DeepMind's Brown Uses Equivalence Principle to Reframe Relativity

DeepMind研究员Brown借助等效原理重构相对论 广义相对论证明，黑洞能在事件视界将物体的全部静止质量转化为功，从而成为宇宙中效率最高的能量提取器。 Adam Brown详细解析了Albert Einstein从狭义相对论迈向广义相对论的革命性飞跃，展示了引力质量与惯性质量的严格等效性如何揭示引力实质上是弯曲时空的效应，而非传统的牛顿力。对话探讨了黑洞的机制，说明了黑洞如何作为终极能量提取器运作、引发严重的引力时间膨胀，并在坠入者与遥远观测者之间产生巨大的视角差异。最后，Brown探讨了广义相对论的实验证明（包括1919年日食期间的引力透镜效应），并思考了先进AI是否能通过探索理论决策树来发现新的物理定律。 广义相对论消解了牛顿引力瞬时超距作用与狭义相对论光速上限的根本冲突，通过爱因斯坦场方程确立了“物质告诉时空如何弯曲，时空告诉物质如何运动”的物理图景。 (3:50) 爱因斯坦敏锐捕捉到惯性质量与引力质量严格等效的线索，指出引力本质上并非外力，而是物体在弯曲时空中沿测地线运动时所表现出的惯性力。 (20:00) 卡尔·施瓦西于1916年求解出静态球对称精确解，预言了逃逸速度等同光速的事件视界；而在三个施瓦西半径之内，物体轨道动能自身产生的引力将压倒离心力，使所有稳定轨道走向崩溃。 (31:46) 引力时间膨胀具有单向不对称性，位于引力势阱深处的时钟真实流逝速度更慢，这一效应直接导致引力红移，并已在庞德-雷布卡实验及现代 GPS 卫星定位系统中得到精确验证。 (55:00)

Theoretical physicist Adam Brown explains how Albert Einstein resolved the conflict between Newtonian gravity and special relativity by redefining gravity as an inertial force produced by curved spacetime. This geometric reinterpretation resolves classical energy paradoxes, predicts black holes, and demonstrates the power of deductive thought—a process Brown argues artificial intelligence can extend across modern physics. Gravity as an Inertial Force in Curved Spacetime (3:50) Sir Isaac Newton’s law of universal gravitation dictated that gravitational force depended instantaneously on distance, violating special relativity’s fundamental principle that no signal can travel faster than light. Unlike electrostatics—where charges can repel and electric charge is independent of mass—gravitational mass is identical to inertial mass, an equivalence verified experimentally to one part in 10¹⁵. Einstein recognized that this exact equivalence is the defining signature of fictitious or inertial forces, such as the centrifugal force holding water inside a spinning bucket. Because an inertial force's strength is strictly determined by inertial mass, gravity could itself be understood as an inertial effect. This insight required redefining straight-line motion. In general relativity, an object in free fall experiences no net force and traces a natural "straight line" or geodesic through curved spacetime. Conversely, a person sitting stationary on Earth is continuously accelerated away from a geodesic by mechanical force from the ground. Einstein formalized this principle in his 1915 field equations, which describe how the stress-energy of matter and radiation dictates the geometric curvature of spacetime. Energy Limits, Time Dilation, and Event Horizons (35:00) In Newtonian gravity, slowly lowering a mass on a tether extracts gravitational potential energy. Dividing this energy by rest-mass energy (mc²) yields a dimensionless fraction, GM/(c²R). At Earth’s surface, this ratio is roughly 7 × 10⁻¹⁰. By pure coincidence, this value closely tracks the chemical energy released by liquid hydrogen-oxygen fuel (~1.5 × 10⁻¹⁰), requiring chemical rockets to burn nearly their entire mass in fuel just to escape Earth's gravity well. If an object is compressed past R ≤ GM/c², classical physics paradoxically predicts extracting over 100% of rest-mass energy. General relativity resolves this paradox: the Schwarzschild metric shows that gravitational acceleration diverges at the Schwarzschild radius (r = 2GM/c²), snapping cables and establishing an event horizon where staying static requires infinite acceleration. Near the event horizon, lowering a mass allows an external observer to theoretically extract nearly 100% of its rest-mass energy—far exceeding chemical reactions (10⁻¹⁰) or nuclear fusion (10⁻²). The metric also dictates gravitational time dilation: clocks deeper in a gravitational well run slower relative to clocks far away, an asymmetry confirmed by the 1950s Pound-Rebka experiment and required for operational GPS navigation. Furthermore, because all energy gravitates in general relativity, an orbiting object’s kinetic energy increases its gravitational attraction, causing orbital stability to collapse entirely inside r = 3GM/c². Empirical Proof from Solar Eclipses to Gravitational Waves (78:51) Karl Schwarzschild derived the exact solution for a spherical mass in 1916 on the Russian front during World War I, but physicists long dismissed black holes as unphysical coordinate artifacts. Acceptance grew after Roger Penrose and Stephen Hawking proved that black holes are generic outcomes of gravitational collapse. Decisive empirical evidence followed: decades of monitoring stellar orbits around Sagittarius A*, radio imaging of accretion disks by the Event Horizon Telescope, and LIGO’s 2015 detection of gravitational waves from colliding 30-solar-mass black holes a billion light-years