也能看到这一点，在广义相对论中，粉笔被抛起时的抛物线弧，它处于自由落体状态，在广义相对论中它就是直线。正如在广义相对论中一样，你对什么是直线、什么不是直线感到困惑的原因，是你试图用这张图假装自己处于平直时空。而实际上，你处于弯曲时空中。

You see this on the Earth, where this line that goes up and then comes down, is in fact the straight line. You see it also in spacetime with general relativity, where this parabolic arc of the chalk as it's thrown up, it is in free fall, it is the straight line in general relativity. And just like in general relativity, the reason you are confused about what's straight and what's not straight is that you are trying to pretend with this graph that you are in a flat spacetime. And in fact, you are in a curved spacetime.

所以，在爱因斯坦的理论中，物质的作用就是弯曲时空。通过弯曲时空，它会改变什么是直线、什么不是直线。然后，那些沿着他们错误地认为是直线的路径运动的人，将会感受到引力。而宇航员则不会感受到引力。这里唯一缺失的部分，是用数学来描述时空弯曲的方式。

So, in Einstein's theory, the effect of matter is going to be to curve spacetime. And through curving spacetime, it's going to change what's a straight line and what's not a straight line. And then, people who are going along what they incorrectly think of as straight lines are going to experience the gravitational force. Whereas astronauts are going to not experience the gravitational force. The only missing piece here is to mathematically characterize the way in which spacetime is curved.

你知道在牛顿物理学中，牛顿力是由质量的存在引起的。在爱因斯坦的广义相对论中，时空的曲率将由质量引起。他在 1907 年（当时他大致勾勒出这个图景）到 1915 年（他写下广义相对论的完整形式）之间挣扎了八年。这八年的最终成果，就是他那个著名的公式，我不会解释它，但会写下来，它精确地捕捉了他的直觉。我会带你逐步了解这个公式。

You know that in Newtonian physics, the Newtonian force is caused by the presence of mass. In Einstein's general theory of relativity, it will be the curvature of spacetime that is caused by the mass. And he struggled for eight years between 1907 when he had this picture approximately mapped out and 1915 when he wrote down in its finished form his general theory of relativity. And the final output of that eight years was his famous formula that I will not explain but will write down that exactly captures his intuition. And I will walk you through this formula.

所以，这只是一个优美的公式，左边是一些东欧人发明的数学，用来描述时空的曲率。它表示时空弯曲的程度。这是一个张量，如果时空是平坦的，这个张量将为零。当时空不平坦时，它非零，它以特定方式弯曲。右边不再是时空。右边是物质。有一些常数，比如我们的老朋友牛顿常数，更老的朋友 π，以及光速。然后是这个量 Tμν。Tμν 就像是牛顿力方程右侧质量的相对论推广。

And so this is just a beautiful formula and the left-hand side is some mathematics invented by some Eastern Europeans that characterizes the curvature of spacetime. This says how much spacetime is curved. This is some tensor and the tensor will be zero if spacetime were flat. And it's non-zero when spacetime is not flat, it's curved in a particular way. On the right-hand side is not spacetime anymore. On the right-hand side is matter. There's some constants, just our old friend Newton's constant, pi, an even older friend, and the speed of light. And then this quantity T mu nu. T mu nu is like a relativistic generalization of the mass that sits on the right-hand side of Newton's force equation.

所以它说的是，质量的存在，实际上不仅是质量，而是右侧所有形式的质量和能量，导致了左侧时空的弯曲。或者用一句口号来说：物质告诉时空如何弯曲。然后，一旦质量告诉时空如何弯曲，时空的曲率就告诉物质如何运动，这是口号的后半部分：时空的曲率告诉物质沿着弯曲空间的直线运动，因此如果你试图假装时空是平坦的，就会经历虚拟力。这就是爱因斯坦广义相对论的概要。

And so it is saying that the presence of mass, and in fact not just mass, but all forms of mass and energy on the right-hand side causes the curvature of spacetime on the left-hand side. Or in a slogan, matter tells spacetime how to curve. And then once mass has told spacetime how to curve, the curvature of spacetime tells matter how to move in the second half of the slogan, where the curvature of spacetime tells matter to move along straight lines of the curved space, and so experience fictitious forces if you try and pretend that spacetime is flat. And that is Einstein's general theory of relativity in a nutshell.

回溯一下，牛顿引力一个了不起的地方在于，他发明了它，据说是源于一个关于苹果从树上掉下来的思想实验，而它不仅描述了苹果从树上掉下来，还描述了天体的运动。这是一个巨大的跨界：它既描述了行星运动，也描述了苹果从树上掉下来。这是牛顿统一了天与地的惊人成就，用一个公式同时适用于两者。广义相对论做到了这一切，并且更进一步。它描述了苹果从树上掉下来的运动，描述了水星和太阳系行星的运动，还描述了整个宇宙的膨胀。这覆盖了数量级上极其惊人的范围。

Backing up, an amazing thing about Newtonian gravity is that he invented it, allegedly due to a thought experiment about an apple falling off a tree, and it describes not only an apple falling off a tree, but the motion of the objects in the heavens. That's a massive cross hit: it describes planetary motion and also an apple falling off a tree. This is an amazing thing that Newton unified the heavens and the earth, having one formula that applied to both. General relativity does all of that and goes one step further. It describes the motion of apples falling off trees, the motion of Mercury and the planets in the solar system, and the expansion of the entire universe. That's a crazy number of orders of magnitude that it covers.

你刚才说，这个理论的一个美妙之处在于，它以各种有趣的方式触及了原本未曾预料到的领域，解决了爱因斯坦最初的观测。其中之一显然是黑洞。所以，我希望获得比高中版本（光掉进去就出不来）更深入的见解，了解黑洞为何如此运作。

You were saying a moment ago one of the beautiful things about this theory is that it has reach in all these interesting ways that was not originally anticipated to solve this original observation that Einstein had. And one of them obviously is the black hole. So, I would love to get more insight than the high school version of light falls into it and can't get out of why black holes work the way they do.

是的，黑洞是广义相对论中迷人的物体，也是广义相对论中真正典型的存在，在牛顿物理学中并不以同样的方式存在。这个故事有点疯狂。爱因斯坦写下了他的场方程，就是我们在黑板上写的那个，描述了曲率与系统中能量之间的关系，他认为这些方程如此复杂，没有人能得出精确解，我们只能一直做近似。但事实并非如此。然而，史瓦西，一位一战中的普鲁士炮兵军官，在计算他们向敌人方向发射的炮弹轨迹的间隙，他发现爱因斯坦的方程，在爱因斯坦写下后几个月内，实际上就有一个精确解。这个精确解，现在被称为史瓦西方程，我们现在理解它描述的是黑洞。这是一个除了可能在最中心（我们不会描述的方式）之外没有物质的解。它是一个中心点状的物质，描述了周围时空的样子。它被称为史瓦西解，描述了一个黑洞。当时它们并不叫黑洞。事实上，人们对这个解的含义感到极度困惑。大约半个世纪里，人们写下了关于它含义的错误内容。也许最糟糕的违规者是爱因斯坦，他对它感到极度困惑，尤其对我将描述的事件视界感到困惑，并说了各种错误的话，比如物体可能会从事件视界弹开。完全困惑，但从现代视角来看，理解发生了什么极其简单。所以，让我告诉你什么是黑洞。广义相对论是关于引力与光速有限性之间的碰撞。最简单的碰撞，实际上在 18 世纪，甚至在我们有狭义相对论之前，人们就已经注意到了。他们只是问了一个非常简单的问题。如果你想从地球发射东西，你需要以一定的速度发射，即逃逸速度。你需要发射得足够快才能逃离地球，使得你发射的物体的动能等于地球表面的引力束缚能。M 是地球的质量，R 是地球表面。对于地球，逃逸速度大约是 11 公里/秒。但对于更重或更致密的物体，逃逸速度更大。例如，木星会是每秒数百公里。你需要从地球发射它。你可以想象如此重或如此致密的物体，以至于逃逸速度变得等于光速。所以人们理想化地想知道那时会发生什么。他们没有工具来解决，但在 18 世纪他们想知道会发生什么。你可以计算出速度的临界值。只需将速度设为光速，这给出一个临界半径 GM（物体质量抵消）除以 c 的平方。所以这有点暗示，如果你有一个如此致密且具有这种质量的物体，逃逸速度将是 c，即光速。

Yeah, so black holes are fascinating objects in general relativity and really the quintessential object in general relativity that doesn't really exist in the same way in Newtonian physics. And the story is kind of wild. So, Einstein wrote down his field equations, the field equations we wrote on the board, describing the relationship between curvature and the amount of energy in the system, and he thought that those equations were so complicated no one would ever come up with exact solutions to them. And we'd just always be having to do approximations. And that turned out not to be correct. But Schwarzschild, who was a Prussian artillery officer in the First World War, in between calculating the trajectories of artillery that they were lobbing over in the direction of their enemy, he figured out that in fact Einstein's equations, pretty much immediately after Einstein wrote them down within a matter of months, have an exact solution. An exact solution, a solution now known as the Schwarzschild equation, and that we now understand describes a black hole. It is a solution in which there is no matter except possibly at the very very center in a way we will not describe. It's a central point-like amount of matter, and it describes what the space-time around that looks like. It's called the Schwarzschild solution, and it describes a black hole. They weren't called a black hole at the time. In fact, people were extremely confused about what the solution even meant. People wrote down wrong things for about half a century about what this meant. And perhaps the worst offender was Einstein, who got extremely confused about it, particularly confused about what I will describe as the event horizon, and said all sorts of wrong things about how objects would maybe bounce off the event horizon. Just totally confused that from a modern perspective it's extremely simple to understand what's going on. So, let me tell you what a black hole is. General relativity is set up about the collision between gravity and the finite speed of light. And the simplest collision you could do was actually noticed by people even in the 18th century before we had special relativity or anything like that. And they just asked a very simple question. If you want to shoot something off the Earth, you need to shoot it with a certain velocity, the escape velocity. You need to shoot it fast enough if you want to escape far away from the earth so that the kinetic energy of the object you're shooting is equal to the gravitational binding energy of the earth's surface. M is the mass of the earth and R is the surface of the earth. And for earth, that turns out it's about 11 km/s is the escape velocity. But for objects that are heavier or more compact the escape velocity is larger. So for example, Jupiter would be hundreds of kilometers a second. You need to shoot it off the earth. And you can imagine objects that are so heavy or so compact that in fact the escape velocity becomes equal to the speed of light. And so people ideally wondered what would happen then. They didn't have the tools to address it, but in the 18th century they wondered what would happen. And you can calculate what the critical value of the velocity is. And just putting the velocity equal to the speed of light, this gives a critical radius of GM, the mass of the object cancels, divided by c squared. And so that's somewhat suggestive that if you had an object that was this compact and had this mass the escape velocity would be given by c, the speed of light.

你指出的这个联系是牛顿式的。之前有人建立过这种联系吗？

The connection you're pointing out is a Newtonian one. Did anybody make this connection before?

当然。所以 18 世纪晚期的人们写下了这个公式。我认为米切尔和拉普拉斯在 18 世纪都有这个。他们说，如果你有一个如此巨大且致密的物体，光将无法逃脱。这种推理在现代标准下并不特别有说服力，但事实证明它完全正确，甚至疯狂的是，这个因子 2 甚至也是正确的，完全是出于巧合的原因。让我给你一个更有说服力的论证，说明在这个半径附近会发生一些有趣的事情。为此，让我们考虑通过将物体向中心质量降低来提取能量。所以，让我们也许从地球开始。在这里。我将从离地球很远的地方开始，拿着一块砖。一块质量为 M 的砖。我将拿起这块砖，把它连接到滑轮系统上。然后我将慢慢地把砖向地球表面降低，并以零速度将其放置在地球表面上。这样做，我可以从砖中提取能量。我从砖中提取了能量，因为有一个力在向下拉砖。这个力我作用了一段距离，这给了你能量。我们至少知道在牛顿物理学中，你可以从砖中提取的能量的公式是什么。你可以从砖中提取的能量是 G 乘以地球质量乘以砖的质量除以 R，即距离。所以，这是能量，不是力，所以是反距离定律，而不是反平方距离定律。

Absolutely. So people in the late 18th century wrote this formula down. I think both Mitchell and Laplace had this in the 18th century. And they said that if you had an object that was that massive and compact light would not be able to escape. That reasoning is not particularly compelling by modern standards, but it does turn out to be exactly correct even including crazily this factor of two is even correct for completely coincidental reasons. Let me give you a more compelling argument that something funny is going to happen around this radius. And to do that, let's think about trying to extract energy from objects by lowering the objects down towards a central mass. So, let's start off perhaps with the Earth. Here it is. And I'm going to start off a long way away from the Earth with a brick. A brick of mass M. And I'm going to take this brick and attach it to a pulley system. And then I'm going to slowly lower the brick down towards the surface of the Earth and deposit it with zero velocity on the surface of the Earth down there. And doing so, I can extract energy from the brick. I've extracted energy from the brick because there's a force pulling the brick down. That force I'm doing for a certain distance and that gives you an energy. And we know at least in Newtonian physics what the formula for the amount of energy you can extract from the brick is. The amount of energy you can extract from the brick is G times the mass of the Earth times the mass of the brick divided by R, the radius away. So, it's an energy, not a force, so it's an inverse distance law, not an inverse square distance law.

这就是你把砖块降到离地球 R 距离处所能提取的能量。当然，如果你试图把它降到地球表面以下，这个公式就会改变。但我们就把它放在地球表面吧。所以这就是我在很远很远的地方从砖块中提取出来的能量。

That's the energy you can extract from the brick by lowering it down to a distance R away from the Earth. And of course, if you try and lower it beyond the surface of the Earth, this formula changes. But let's just put it on the surface of the Earth. So this is the amount of energy I've got out here a long way away and I've extracted from the brick.

你可以问，我提取了砖块静质量能量的多少比例？这个问题只有在你发明了狭义相对论、知道静质量能量等于 mc 平方之后，才会自然地问出来。所以我们可以直接计算，至少在这个近似下，你提取的能量比例就是那个能量除以你一开始的静质量能量。

You can ask what fraction of the rest mass energy of the brick have I extracted? This is a question that you would only naturally ask once you've invented special relativity and know that the rest mass energy is given by mc squared. So we can straightforwardly calculate, at least in this approximation, that the fraction of the energy that you've extracted is that divided by the rest mass energy you started with.

砖块的质量当然会抵消掉，但地球的质量不会。这个比例等于 G 乘以地球质量除以 c 平方再乘以你停下来的地方离地球的距离，也就是地球半径。那么我提取了多少能量、多大比例呢？如果你降到地球表面，答案是：你并没有真正从砖块中提取出多少。你提取了砖块原始静质量能量的 7×10^-10 这个比例，在很远的地方做了有用功。

The mass of the brick, of course, is going to cancel, but not the mass of the Earth. And this is going to be given by G times the mass of the Earth divided by c squared times the radius away from the Earth at which you stop, the radius of the Earth. So how much energy, what fraction have I got out of it? If you lower down to the Earth's surface, then the answer is you haven't really extracted that much from the brick. You've extracted a fraction 7 * 10^-10 of the original rest mass energy of the brick, doing useful work a long way away.

有趣的是，嗯，第一点观察：这个数很小。换句话说，地球表面物体的引力束缚能在自然单位下非常小。这就是为什么我们直到做了非常灵敏的实验才在地球表面真正注意到广义相对论，因为广义相对论在某种意义上就是这个数的泰勒展开，其中相对论效应的一阶项就是牛顿项，然后下一阶项会给出对牛顿答案的广义相对论修正。

Interestingly, well, first observation, this is small. In other words, the gravitational binding energy of something on the Earth's surface is quite small in natural units. 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, where the relativistic effects' first-order term is just Newtonian, and then the next-order terms will give you the GR corrections to the Newtonian answer.

第二点观察，这有点跑题，但第二点观察是：纯属巧合，这个数非常接近火箭燃料的化学结合能。所以如果你取一种火箭燃料，比如氧氢混合物，把火箭结合在一起的化学能——也就是你燃烧它让火箭飞起来时提取的能量——除以你要混合的氧和氢的 mc 平方，结果是 1.5×10^-10。

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. So if you take a rocket fuel like an 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 mc^2 of the oxygen and hydrogen you're going to mix together, is given by 1.5 * 10^-10.

第一点观察：这两个数彼此接近，尽管它们来自完全不同的计算。一个是与地球有关的引力计算，一个是氢和氧的化学性质。这个数也非常小。它之所以非常小，是因为氢氧中几乎所有的能量都不储存在这些东西结合在一起的化学结合能中。绝大部分能量储存在质子和中子的静质量能量中，而化学燃烧根本不会影响这些。

First observation, these two are close to each other even though they came from completely different calculations. This was a gravitational calculation that has something to do with the Earth. This is a chemical property of hydrogen and oxygen. This is also very small. The reason it's very small is that almost all of the energy in hydrogen-oxygen is not stored in the chemical binding energy of these things going together. 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.

第二大的能量储存在质子和中子之间的核结合能中，由强力和弱力提供，而化学反应同样完全不会触及这些。这个数很小，因为化学键与我们考虑的物体的静质量相比非常弱。这两个小数几乎完全相等，这就是为什么我们可以用化学火箭进入太空，但很困难。

The second largest amount is stored in the nuclear binding energy of the protons and the neutrons to each other, given by the strong force and the weak force, which again chemical reactions don't touch at all. This is a small number because chemical bonds are very weak compared to the rest mass of the things we're considering. 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，这个数比那个数大几倍，这意味着当你试图用化学火箭进入太空时，绝大部分的有效载荷比例非常小，因为你的大部分燃料无法进入轨道。你必须付出一个火箭因子，它告诉你：发射台上那个有效载荷的大部分，在到达太空之前就必须被烧掉，才能让火箭的一小部分进入太空。

In particular, this number is a few times bigger than this number, which means that almost all of the vast majority of payload fraction is quite small when you're trying to use chemical rockets to get to space, because most of your fuel cannot get to orbit. You have to pay a rocket factor that's going to 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 rocket up to space.

换句话说，我们可以用化学火箭进入太空，这种方式如果我们试图从太阳表面出发是完全不可能的，但确实很难。好了，这是地球上的比例，但这个公式告诉你：如果你有一个更重或更致密的物体，你把物体降到表面所提取的能量比例会更大。

In other words, we can use chemical rockets to get to space in a way that would be totally impossible if we tried to do it from the surface of the sun, but it's hard. Okay, that's the fraction on the Earth, but this formula tells you that if you have an object that is heavier or more compact, the fraction of energy that you extract by lowering the object down to the surface is going to be larger.

所以，举个例子，如果你不是降到地球表面，而是降到太阳表面，这个数会更大。你知道，大一百万倍，因为太阳的质量是地球的几百万倍，但它也更大，所以这又抵消了一点，最终得到 2×10^-6，这就是著名的太阳表面红移。

So, for example, if you lower it not down to the Earth's surface, but down to the sun's surface, this will be larger. You know, a million times larger because the sun is a few million times the mass of the Earth, but then it's also bigger, so that takes it away a little bit, and you end up with 2 * 10^-6, the famous redshift from the sun's surface.

而且你可以从这里继续升级。你可以想象把太阳质量的物质塞进地球大小的半径，让这个公式变得更大。事实上，这几乎正是像天狼星 B 这样的白矮星里发生的情况。这个比例会再次变大，你通过把物体降到表面所提取的物体质量比例会更大。

And you can escalate from there. You can imagine cramming a sun-like mass into an Earth-like radius to make this formula even bigger. In fact, that's pretty much exactly what happens in a white dwarf like Sirius B. And this will get even bigger again, a larger fraction of the mass of the object you'd be extracting by lowering it down to the surface.

但真的感觉在我们制造出一个过于巨大、过于致密的物体之前，有些事情必须让步。特别是，如果你看这个公式，当 R 小于或等于 GM 除以 c 平方时会发生什么？如果这个物体如此致密、如此重，以至于它的半径小于物体质量除以 c 平方，那看起来你确实可以得到超过 100%。比例会大于 1。你可以通过把砖块降到这个物体表面，拿回超过砖块质量 100% 的能量。

But it really feels like something has to give before we make an object that is too massive and too compact. In particular, if you look at this formula, what happens for R less than or equal to GM over c squared? 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 would be bigger than one. You could 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。事实上，这感觉比这里发生的情况更不对，因为现在你在很远的地方得到了所有这些能量，你也许可以用它来做一块全新的砖。你知道，你在那里得到了超过 mc 平方的能量。把那一块降下来，感觉就像我们找到了一种方法，在没有能量的地方制造出大量能量。

And that feels wrong. That feels in fact sort of more wrong than what's going on here, because now you've got all this energy a long way away, and you could perhaps use it to make a whole new brick. You know, you've got all this more than mc squared 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.

这个论证相当有力地表明，当你降到那个半径时，一定有什么地方出错了。确实，当你做计算时，你知道，这是一个牛顿计算，所以它只是暗示性的，但当你用完整的广义相对论做计算时，确实有什么地方出错了。出错的地方就是你形成了一个黑洞。

This argument is pretty suggestive that something has to go wrong by the time you get down to that radius. 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. And the thing that goes wrong is that you form a black hole.

你可以想象两种避免这个结论的方法。一种是以某种方式，当你离一个巨大物体那么近时，引力变得非常弱，比牛顿定律预测的还要弱。

You can imagine two ways that you could avoid this conclusion. One would be somehow that gravity becomes very weak when you get that close to a massive object, weaker than the Newtonian law would predict.

如果你在电磁学里尝试同样的把戏——把一个电荷向另一个电荷降低，试图提取它们之间的静电能量——那正是能救你的东西。实际上发生的是，由于量子效应，当一个电荷离另一个太近时，它们都开始模糊化，能量随 1/R 变化的关系被软化，你无法提取更多能量，因为它们不再那么强烈地相互吸引。所以，一种可能是，当你接近另一个物体时，力会比牛顿定律预测的更弱。

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. What happens is essentially due to quantum effects, when one gets too close to the other, they all start to fuzz out, and the energy going like inverse R gets softened, and you can't extract more energy because they stop attracting each other so hard. So, that's one possibility is that the force gets weaker than Newtonian law would predict as you approach the other object.

这实际上与广义相对论解决这个问题的方式相反。广义相对论通过让力比牛顿定律预测的更强来解决这个悖论。特别是，当你试图进入这个半径以内时，力变得如此之强，以至于你实际上无法把砖块慢慢降到表面，因为你已经形成了一个黑洞。引力在有限距离处变得无限大，不是在 R 等于零处，而是在某个有限的 R 值处，砖块会直接从你手中被扯走，你无法再从中提取任何能量。这就是广义相对论对这个悖论提供的解答，特别是，你会发现你已经形成了一个黑洞。

That's actually the opposite of how general relativity resolves this. General relativity resolves this paradox by the force getting stronger than Newtonian law would predict. 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 because you've formed a black hole. The gravitational force becomes infinite at a finite distance this way, not at R equals zero, but at some finite value of R, and the brick simply gets ripped out of your hand, and you're unable to extract any more energy out of it. And that's the resolution that general relativity provides to this paradox, and in particular, you will find that you've formed a black hole.