away. General relativity's initial public triumph occurred during the 1919 solar eclipse expedition led by Sir Arthur Eddington. Because mass warps spacetime, light passing the Sun bends by exactly twice the deflection predicted by Newtonian mechanics. Weather disruptions in 1911 and the outbreak of World War I in 1914 had thwarted earlier measurement attempts. Brown points out this delay proved fortunate for Einstein, whose pre-1915 calculations had relied on an incomplete equivalence model that mistakenly matched the halved Newtonian prediction. Deductive Physics and AI-Driven Frontier Exploration (90:00) General relativity was formulated with minimal empirical prompts beyond the constancy of light speed, Lorentz symmetry, and the equivalence principle. Modern endeavors such as string theory adopt a similar deductive strategy, relying on mathematical consistency because direct empirical tests of quantum gravity would require galactic-scale particle colliders. Brown suggests that autonomous AI models could systematically evaluate branches of theoretical physics trees where human intuition struggles, testing low-probability hypotheses without cognitive bias. Addressing mathematician Terence Tao’s concern that AI could cause mathematical "indigestion" by outputting inscrutable, billion-line formal proofs, Brown argues advanced models will also serve as superhuman explainers. He notes that AI solutions to an Erdős problem and the unit distance conjecture generated intuitive, human-comprehensible concepts that mathematicians subsequently adapted to prove separate theorems. While general relativity demonstrated that pure thought and strict physical principles could deduce the large-scale mechanics of spacetime, contemporary physics faces the challenge of unifying gravity with quantum mechanics without direct collider data. Brown frames the central question as whether mathematical consistency alone, accelerated by parallel AI exploration, will prove sufficient to pinpoint the next fundamental laws of nature.

前斯坦福理论物理学家、现Google DeepMind科学团队负责人Adam Brown阐释指出，广义相对论将引力从牛顿式的超距作用重塑为弯曲时空中的惯性力，仅凭光速上限与等效原理极简的逻辑约束，就推导出了黑洞、时间膨胀与动态宇宙的存在。 牛顿引力与狭义相对论的冲突及引力的惯性本质 (0:00) 狭义相对论确立了宇宙中没有任何物质或信号的传播速度能超越光速（$c$）。然而牛顿在1687年提出的万有引力定律中，引力大小仅仅取决于物体间的瞬时距离（$r$）。这意味着晃动太阳会导致地球上的引力瞬间改变，从而违反了约8分钟的光速传播延迟限制。虽然麦克斯韦电磁学成功将静电库仑定律拓展为符合光速限制的动力学方程，但该机制无法套用到引力上：同种电荷相互排斥，而质量总是相互吸引；更关键的是，电荷与惯性质量互不相干，但引力质量与抗拒加速的惯性质量在实验中表现出惊人的一致（从牛顿时代的千分之一到现代检验的$10^{-15}$精度）。这一精确重合使真空中的羽毛与砖块以完全相同的速率下落。 Brown解释，这种现象只存在于水桶旋转实验中的离心力等“惯性力”（虚构力）之中——在离心力公式中，决定受力大小的“荷”严格等同于物体的惯性质量。爱因斯坦在1907年迎来其“最幸福的思想”，意识到引力本身正是一种由于物体沿着弯曲时空运动而产生的惯性力。在广义相对论中，经典力学中“直线”的概念被彻底颠覆：自由落体（如空中飞行的粉笔或失重宇航员）沿时空测地线运动，不受任何合力，属于真正的“直线运动”；反而是静止坐在地面椅子上的人，正受到地面施加的向上机械力，从而偏离了测地线。 时空弯曲、场方程与能量提取极限 (25:00) 通过将旧金山飞往伦敦在平面地图上显示为北极弧线的“大圆航线”作类比，Brown指出，在弯曲时空中，自由落体测地线在假定时空平坦的观察者眼中看似弯曲。爱因斯坦历经八年推导，在1915年发表了引力场方程（$G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$），左侧描述由黎曼几何刻画的时空弯曲，右侧则是能量-动量张量，精准概括为“物质告诉时空如何弯曲，时空告诉物质如何运动”。 以滑轮绳索将质量为$m$的物体缓慢放入引力井的思想实验展示了广义相对论的修正。牛顿力学中下落至半径$R$提取的能量占比为$\frac{GM}{c^2 R}$，在地球表面该无量纲比值仅为约$7 \times 10^{-10}$，与氢氧液态火箭燃烧释放的化学能量占比（约$1.5 \times 10^{-10}$）高度巧合，使人类用化学火箭脱离地球引力处于物理可行性的极限边缘。当物体被压缩至理论极限$r \le \frac{GM}{c^2}$时，牛顿公式甚至会荒谬地得出超过$100\%$静质量能的提取率。广义相对论通过使引力在有限半径处趋向无穷大而非发散于中心，化解了这一佯谬。 史瓦西度规与黑洞物理学的反直觉特征 (45:00) 1916年一战前线，卡尔·史瓦西求出了爱因斯坦场方程的首个精确解。史瓦西度规表明，在史瓦西半径（$r = \frac{2GM}{c^2}$）定义的事件视界处，任何物体维持静止所需的固有加速度都将趋近于无穷大。在$r = \frac{3GM}{c^2}$以内，由于物体的轨道动能本身也参与引力相互作用，角动量带来的离心效应失效，所有稳定的弹道轨道都会崩溃并落向中心奇点。同时，引力场造成非对称的时间膨胀（$\Delta t_{\text{local}} = \Delta t_{\infty} \sqrt{1 - \frac{2GM}{c^2 r}}$），这一效应已被哈佛大学塔楼的庞德-雷布卡实验和日常运行的GPS卫星精确定位所证实。 时间膨胀的“汇率”直接映射为能量价值：从视界附近向无穷远处发射的光子会经历极度引力红移并损失能量。若在事件视界处将物体放入黑洞，理论上可提取该物体近$100\%$的静质量能（$mc^2$），远超过化学能（$10^{-10}$）与核聚变（$10^{-2}$）。对于下落者自身而言，穿过视界并无物理阻碍，在超大质量黑洞中潮汐力极弱，甚至可平静跨过视界；但对远处的观测者而言，下落者的时钟逐渐冻结、红移变暗，永远无法被看到穿越视界的一瞬。 实证奠基与AI驱动的现代纯理论探索 (78:51) 黑洞曾长期被视为纯数学怪胎，但彭罗斯与霍金的奇点定理证明其形成是相对论的必然结果。银河系中心人马座A*周围恒星轨道观测、LIGO在2015年捕捉到相隔十亿光年的双黑洞碰撞引力波，以及事件视界望远镜的直接射电成像，最终确立了黑洞的物理真实性。此外，爱德华·爱丁顿在1919年日食观测中测出星光偏折角度恰好为牛顿预测值的两倍，验证了爱因斯坦的理论并引发全球关注；而早年因天气与一战爆发导致的观测流产，恰好掩盖了爱因斯坦早期基于不成熟假设计算出的错误数值。 在当代，像弦理论这样的前沿研究由于验证量子引力需要银河级对撞机，只能极度依赖数学一致性与逻辑自洽。Brown认为，爱因斯坦凭极少实验事实单枪匹马推导相对论的纯思想模式虽难以被人类学者复制，但AI能够并行模拟海量推理分支，全面检索理论决策树。Brown反驳了陶哲轩等学者关于AI将产生难以理解的代码级冗长证明（如Lean代码）引发“数学消化不良”的担忧；他指出AI在埃尔德什问题上的表现证明，AI不仅能克服人类认知偏差探寻低概率路径，还能充当“超级解释器”，提供具备启发性、可被人类数学家完全理解并迁移的新思想。 广义相对论展现了以极少数物理公理通过纯逻辑构建基础理论的极致力量；现代物理学在缺乏超级对撞实验的境况下，正寄望于将这一理性探索传统与AI的超强并行推理及概念解释能力相结合，以突破量子引力等未决疆界。