Crusoe 让我们提前体验了他们的无服务器微调产品，它让你无需处理基础设施或资源配置就能微调开放模型。我觉得用我以前采访的转录稿来微调一个提问生成器会很有趣。而且模型已经变得非常好，如果它们掌握了我所有的研究和准备，并能看到目前的对话，它们提出的下一个问题可能会比我自己提的更好。Crusoe 让实现变得非常简单。我只需上传数据，选择一个开放模型，然后开始运行。我完全不需要碰任何超参数。Crusoe 的应用 AI 团队为每个模型维护了最优配方，所以我只需把所有设置设为自动。运行完成后，我把它部署为一个自助服务端点，并为我的团队构建了一个评估。我让他们从三个匿名选项中选择最佳的下一个问题：一个由基础模型生成，一个由微调模型生成，一个是我实际提出的。幸运的是，我的团队大约三分之二的时间更喜欢我实际提出的问题。希望这个基准不会饱和。在其余情况下，他们几乎总是更喜欢微调模型而不是基础模型。无服务器推理现已上线，无服务器微调将于下周上线。更多信息请访问 crusoe.ai/doorkash。

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. I thought it'd be cool to try fine-tuning a question generator using the transcripts of my old interviews. And the models gotten so good that if they had all my research and prep, and they could look at a conversation so far, they could ask a next question better than I would. Crusoe made the implementation super straightforward. I just uploaded the data, picked an open model, and started the run. I didn't have to touch any of the hyper parameters. Crusoe's applied AI team maintains optimal recipes for each model, so I just set everything on auto. When the run finished, I deployed it as a self-serve endpoint and built an eval for my team. I had them choose the best next question out of three anonymized choices. One that was produced by the base model, one that was produced by the fine-tuned model, and one that I actually asked. Fortunately, my team preferred my actual questions about two-thirds of the time. Hopefully, this benchmark doesn't saturate. And in the remaining cases, they almost always preferred the fine-tuned model over the base model. Serverless inference is live now, and serverless fine-tuning goes live next week. Learn more at crusoe.ai/doorkash.

到目前为止，我们在黑板上写下的都是牛顿力学。这是牛顿力学，你一旦开始代入光速，就会开始困惑。要真正回答我们提出的这些问题，你需要用到广义相对论，这个理论正确地将光速与引力统一起来。这在黑洞背景下最早由 Schwarzschild 完成，他写下了 Schwarzschild 度规，描述了中心质量周围的引力场，包括可能围绕黑洞的引力场。让我写下一些由此得出的公式。实际上，我想我会写下三个公式，它们是 Schwarzschild 度规的三个直接推论，将给我们提供关于黑洞外部乃至内部情况的直觉。

So far, everything we've written down on the board is Newtonian. This is Newtonian, and you just start plugging in the speed of light, and you start getting confused. To actually answer some of these questions that we're asking, you need to go to general relativity, the theory that correctly unifies the speed of light with gravity. 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. And let me just write down some of the formulas that emerge. In fact, I think I'm going to write down three formulas, the three direct consequences of Schwarzschild metric, that are going to give us intuition for what it's like outside and indeed inside a black hole.

所以我要写下的第一个公式是，如果你试图在中心质量外部保持静止，你所经历的引力场的公式。那么，让我们只讨论静态观察者，这样我可以讨论这些公式如何为移动的观察者升级。但现在，我只是想象你试图坐在这里，在某个半径 R 处，离黑洞某个固定的半径 R。你保持静止的原因，你没有掉进黑洞的原因，也许，你知道，我用滑轮把你放下来，你只是坐在这里抓住滑轮。问题是，你需要多大的力才能阻止自己掉下去？你在非常缓慢地下降。你是静止的。你经历的局部引力是多少？或者你可以想象你坐在这里，你保持静止的原因是你非常用力地发射火箭。那么，你局部感受到的加速度是多少？所以，无论你通过什么机制保持静止，你感受到的局部引力是多少？

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. And so, let's just talk about for static observers, so I can discuss how these will get upgraded for observers who are moving around. 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. 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? You're abseiling down very slowly. You're static. What is the local force of gravity that you experience? Or you can imagine that you're sitting here. The reason you're static is you're firing a rocket very hard. And, you know, what is the how much acceleration do you locally feel? 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，嗯，在牛顿物理学中，你知道这个问题的答案。引力是 GM 除以 R 的平方，这是牛顿著名的平方反比定律。但广义相对论对此进行了修正。修正项是 1 - 2 GM 除以 c 的平方乘以 R。所以，这个 2 GM 除以 c 的平方我们到处都能遇到。这告诉你什么，嗯，首先，如果你离黑洞非常远，这里基本上等于 1。R 非常大，你又得到了牛顿的力定律。而且你知道，对于地球来说，这个非常小。正如我们讨论过的，这大约小了 10 的负 10 次方倍。然后你取平方根。所以你不会真正注意到它，但你可以在大 R 处进行泰勒展开，你会发现你得到了修正，你得到一个平方反比定律加上一个立方反比修正再加上一个四次方反比修正。你会发现，在短距离上，引力比牛顿物理学中的更强。这是广义相对论修正，它使引力场更强。你必须更努力地加速才能不掉进黑洞。

The local force of gravity that you feel, well, in Newtonian physics, you know what the answer to that question would be. The force of gravity is GM over R squared, which is Newton's famous inverse square law. But this gets a correction from general relativity. And the correction is 1 - 2 GM over c squared times R. So, this same 2 GM over c squared that we find all over the place. 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. R is very big and you get Newton's force law back again. And you know, for the Earth, this is very small. This, as we discussed, is down by a factor of 10 to the minus 10 or so. And then you take the square root. So you don't really notice it, but you can Taylor expand this at large R and you find out that you get corrections, you get an inverse square law plus an inverse cube law correction plus an inverse 4th law correction. And you find that gravity at short distances is stronger than it would have been in Newtonian physics. This is the general relativity correction and it's making the gravitational field stronger. You have to accelerate harder to not fall into the black hole.

特别是，一旦 R 等于 2GM 除以 C 的平方，这就是所谓的 Schwarzschild 半径，你必须无限加速。为了在 R 上不移动所需的固有加速度趋于无穷大。所以，事实上，如果我们现在把这个转换到地球到黑洞的情况，这是一个非常重要的半径，2GM 除以 C 的平方。它被称为事件视界。它被称为事件视界，因为如果你想在事件视界之外保持静止，离事件视界更远，你只需要以某个有限速度加速以保持静止。你需要有一个有限的引力场，但当你接近事件视界时，引力场变得无限大。所以，一旦你处于事件视界或更远，就不可能保持静止。

And in particular, once R is equal to 2GM over C squared, this, what's called the Schwarzschild radius, you have to accelerate infinitely. The proper acceleration required to not move in R goes to infinity. So, in fact, if we now convert this to an Earth to black hole, this is a very significant radius over here, 2GM over C squared. It's called the event horizon. 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 finite velocity in order to remain static. You need to have a finite gravitational field, but the gravitational field as you approach the event horizon becomes infinite. So, once you're at or beyond the event horizon, it is impossible to remain static.

无论你如何用力发射火箭，你都不可避免地会被吸入黑洞。现在，这只是静态公式。你可能会想，好吧，在比这更近的地方保持静止是不可能的，但也许我可以非常非常快地绕轨道运行，从而不会掉进黑洞。如果我绕轨道运行得非常快，我就会有巨大的离心力把我从黑洞推开，我就能以这种方式保持在黑洞之外。这实际上行不通，而它行不通的原因对于理解广义相对论中引力吸引的方式有些启发意义。

You will inevitably get sucked into the black hole, no matter how hard you fire your rocket. Now, this is just the static formula. 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. And if I orbit really, really fast, I have a huge centrifugal force that pushes me away from the black hole, and I can stay out of the black hole that way. That actually doesn't work, and the reason it doesn't work is somewhat instructive for the way gravitational attraction happens in general relativity.

当然，如果你想想国际空间站，它为什么不朝地球坠落？恰恰是因为它在绕轨道运行，而绕轨道运行这一事实给了它一种离心力，将宇航员推离地球，并恰好平衡了宇航员的引力场，这就是为什么他们在那里感到失重。所以，如果你离黑洞很远，轨道角动量有助于你远离黑洞，阻止你掉进去。

Of course, if you think about the International Space Station, why doesn't it fall towards the Earth? It is precisely the fact 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 gravitational field of the astronauts, which is why they feel weightless there. So, orbital angular momentum, if you're a long way away from the black hole, helps you escape from staying away from the black hole. Stops you falling in.

有一种科幻观念认为黑洞会吸进周围的一切。这不是真的。如果你离黑洞很远，你完全可以绕它运行，就像绕任何中心质量运行一样。你并非不可避免地掉进黑洞。你可以很好地绕轨道运行。但是，当你离黑洞太近时，绕轨道运行就不再起作用了。

There is this kind of sci-fi notion that black holes just suck in everything around them. Not true. 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. You are not inevitably falling into the black hole. You can orbit just fine. But, orbiting stops helping when you get too close to the black hole.

我们说过事件视界是 2 GM 除以 c 的平方。事实上，一旦你进入 3 GM 除以 c 的平方以内，如果你想远离黑洞，绕轨道运行就会适得其反。那是因为绕轨道运行有两个效应。一个效应帮助你远离黑洞，那就是离心效应。由于离心效应，轨道角动量把你推开。如果我们写下适用于有角动量情况的这个公式的版本，你会看到它把你推开。

We said that the event horizon is 2 GM over c squared. In fact, already once you're within 3 GM over c squared, orbiting is counterproductive if you're trying to stay away from the black hole. And that's because there are two effects of orbiting. One effect of orbiting helps you stay away from the black hole. That's the centrifugal effect. Orbital angular momentum pushes you away from the black hole due to the centrifugal effect. And if we wrote the version of this formula that applies when you have angular momentum, you would see that pushing you away from the black hole.

但是，还有另一个效应把你拖向黑洞。那就是在广义相对论中，所有能量都会产生引力。不仅仅是静质量能量产生引力，动能也产生引力。所以绕轨道运行的效应是，由于黑洞质量与你的轨道角能量之间的耦合，你会受到一个额外的向黑洞的拉力。当你远离黑洞时，离心力是更重要的项。当你靠近黑洞时，那个耦合是更重要的项。事实上，一旦你进入 3 GM 以内，轨道角动量就不再帮助你，反而开始伤害你。没有任何弹道轨道能进入 3 GM 以内并再次逃脱。

But, there's another effect which drags you towards the black hole. And that is the fact that in general relativity, all energy gravitates. Not just rest mass energy gravitates, kinetic energy also gravitates. 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. And when you're far away from the black hole, the centrifugal force is a more important term. When you're close to the black hole, that coupling is the more important term. And in fact, once you get within 3 GM, orbital angular momentum stops helping and starts hurting. There are no ballistic orbits that go within 3 GM and that manage to escape again.

好的，这就是公式一。它告诉你距离黑洞 r 处的引力场是什么。特别是，它表明一旦你到达这个临界半径，引力场就变得无限大，如果你越过它，无论你如何用力发射火箭，你都必然走向黑洞的中心。那叫做事件视界。在事件视界，你还没有死。然而，你注定要死。如果你越过事件视界，你将永远无法逃脱。即使你把自己变成光并试图射出去也不行。即使你无限用力地发射火箭也不行。

Okay, so that's formula number one. It tells you what the gravitational field is a distance r away from a black hole. 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 center of the black hole, no matter how hard you fire a rocket. That's called the event horizon. At the event horizon, you are not yet dead. You are, however, doomed. If you cross the event horizon, you will never be able to escape. Not if you convert yourself to light and try and shoot yourself out. Not if you fire your rocket infinitely hard.

当然，另一个地方是 r 等于零，那是你真正死亡的地方。那就是奇点。我们稍后会描述一下。在牛顿物理学中，引力只在这里变为无限。在广义相对论中，它已经在事件视界变为无限。如果你试图抵抗引力的话。好的，这就是公式一。

The other place, of course, is r equals zero, which is where you actually die. And that's at the singularity. And we'll describe that a little bit in a moment. In Newtonian physics, the gravitational force only becomes infinite here. In general relativity, it becomes infinite already at the event horizon. If you try and resist the force of gravity. Okay, that's formula number one.

现在让我们来看公式二。公式二，以及我将要写下的所有三个公式，都只是彼此密切相关。它们实际上是彼此的重新表述。它问的是引力时间膨胀。所以，让我们再次想象你坐在这里，你知道，这是 Walker 坐在离黑洞某个半径 R 的地方。而我坐在外面，你知道，在无穷远处，只是看着你。我们相对静止。没有相对运动。你只是被你的滑轮系统悬挂在这里。问题是，你的手表相对于我的手表走得有多快？

Now let's do formula number two. Formula number two, and all three formulas I'm going to write down are just going to be heavily related to each other. They're really going to be reformulations of each other. It asks about gravitational time dilation. So, let's again imagine that you're sitting here, you know, this is Walker sitting here some radius R away from the black hole. And I'm sitting, you know, out here, way off at infinity, just watching you. And we're static relative to each other. There's no relative motion. You're just suspended here by your pulley system. And the question is, how fast does your watch go relative to mine?

当然，就你而言，你的手表每秒滴答一次。就我而言，我的手表每秒滴答一次。但是，如果我看着你，如果我看着你，我看到你的手表走慢了。如果你看着我，你看到我的手表走得快。所以，第二个公式使这一点定量化。你，靠近黑洞的人，你的手表比我慢多少。

Of course, as far as you're concerned, your watch is ticking at 1 second per second. As far as I'm concerned, my watch is ticking at 1 second per second. But, if I watch you, if I look at you, I see your watch as running slow. If you look at me, you see my watch is running fast. And so, the second formula makes that quantitative. How much slower you, who is close to the black hole, how much slower your wrist watch runs than mine.

它说，你的手表测量的时间间隔等于远处我的手表测量的时间间隔乘以这个到处出现的完全相同的平方根因子，即 1 - 2 GM 除以 R c 的平方的平方根。所以，这里的这个因子小于 1。因此，如果我认为 1 秒过去了，你认为不到 1 秒过去了。

And it says that the time interval as measured by your wrist watch is given by the time interval as measured by my wrist watch a long way away times this exact same square root factor that's showing up all over the place times the square root of 1 - 2 GM over R c squared. And so, this factor here is less than 1. So, if I think 1 second has passed, you think less than 1 second has passed.

换句话说，如果我慢慢把你降到黑洞附近，你在离黑洞有限距离的地方待上感觉像一年的时间，然后把你拉回远离黑洞的地方，你将回到一个比你老得多的世界。这个公式使这一点精确化。

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. And this formula makes that precise.

我观察到你的手表走慢了。你观察到我的手表走得快。这里的时间比这里上面过得更慢。这是一个事实，至今已在实验中得到了极好的验证，在 20 世纪 50 年代哈佛物理系。他们把两个原子钟放在大楼里两个不同的高度，注意到较高的那个比低的那个走得快。

I observe your wrist watch to be running slow. You observe my wristwatch to be running fast. Time passes slower down here than it does up here. This is a fact that has by now been extremely well observed experimentally in the 1950s in the Harvard physics department. They put two atomic clocks at two different heights in the building and noticed that the one that was higher was running faster than the one that was lower.

这是一个现在已经在例如 GPS 的精度范围内的效应，GPS 必须减去那个效应。否则，一切都会到处漂移。地球表面的 GPS 时钟比在轨道上发送信号的原子钟走得慢，你必须减去那个差异才能获得准确的读数。这被称为引力时间膨胀。

This is an effect that is now considerably within the precision of, for example, GPS that just has to subtract that effect. Otherwise, everything would drift all over the place. 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 subtract off that difference in order to get an accurate read. This is known as gravitational time dilation.

我注意到这与你在狭义相对论中看到的相对论时间膨胀很不一样，那个是由两个物体相对运动引起的。在这里，我们并没有相对运动。我们都是静止的，固定的。这个是由我们处于引力势的不同位置引起的。你比我更深地处在引力势中。所以，这是时间膨胀的两个不同来源，它们会叠加。那么，假设我们不是静止在这里，而是你在轨道上。你离得足够远，可以绕黑洞运行。我问，我看到你运动得多慢？现在有两个贡献，两者都让你看起来比我慢。一个贡献是引力时间膨胀，由这个公式给出。第二个贡献是经典的狭义相对论修正，即运动观察者看起来时间变慢，我们会有这两种效应。所以，你看起来会比绕黑洞运行时本来应该的速度更慢。

I 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. Here, we're not in motion relative to each other. We're both static. We're fixed. This is caused by us being at a different place in the gravitational potential. You deeper in the gravitational potential than me. So, those are two different sources of time dilation and they stack. So, let's say instead of being static here, you're in orbit. You're far enough away that you can orbit the black hole. And I ask, how slow do I see you as moving? There are now two contributions, both of which make you look slow relative to me. The one contribution is the gravitational time dilation given by this formula. A second contribution is the good old special relativity correction where moving observers look like they're going slow, and we'll have both of those effects. So, you'll look like you're going even slower than you would have done otherwise as you go around the black hole.

所以，这与狭义相对论的一个不同之处在于没有对称性。在狭义相对论中，两个观察者都会觉得对方比自己老得慢，因为他们以相同的速率相对运动。而且没有真正的惯性路径，但在这里，似乎确实有一个全局意义上的更正确、更相关的惯性参考系。

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

你说得完全正确。是的，在狭义相对论中，如果你和我相对运动，我认为你的表走得慢，你认为我的表走得慢。我们谁都不比谁更正确。相对性原理告诉你，我们两个的视角同样有效。在这里，我们两个的视角并不同样有效，因为没有狭义相对论中的那种对称性。特别是，对称性被黑洞打破了。我们都同意你比我更深地处在引力井中。

You're exactly right. Yeah, so in special relativity, if you and I are moving relative to each other, I think your watch is moving slow, you think my watch is moving slow. Neither of us is more correct than the other. The principle of relativity tells you that both of our perspectives are equally valid. Here, both of our perspectives are not equally valid because there is not the symmetry that there was in special relativity. In particular, the symmetry is broken by the black hole. We both agree that you are deeper in the gravitational well than I am.

嗯。

Mhm.

我们都同意这一点，而且你的钟比我的钟走得慢。你并不会看到我的钟反过来变慢。事实上，你看到我过得加速了。如果你在观察我，你会看到我的生活在快进。

We both agree on that, and your clock runs slower than mine does. You do not see my clock reciprocally running slow. You in fact see me living sped up. If you are observing me, you see me living my life in fast forward.

有趣。

Interesting.

所以，这是第二个公式。它说明了我们的手表相对彼此走得多快。现在，让我们想象你在这里，戴着你的慢速手表，向我射出一束光。假设光有特定的频率。你通过钠跃迁产生它，例如，特定频率的光。当光向上传播时，当它到达我时，我会认为它的频率比你发送时认为的要低。为什么？因为频率是关于它振荡得多快，而我只是认为你做的所有事情相对于我都是慢的。你认为它振荡得更慢。它有更低的频率，这意味着它向光谱的红色部分移动。我们使用的词是红移，引力红移。它被红移，频率更低，因此能量更少。如果你发送一个光子向上，光子的能量由频率给出。它到达我时会更加红移，能量比离开你时更低。相反，如果我在上面，我向你发送一个由钠跃迁产生的光子，正如你所观察到的，当光到达你时，你看到我在快进。所以，你认为它的频率比我离开你时更高。它向光谱的蓝色部分移动。我们说它是蓝移的。所以，这个思想实验告诉你，知道不同高度时间流逝的兑换率，直接给出了不同高度能量价值的兑换率。如果你试图向我发送一些能量，当它到达我时，对我来说它比你感知到的价值更少。而减少的量将精确地由控制其他一切的同一个平方根公式给出。所以，这给了我们第三个方程。

So, this is the second formula. It says how fast our wristwatches move relative to each other. Now, let's imagine that you're here with your slow-moving wristwatch and you shine a light towards me. And let's say the light has a particular frequency. You made it with a sodium transition, for example, a particular frequency of light. 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 you thought it was when you sent it. Why? Because frequency is about how rapidly it oscillates, and I just think that everything you do is moving slow relative to me. You think it's oscillating slower. It has lower frequency, which means it gets shifted towards the red part of the spectrum. The word that we use is red shift, gravitational red shift. It gets red shifted, lower frequency and therefore less energy. If you send one photon up, the energy of the photon is given by the frequency. It'll arrive at me more red shifted and with lower energy than it had when it left you. 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 fast forward. So, you think that it has higher frequency than it had when I left you. It's moved towards the blue part of the spectrum. We say that it is blue shifted. 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. 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. And the amount it's less is going to be precisely given by the same square root formula that's controlling everything else. And so, that gives us our third equation.

所以，第三个公式说，假设你，德沃金，有一个质量为 mc 平方的物体，和你一起坐在下面那个固定半径处。从我这个远离黑洞的地方测量，那个物体对我来说值多少能量？当然，如果它在我这里，它值 mc 平方的能量。但它不在我这里。它不幸地和你一起深陷在引力势中。所以，对我来说它值少于 mc 平方。事实上，这正是同一个公式。当它到达我时，它对我来说值的能量是 gm 除以 r c 平方。有几种方法可以看到这一点。一种是我们刚才说的方式。假设你拿你的质量为 m 的物体，你知道，它只是阿伏伽德罗数的碳原子，或者假设它是半个阿伏伽德罗数的碳原子和半个阿伏伽德罗数的反碳原子。你向我发送能量的一种方式是把它们撞在一起，剧烈爆炸，你把所有能量转化为光，然后试图把光能射向我。但你会发现，正是由于这种引力时间膨胀，当它到达我时，我得到的不是 mc 平方。根据我们刚才的论证，我得到的是少于 mc 平方。我得到的是 1 - 2 gm 除以 r c 平方。这里的质量在上升时经历这种红移，当它到达无穷远时，能量比开始时少。还有另一种方法你可以把能量传给我。不是以光的形式射上去，而是把你的质量物体绑在滑轮上，让我把它拉出来。当我把它拉出来时，我现在在远离黑洞的地方有了 mc 平方。所以，我确实有 mc 平方，完整的 mc 平方能量。但要得到它，我需要付出代价。我需要付出的正是把它从引力势中拉出来。所以，从那种思考方式来看，这就是为什么我剩下少于 mc 平方的能量，因为我必须付出把它从势中拉出来的代价才能获得那个质量。所以，这个公式告诉你，如果我有一块质量为 mc 平方的砖头，放在离黑洞某个半径 R 处，如果我在远离黑洞的地方，我能从这块砖头中提取多少能量？所以，如果我们知道那个公式，那么我们实际上可以精确计算这个公式。

And so, the third formula says, suppose that you, Dworkin, have an object of mass mc squared sitting with you at this fixed radius down there. How much energy, as measured by me a long way away from the black hole, how much energy is that worth to me? Of course, if I had it with me, it would be worth mc squared worth of energy. But I don't have it with me. It's unfortunately sitting with you deep in a gravitational potential. So, it's worth less than mc squared to me. In fact, it's just the exact same formula. The amount of energy that it's worth to me by the time it reaches me is gm over r c squared. And there are a couple of ways to see that. One is the way that we just said. Suppose you take your object of mass m, you know, it's just Avogadro's numbers of carbon atoms, or let's say it's half an Avogadro's numbers of carbon atoms and half an Avogadro's numbers of anti-carbon atoms. And one way you could send me the energy is by smashing them together, violent explosion, you convert all of that energy to light, and you try and beam that light energy up to me. But what you find, precisely because of this gravitational time dilation, is that by the time it reaches me, I'm not getting mc squared worth out. I'm getting, by the argument we just gave, less than mc squared worth out. I'm getting 1 - 2 gm over r c squared out. Mass down here suffers this redshifting as it goes up and has less energy by the time it reaches infinity than it did to begin with. There is another way that you could have got the energy to me. Not by beaming it up as light, but by just taking your mass object, attaching it to the pulley, and having me pull the object out. By the time I pulled it out, I've now got mc squared sitting out here a long way away from the black hole. So, I do have mc squared, the full mc squared worth of energy. But to get it, I needed to pay. And what I needed to pay was precisely pulling it out of the gravitational potential. So, from that way of thinking about it, that's why I have less than mc squared worth of energy left, because I had to pay the pulling it out of the potential in order to accrue that mass. So, this formula tells you, if I have a brick of mass mc squared sitting at some radius R away from the black hole, how much energy can I extract from that brick if I'm a long way from the black hole? And so, if we know that formula, then we can in fact calculate exactly this formula.