## Background 本期节目由 Dwarkesh Patel 于 2026 年 7 月 10 日在其同名播客频道（RSS / Web / YouTube）发布。节目嘉宾为前斯坦福大学理论物理学家、现 Google DeepMind Blueshift 团队负责人 Adam Brown；根据节目介绍，本期对话将原本长达 20 讲的研究生级广义相对论课程提炼为直观易懂的核心洞见，深入探讨了爱因斯坦的“最幸福思想”、黑洞对无限能量提取的遏制机制、坠入黑洞的观测视角差异，以及 AI 从零重构物理学规律的前景。 ## TL;DR 对话从牛顿瞬时超距引力与狭义相对论光速上限的根本冲突切入，阐明了爱因斯坦如何通过惯性质量与引力质量的严格等同提出等效原理，并将引力重构为弯曲时空中的惯性运动（测地线）。随后，Adam Brown 详细剖析了黑洞的史瓦西几何、引力时间膨胀以及黑洞作为理论极限能量源的机制，消解了超致密天体提取能量超过静质量的永动机佯谬，并对比了下落者与远方旁观者眼中的极端反差。最后，二人结合 LIGO 引力波与爱丁顿日食等实验实证，深入探讨了仅凭纯粹思维构建物理理论的边界，以及大语言模型未来作为“超级解释者”消化复杂前沿科学的潜力。 ## Atmosphere & style 整场对谈兼具严谨的学术硬核感与探索宇宙奥秘的纯粹兴奋感，Adam Brown 甚至在现场挥舞旋转水桶演示离心力，将抽象复杂的微分几何拆解为一个个直观精妙的思想实验。Dwarkesh 的提问敏锐且极具代入感，不断追问直觉与理论冲突的临界点，两人在黑板数学推演与科学哲学思辨之间张弛有度地穿梭。聆听过程宛如置身于斯坦福顶尖学者的私人研讨课，既能领略广义相对论严丝合缝的极致美感，又充满对理论物理浪漫主义与未来人工智能认知的深邃反思。 ## Bulletin - `12:35` 广义相对论的诞生源于爱因斯坦将“惯性质量与引力质量完全相等”这一巧合升级为基本法则，将引力重新诠释为非惯性系中的惯性力与弯曲时空下的几何效应，消除了牛顿超距引力与狭义相对论光速上限的根本冲突。 - `45:00` 在强引力场中，维持静止所需的加速度和引力修正项在史瓦西半径处发散至无穷大，这使得事件视界成为不可逆转的单向分界线，并且从致密天体缓慢抽取引力势能的极限效率精准收敛于100%静质量能，彻底避免了超额提取的永动机悖论。 - `01:13:50` 自由落体穿越事件视界本身在局域上毫无物理突变，但视界具有目的论性质，使落入者注定滑向中心奇点；而外部远距离观察者由于极端引力时间膨胀与红移，只能看到落入者无限渐近并在红移至黑中逐渐消失。 - `01:32:41` 理论物理是否能单靠AI与纯思维取得突破，取决于学科的分支因子与自洽理论的数目；同时AI不仅能充当超人类的数学证明器，还展现出提炼直观解释与跨领域迁移概念的超级解释者潜力。 ## Key takeaways - `05:00` 牛顿万有引力定律中即时的超距作用与狭义相对论关于光速不可逾越的公理直接冲突，迫使爱因斯坦必须彻底重写引力理论。 - `10:00` 引力无法直接套用麦克斯韦电动力学的向量规范场推广，因为同种质量表现为吸引而非排斥，这在现代场论中对应着自旋为2介导粒子与自旋为1粒子的本质差异。 - `20:00` 惯性质量与引力质量分毫不差的严格等同（等效原理），启发了爱因斯坦提出引力本质上是一种惯性力，自由落体状态才是没有受力的真正惯性测地线运动。 - `23:45` 在弯曲时空中，自由落体的粉笔沿着测地线做最短路径运动，而静坐椅子上感受重力的人实际上受到了地面推力被迫脱离了真正的测地线。 - `26:50` 爱因斯坦场方程将黎曼曲率几何与应力-能量张量统一起来，不仅能解释地表重力，还能无缝延展至水星轨道进动以及整个宇宙的大尺度演化。 - `34:00` 纯牛顿力学推导出的光速逃逸临界半径与广义相对论导出的史瓦西视界半径在公式与系数上完全一致，是科学史上一场惊人的数学巧合。 - `36:10` 地球引力势能与静质量之比（约7×10⁻¹⁰）与化学火箭燃料反应能量占比（约1.5×10⁻¹⁰）在数量级上接近，造成地表发射化学火箭的运载系数极其极限。 - `45:00` 若引力在近距离不发散，将物体降至超致密天体表面提取的引力能将超过其静质量能量mc²形成永动机；广义相对论通过黑洞视界使引力在有限半径处发散，解决了这一能量佯谬。 - `52:15` 在广义相对论中所有形式的能量都产生引力，轨道角动能引起的自引力在r ≤ 3GM/c²以内压倒向外的离心效应，使近距离公转无法避免坠入黑洞。 - `01:00:00` 引力时间膨胀打破了狭义相对论的对称性：处于引力势阱深处的手表在全局意义上绝对变慢，对应深处向外辐射的光子发生引力红移并损失能量。 - `01:05:00` 将物质缓慢下放至事件视界边缘释放，理论上能将接近100%的静质量转化为可用能量，使黑洞成为远超化学能和核能的终极能量提取机制。 - `01:10:00` 在量子引力中，黑洞通过霍金辐射蒸发几乎只释放光子和引力子，打破了微观粒子物理中质子与中子总数的核子数守恒，印证了量子引力不尊重全局对称性的猜想。 - `01:15:45` 穿越大质量黑洞事件视界时局域潮汐力可以极小，视界是一个无法在局部感知的目的论分界，外部旁观者由于无限红移永远无法看到坠入者穿过视界的一瞬。 - `01:18:00` 彭罗斯奇点定理严格证明了黑洞的形成是引力塌缩的一般普遍特征而非特殊微调，银河系中心恒星轨道追踪、LIGO引力波及EHT阴影成像提供了三重不可辩驳的观测证据。 - `01:24:21` 广义相对论预言光线经过太阳边缘的弯曲角恰好是牛顿力学预言的两倍，1919年爱丁顿日全食远征的实测验证使爱因斯坦的理论迅速确立为全球共识。 - `01:32:41` 纯粹通过数学自洽性与极少经验数据发现真理的模式仅在理论分支因子极低时可行，凝聚态物理等高分支领域必须依赖海量实验进行硬性剪枝。 - `01:35:00` AI在先进科学中的角色不仅是吐出复杂代码的形式化机器，在埃尔德什问题与单位距离猜想反例等实践中，模型已经开始扮演提供直观洞见的超级解释者角色。 ## Detailed walkthrough ### `00:00` 广义相对论的起源与“最美理论”的挑战 访谈由 Dwarkesh Patel 对嘉宾 Adam Brown 的介绍拉开帷幕。Adam Brown 目前在 Google DeepMind 领导致力于科学推理与前沿突破的 Blueshift 团队；在此之前，他是一位成果丰硕的理论物理学家，曾在斯坦福大学任教，研究领域横跨宇宙学、弦论以及广义相对论。Dwarkesh 提出，物理学界普遍将广义相对论誉为“人类大脑构想或见证过的最美丽的产物”。对于绝大多数普通人而言，不可能像物理系研究生那样去听 Adam 讲授的整整 20 讲研究生核心课程，因此这场对谈的核心目标就是探讨：普通大众能否无需高深数学，也能直观理解广义相对论的本质及其摄人心魄的美感。 Adam 欣然应允，并指出广义相对论与量子力学并列为 20 世纪物理学的两大支柱，但它与量子力学有着根本的不同：量子力学是众多顶尖科学家共同拼凑出的集体成果，而广义相对论几乎全凭爱因斯坦一人之力，在长达十年的执着追求中独立构建而成。这一理论不仅精确描述了太阳系中行星的公转轨迹，更揭示了整个宇宙的起源与最终宿命。在斯坦福大学，Adam 用 10 周课程即可让学生建立起超越当年爱因斯坦本人长达 10 年探索历程的深刻认知。这种后发优势源于后来的物理学家剥离了迷雾、去除了弯路，将原本被认为只有“爱因斯坦级大脑”才能领悟的极其复杂的思想，提炼浓缩到了最核心的本质。虽然在十几分钟内无法替代十周的高强度学习，但完全可以直击爱因斯坦本人所称的“最幸福的思想”（happiest thought），剖析理论的内核。 从科学史的脉络来看，爱因斯坦在 1905 年的“奇迹年”（*Annus Mirabilis*）率先提出了狭义相对论（Special Relativity）。所谓“狭义”，意味着其适用范围有限——它完美适用于电磁学，并且后来证明也直接适用于爱因斯坦当时尚不知晓的强核力和弱核力，但它却与万有引力格格不入。狭义相对论的核心口号可以归结为：**任何事物的传播速度都无法超越光速**。爱因斯坦将这一定律提升至基本公理的高度，并以此重构了时空观。经历了整整十年（直到 1915 年）的顽强攻坚，爱因斯坦提出了广义相对论，将引力纳入这一框架，完成了对已知所有基本作用力的包容。如果也要为广义相对论提炼一句口号，那就是：**“连引力也不例外——光速不可逾越”**。 ```mermaid flowchart TD accTitle: 经典力学到相对论的发展脉络 accDescr: 展示从牛顿力学、狭义相对论到广义相对论的思想演进与相互矛盾解决过程。 subgraph Classical["经典物理学基础"] N1["牛顿运动定律 (1687)<br>ma = F 及惯性定律"] NG["牛顿万有引力定律<br>超距作用，瞬间传播"] EM["麦克斯韦电磁学 (19世纪)<br>光速恒定，场论传播"] end subgraph Relativistic["相对论革命"] SR["狭义相对论 (1905)<br>光速为宇宙速度极限<br>电磁学天然自洽"] GR["广义相对论 (1915)<br>消除超距作用<br>连引力速度也不超光速"] end NG -->|"与光速上限直接冲突"| SR EM -->|"启发狭义相对论"| SR N1 -->|"修正力和加速度定义后保留"| GR SR -->|"引入引力后升级推广"| GR ``` --- ### `05:00` 牛顿引力与光速上限的根本冲突 为了理清爱因斯坦为何必须重构引力理论，需要将目光回溯到 17 世纪晚期牛顿在 1687 年出版的《自然哲学的数学原理》（*Principia*）。牛顿力学体系中有两个核心支柱与引力密切相关： 1. **牛顿第二定律**（$F = ma$）：力决定了物体的加速度 $a$。质量 $m$ 在此充当“惯性”的角色，代表物体对抗加速的阻力；施加同样的加速度，质量越大的物体所需的力越大。这个定律在广义相对论中依然得以保留，只是物理学界必须对何谓“力”与何谓“加速度”建立更加高阶且深刻的定义。其特殊情况即牛顿第一定律（惯性定律）：外力为零时，物体将永远保持匀速直线运动。广义相对论同样承认这点，但对“直线”（测地线）的定义进行了时空几何层面的升级。 2. **牛顿万有引力定律**：描述了两个天体之间的引力大小。其数学形式为反比平方律： $$F = -G \frac{m_1 m_2}{r^2}$$ 力的大小与两物体质量乘积成正比，与它们之间距离的平方成反比，方向沿着两者的连线且始终表现为吸引力。 正是在万有引力定律上，牛顿经典力学与狭义相对论爆发了无法调和的致命冲突。根据牛顿的公式，引力是即时的超距作用。如果有人在太阳上“晃动”一下太阳的位置，按照牛顿定律，地球感受到的引力距离 $r$ 瞬间发生变化，地球上的引力测量计会**立即**探测到扰动，而完全不需要等待光从太阳传播到地球所需的 8 分多钟。这就意味着，利用牛顿引力人们可以制造出一种“超光速引力电话”（faster-than-light telephone），实现信息的瞬间跨空间传输。 这与狭义相对论中“光速为一切因果与信息传递的绝对上限”的不可动摇的基石彻底背道而驰。当时理论物理界面对两条路：要么承认引力是超光速的特例，保留超光速信号的可能性；要么彻底推翻并修正牛顿的引力公式。