我把砖块降到半径 R 处，从中提取了多少能量？我们知道这个问题的答案。它一开始的能量是 mc²。它现在的能量是这个。所以，我用滑轮系统慢慢降下砖块时提取的能量，一定等于初始能量 mc² 减去它现在的能量。换句话说，我把砖块降到半径 R 处所提取的能量比例是 mc² 减去这个，再除以 mc²，即 1 减去根号下 1 减 2GM/c²R。这就是提取能量比例的精确正确答案。它看起来不太一样，因为这只是牛顿极限下的结果。我们是用牛顿物理推导出来的。但这个公式不仅在牛顿极限下精确，而且在广义相对论效应重要的整个范围内都精确。

How much energy have I extracted from the brick by lowering it down to a radius R? Well, we know the answer to that question. The energy it started with is mc squared. The energy it now has is this. 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 mc squared minus the energy it now has. Or in other words, the fraction of the energy that I've extracted by lowering it down to a radius R is mc squared minus this all divided by mc squared, 1 minus root 1 minus 2 gm over c squared R. And this is the exactly correct answer for the fraction of the energy extracted. It doesn't look exactly like this because this is only correct in the Newtonian limit. We derived this using Newtonian physics. This is exactly correct not just in the Newtonian limit, but all the way to where the effects of general relativity are important.

嗯。

Mhm.

现在，如果你离黑洞非常非常远，R 远大于 2GM/c²，那么你可以对这个公式做泰勒展开，一阶项就是旧的牛顿公式。广义相对论的长距离极限必须恢复我们最初发现的牛顿物理。但随着你越来越靠近黑洞，这个公式开始偏离牛顿答案，而正是这个偏离最终解决了我们最初关于把砖块降到黑洞附近的思想实验。

Now, if you are very, very long way away from the black hole, R is much, much bigger than 2GM over c squared, then you can Taylor expand this formula, and the first order term is just the old Newtonian formula. It better be that the long distance limit of general relativity recovers the Newtonian physics that we originally discovered. But as you get closer and closer to the black hole, this starts to deviate from the Newtonian answer, and that deviation is exactly what is going to end up resolving our original thought experiment to do with lowering a brick down towards a black hole.

那么，看着这个公式，当我把它降到黑洞附近时，我到底提取了多少能量？如果 R 等于无穷大，如果砖块离黑洞还很远，那么我提取的是 1 - 1 = 0。我没有从黑洞提取任何能量。当我把它降得越来越靠近黑洞时，起初我得到的只是牛顿公式。所以，实际上，这些在广义相对论中也相当接近正确，因为只有当这一项变得量级为 1 时，修正才会开始变大，而在这里它仍然非常小。所以，这些基本上都是正确的。但一旦我越来越靠近黑洞，它们就不再正确了。我看到的是，当 R 接近黑洞事件视界时，当这个公式趋于零时，我已经从砖块中提取了全部能量。

So, how much then, looking at this formula, have I extracted from the brick as I lower it down towards the black hole? If R equals infinity, if the brick is still a long way from the black hole, then I've extracted 1 - 1 = 0. I haven't extracted any energy from the black hole. As I lower it closer and closer to the black hole, initially I just get the Newtonian formula. 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. So, these are all essentially correct. But once I get closer and closer to the black hole, they stop being correct. 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.

所以，我从离黑洞很远的地方开始，把砖块系在绳子上，慢慢把它降到事件视界附近。当然，我不能把它降到事件视界以下，否则我会失去对砖块的控制，但我把它降到事件视界正上方，也就是我能降到的最低位置，然后以零速度放手。砖块落入黑洞，而我已经在我的滑轮系统中提取了砖块原本拥有的全部 mc² 能量。所以，这正好解决了我们之前的疑问。有没有可能从砖块中提取超过 mc² 的能量？不可能。有没有可能利用黑洞从砖块中提取完整的 mc² 能量？是的，可能。这其实非常巧妙，也是为什么人们谈论把黑洞当作发电厂。

So, I start off with a brick a very long way from the black hole, attach it to a rope, slowly lower the brick down towards the event horizon. Of course, I can't lower it past the event horizon, otherwise I'll lose control of the brick, but I lower it as, you know, right above the event horizon, the last possible place I can lower it to, and then just let go of it with zero velocity. 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. And so, it exactly resolves this question we had. Is it possible to extract more than mc² from the brick? No. Is it possible to extract the full mc² from the brick using a black hole? Yes, it is. And that's actually pretty neat and why people talk about using black holes as power plants.

你知道，如今大多数发电厂都是通过燃烧化学能来运作的。那效率并不高。你不得不付出 10 的负 10 次方量级的代价，因为化学键相对于物体的静质量来说非常弱。你实际上只提取了所考虑燃料静质量的极小一部分。你可以从那里升级，转向核能，它不再处理原子之间微弱的电磁键，而是开始关注原子核内质子和中子之间的核力。所以你可以从大约 10 的负 10 次方提升到裂变的 10 的负 3 次方，或者聚变的 10 的负 2 次方。但即使有裂变和聚变，这也就是你能达到的最好水平了，因为即使你能从强核力中提取能量，裂变和聚变都不会改变你过程中质子加中子的总数。而大部分能量，99% 的能量，不是储存在电磁相互作用中，也不是储存在强相互作用中，而是储存在质子和中子的静质量能中，这是化学反应和核反应都无法触及的。但引力可以触及它们。如果我一开始有一个质量为 M 的物体，我基本上可以提取，除了量子修正之外，基本上可以提取投入的静质量能量的 100%。这是效率最高的发电厂，因为通过建造这样的装置，原则上我可以提取我一开始拥有的任何能量的 100%。

So, you know, most power plants today operate by burning chemical energy. That is not very efficient. You have to 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 considering. You can level up from there by going to nuclear energy, which instead of dealing with the feeble electromagnetic bonds between atoms, starts to concern itself with the nuclear forces between the protons and the neutrons within the nucleus. 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. But that's about as good as you can go even with fission and fusion, 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. And the bulk of the energy, 99% of the energy is stored not in the electromagnetic interaction, not in the strong interaction, but in the rest mass energy of the protons and neutrons, something that neither chemical reactions nor nuclear reactions can touch. But gravity can touch them. If I start off with a mass object of mass M, I can extract up to the quantum corrections essentially, I can extract essentially 100% of the rest mass energy that have gone in. It is the most efficient possible power plant because by building an apparatus like this, in principle, I can extract 100% of the energy of whatever I started with.

我直觉上理解能量等于质量，然后像这些化学键，它们被破坏时会释放能量，如果这些键被释放，物体就会变轻。我甚至理解如果质子和中子之间的键被打破，会释放能量，使物体质量变小。但如果一个带有质子和中子的物体刚好在事件视界上方，是不是可以解释为那些质子和中子就在那个点停止存在了？它们拥有原来质量的 1%、2% 或 5% 到底意味着什么？

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 those bonds are released. 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. 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? What does it even mean for them to have 1% or 2% or 5% of their original mass?

是的，这是个很好的问题，而且一旦你引入量子力学，它就变得非常相关，不过这超出了今天讨论的范围。但在经典情况下，黑洞只是永远待在那里，所以你可以说：“好吧，质子和中子怎么了？”你说：“它们现在住在黑洞里。”而宇宙中质子加中子的总数仍然是守恒的。只是你需要给黑洞本身赋予一个所谓的核子数。在经典层面，这没问题。在量子力学层面，这远远超出了今天讲座的范围，霍金和贝肯斯坦发现黑洞会辐射能量，最终黑洞会消失，如果你计算的话，所有能量最终会变成引力子和光子，也许还有一些中微子。其中没有或几乎没有变成质子和中子。所以，一旦你引入量子引力，一个非常有趣的事实是，黑洞会吞噬核子数。这个看似至少在微扰论上被电磁力和核力都守恒的东西，最终却被引力吞噬了。人们喜欢把这一点从仅关于量子引力提升为一般原理：量子引力不尊重任何全局对称性。它不尊重核子数对称性。

Yeah, this is a great question and really becomes relevant once you turn on quantum mechanics, which is beyond the scope of today's discussion, but classically the black hole just sits there forever and so you can just say, "Well, what happened to the protons and neutrons?" You say, "Well, they now live inside the black hole." And the number of protons plus neutrons is still conserved out there in the universe. It's just you need to assign a what's called a nucleon number to the black hole itself. That's fine as far as it goes classically. Quantum mechanically, way beyond the scope of today's lecture, Hawking and Bekenstein discovered that black holes radiate away energy and 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. None of it or almost none of it ends up in protons and neutrons. So, it is a very interesting fact once you turn on quantum gravity, that black holes eat nucleon number. This thing that seems like it's conserved at least perturbatively both by electromagnetism and by the nuclear forces, ends up being eaten by gravity. And people like to promote this with only about quantum gravity now to a general principle: quantum gravity doesn't respect any global symmetries. It doesn't respect nucleon number symmetry.

它不遵守这些对称性。那是另一类问题，我们改天再展开。

It doesn't respect any of these symmetries. And that's a whole other kind of worms that we can open some other day.

我最近写了一篇博客，推测训练过程中的样本效率实际上在过去几年并没有提升太多，而是我们大幅改进并拓宽了数据分布。最近我和朋友吃晚饭时，想到了一个方法，可以获取关于这个问题的经验性信息。有个 nano GPT 速通比赛，人们竞相用越来越少的算力把 Karpathy 的 GPT-2 基线训练到固定损失。训练数据是固定的，所以我想，每个纪录的损失曲线随时间的变化，或许能大致反映样本效率提升的速度。于是我掏出手机，把这个想法录成语音备忘录，存进 Cursor 应用，然后回去继续吃饭。大约 50 分钟后，我收到通知，Cursor 智能体已经克隆了修改版的 nano GPT 仓库，分析了所有纪录的损失曲线，并估计样本效率每年提升约 2 到 5 倍。当然，这是非常粗略和间接的证据，但它启发我和一位朋友开始写一篇完整的文章，用多种不同方法研究这个问题。这里的摩擦成本真的很重要。如果我不能当场用 Cursor 应用启动调查，这个想法可能就飘走了。如果你想试用 Cursor 的 iOS 应用，请访问 cursor.com/the war cache。好了，Adam。我喜欢认为这个播客不仅传授理论知识，也传授实用知识。

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. 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. There's this nano GPT speed run where people compete to train Karpathy's GPT-2 baseline to a fixed loss with less and less compute. 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. So, I pulled up my phone, I dubbed this idea into a voice note in the Cursor app, and I went back to dinner. And then I got a notification about 50 minutes later, the Cursor agent had cloned the modded nano GPT repo, it had analyzed all the loss curves for all the records, and it had estimated that sample efficiency had been improving about two to five x every single year. Of course, this 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. And the friction really mattered here. 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. If you want to try Cursor's iOS app, go to cursor.com/the war cache. Okay, Adam. I like to think on this podcast we impart not only theoretical but practical knowledge as well.