毕生致力于排除任何超光速可能性的爱因斯坦，以及当时的大多数物理学家，都坚信必须退让和被彻底修改的只能是牛顿的引力公式。 实际上，物理学史上此前就存在过类似的先例：库仑在 18 世纪提出的静电反平方律（$F \propto \frac{q_1 q_2}{r^2}$）形式与万有引力完全一致，同样表面上看似违背了狭义相对论。然而，静电学只是完整电磁场理论在静态极限下的一个特例。在 19 世纪中叶，麦克斯韦（James Clerk Maxwell）写下了完整的麦克斯韦方程组，引入了磁场、电磁感应以及动态修正。当电荷运动时，修正效应保证了电磁相互作用绝不会超光速，而是以光速通过电磁波的形式向外弥散。在历史上，物理学甚至走了一条相反的道路：麦克斯韦首先提出了电磁方程组，爱因斯坦随后才发现这些方程天生就与狭义相对论完美融合。引力理论正是缺少这样一个类似麦克斯韦电磁场的革命性跃迁。 ### `10:00` 引力不能直接套用电磁学：引力与静电力的正负号差异 麦克斯韦方程组本质上与“没有任何东西能超过光速”完全兼容。这种自洽性体现在麦克斯韦场方程中后来才被发现的对称性——洛伦兹对称性（Lorentz symmetry），而正是洛伦兹对称性启发爱因斯坦创立了狭义相对论。在物理学历史上，存在着“从平方反比定律起步，最终将其改写为完全具备相对论协变性理论”的先例。因此很自然会有人提出：是否可以直接对引力依样画葫芦，仿照静电学拓展为电动力学的方式，构建出一套“引力磁学”（gravitomagnetic theory），将牛顿第二定律作为近似，从而与狭义相对论相容？ 从宏观角度看，这确实是爱因斯坦最终探索的大方向，但引力理论的发展经历了一场远比麦克斯韦电磁学更为激进的思想蜕变。直接对比牛顿引力定律与静电力定律公式，就能发现第一个无法直接照搬的根本差异：符号相反。 同种静电荷之间互相排斥，公式中带有排斥性质的符号；而两个正质量物体（例如太阳与地球）之间却永远相互吸引，公式中带有一个负号。若在数学上对引力完全照搬电磁场的推广技巧，推导出来的将是“同类质量相斥”的荒谬结论。从现代场论的微观机制来看，静电力是由自旋为 1 的玻色子（光子）传递的，而引力是由自旋为 2 的玻色子传递的，自旋的差异决定了作用力正负号的逆转。因此，爱因斯坦必须另辟蹊径，寻找其他理论线索来完成相对论推广。 ```mermaid flowchart TD accTitle: 比较静电力与引力向相对论推广的路径差异 accDescr: 对比麦克斯韦电动力学与广义相对论在对称性、介导粒子自旋与符号机制上的不同。 subgraph EM["静电力 (库仑定律) 推广路径"] A["平方反比定律"] --> B["自旋为 1 媒介粒子 (光子)"] B --> C["同种电荷相斥 (+)"] C --> D["洛伦兹协变场论 (麦克斯韦方程组)"] end subgraph GR["牛顿引力 推广困境与转折"] E["平方反比定律"] --> F["自旋为 2 媒介粒子"] F --> G["同种质量相吸 (-)"] G --> H["无法直接套用向量规范场，须寻找全新线索"] end ``` --- ### `12:35` 惯性质量与引力质量的神秘巧合：等效原理 爱因斯坦的核心天赋在于敏锐地抓住了一个在牛顿力学中被视作“纯粹巧合”的重大线索，这一线索后来被他称为“一生中最幸福的思想”。这个线索隐藏在引力与静电力的第二个关键差异中：充当“引力荷”的物理量，恰好就是质量本身。 在静电学与一般力学中，物体的质量只承担单一角色——即牛顿第二定律 $F = ma$ 中的“惯性质量”（inertial mass），代表物体抵抗被加速的惰性大小。电荷与质量毫无关系，重粒子可以完全不带电（如中子），而极轻的粒子却可以带有单位电荷（如电子），电荷与惯性质量之间不存在必然的数学联系。 而在牛顿引力中，决定物体抵抗加速度的“惯性质量”，竟然分毫不差地等于牛顿万有引力定律中决定物体感受引力大小的“引力质量”（gravitational mass）。在牛顿力学框架内，这两个物理量的完全等同是一个毫无道理的宇宙巧合： - 牛顿本人便注意到了这一巧合，并设计了摆动实验来验证两者一致性，精度达到千分之一（$1/1000$）。 - 到了爱因斯坦时代，实验精度已提升至十亿分之一（$1/10^9$）。 - 如今，两者的等同性检验精度已高达 $10^{-15}$。 这种精确等同即为“弱等效原理”（Equivalence Principle）。若在真空舱中同时释放一根羽毛与一块砖头，二者会完全同步着地。尽管砖头受到的万有引力远大于羽毛，但由于砖头抵抗加速度的惯性阻力也以完全相同的倍数增加，二者精确抵消，最终产生完全相同的下落加速度。爱因斯坦将这个惊人巧合作为推翻牛顿理论、建立新时空观的核心跳板。 --- ### `16:00` 水桶实验与惯性力：非惯性系中的“引力荷”本质 为什么爱因斯坦将质量的完全等同视为解开引力之谜的核心？因为物理学中存在另一类力——它们不是像电磁学那样的基本相互作用，而是所谓的“涌现力”或“虚构力/惯性力”（fictitious or inertial forces），而在这些力的数学形式中，等效性是天生注定且完全不可避免的。 为了直观展示这一点，亚当在录制现场拿出了一个装满水的水桶，甚至开玩笑让主持人伸手指摸摸水以证明“绝无魔术机关”，主持人打趣提醒：“亚当，你最好精通物理，否则你今天就要把录音室全砸了。”紧接着，亚当手提水桶在空中做起了垂直圆周旋转（loop-the-loop）。令人直观感到惊奇的是，即便水桶运动到最高点、完全倒扣过来，水也没有洒出来。 解释水不落下的现象有两种等价视角： 1. **外部惯性参考系视角**：当水处于轨迹顶端时，它的惯性使得它想沿着切线方向继续向前运动。在水来得及在重力加速度下完全落出桶外之前，高速转动的水桶本身已经移动到了下方，水被桶底“兜住”而一同转动。这与宇航员在近地轨道绕地球自由落体而不坠落是同一个道理。 2. **随水旋转的参考系视角**：如果观察者坐在水桶里，从这个非惯性系来看，会感到有一种强大的推力将水死死按在桶底，这就是离心力（centrifugal force）。 $$F_{\text{centrifugal}} = m \frac{v^2}{R}$$ 在离心力公式中，决定物体感受离心力强弱的“离心力荷”，再次精确地由其“惯性质量” $m$ 决定。但在此处，质量的出现绝非神秘的宇宙巧合：之所以离心力正比于惯性质量，正是因为离心力本身就是物体试图保持直线匀速运动的惯性倾向与参考系被迫转弯之间的几何错位。正是这种“惯性表现为力”的机制，彻底启发了爱因斯坦对真实引力本质的重构。 ### `20:00` 爱因斯坦的“最幸福思想”：引力其实是一种惯性力 在经典力学中，凡是源于惯性的“惯性力”（如离心力或科里奥利力），物体在该力场下感受到的“荷”（charge）必然由其惯性质量决定。引力也具有一种“引力荷”，而实验表明物体的引力荷与惯性质量完全相等。这激发了爱因斯坦在 1907 年跳跃性地提出他一生中“最幸福的思想”（happiest thought）：引力会不会根本就不是传统意义上的力，而正是一种惯性力？ 对于电磁力而言，这种猜想从根本上就是不成立的，因为电磁荷与惯性质量之间没有必然联系；但引力质量与惯性质量的严格相等，为这一假设扫清了障碍。在牛顿体系下，两者的重合仅仅是一个巧合、一种经验上的偶然事实；但若引力本质上就是惯性力，这个相等性就变成了宇宙运行的必然法则。 然而，这个设想在直觉上显得极其不可思议，因为它强迫我们推翻对“直线”概念的原有理解。惯性力是物体在**不做直线运动**时才会感受到的力。如果一个人沿着真正的直线匀速运动，他是感受不到任何惯性力的： - 在太空中自由漂浮或自由下落的宇航员感受不到引力，因此按照这一新逻辑，他们才是在沿着“直线”运动； - 反之，静坐在椅子上、看似静止不动的我们，感受到了椅子向上的支撑力与引力的拉扯，这意味着我们实际上**并没有**沿着直线运动。 ```mermaid flowchart TD accTitle: 经典力学与爱因斯坦广义相对论对受力与运动路径的对照 accDescr: 对比自由落体与静坐地面两种状态在牛顿视角与爱因斯坦视角下的本质区别 subgraph Newton["牛顿视角 (平直时空)"] N1["静坐椅子上: 静止 (沿时间轴直线)"] N2["自由落体/抛掷粉笔: 受引力作用 (曲线抛物线)"] end subgraph Einstein["爱因斯坦视角 (弯曲时空)"] E1["静坐椅子上: 受支撑力推离测地线 (非直线运动)"] E2["自由落体/抛掷粉笔: 纯惯性运动 (时空中的真正直线/测地线)"] end Newton -.->|"颠覆常识"| Einstein ``` --- ### `22:20` 赞助商插播：Jane Street 与物理学直觉 在深入讲解时空几何之前，节目插入了一则赞助介绍。主持人 Dwarkesh Patel 提到了与知名量化交易机构 Jane Street 的交易员 Jed Thompson 的交流。Jed 在转型为交易员之前是一名粒子物理学家。他谈到，在物理学中养成的一种思维习惯对交易大有裨益——那就是“在真正动手计算之前，心里往往已经对答案有了相当准确的猜测”。这种通过反复观察模式构建的良好直觉，能让人从一开始就提出正确的问题，从而大幅缩短弯路。 --- ### `23:45` 谁在走直线？航线图与弯曲空间的直观比喻 回到黑板前，Adam Brown 用粉笔画出了两张运动轨迹图。纵轴代表距离地心的垂直高度，横轴代表时间： - 第一条轨迹是静坐在椅子上的 Dwarkesh，其高度随时间是一条水平的直线； - 第二条轨迹是被抛向空中的粉笔，其高度随时间上升再落下，是一条抛物线曲线。 直观上看，静坐者的轨迹是一条直线，而粉笔的轨迹是一条曲线。但在自由落体状态下，粉笔感受不到任何合力；如果引力是一种惯性力，那么无受力感的粉笔轨迹必须被判定为“直线”，而静坐着的人反倒在走“弯路”。这种认知冲突源于我们把弯曲的空间强行画在了平直坐标系中。 Adam 举了一个所有人都经历过的例子：乘坐越洋飞机时椅背屏幕上的航线地图。 - 如果你要从旧金山（San Francisco）飞往伦敦（London），在平直的二维世界地图投影上，最短路径看起来理应是一条平直切过去的横线（等角航线/恒向线，rhumb line）； - 但现实中的飞机会沿着一条高高隆起、擦过格陵兰岛南部边缘再向下的弧线飞行，看起来就像绕了一个巨大的远路； - 实际上，地球表面是一个曲面，从旧金山到伦敦的大圆航线（两点之间最短路径）在三维球面上本来就是经过格陵兰岛的真正测地线（直线）。平直投影地图之所以让人产生“绕路”的错觉，是因为它强行把曲面展平成平面，必然带来几何失真。 当你试图假装一个弯曲的对象是平直的时，你必然会在“什么是直线、什么不是直线”这个问题上产生根本性的误判。 --- ### `26:50` 爱因斯坦场方程：物质如何弯曲时空 飞机航线的失真完美投射到了广义相对论中：在三维空间加上时间构成的“时空”（spacetime）中，抛物线上升下落的粉笔其实正沿着弯曲时空中的“最短路径”（测地线，geodesic）前行，它是在弯曲时空中所能走出的一条真正的直线；而坐在椅子上的人，因为受到椅子的推力，被迫脱离了这条测地线。 因此，爱因斯坦广义相对论的核心图景得以确立： - **物质的存在导致了时空的弯曲**； - 时空的弯曲改变了局部直线的定义与轨迹； - 沿着所谓“错误直线”运动的物体感受到了加速度与引力，而在自由落体轨道上的宇航员则体验到彻底的失重。 为了把这种直觉升华为严密的数学理论，爱因斯坦从 1907 年构思出这一物理图像开始，整整苦苦探索了八年，直到 1915 年底才最终写下描述引力本质的“爱因斯坦场方程”（Einstein Field Equations）： $$G_{\mu\nu} = 8\pi G \, T_{\mu\nu}$$ 这一公式的结构呈现出极高的对称性与美感： - **等式左边（几何部分）**：由东欧数学家（如黎曼、里奇等人）创立的张量数学构成，用来精确刻画时空的弯曲程度（若时空平直，该张量为零；若时空弯曲，则为非零）； - **等式右边（物质部分）**：描述分布在时空中的能量与动量密度（由应力-能量张量 $T_{\mu\nu}$ 描述），并附带牛顿引力常数 $G$ 等常数项。 这行简洁的方程用最精确的数学语言证实了爱因斯坦的洞见：物质决定时空如何弯曲，时空决定物质如何运动。 ### `30:00` 爱因斯坦场方程的精髓与惊人的跨尺度适用性 爱因斯坦场方程将曲率张量与物质能量张量 $T_{\mu\nu}$ 紧密联系在一起，方程中还包含了牛顿引力常数 $G$、圆周率 $\pi$ 和光速 $c$。其中，$T_{\mu\nu}$ 是牛顿力学中质量概念在相对论下的自然推广：它表明不仅是静止质量，宇宙中所有形式的质量与能量都会直接导致时空的弯曲。物理学家约翰·惠勒（John Wheeler）曾将此凝练为著名的格言：“物质告诉时空如何弯曲，时空告诉物质如何运动。” 在后半句中，弯曲的时空指引物质沿着弯曲几何中的“测地线”（即弯曲空间中的直线）前行；若人类强行假定时空是平直的，就会将这种沿测地线的自然惯性运动误认作“虚构的引力”。 牛顿万有引力的伟大之处在于其跨维度的统一性：传说中一颗从树上落下的苹果，其背后的物理法则不仅支配着地表下落的物体，同时也完美描述了天穹之中行星的公转轨道。牛顿凭一己之力统一了天上与人间的运动法则。然而，广义相对论在此基础上更跨越了不可思议的尺度层级： - 它同样能描述苹果落地的地表力学； - 它精确预测了水星近日点的进动以及太阳系各行星的轨道偏离； - 它甚至能够描述整个宇宙大尺度下的膨胀与演化历程。 这种跨越数个数量级的理论延展力（reach），远远超出了爱因斯坦最初为了解决某个经验巧合时的预想。 --- ### `32:15` 史瓦西解的诞生与半个世纪的认知混乱 广义相对论衍生出了一批牛顿力学中绝不存在的奇特实体，其中最具代表性的便是黑洞。爱因斯坦在1915年写下场方程时，本以为这些复杂的非线性偏微分方程永远无法得到严格的解析解，人类往后大概只能依赖近似计算。然而，一战期间身处前线的普鲁士炮兵军官卡尔·史瓦西（Karl Schwarzschild），在炮火连天、忙着为火炮计算弹道轨道的间隙，仅用了短短几个月便找到了场方程的第一个精确解。 这个解如今被称为“史瓦西解”（Schwarzschild solution）。它描述了一个中心点状质量周围真空时空的精确几何结构，其物理本质正是我们今天所熟知的无自旋黑洞。 然而在当时，根本没有“黑洞”这个词。甚至在史瓦西解提出后的半个世纪里，物理学界对这个解的物理意义陷入了极大的混乱。就连爱因斯坦本人也是最严重的迷茫者之一：他对如今被称为“事件视界”（Event Horizon）的概念困惑不已，甚至撰写论文提出过诸如“下落物体可能会在视界处反弹”等完全错误的物理推论。直到半个世纪后，现代广义相对论才彻底澄清了视界的本质。 --- ### `34:00` 逃逸速度与经典力学的惊人巧合 理解黑洞最朴素的角度，是探讨引力与有限光速之间的冲突。这种思想实验甚至早在18世纪就已经由约翰·米歇尔（John Michell）和皮埃尔-西蒙·拉普拉斯（Pierre-Simon Laplace）各自独立提出。在牛顿引力体系下，若想让一个物体彻底摆脱天体的引力束缚飞向无穷远，其初始动能必须克服天体表面的引力势能，即达到逃逸速度 $v_{\text{esc}}$： $$\frac{1}{2} m v^2 = \frac{G M m}{R} \implies v_{\text{esc}} = \sqrt{\frac{2GM}{R}}$$ - 地球的逃逸速度约为 $11\text{ km/s}$； - 木星由于质量更大且更为致密，其逃逸速度高达几十甚至上百千米每秒； - 18世纪的思想家们便推论：若一个天体足够大或足够致密，使其逃逸速度达到光速 $c$，那么即便是光也无法逃逸出其表面。 将逃逸速度设为光速 $c$ 求解，可得到对应的临界半径： $$R_c = \frac{2GM}{c^2}$$ 尽管牛顿力学推导光线受引力束缚的逻辑在现代物理看来并不严密，但不可思议的是，这个纯经典力学推导出的临界半径公式，竟然与现代广义相对论计算出的史瓦西视界半径完全一致，连常数系数“2”都丝毫不差——这在物理学历史上是一场令人惊叹的纯粹数学巧合。 ```mermaid flowchart TD accTitle: 从经典逃逸速度到史瓦西视界 accDescr: 展示牛顿经典力学与广义相对论在逃逸速度达到光速时对黑洞临界半径推导的巧合一致性。 subgraph Classical["18世纪经典思考 (Michell / Laplace)"] A["动能与引力结合能平衡"] --> B["逃逸速度 v = sqrt(2GM/R)"] B --> C["令逃逸速度等于光速 c"] C --> D["临界半径 R = 2GM / c^2"] end subgraph GR["20世纪广义相对论 (Schwarzschild)"] E["爱因斯坦引力场方程"] --> F["史瓦西精确真空解"] F --> G["事件视界半径 R_s = 2GM / c^2"] end D -.->|"数值与系数完全巧合一致"| G ``` --- ### `36:10` 提取引力势能的思想实验与能量效率 为了更深刻地理解为什么在临界半径附近必然会发生剧烈的反常现象，可以通过一个从物体中提取引力能量的思想实验来探讨。 假设在距离地球无限远处有一块质量为 $m$ 的砖块，将其连接到滑轮缆绳系统上，缓慢地下放至半径为 $R$ 的地球表面，最终以零速度平稳放置在地面上。在缓慢下放的过程中，引力拉着砖块向下做功，这部分能量被远端的缆绳系统捕获并输出到无穷远处。 根据经典力学，能够从这块砖头中提取出的引力势能为： $$\Delta E = \frac{G M_{\text{Earth}} m}{R}$$ 若结合狭义相对论质能等价理论 $E_0 = m c^2$，便能计算出所提取的能量占砖块本身全部静止质能的比例： $$\text{提取效率} = \frac{\Delta E}{m c^2} = \frac{G M_{\text{Earth}}}{c^2 R}$$ 在这个无量纲比例中，砖块本身的质量被约去，结果完全取决于天体的致密程度： - 当把砖块降至地球表面时，所能提取的能量比例极小，仅约为 $7 \times 10^{-10}$； - 这正是为什么人类在日常地表生活中难以察觉相对论效应的原因——广义相对论对牛顿力学的修正本质上是以该极小数值为展开参数的泰勒级数（Taylor expansion），在地球表面，一阶项牛顿力学已极为精准，高阶相对论效应微乎其微； - 还有一个极其绝妙的宇宙巧合：地球引力势能占比（约 $7 \times 10^{-10}$）与航天火箭化学燃料（如液氢液氧混合物）燃烧时化学结合能占反应物总静止能量的比例（约为 $1.5 \times 10^{-10}$）在数量级上惊人地接近。这也是人类在地球上制造化学火箭发射极其困难、运载系数极其极限的地缘物理巧合。 ### `40:00` 地球引力与化学火箭的巧合，以及超致密天体的能量悖论 Adam 首先指出两个截然不同计算结果之间的惊人巧合。一个是从地球引力势阱中提取能量的引力计算比例，另一个是氢氧结合产生化学能的化学性质比例。这两个数字不仅同源完全不同，而且都极其微小。化学反应所释放的能量之所以极小，是因为氢和氧中绝大部分能量并非存储在化学结合能中，而在于质子和中子的静止质量能量（$E=mc^2$），化学燃烧根本动摇不了这部分能量；第二大能量部分则存储在强力和弱力维系的核结合能中，化学反应同样无法触及。相比于物质自身的静止质量，化学键极其脆弱。 正是由于这两种极小数值在数量级上恰好相近（引力所需的逃逸比值比化学燃烧释放的比值略大几倍），人类才能制造化学火箭进入太空，但这过程极具挑战性。火箭方程意味着地面发射架上绝大多数质量必须是推进剂，只能将极小比例的有效载荷（payload fraction）送入轨道。倘若在太阳表面，由于引力井深得多，用化学火箭进入太空完全是不可能的。 紧接着，根据能量提取公式，天体越重、半径越小（越致密），将物体缓慢下放至表面所能提取的能量比例就越高： - **太阳表面**：太阳质量是地球的数百万倍，但半径也更大，最终能量提取比值约为 $2 \times 10^{-6}$（即著名的太阳表面引力红移效应）。 - **白矮星（如天狼星B，Sirius B）**：将相当于太阳的质量压缩到地球大小的半径，提取的质量能量占比会变得显著增大。 由此引出一个极其尖锐的佯谬：如果继续压缩天体，使得半径满足 $r \le \frac{GM}{c^2}$，提取的能量比例就会超过 100%（大于 1）。这意味着将一块砖头垂降到天体表面所回收的能量，竟比砖头本身的全部静止质量能量 $mc^2$ 还要多。你甚至可以用提取出来的多余能量凭空制造一块全新的砖头，再把新砖头降下去，形成凭空产生无限能量的永动机。这种违反物理学基本直觉的荒谬结论表明：一旦天体致密到接近这一临界半径，必定有某种底层机制崩溃了。 ```mermaid flowchart TD accTitle: 致密天体能量悖论机制 accDescr: 展示将质量下放到不同天体表面时提取能量比例的增长趋势及最终引发的佯谬与黑洞形成 subgraph Objects["天体致密程度与能量提取"] A["地球表面: 极小提取比例<br/>与化学能相近但需高燃料比"] B["太阳表面: 红移效应 (~2×10⁻⁶)"] C["白矮星 (如天狼星B): 太阳质量/地球半径<br/>能量提取比例显著提升"] D["极端压缩: 半径 r ≤ GM/c²<br/>能量提取比例计算值 > 100%"] end subgraph Paradox["佯谬与广义相对论解答"] E["经典佯谬: 提取能量大于 mc²<br/>可循环制造新质量（永动机）"] F["广义相对论实解: 形成黑洞<br/>引力在有限半径发散，砖块被撕碎剥离"] end A --> B B --> C C --> D D --> E E --> F ``` --- ### `45:00` 电磁类比、广义相对论的解法与史瓦西度规 如何化解这种“提取超过 100% 质量能量”的佯谬？