那么，假设一个人学会了所有这些方程，但随后不幸落入黑洞，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。实际上，你可以从两个不同的视角来看。一个是我看着你掉进黑洞的视角，另一个是你自己掉进黑洞的视角。这两个视角是自洽的，但有趣的是它们截然不同。所以，也许我应该把两者都描述一下。首先，我们问：当你掉进黑洞时，我看到了什么？我看到的是——当然，这取决于你需要多努力地发射火箭来避免掉进黑洞，但你不会这么做。你只会坐在离黑洞很远很远的地方，关掉火箭，接受将要发生的一切。接下来，你会慢慢加速向黑洞靠近，起初遵循牛顿公式，然后当你接近黑洞时，开始出现修正，即广义相对论对牛顿平方反比定律的修正。所以，当我看着你掉向黑洞时，首先你会越来越快、越来越快地掉向黑洞，因为你正沿着黑洞的引力势下落。越来越快、越来越快，但随后奇怪的事情发生了：你不再加速，反而开始减速。你减速的原因是，在我观察你时，你开始受到引力时间膨胀的影响，我看到你的动作变慢，你的时钟变慢。这是如果你静止时的公式，但你不是静止的，你在运动，所以有另一个公式，但效果相同：当你越来越接近黑洞时，你的手表会越走越慢。事实上，如果你做适当的积分，我永远不会看到你穿过事件视界。我只看到你越来越接近事件视界，但越来越慢、越来越慢。而且，当我看着你时，我大概是用光来看你的，那光会越来越红移，波长越来越长。光的波长越长，就越难真正看到你。你开始被光的波长去局域化。最终，我完全看不到你了。我看到你发出的最后一个光子，然后你就淡入黑暗，从红色淡入黑色。我从未看到你穿过事件视界。这在广义相对论早期就被人们注意到，并让他们大为困惑。他们开始认为，你自己在穿过事件视界时会经历某种奇怪的事情。但事实并非如此。事实上，如果我采用你的视角，从你的角度看，你的时钟并没有变慢，它以每秒一秒的速度运行。如果你回头看我，可能会看到一些奇怪的事情，比如我可能跑得很快，但对你来说，一切完全正常。你加速向黑洞靠近，越来越快、越来越快，然后你完全正常地滑过事件视界。事件视界对你来说并不是一个特别暴力的地方。你可以计算潮汐力，当你接近并穿过事件视界时，它们并不特别大。或者说，对于大质量黑洞，它们并不特别大。对于太阳质量的黑洞，潮汐力会相当大，对你来说会很痛苦。你会发现你的脚被黑洞吸引的力比头大得多，因为脚更近，你最终会被拉长。但如果我选一个足够大的黑洞，你不会注意到任何奇怪的事情。黑洞越大，潮汐效应越小。如果我选一个星系质量的黑洞，你穿过事件视界时基本上没事。如果我选一个更大的黑洞，你可以在穿过事件视界后度过余生，然后才撞上奇点，那才是致命的。当你穿过事件视界时，你注定要完蛋。你注定要完蛋，因为一旦你穿过事件视界，你必须走向奇点。你无法通过发射火箭来阻止自己撞上奇点。你注定要完蛋，但你还没死。只有当你撞上奇点，被潮汐力撕成意大利面状时，你才必死无疑。但对于足够大的黑洞，你可以注定要完蛋却浑然不知。事件视界实际上不是一个局部可测量的量。它是一个目的论事实。它说一旦你穿过事件视界，你必须走向奇点，但对于足够大的黑洞，到达那里可能需要很长时间。

Great question. So, there's actually two different perspectives you could take. One is the perspective of me watching you falling into the black hole. The other is the perspective of you falling into the black hole. And those two perspectives are consistent with each other, but interestingly different. So, maybe I should describe them both. So, first let's ask the question, what do I see as you fall into the black hole? 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 into the black hole, but you're not going to do this. You're just going to sit here, a long, long way away from the black hole. You're going to turn off your rocket and accept what comes. 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. Um 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. Uh 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. 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. This is the formula for if you were static, you're not static, you're moving, so there's a some other formula, 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. And in fact, if you do the appropriate integral, I never see you cross the event horizon. 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. Uh and as I watch you, I'm presumably using light to watch you, that light gets more and more redshifted. The wavelength gets longer and longer. Uh the longer the wavelength of light, the harder it is even to really see you. I start getting You start getting delocalized by the wavelength of the light. And eventually I just stop seeing you entirely. There's the a final photon that I see that that you emit, and then you just fade to black, fade through red to black. I never see you cross the event horizon. Uh 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. That is that is not true. In fact, if I instead adopt the perspective of you, from your point of view, your clock isn't running slow, it's running at 1 second per second. If you look back at me, there's some some funny stuff going on to do with me me running fast, perhaps, but as far as you're concerned, everything's totally normal. You accelerate towards the black hole. 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. The event horizon is not a particularly violent place for you. You can calculate the tidal forces um as you cross the event horizon or as you approach and then across the event horizon, and they're not particularly big. Uh or rather for large black holes, they're not particularly big. For a solar mass black hole, they they would be pretty big and would be pretty painful for you. You'd find that the your feet are being attracted to the black hole uh much more vigorously than your head is cuz it's cuz it's cuz they're closer, uh and you end up getting getting stretched. But if I take a big enough black hole, uh you wouldn't notice anything funny happening whatsoever. Um the the bigger the black hole, the smaller the the tidal effects. And if I took a black hole the mass of the galaxy, uh you'd be basically fine as you cross the event horizon. If I took an even bigger black hole than that, uh you could live out your entire life uh having crossed the event horizon of the black hole um before you hit the singularity, which is fatal. Um When you cross the event horizon, uh you are doomed. You are doomed because uh once you cross the event horizon, you must proceed to the singularity. There's no way you can fire a rocket to stop yourself uh hitting the singularity. You are doomed, but you are not dead. Uh you are only for sure dead once you hit the singularity and get spaghettified, get mangled by the by the tidal forces. But for a large enough black hole, uh you can be doomed and not even know it. Uh the event horizon is is really a not locally measurable quantity. It is a teleological fact. 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.

原则上，对于一个足够大的黑洞，如果它横跨许多光世纪，你可以在里面度过一生。你可以活一辈子，你的后代也都可以生活在黑洞内部，只有当你真正接近奇点时，潮汐力才会变得强大并杀死你。

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. You could live out your life, you could have descendants, all of whom live inside the black hole, 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，有些我们认为是对的，有些我们不知道对不对。为什么你认为黑洞是对的，而虫洞不是？

As you're saying, GR explains or predicts a lot of phenomena, and some we think are correct, some we don't know are correct. Why do you think black holes are correct, but not wormholes?

是的，这是个好问题，而且一开始人们并不相信。施瓦西在爱因斯坦写下场方程后几乎立刻就得出了他的解。人们认为那个方程在某种程度上是有问题的。那是一个测度为零、永远不会发生的事情。那是个数学怪物，但在真实宇宙中不可能自然形成黑洞。他们错了，因为黑洞确实存在。我们现在非常确信。有理论发展，也有实验证据表明黑洞存在。最大的理论发展是彭罗斯，后来还有霍金和彭罗斯，彭罗斯因此获得了诺贝尔奖，他们在理论上证明了黑洞的形成是广义相对论的一个普遍特征。它不仅仅是在微调初始条件时才会发生的某种病态现象，而是只要从一般的初始条件出发，黑洞的形成就是其中的普遍特征。那是一个巨大的发展，然后是实验方面。我们现在拥有的黑洞实验证据已经非常多了。在爱因斯坦时代并不存在，在爱因斯坦之后 50 年里，人们对黑洞非常困惑，认为它们不存在。但证据有很多。我认为最直观的证据就是观察我们银河系的中心。如果你看银河系的中心，剧透警告，那里有一个黑洞。我们称之为人马座 A*，一个巨大的黑洞，质量是太阳的数百万倍。你看不到黑洞，至少不能直接看到，因为它是黑的。你能看到的是它周围的恒星。如果你在几十年的时间里观察这些恒星，我们现在已经有几十年的观测数据了，你会看到这些恒星不是沿着我们所说的直线运动，而是沿着漂亮的小椭圆或进动椭圆运动，这些椭圆看起来像是在围绕某个东西公转。你看不到那个东西，但你可以看到围绕它公转的恒星，你可以计算出它有多大、有多重，你会发现它确实非常巨大，而且也非常小。你知道它小是因为恒星非常靠近它，但似乎没有与它相撞。所以通过追踪这些轨道，你可以判断出在那个星系中心有一个超重、超暗、超致密的东西，那就是人马座 A*，那个星系中心的黑洞。这是一个令人信服的证据。另一个令人信服的证据大约在十年前，我们不仅看到了黑洞，还感受到了它们。LIGO 是一个巨大的干涉仪，激光干涉仪，我们建在多个地点，对时空本身的振动超级敏感。有一个著名的事件，几乎在我们 2015 年底开启它之后立刻发生，我们感受到了时空的震动。你知道那是时空在震动，而不仅仅是地球在震动，因为我们有一堆这样的探测器，当时两个，现在四个，分布在地球不同位置，它们都以完全相同的方式震动。所以不能仅仅用一辆卡车经过一个探测器而不是另一个，或者一个地方发生地震而另一个没有来解释。它们都以完全相同的方式震动，我们能够反推计算出导致它们震动的原因是两颗巨大黑洞的碰撞，两颗黑洞的质量都约为太阳的 30 倍，在宇宙的另一边，大约 16 亿光年之外。那次碰撞发生在大约 16 亿年前，恰好在我们开启 LIGO 探测器后的几周内到达地球。我们现在已经看到了，更准确地说，感受到了数千次这样的震动，对应着数千次黑洞合并。然后还有更多证据。后来，我们有了所谓的事件视界望远镜，这是一个巨大的射电望远镜阵列，分布在地球各地，能够非常近距离地观察我们银河系中心的黑洞人马座 A*，以及我们邻近星系中心的更大的黑洞，并非常微弱地看到物质落入这些黑洞时发出的射电辐射，这些物质在坠落时发出极其明亮的光，我们能够通过射电辐射看到这些。所以，我们感受到了它们。我们看到了它们。我们看到了它们对轨道恒星产生的引力效应。在这个阶段，我们非常确信黑洞存在。

Yeah, that's a great question, and people did not believe it to begin with. Schwarzschild wrote down his solution almost immediately after Einstein wrote his field equations. People thought that that equation was sick in some way. It was a measure zero thing that would never happen. It was some mathematical monstrosity, but it was impossible to make black holes naturally in the real universe. And they were wrong, because black holes do exist. We're extremely confident now. There were theoretical developments, and there was experimental evidence that black holes exist. The biggest theoretical development was Penrose, and then later Hawking and Penrose, for which he won the Nobel Prize, who showed theoretically that the formation of black holes is a generic feature of general relativity. It's not just some sick thing that happens if you fine-tune the initial conditions, but if you just start off with generic initial conditions, the development of black holes is a generic feature of that. That was a huge development, and then there was the experimental side. And the pieces of experimental evidence we have for black holes is now huge. It did not exist in Einstein's day, and for 50 years after Einstein, people were extremely confused about black holes and thought they didn't exist. But there are numerous pieces of evidence. I think the most visually appealing piece of evidence is just observing the center of our galaxy. So, if you look at the center of the galaxy, there is, I mean, spoiler alert, there is a black hole there. We call it Sagittarius A*, a huge black hole, weighs many millions of times the mass of the sun. You can't see the black hole, or at least you can't see it directly, because it's black. What you can see is the stars around it. 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 nice little ellipses or precessing ellipses, and those ellipses look like they are orbiting something. You cannot see the something, but you can see the stars that are orbiting that something, and you can calculate how big it is, how massive it is, and what you find is that it's very massive indeed, and it's also very small indeed. You know it's small because the stars get super close to it, but don't seem to collide with it. 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 that is Sagittarius A*, the black hole at the center of that galaxy. That's one compelling piece of evidence. Another piece of compelling evidence was about a decade ago, we not only saw black holes, we felt them. 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. And there's a famous event pretty much immediately after we turned it on in late 2015 where we felt spacetime shaking. And you know it was spacetime shaking, not just the earth shaking, because we had a bunch of these detectors, then two, now four, at different points on the earth, and they all shook in exactly the same way. So it couldn't just be explained by a truck passing one and not the other or a seismic event on one and not the other. 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 weighed about 30 times as much as the sun on the other side of the universe, about 1.6 billion light years away. That collision happened about 1.6 billion years ago and just happened to reach the earth within weeks of us turning on the LIGO detectors. We've now seen, we've now felt, more accurately, thousands of such shakings corresponding to thousands of black hole mergers. And then there's more evidence. 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*, and the even bigger black hole at the center of 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're able to see that in the radio emissions of these things. So, we felt them. We've seen them. We've seen their gravitational effects on orbiting stars. We're extremely confident at this stage that black holes exist.