理论上有两种可能的物理逻辑： 1. **力在近距离变弱**：正如电磁学中的情况，将相反电荷缓慢靠近试图抽取静电势能时，量子效应会使电荷发生弥散（fuzz out），能量随 $1/r$ 增长的趋势被软化平滑，两电荷不再以原有的强烈方式相互吸引，从而避免能量无限释放。 2. **力在近距离变得更强**：广义相对论选择的恰恰是完全相反的道路。当物体压缩至临界尺度时，引力非但没有变弱，反而远比牛顿定律预测的还要剧烈。在达到 $r = 0$ 之前的一个有限半径处，引力就已经发散至无穷大。此时物体形成了一个黑洞，悬挂砖头的绳索会被无情拽裂，你根本无法继续缓慢下降物体来抽取能量，从而彻底封死了永动机的漏洞。 随后，Dwarkesh 插入了一段赞助广告，介绍了 Crusoe 的 Serverless 微调产品。他利用自己过往采访的转录文本对开源模型进行微调，制作了一个能够预测并提出“Dwarkesh 风格”后续问题的生成器。在团队的双盲评测中，大家在约三分之二的情况下依然更青睐他本人真实提出的问题，而在其余三分之一中，微调模型则显著击败了基础模型。 回到物理正题，Adam 指出之前在黑板上写的公式本质上都是牛顿物理加上光速 $c$ 拼凑出的启发性结果；要严谨回答超致密天体的问题，必须依赖将引力与光速统一的广义相对论。卡尔·史瓦西（Karl Schwarzschild）率先解出描述中心质量及黑洞周围引力场的“史瓦西度规（Schwarzschild metric）”。 为了建立黑洞内外的直观物理图像，Adam 着手推导史瓦西度规带来的三个直接推论。首先是**静态观察者（static observer）所感受到的局部引力**： - 假设观察者停留在离黑洞固定半径 $r$ 的位置（无论是悬挂在坚固的滑轮绳索上，还是下方开足火箭发动机提供推力）。 - 在牛顿物理中，该处引力仅为经典的平方反比律：$\frac{GM}{r^2}$。 - 在广义相对论中，引力公式得到修正，分母引入了带有史瓦西半径特征的修正项： $$\frac{GM}{r^2 \sqrt{1 - \frac{2GM}{c^2 r}}}$$ 当处于远离黑洞的区域（$r$ 极大）时，修正因子趋近于 1，公式退化回熟悉的牛顿引力定律；在地球这种弱引力场环境下，修正项微乎其微。然而，一旦 $r$ 不断靠近临界半径 $\frac{2GM}{c^2}$，分母便急速趋近于零，维持静态所需的加速度和引力将直奔无穷大。 ### `50:00` 广义相对论下的引力修正与视界的诞生 Adam Brown 解释道，在弱场或大距离 $r$ 处，广义相对论公式中的根号项 $\sqrt{1 - \frac{2GM}{rc^2}}$ 对引力产生的影响极小（通常只有约 $10^{-10}$ 数量级的微弱修正）。然而，如果对其进行泰勒展开（Taylor expansion），就会发现引力不仅仅包含牛顿物理中的平方反比律（$\frac{1}{r^2}$），还包含了立方反比项（$\frac{1}{r^3}$）以及四次方反比项（$\frac{1}{r^4}$）的高阶修正。在短距离下，广义相对论预言的引力实际上比牛顿力学预测的要更加强烈。为了不在引力场中下落，飞船必须以更大的固有加速度（proper acceleration）向上推射。 当径向距离减小到 $r = \frac{2GM}{c^2}$ 时，这一特定的临界距离被称为“史瓦西半径”（Schwarzschild radius），其所定义的闭合曲面就是“事件视界”（event horizon）。在这个位置上，分母趋近于零，使得保持位置静止所需要的固有加速度直接飙升为无穷大。这意味着，在视界之外较远的地方，你只需要有限的加速度就能维持静态悬停；但一旦越过或处于事件视界之上，无论你的火箭引擎推进得多么剧烈，都不可能再保持静止，你将被无可挽回地拖向黑洞深处。 - **广义相对论引力修正**：在近距离下，引力场强于牛顿力学预期，包含比平方反比更陡峭的高阶修正项。 - **史瓦西半径公式**：$r_s = \frac{2GM}{c^2}$，对应静止加速度趋于无穷大的分界线。 - **事件视界的物理本质**：维持静止所需的外力变为无限大，因此任何物体在视界内都必然向内坠落。 --- ### `52:15` 为什么近距离高速环绕也无法拯救坠入黑洞的命运 常有人设想：既然静止悬停需要无穷大的推力，那是否可以通过以极高的速度环绕黑洞公转，借助巨大的离心力来抵抗黑洞引力？Adam 指出科幻作品中常有“黑洞会把周围一切都无条件吸进去”的误解——在远距离上，轨道角动量（orbital angular momentum）确实能够很好地支撑物体，正如国际空间站（ISS）环绕地球运行时，离心效应正好抵消重力，让宇航员体验到失重一样。 然而在黑洞周围，当轨道半长轴缩短至 $r \le \frac{3GM}{c^2}$ 时，环绕轨道不仅不再起保护作用，反而会加速坠向黑洞的进程。这是因为在广义相对论中，**所有的能量都产生引力（all energy gravitates）**，包括动能和角动能。快速轨道公转产生两股相互竞争的效应： ```mermaid flowchart TD accTitle: 黑洞强引力场下的角动量双重效应 accDescr: 展示环绕黑洞时离心效应与能量产生引力效应的竞争关系 subgraph OrbitEffects["环绕黑洞的物理效应"] A["轨道高速运动 (角动能)"] --> B["向外的离心效应 (推离黑洞)"] A --> C["轨道能量的自引力耦合 (拉向黑洞)"] end subgraph RegimeComp["不同距离下的主导机制"] B -.->|"在 r > 3GM 时主导"| D["成功维持稳定轨道"] C -.->|"在 r ≤ 3GM 时主导"| E["引力压倒离心力，失稳坠入黑洞"] end ``` 在距离较远时，向外的离心效应占据主导；但一旦进入 $3GM$（以几何单位制计，即 $\frac{3GM}{c^2}$）的范围内，轨道角能量与黑洞质量之间的引力耦合项将占据绝对上风，把物体更快地拽向中心。在 $3GM$ 之内，不存在任何可以逃逸出来的弹道轨道（ballistic orbits）。 - **动能产生引力**：广义相对论中引力的来源不仅是静质量，动能同样会加剧引力吸引。 - **光子球层临界点**：在 $r = \frac{3GM}{c^2}$（光子轨道半径）以内，角动量变成拖累而非助力。 - **视界 vs 奇点**：在事件视界处你“尚未死亡，但已注定毁灭（not yet dead, but doomed）”，即便把自己完全转化成光射出也无法逃离；而直到中心 $r = 0$ 的奇点（singularity）处，潮汐力才会彻底撕裂一切导致真正死亡。牛顿引力只在原点 $r = 0$ 处发散，而广义相对论中尝试抵抗引力的力在视界处就已发散为无穷。 --- ### `55:00` 第二个核心公式：引力时间膨胀与时间的不对称性 Adam 引入了描述黑洞物理的第二个关键公式——**引力时间膨胀（gravitational time dilation）**。他与 Dwarkesh 设想了一个思想实验：Dwarkesh 悬挂在黑洞外半径为 $r$ 的缆绳上保持静止，而 Adam 留在无穷远处的平直时空中观察他，两人之间没有相对运动。对彼此而言，各自的手表都在以“每秒一秒（one second per second）”的本征节奏走动，但如果 Adam 观察 Dwarkesh 的手表，会发现他的表走得极其缓慢；反过来，Dwarkesh 观察 Adam，会觉得远方宇宙的时间流逝得飞快。 两者之间的走时关系严格遵循之前出现过的同一个根号因子： $$\Delta \tau_{\text{近}} = \Delta \tau_{\text{远}} \sqrt{1 - \frac{2GM}{rc^2}}$$ 由于根号项始终小于 1，如果 Adam 测得经过了 1 秒，Dwarkesh 测得的经历时间必然不足 1 秒。如果你被缓缓降落到黑洞边缘待上自感一年的时间再被拉回远处，你所返回的母星世界早已历经千百年沧桑。这一效应并非空想理论，20 世纪 50 年代哈佛大学物理系（Pound-Rebka 实验）就在同一栋楼的不同高度放置了原子钟，成功测出了上方时钟比下方走得更快的引力红移效应。现代的 GPS 导航系统也必须对地球表面的时间膨胀与轨道卫星原子钟之间的走时差进行硬性校准，否则整个定位系统会在短时间内彻底失准漂移。 Dwarkesh 指出，这与狭义相对论有着根本性的哲学区别：狭义相对论中由于惯性参考系的对称性，互有相对速度的双方都会认为对方的钟走慢了；但在引力场中，这种对称性被打破了——处在引力势井深处的观察者确实具有全局意义上更慢的时间流逝。此外，如果该观察者既处于引力场深处又同时保持高速轨道运动，两种时间膨胀效应将会发生叠加（stack）：引力时间膨胀与狭义相对论动钟变慢共同作用，使得远处观察者眼中的他显得格外迟缓。 ### `01:00:00` 引力红移与不同高度下的“能量汇率” 在狭义相对论中，如果两个人相对运动，对称性决定了双方的参考系是平等的：我看你的表走慢了，你同样看我的表走慢了，不存在谁比谁更“正确”。但在黑洞附近，这一对称性被彻底打破。Adam 指出，双方都会毫无争议地承认：你比我更深地处于引力势阱之中。因此，在势阱深处的你，时钟确实走得比远处的我慢；而深处的人看远处的观察者，看到的不是时钟变慢，而是被加速的景象——“你会看到我像在按快进键一样生活”。 ```mermaid flowchart LR accTitle: 引力红移与蓝移示意图 accDescr: 展示光子从引力势阱深处向外传播发生红移，以及从远距离向内传播发生蓝移的能量变化。 subgraph DeepInWell["强引力深处 (Dwarkesh)"] A["慢速时钟 / 低势能"] end subgraph FarAway["遥远平坦时空 (Adam)"] B["快速时钟 / 高势能"] end A -->|"光子向外发出：频率降低 / 引力红移 / 能量减少"| B B -->|"光子向内发出：频率升高 / 引力蓝移 / 能量增加"| A ``` 由此引出了第二与第三个关键物理结论：时间的“流动汇率”直接决定了能量在不同高度的“兑换汇率”。设想深处的人利用钠跃迁（Sodium transition）向远方发射一束特定频率的光： - **引力红移（Gravitational Redshift）**：由于远方观察者眼中深处的一切都在“慢放”，光波的振荡频率看起来变低了，光谱发生红移；根据光子能量公式，频率降低意味着光子能量减少。