太美了，不仅一个头脑能想出这个理论，而且这个理论有如此大的影响力，我们还能想出各种机制来评估、扰动和理解它在许多不同疯狂方面的含义。

It's so beautiful that not only can a single mind come up with this theory, but then the theory has so much reach and that we can come up with the machinery to evaluate and perturb and understand its implications in so many different wild ways.

它覆盖的自由度数量、数量级跨度真是疯狂。它涵盖，你知道，你最初通过做关于在电梯里跳上跳下的思想实验来思考它，然后它延伸到描述水星的轨道和太阳系内可探测的轨道动力学扰动，然后是光的弯曲，然后它描述整个星系的旋转，然后它描述整个宇宙的膨胀和潜在的命运。那确实是许多数量级，而且令人印象深刻的是，这几乎是单一个体的工作，坦率地说，我们的宇宙应该感到荣幸，能被如此美丽的理论所描述。

It's crazy the number of degrees of freedom, the number of orders of magnitude that it covers. It covers, you know, 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, and then it describes the rotation of the entire galaxy, and then it describes the expansion and potential, you know, fate of the entire universe. That's many orders of magnitude indeed, and it's pretty impressive that it was the work of almost a single mind, and frankly, our universe should be honored to be described by such a beautiful theory.

你能讲讲广义相对论如何从爱因斯坦的一个理论变成世界相信为真的东西的故事吗？

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，是的，那就是光的弯曲。之前牛顿物理学就有已知的异常，比如我们无法精确计算水星的轨道。而广义相对论早期的一个很好的检验就是它确实精确地得到了水星的轨道。所以，那是一个相当好的确认。但同时，这并不太令人满意，因为那是一个已知的数字。

Oh, yeah, that would be the bending of light. So, there were known anomalies with Newton's physics beforehand, like we couldn't get the orbit of Mercury exactly right. And one of the very nice early tests of general relativity is that it did get the orbit of Mercury exactly right. So, that was a pretty good confirmation. But, at the same time, that's not quite so satisfying because it was a number that's already known.

但你发明一个理论，然后它正确预测了——如果你在事先不知道正确答案的情况下得到正确答案，那就更令人印象深刻。所以那就是光的弯曲。当然，从历史上看，那是最有影响力的。根据广义相对论，所有能量都有引力，所有能量都受引力影响。所以光在经过像太阳这样的大质量天体时，会朝太阳方向弯曲。实际上，牛顿物理学中也会发生同样的情况。如果你假设有一个粒子在运动，你知道它经过太阳时弯曲多少取决于它的碰撞参数，但也取决于它的速度。速度越快，弯曲越小。所以你取那个牛顿公式，代入速度等于光速，你会得到牛顿物理学中的一定弯曲量。而在广义相对论中，你可以做同样的计算，实际上你会得到牛顿答案的两倍。

But you invent a theory and then it correctly predicts—it's considered more impressive if you get the right answer without knowing what the right answer is in advance. So that would be the bending of light. Certainly historically, that was the most influential. According to general relativity, all energy gravitates and all energy is affected by gravity. So light, as it passes a massive object like the sun, gets bent in the direction of the sun. Actually, the same happens in Newtonian physics. If you suppose you have a particle going along, you know how much it gets bent as it passes the sun depending on its impact parameter, but also its velocity. The faster it goes, the less it gets bent. So you take that Newtonian formula, plug in velocity equals the speed of light, and you get a certain amount of bending through Newtonian physics. And in general relativity, you can do the same calculation and you actually get double the Newtonian answer.

所以这是个大事。

So this was a big deal.

我的意思是，这段历史有点奇怪。在他完成广义相对论之前，爱因斯坦基于他认为是等效原理的东西，基于他对等效原理的理解，对答案应该是什么做出了预测。所以他写下了答案，而作为回应，他和其他一些对此感兴趣的人，有人派出远征队去尝试测量。实际上，我认为爱因斯坦做的第一件事是打电话给天文台，说：“你能测量太阳后面的遥远恒星，并测量光经过太阳时如何弯曲吗？”这是真正的理论家之举，因为威尔逊山天文台的台长说：“绝对不行，我们不能那样做。如果你把望远镜对准太阳，你会失明。如果你把它对准太阳旁边，你会被太阳的日冕淹没，什么也看不见。”但有一种情况你不会被太阳淹没，那就是在日全食期间，月球挡住太阳，你就能看到非常靠近太阳的恒星，并测量它们背后光的弯曲。

I mean, slightly strange history of it. 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. So he wrote down the answer, and in response to him and a number of other people being interested in this, there were people sending out expeditions to go and try to measure it. Actually, I think the very first thing that Einstein did was he phoned up the observatory and said, "Can you measure distant stars behind the sun and measure how light bends as it passes the sun?" This was the true theorist move, because the director of the Mount Wilson Observatory said, "Absolutely not, we cannot do that. If you point a telescope at the sun, you'll go blind. If you point it just next to the sun, you'll just get washed out by the corona of the sun and you won't see anything." 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 1910 年代，有一批远征队被派去测量光的偏转。

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

他们会停在全食路径上，用望远镜观察紧挨着太阳的恒星，看它们是否在天空中移动，如果移动了，移动了多少。我认为他们做的第一次是在 1911 年。他们去了阿根廷观测日食，然后一切都准备好了。我的意思是，这就是问题所在——你一路到阿根廷，在那些日子里非常远，然后他们被云层淹没，什么也看不见，非常令人沮丧。然后下一个去的是由军火制造商克虏伯赞助的德国远征队，他们去了克里米亚，试图在那里测量。就在日食发生之前，第一次世界大战爆发了，现在德国和俄罗斯在交战，他们都被逮捕并关押到战争结束，所以那也失败了。

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. I think the first one they did was in 1911. They went to Argentina for an eclipse, and then everything was set up. I mean, this is the problem with this thing—you get there all the way to Argentina, a very long way in those days, and they're just washed out by the clouds and you don't see anything, and it's very frustrating. Then the next one that went along was a German expedition sponsored by the arms manufacturer Krupp, who went to the Crimea and tried to measure it there. And just before the solar eclipse happens, World War I breaks out, and now Germany and Russia are at war, and they're all arrested and turned for the rest of the war, so that also fails.

而他们全都失败，对爱因斯坦来说反而是件好事。

And it turned out to be a good thing for Einstein that they all failed.

实际上，他们全都失败对爱因斯坦来说是件好事，因为事实证明，爱因斯坦最初的、在他拥有完整广义相对论之前的等效原理论证是错误的，导致他预测广义相对论中的光弯曲会与牛顿物理学中的相同。所以在战争期间，当一切关闭，没有人考虑日食远征时，他纠正了这个错误，并提出了一个新的预测，实际上将是牛顿预测的两倍。然后在 1919 年，亚瑟·爱丁顿爵士发起了一次英国远征，去世界各地观测日食，并成功返回，宣布确实是爱因斯坦的预测——是牛顿预测的两倍。而这正是让爱因斯坦成为全球名人的原因——这个英国实验证实了一个德国起源的理论，是战后和解的一部分，而且爱因斯坦已经弄明白了一切。我想说，这就是广义相对论成为共识观点的时刻。人们对这个非常令人印象深刻的测试超级信服。

It actually turns out to be a good thing for Einstein that they all failed, because it turned out that Einstein's original, before he had full general relativity, his original equivalence principle argument was wrong and led him to predict that the bending of light in general relativity would be the same as it was in Newtonian physics. So during the war, while everything shut down and no one was thinking about eclipse expeditions, he corrects this mistake and comes up with a new prediction that actually will be double the Newtonian prediction. Then in 1919, Sir Arthur Eddington launches a British expedition to go and observe the eclipses all over the world and successfully comes back and declares that indeed it was the Einstein prediction—that it was double the Newtonian prediction. And that's really what launches Einstein as a global celebrity—this British experiment confirming a sort of German origin theory was part of the post-war reconciliation, and that Einstein had figured out everything. And that is, I'd say, the point at which general relativity became the consensus view. People were sort of super convinced of this very impressive test.

如今，我们做的测试比那多得多。

Nowadays, we've done huge more tests than that.

如今，我们做的测试比那多得多——非常精确的轨道动力学。你可以在水星的轨道上看到它，甚至在其他行星上也能看到。你可以测量光在传播过程中的红移，引力对光传播的影响，光能无处不在。但从历史上看，那是对广义相对论最令人印象深刻的确认。

Nowadays, we've done huge more tests than that—very precise orbital dynamics. You can see it in the orbit of Mercury and indeed even in the other planets. You can just measure the redshifting of light as it goes, the gravitational effect on the propagation of light, the energy of light all over the place. But historically, that was the most impressive confirmation of general relativity.

我想问的是，我们作为一个社会，基本上花费数十亿甚至数百亿美元来建造这些巨大的物理实验。而如果你看看也许是有史以来最美丽、最重要的物理学理论，它似乎只是一个家伙在洞穴里思考。那个理论的经验基础似乎只是知道光有速度。我的意思是，也许你需要通过实验测量 G。实际上不需要。G 是广义相对论中的一个自由参数。它并不需要。所以你说得对，经验基础相当薄弱。你不需要太多。而且理论物理学家相当便宜。有一个巨大的诱惑：我们为什么不把所有钱都花在理论物理学家身上，而不建造这些极其昂贵的实验呢？

What I was going to ask is, we are spending as a society billions, maybe tens of billions of dollars on building these huge physics experiments, basically. And if you look at maybe the most beautiful, the most important theory of physics ever conjured, it seems like a guy is just thinking in a cave. It seems like the empirical basis for that theory is maybe knowing that light has a speed. I mean, maybe you need to measure G experimentally. Not really. G is a free parameter in general relativity. It does not require it. So you're right, the empirical basis is pretty thin. You don't need much. And theoretical physicists are pretty cheap. There's a great temptation: why don't we just spend it all on theoretical physicists and not build these vastly expensive experiments?

公司确实在提高对理论物理学家的需求曲线，但没错。

Companies are really increasing the demand curve on the theoretical physicists, but yeah.

没错。不再那么便宜了。但那能让你走多远？

That's right. Not so cheap anymore. But how far can that get you?

我想说，广义相对论也许是那方面的一个极端。这不是物理学史上通常的情况。这确实更接近某个安·兰德式英雄独自坐着，是单一心智的产物。他以各种方式得到了很多帮助，但这确实只是一个他追求多年的独特愿景。而且它相当——他写下来，很多人几乎立刻印象深刻。它确实需要发起一次相当昂贵的日食远征来确认，然后他才真正成为全球名人，大多数人才被说服，但这也许是这方面最极端的例子之一，就是有人坐下来，努力思考，写下一个真实的理论。在某种意义上，物理学从那以后一直在追逐那种高潮。

I would say that general relativity is perhaps one extreme of that. That is not how it usually works in the history of physics. This really is closer to some Ayn Rand hero just sitting alone, the product of a single mind. He got lots of help in various ways, but it really was just a singular vision that he pursued for years. And it was pretty—he wrote it down and a lot of people were very impressed almost immediately. It did require launching a somewhat expensive eclipse expedition to go confirm it before he really achieved global celebrity and most people were sold on it, but it was perhaps one of the most extreme examples of this where just somebody sits down and thinks very hard and writes down a true theory. And in some sense, physics has been chasing that high ever since.

人们喜欢那种浪漫的自我想象：坐下来，凭很少的经验洞察，只是非常非常努力地思考，做思想实验。但这对其他人来说，通常不像对爱因斯坦那样奏效。事实上，在爱因斯坦职业生涯的后期，这甚至对他本人也不那么奏效。

People love that romantic vision of themselves just sitting down and thinking with very few empirical insights, just thinking very, very hard and doing thought experiments. And it's typically not worked out quite as well for everybody else as it worked out for Einstein. In fact, it didn't even work out that well for Einstein in the later part of his career.

仅靠思考能走多远？要建立广义相对论，你需要什么？

How far could you get just by thinking? What do you need to do general relativity?

你需要光速的有限性。你不仅要说服自己光速是有限的，还要相信存在一种保护它的对称性，也就是爱因斯坦在狭义相对论中提出的对称性。然后你可能需要等效原理，即惯性质量与引力质量对一切物体都相同这一经验事实。但这些信息相当稀疏，仅凭这两点，仍有一些可能的选择。但如果你有大量的大语言模型，你就可以探索整棵决策树。如果选项数量有限，你就可以遍历整棵树，说：“好，聚焦这个。等效原理非常重要，还有那个。”现在，放弃同时性，看看能走多远。所以，那里仍然只有有限数量的东西可以探索。我认为我们在广义相对论上非常非常幸运，它在那种情况下如此强大。但如果你有大量爱因斯坦，给每个不同的选项，你大概可以并行探索。

You need the finiteness of the speed of light. You need to convince yourself not just that the speed of light is finite, but that there's the symmetry that protects that, which Einstein came up with in special relativity. Then you probably want the equivalence principle, the empirical fact that the inertial mass and the gravitational mass are the same for everything. But that's pretty sparse, and from just those two things, there are still a few options. But if you have lots and lots of large language models, you can explore the entire tree. If there's only a limited number of options, you can just explore the entire tree and say, "Okay, focus on this. The equivalence principle is something that's super significant, and this other thing." Now, abandon simultaneity and see how far that takes you. So, there's still only a finite number of things to explore there. I think we got very, very lucky with general relativity, that it's quite so powerful under those circumstances. But if you just had lots and lots of Einsteins and you give each of them various options, you could presumably pursue them in parallel.

在当今物理学的前沿，根据你的经验，是否感觉如果让数百万个这样的模型自主运行，就能带来巨大的发现？还是说我们现在处于一个不同的时代，这种并行性的用处有限？

At the frontiers of physics today, in your experience, does it feel like if you just have millions of them running autonomously, you could have enormous discoveries, or are we in a different era now, and there would be limited usefulness of that parallelism?

我认为是有用的。我确实认为科学的不同分支有不同的分叉率，以及你需要多少实验来剪断那个分支。

I think there is usefulness. I do think that different parts of science have different branching fractions and how much you need experiment to cut off that branch.

嗯。

Yeah.

你知道，我谈到了追寻爱因斯坦的高度。可以说，弦理论真的在这条路上全力以赴。所以，爱因斯坦的理论只是广义相对论。还有量子力学，试图以一致的方式将两者结合，而广义相对论完全没有做到这一点。广义相对论中没有量子力学。如何做到这一点激励了许多人。问题是，要在实验中验证这一点，如果只做量纲分析，你需要巨大的粒子对撞机，绝对是星系级别的粒子对撞机。要看到那些东西非常困难。但这并没有阻止人们。我的意思是，它阻止了许多人，但许多人没有停止，继续尝试。所以，在那里你只能希望它像广义相对论那样成功，仅凭非常非常努力的思考，加上极少的实验输入，就能摸索出正确的答案。而要做到这一点，可能的自洽理论的数量必须很少。你手头的工具是数学自洽性，以及它是否能正确还原到已知极限。所以，如果你要这样做，你最好希望只有一个或极少数自洽理论能够成立。如果结果是有无限多个自洽理论，你永远无法摸索出正确的答案，因为它们都自洽，而你唯一的工具是自洽性，也许还有一些美学观念。但如果只有少数几个，那么也许你可以一路走下去。所以，弦理论有点必须在这条路上全力以赴，我认为，就是试图以极少的实验输入，相信只有一个自洽的引力理论，并且通过足够的自洽性检验，你就能找到它。但对于其他例子，那就困难得多。此外，很明显在凝聚态物理中，你常常只需要去做实验，才能知道哪个是正确的。

You know, I talked about chasing the higher of Einstein. Arguably, string theory has really been going all in on that. So, Einstein's theory is just general relativity. There's also quantum mechanics, and trying to marry those two in a consistent way that general relativity doesn't do at all. There's no quantum mechanics in general relativity. How to do that has motivated a lot of people. The problem is that in order to see that in experiments, if you just do the dimensional analysis, you need ginormous particle colliders, absolutely galactic-size particle colliders. It's just very hard to see any of that stuff. But that didn't stop people. I mean, it stopped many people, but many people didn't stop and kept trying to do it. So there you just have to kind of hope that it works out sort of like it did with general relativity, where just by thinking very, very hard with minimal input from experiment, you can feel your way to the right answer. And for that to be true, it needs to be the case that the number of possible consistent theories is small. The tool you have at your disposal is mathematical consistency, and does it reduce correctly to the known limits. So you better hope that there's only one or a very small number of possible consistent theories that would work out if you're going to do that. If it turns out that there's an unlimited number of consistent theories, you're never going to feel your way to the correct answer because they're all consistent, and the only tool you have is consistency and perhaps some notion of aesthetics. But if there's only a few, then maybe you could do it all the way. And so string theory has kind of got to go all in on that, I would say, just trying with minimal experimental input, believing that there's only one consistent theory of gravity, and just by doing sufficient consistency checks, you can find it. But for other examples, it's much harder. Plus, clearly in condensed matter physics, often you simply need to go and do an experiment to find out which one is correct.

当我们未来的 AI 文明确实提出越来越统一的物理理论或更深层的理论，做出更好的预测时，你认为人类能否跟上？一旦迈出这一步，我们是否真的能够理解我们的 AI 文明所理解的东西？

Whenever our future AI civilization does come up with more and more unified theories of physics or deeper theories, make better predictions, do you think that humans will be able to keep up, as once this sort of step is taken, that we will actually be in a position to understand what our AI civilization understands?

我不知道我们能否完全跟上，但我认为我们会比悲观预测所暗示的要好得多。那么，让我们以数学为例，它比物理更简单。在数学中，许多数学家担心这些大语言模型会变成证明机器。陶哲轩有一个说法叫“消化不良”，指的是这些大语言模型会产生数十亿行难以理解的 Lean 代码，作为某个定理为真的证明，却不提供任何关于为何为真的洞见。数学家们说，那难道不是一个令人沮丧的世界吗？我认为那是一种可能的未来，但我实际上并不认为那是很可能的未来。因为除了作为超人的证明者，我们也期望这些大语言模型成为超人的解释者。也许它们会做的恰恰相反。也许它们会做的是，把那些很难理解的证明，通过坚持不懈地尝试再尝试，最终找到人类能理解的方式。它们将能够把难以理解的证明变得容易理解。我认为经验证据，虽然还早，但相当支持这种更积极的愿景。所以，如果你看一个几个月前被证明的 Erdős 问题，它不仅仅是一套难以理解的 Lean 事实；它甚至不是用 Lean 证明的，尽管它是一个非正式证明。然后有一些人类数学家写了后续论文，采用了机器提出的这些新的、人类可解释的想法来证明 Erdős 猜想，并用它来证明以前没有用过的新定理。所以那恰恰是相反的情况。它提出了一个非常人类可解释的想法，然后人类能够完全理解它，并且理解得如此之好，以至于他们能够在新场景中部署它。我们到处都看到这种情况。我的意思是，单位距离猜想，我认为，对于你一直在讨论的许多主题来说，是一个很好的例子。

I don't know that we're going to be able to keep up entirely, but I think we'll keep up pretty much better than pessimistic forecasts would suggest. So, let's take mathematics as a simpler example than physics. In mathematics, many mathematicians are worried that these LLMs are just going to turn into proof machines. Terry Tao has this phrase "indigestion" he uses, in which these LLMs will produce billion-line inscrutable Lean codes that will serve as a certificate that a particular theorem is true without providing any insight as to why that might be true. And wouldn't that be a depressing world, say the mathematicians. I think that is a possible future, but I actually don't find that to be a very likely future. Because as well as being superhuman provers, we also expect these large language models to be superhuman explainers. And maybe they'll do the exact opposite of that. Maybe what they'll do is they'll take proofs that are very hard to understand and, by doggedly trying and trying and trying, they will be able to come up with ways that are human comprehensible. They will be able to take proofs that are difficult to understand and make them easy to understand. And I think the empirical evidence, it's early days, but it's pretty supportive of that more positive vision. So, if you look at an Erdős problem that was proved a few months ago now, it wasn't just an incomprehensible set of Lean facts; it wasn't even proved in Lean, though it was a proof proved informally. And then there was a follow-up paper by some human mathematicians that took these new ideas, human-interpretable ideas that the machine had come up with to prove this Erdős conjecture, and used it to prove new theorems that had not been used before. So that was the exact opposite of that case. It came up with a very human-interpretable idea, and then humans were able to fully comprehend it, and comprehend it so well they were able to deploy it in a new scenario. We've seen that throughout. I mean, the unit distance conjecture, I think, is a good example here for a number of the themes you've been discussing.

其一，这完全是可以理解的。单位距离猜想的反例出现了，它给出了……我的意思是，对数学家来说——我不是数学家——人类之所以没能推翻单位距离猜想，或许是因为他们错误地相信这个猜想是真的。而大语言模型的好处在于，它们愿意突破这个障碍，愿意像人类理解的那样“浪费时间”，去尝试推翻一个被认为为真的猜想，并最终到达另一端。所以，这是大语言模型让我相当乐观的另一个方面：它们有极大的耐心，即使去做那些看起来成功概率很低的事情。

One is that it's totally comprehensible. The disproof of the unit distance conjecture came up, and it came up with... I mean, to mathematicians—I'm not a mathematician—perhaps the reason that humans haven't disproved the unit distance conjecture is because they erroneously believe the conjecture to be true. And the good thing about large language models is that they're willing to push through that barrier and just waste their time, as a human would understand it, trying to disprove a presumed true conjecture and reach the other end. So that's another aspect of large language models that makes me pretty optimistic: they just have extreme patience, even for doing things that perhaps look like low probability of success.

嗯。

Mhm.

Adam，非常感谢你再次来到节目，在你可以去构建超级智能的时候来做这件事。你在解释 100 年前的物理学，但这非常有趣。

Adam, thanks so much for coming back on and doing this while you could be building super intelligence. You're explaining 100-year-old physics, but it was very interesting.

这是一个非常有趣的话题。我非常乐意分享。

It's a super fun subject. I'm super happy to share it.