当光子抵达远方时，其携带的能量小于出发时的能量。 - **引力蓝移（Gravitational Blueshift）**：反过来，远方观察者向下发射光子，深处的人看外界处于“快进”状态，测得的光子频率升高，光谱发生蓝移，能量也随之增加。 这就引出了第三个公式：如果一个静质量能量为 $mc^2$ 的物体停留在黑洞外部半径 $r$ 的位置，对遥远观察者来说，它究竟值多少能量？答案由完全相同的平方根因子调节： $$E(r) = mc^2 \sqrt{1 - \frac{2GM}{c^2 r}}$$ 无论是将物质与反物质对撞（例如各半个阿伏伽德罗常数的碳原子与反碳原子全湮灭）并把光束发射出来，还是用绳索滑轮将它硬拉回无穷远处，遥远观察者最终能够保留或接收到的净能量都严格受到这一因子的限制。在向外提拉的过程中，你必须支付克服引力势阱所做的功，两相抵消后，留在手中的能量必然小于 $mc^2$。 --- ### `01:05:00` 提取全部质能：终极黑洞动力源 如果知道了处于半径 $r$ 处的砖块对无穷远处观察者所值的剩余能量，我们就能精确计算在将它用滑轮缓缓下放至半径 $r$ 的过程中，究竟提取出了多少功。起始时砖块在无穷远处拥有能量 $mc^2$，下放后系统剩余的能量为 $mc^2 \sqrt{1 - \frac{2GM}{c^2 r}}$。因此，提取出的能量分数即为： $$\text{提取效率} = \frac{mc^2 - mc^2 \sqrt{1 - \frac{2GM}{c^2 r}}}{mc^2} = 1 - \sqrt{1 - \frac{2GM}{c^2 r}}$$ 在距离黑洞极远处（$r \gg \frac{2GM}{c^2}$），对该式做泰勒展开，一阶近似刚好还原为牛顿力学公式 $\frac{GM}{c^2 r}$。“广义相对论的远距离极限必须还原为最初发现的牛顿物理学，事实也确实如此。”但随着砖块越来越接近黑洞事件视界，相对论修正项占据主导，彻底解决了此前关于“提取能量是否会超过 $mc^2$ 从而造出永动机”的悖论： - 当 $r \to \infty$ 时，提取的能量为 $1 - 1 = 0$。 - 当 $r$ 逐渐逼近视界半径 $r_s = \frac{2GM}{c^2}$ 时，根号项趋近于 $0$。 - 这意味着提取出的能量分数精准地逼近 $100\%$（即整整 $mc^2$）。 将砖块缓缓下放至紧贴事件视界上方的极限位置并以零初速度释放，砖块落入黑洞，而绞盘系统刚好完整榨取了砖块全部的静质量能量。这一惊人特性正是物理学家热衷于探讨将黑洞作为“终极发电厂”的原因： | 能源类型 | 作用机制 | 静质量转化效率（提取比例） | |:--- |:--- |:--- | | **化学能**（煤、石油等燃烧） | 分子间微弱的电磁键重组 | 约 $10^{-10}$ | | **重核裂变**（核电站） | 原子核内强核力的重组 | 约 $10^{-3}$（千分之一） | | **轻核聚变**（氢弹、太阳核心） | 强核力更高效的紧密结合 | 约 $10^{-2}$（百分之一） | | **黑洞引力能量提取** | 缓慢下放至事件视界边缘释放 | 理论极限高达 $\sim 100\%$ | ### `01:10:00` 黑洞作为终极能量源与量子引力打破核子数守恒 无论是化学反应还是核反应，都无法改变反应体系中质子加中子的总数。物质约 99% 的质量都储存在质子和中子的静止质量能中，既非电磁相互作用能，也非强相互作用能；而这两大基本力都无法触及并释放这部分最庞大的静质量。引力却展现出截然不同的力量：如果从一个静质量为 $M$ 的物体开始，通过黑洞装置，原则上在扣除量子修正后，可以提取几乎 100% 的静止质量能量。这使得黑洞成为物理学定律所允许的最极端、最高效的“终极发电厂”。 ```mermaid flowchart TD accTitle: 能量提取机制对比 accDescr: 对比化学/核反应与黑洞引力提取静质量能的原理及核子数守恒差异 subgraph Conventional["常规反应机制 (化学与核反应)"] A["化学键 / 核子结合能"] --> B["仅释放少于 1% 的能量"] B --> C["质子 + 中子总数严格守恒"] end subgraph BlackHole["黑洞极端引力机制"] D["接近事件视界下落"] --> E["理论上提取近 100% 静质量能"] E --> F["经典视角：核子数储存于黑洞内部"] E --> G["量子引力视角 (霍金辐射)：黑洞吞噬核子数，辐射纯引力子/光子"] end ``` Dwarkesh 提出疑问：既然化学键或核结合能断裂时物体变轻，那如果一个包含质子和中子的物体无限贴近事件视界，是否意味着这些核子在外界看来消失了？它们的质量缩减到仅剩 1% 或 2% 究竟代表什么？Adam 解释道，在经典广义相对论中，黑洞永远存在，这些核子只是落入黑洞内部，只要将相应的核子数归属于黑洞本身，宇宙中的质子与中子总数在外部视角下依然是守恒的。 然而一旦引入量子力学，局面便发生剧变。根据贝肯斯坦（Bekenstein）和霍金（Hawking）的发现，黑洞会因霍金辐射而不断损耗质量，直至最终彻底蒸发殆尽。计算表明，黑洞蒸发释放出的全部能量几乎全都以引力子、光子以及极少量的中微子形式耗散，几乎完全没有质子或中子。Adam 指出，在量子引力框架下，黑洞会直接“吃掉核子数”（black holes eat nuclear number）。原本在电磁力和强弱核力中被微扰严格保护的全局对称性，在引力面前荡然无存——这也催生了现代量子引力的一条核心共识：“量子引力不尊重任何全局对称性”（quantum gravity doesn't respect any global symmetries）。 --- ### `01:12:35` 中场插播：利用 Cursor 估算 AI 样本效率与灵感捕捉 Dwarkesh 在节目中分享了赞助商 Cursor 的实际使用案例。他此前在一篇博文中提出猜测：过去几年 AI 训练中的样本效率（sample efficiency）或许并没有根本性的飞跃，人们观察到的提升更多来自数据分布的拓宽与质量的爆发。在一次与朋友聚餐时，他突然想到可以通过检验社区的“nano-GPT 极速挑战赛”（nano-GPT speedrun，即在冻结的训练集上，以越来越少的算力将 Karpathy 的 GPT-2 基线训练至固定损失值）的历史损失曲线，来估算样本效率的实际演进速度。 由于当时正处于社交场合，Dwarkesh 掏出手机直接在 Cursor iOS 应用中录制了一段语音备忘录便放回口袋。仅 15 分钟后，手机便收到了自动化完成的通知：Cursor 的智能体已经自动克隆了修改版 nano-GPT 的代码仓库，解析了历次纪录的损失曲线，并初步估算出模型样本效率每年大约有 2 到 5 倍的提升。Dwarkesh 强调，若不是工具消除了将灵感转化为实际行动的摩擦力，这个想法很可能在聚餐结束后便随风而逝。 --- ### `01:13:50` 坠入黑洞的两种视角：远方旁观者眼中的永恒红移与定格 Dwarkesh 幽默地切入一个实操物理场景：假设有人学完了广义相对论的所有高深方程，却不幸失足跌向黑洞，他会看到什么？Adam 强调，研究这个问题必须区分两个自洽却截然相反的观测参考系：一个是静止在无限远处观察落入者的旁观者视角，另一个则是坠落者自身的视角。 站在远方的旁观者看来，黑洞周围发生的情景极其怪诞： - **加速与减速的表象矛盾**：起初，下落者在黑洞引力势阱中下落，速度越来越快，先符合牛顿引力反平方律，随后显现出广义相对论的高阶修正。但随着越来越贴近黑洞，下落者的动作反而开始慢了下来。 - **引力时间膨胀的极致**：由于强引力场中的时间膨胀效应，下落者手腕上的时钟在旁观者眼中变得越来越慢。在数学积分上，外部观察者在有限时间内永远看不到下落者真正穿过事件视界，只会看到其以无限渐近的方式停留在视界边缘。 - **红移与“褪色变黑”**：观察下落者必须依赖光线，而光子从强引力场向外攀爬会发生极其强烈的引力红移，波长被拉得无限长。由于长波长导致的空间不确定性，落入者的轮廓会逐渐弥散，在发射出最后一个可被探测到的红移光子后，彻底隐入漆黑之中（"fade through red to black"）。 ```mermaid flowchart LR accTitle: 坠入黑洞的两种观测视角对比 accDescr: 对比外部静止旁观者与自由下落观察者穿越视界时的经历 subgraph Remote["远方旁观者视角"] R1["看到坠落者加速"] --> R2["引力时间膨胀显现"] R2 --> R3["速度看似变慢、定格在视界外"] R3 --> R4["光波无限红移，最终消失为黑"] end subgraph Freefall["自由下落者视角"] F1["自身时钟每秒一秒正常跳动"] --> F2["感觉完全平淡地平缓穿过视界"] F3["视界处毫无剧烈物理突变"] F3 --> F4["注定滑向中心奇点并被潮汐撕裂"] end ``` --- ### `01:15:45` 坠入者的亲身体验：“注定毁灭”与事件视界的非局域本质 然而，从自由下落者自身的视角来看，广义相对论呈现了完全不同的一幅图景。早期物理学家曾误以为穿越视界本身会让人遭受什么物理上的异象，但事实是：下落者自身的手表每秒依然精准地走过一秒（"running at one second per second"），下落者以不断加快的速度呼啸而过，毫无阻碍地“平滑穿过”事件视界。 - **潮汐力与黑洞质量的反比关系**：事件视界本身在局域上没有任何剧烈的物理撕扯。撕碎物体的真正力量是潮汐力（由于脚部比头部离引力中心更近而产生的差异引力拉伸）。如果是一个太阳质量大小的小黑洞，视界外的潮汐力就足以让人遭受致命的痛苦拉伸，将人“意大利面化”（spaghettified）；但黑洞质量越大，事件视界处的引力梯度与潮汐效应反而越弱。 - **星系级黑洞中的从容漫步**：如果面对一个相当于银河系质量的超大质量黑洞，穿过视界时下落者根本感觉不到任何异常；如果黑洞跨度达到数光世纪，从穿过事件视界到最终坠入中心奇点，其间的客观时间长到足以让一个人在黑洞内部过完一生，甚至繁衍后代。 - **目的论性质（Teleological Fact）**：Adam 深刻地指出，事件视界并不是一个可以在局域空间中被仪器直接测量的物理屏障，而是一个“目的论性质的事实”。一旦穿过视界，火箭如何喷火都无法阻止自己走向奇点——“你注定毁灭，但你尚未死亡”（"You are doomed, but you are not dead"）。只有当真正逼近最中心的引力奇点时，暴涨的潮汐力才会最终将一切物质撕得粉碎。 --- ### `01:18:00` 为什么物理学家相信黑洞真实存在而非虫洞？彭罗斯的理论突破 既然广义相对论的场方程预测了形形色色的奇特几何结构，为什么现代物理学界坚信黑洞必然真实存在，却普遍认为虫洞只停留在科幻与纯数学猜测之中？ Adam 回顾了科学史上的认知逆转： - **施瓦西解的早期争议**：1915 年爱因斯坦刚发表场方程不久，卡尔·施瓦西（Karl Schwarzschild）就给出了这一精确球对称解。当时几乎所有顶级物理学家（包括爱因斯坦本人在内）都认为这个解存在“病态”（sick），只是一个测度为零的数学怪胎，在拥有扰动和非完全对称的真实宇宙中不可能自然形成。 - **彭罗斯奇点定理的理论革命**：彻底扭转这一观念的是罗杰·彭罗斯（Roger Penrose，以及随后与霍金合作的拓展研究，彭罗斯也因此荣获诺贝尔物理学奖）。彭罗斯在理论上严格证明了：引力塌缩形成奇点和黑洞并非依赖精细微调的特例，而是广义相对论在一般初始条件下的“通用必然特征”（generic feature）。无论初始恒星物质的分布多么不规则，塌缩一旦启动，黑洞的诞生便不可避免。 - **观测证据的铁证如山**：理论奠基之后，天文学界又积累了排山倒海般的观测与实验证据，使得黑洞的存在从最初受人怀疑的假说，变成了现代天体物理学最无可争议的基石。 ### `01:20:00` 我们如何确知黑洞真实存在：从视线观测到引力波感知 在爱因斯坦所处的时代，“黑洞”这个词及其背后的物理实体根本没有被学界正式确立；甚至在爱因斯坦提出广义相对论后的整整五十年里，物理学家们依然对黑洞感到极度困惑，许多人一度笃信这种奇异的结构在现实宇宙中不可能存在。然而时至今日，人类已经积累了极为丰富且不容置疑的观测证据。 最直观、视觉上最震撼的证据来自对银河系中心的长期追踪。银河系中心盘踞着一个名为“人马座 A*”（Sagittarius A*）的超大质量黑洞，其质量高达太阳的数百万倍。黑洞本身是完全漆黑的，任何光学望远镜都无法直接看到它；但天文学家可以通过跟踪它周围恒星的运行轨迹来揭开它的面纱。经过数十年的连续精密监测，科学家们发现那里的恒星并不是沿直线穿行，而是在围绕着一个不可见的中心做着极其规则的小椭圆运动或发生进动的椭圆轨道运动。恒星们飞速掠过某个极小的区域却从未与之发生碰撞，由此计算可知，那个不可见的中心不仅极其沉重、极度黑暗，而且体积异常紧凑致密。 除了用眼睛“看”，大约十年前人类甚至真正“感受”到了黑洞的存在。激光干涉引力波天文台（LIGO）通过在不同地点建造的大型激光干涉仪，能够灵敏地捕捉到时空本身极其微弱的震颤。2015 年底刚开机后不久，LIGO 就迎来了一个历史性时刻：分布在不同地点的探测器（当时为两处，如今全球已有四处）在完全相同的时刻以完全相同的方式发生了震颤。这直接排除了局部路过一辆卡车或偶发地质地震的干扰，证明是时空本身在剧烈颤抖。科学家逆向推算确认，这次震颤源自约 10 亿光年之外、两颗质量各自约为太阳 30 倍的超级黑洞发生的正面碰撞。那场 10 亿年前的宇宙大碰撞产生的引力波涟漪，在 LIGO 开机几周内恰好抵达了地球。如今，人类已经探测并“触摸”到了数千次这类代表黑洞并合的时空震颤。 在此之后，事件视界望远镜（Event Horizon Telescope, EHT）通过联合全球各地的射电望远镜构建起一个口径相当于地球大小的虚拟观测网。EHT 细致入微地观测了银河系中心的人马座 A* 以及邻近星系（M87）中心更加庞大的超大质量黑洞，成功捕捉到了物质在被吞噬前剧烈辐射出的微弱射电波光晕，直接勾勒出黑洞视界的暗影。 ```mermaid flowchart TD accTitle: 三重独立证据链证实黑洞存在 accDescr: 展示证实黑洞存在的三大独立观测支柱：恒星轨道追踪、LIGO引力波感知与EHT直接阴影成像。 subgraph DirectEvidence["证实黑洞存在的观测支柱"] A["银河系中心恒星轨道追踪<br/>（Sagittarius A*）"] --> D["确立结论：<br/>超大质量、超高密度且不发光的紧凑天体绝对存在"] B["LIGO 激光干涉引力波探测<br/>（双黑洞并合的时空震动）"] --> D C["事件视界望远镜 EHT<br/>（吸积盘高能射电辐射与视界暗影）"] --> D end ``` - **三大观测支柱**： - **人马座 A* 周边恒星运动轨迹**：长达数十年的红外椭圆轨道监测直接锁定了数百万倍太阳质量的致密中心天体。 - **LIGO 引力波直接探测**：2015 年底首次捕捉到 10 亿光年外双黑洞（各约 30 倍太阳质量）并合所造成的时空真实震颤，目前已记录数千次并合事件。 - **事件视界望远镜（EHT）**：全球跨大洲射电望远镜协同干涉，拍下了吸积盘落入黑洞时产生的发光视界阴影。 --- ### `01:23:45` 跨越数十个数量级的理论壮举 广义相对论令人叹为观止之处，不仅在于它是近乎由爱因斯坦一己之力构想出来的宏伟理论，更在于它展现出的惊人解释力与跨度。正如对话中所感叹的：“真正不可思议的是它所涵盖的自由度与数量级。” 思考广义相对论的起点，仅仅是爱因斯坦在脑海中想象人们在封闭电梯里跳上跳下的思想实验；但这个理论随后一路延伸，精确解释了水星轨道的微小进动，揭示了太阳系内引力摄动的精细动力学，预言了光线掠过大质量天体时的弯曲，接着能够描述整个星系的旋转，最终甚至跨越至整个宇宙的膨胀模式与终极命运。几乎只凭一个人的大脑，就推导出了统摄如此多数量级尺度的优美法则，以至于“坦白讲，我们的宇宙能被这样一个优美的理论所描述，本身就是宇宙的荣幸”。 --- ### `01:24:21` 引力弯曲光线：从理论假说到世界共识的曲折传奇 爱因斯坦的理论是如何从一个孤独思考者的脑中假说，转变成全世界科学界信服的真理的？答案的核心就在于引力导致的光线偏折实验。 在广义相对论出现之前，牛顿经典力学就存在已知的反常观测——例如水星近日点的进动始终无法用牛顿理论完美解释。广义相对论一出世就极其精确地计算出了水星轨道的进动值。然而，用新理论解释一个已知的既有常数，在说服力上总缺少了一点戏剧性，科学界往往认为：唯有在事先完全不知道真实数值的情况下做出全新且准确的预测，才能带来无可辩驳的震撼。这便是光线偏折预言发挥作用的历史舞台。 在广义相对论中，所有的能量都会产生引力，也都会受引力影响；因此光线掠过像太阳这样的庞然大物时必然会发生偏折。其实在牛顿力学框架下，如果将光子看作以光速运行的粒子，根据撞击参数与速度，光线也会弯曲（速度越快弯曲越小）；但广义相对论通过计算时空弯曲得出的偏折角度，恰好是牛顿力学计算结果的整整两倍。 然而这段历史充满了不可思议的戏剧性转折： 1. **爱因斯坦初期的“幸运错误”**：在正式完成完整的广义相对论场方程之前，爱因斯坦仅凭早期的等效原理推算光线偏转，得出的数值竟然恰巧等于牛顿力学的值。 2. **天文学家的无情泼冷水**：爱因斯坦当时致电天文台询问能否直接观测太阳背后恒星的光线偏转。威尔逊山天文台（Mount Wilson Observatory）台长像对待一个“天真的纯理论家”那样回应他：这绝不可能，把望远镜对准太阳会瞎掉，对准太阳边缘也会被耀眼的日冕完全淹没——除非发生日全食，月球完全遮挡住太阳光辉，才能在黑暗中看到掠过太阳边缘的恒星位置是否发生了偏移。 3. **1910 年代的探险噩运**： - 1911 年，观测队跋山涉水远赴阿根廷观测日全食，结果遭遇阴云密布，大雨彻底浇灭了观测希望； - 随后，德国军火巨头克虏伯（Krupp）资助了一支德国考察队前往克里米亚半岛架设望远镜。就在日全食即将发生前夕，第一次世界大战猝然爆发，德俄成为交战国，考察队全体科学家当场被俄军逮捕并关押拘留，实验再次告吹。 4. **塞翁失马**：这些早期的失败对爱因斯坦来说其实是天大的幸运。因为如果当时真的测出了数据，就会发现它与爱因斯坦最初根据早期错误推导发布的“牛顿数值”不符。在一战期间全球日食观测完全停滞的几年里，爱因斯坦得以静下心修正了错误，并于完整的广义相对论中确立了“偏折角是牛顿预言的两倍”的最终结论。 5. **1919 年爱丁顿的历史性日食观测**：一战结束后不久，英国天体物理学家阿瑟·爱丁顿爵士（Sir Arthur Eddington）率领英国远征队奔赴普林西比岛等地观测日全食。当他成功带回观测数据并宣布测量结果完美符合爱因斯坦两倍于牛顿的偏折预言时，举世轰动。一位英国科学家在战后通过实验验证了一位德国科学家的激进新理论，不仅成为一战后科学界跨国和解的伟大象征，更一夜之间将爱因斯坦推上了全球顶流科学偶像的王座。 自此之后，广义相对论正式确立为现代物理学的公认支柱。如今，科学家已通过极高精度的轨道动力学、全宇宙范围的光线引力红移以及更复杂的引力透镜现象，对该理论进行了无数次高精度验证。 - **牛顿预言 vs. 广义相对论预言**： - 牛顿力学（假设光速粒子受引力牵引）：预言光线偏折角度为基准值 $\theta_{\text{Newton}}$。 - 广义相对论（能量引力耦合 + 空间曲率贡献）：得出偏折角度恰好为 $2 \times \theta_{\text{Newton}}$。 - **验证历史里程碑**： - 1911 年阿根廷日食：因云层遮挡彻底失败。 - 克里米亚半岛日食：受一战爆发影响，德国考察队全员被捕扣押。 - 1919 年爱丁顿非洲日食观测：实测偏折角吻合两倍牛顿预言，广义相对论自此成为科学界共识。 --- ### `01:28:45` 纯粹思维 vs 宏大实验：科学发现模式的反思 面对现代科学为了探寻基础物理规律动辄耗资百亿美元建造巨型粒子对撞机或天文阵列的现状，德瓦尔凯什提出了一个发人深省的问题：回顾人类历史上最深邃、最壮丽的物理理论——广义相对论的诞生，它几乎就像是一个天才独自躲在洞穴里纯靠深度思考创造出来的产物。爱因斯坦提出广义相对论时所依赖的经验基础少得惊人：也许仅仅是知道光速是有限常数这一事实；至于引力常数 $G$，在广义相对论的结构中也只是一个自由参数，理论本身的数学与哲学自洽性并不严重依赖实验对常数的预先标定。这种极端依托直觉与纯粹理性构建宏伟理论的范式，与如今高度依赖工业级巨额预算实验的物理学形成了鲜明而迷人的对比。 ### `01:30:00` 纯思维的浪漫：爱因斯坦奇迹与理论物理的局限 关于广义相对论的诞生，Dwarkesh Patel 与 Adam Brown 探讨了理论物理学对经验事实的极低依赖度。从成本上看，理论物理学家本身非常廉价，人们总有一种强烈的诱惑：与其花费巨资建造昂贵的大型实验装置，不如把钱全投给坐在椅子上沉思的理论学家。尽管如今 AI 公司对理论物理学家的争夺极大推高了这一需求曲线，但仅凭纯粹思考究竟能走多远？ Adam 指出，广义相对论在这方面算是一个极端特例，物理学史的大多数时候并非如此运作。爱因斯坦的形象就像安·兰德（Ayn Rand）笔下孤身奋战的英雄，凭借单一的心智追求多年，最终拿出了纯粹由个人愿景凝结而成的杰作。虽然他在数学等方面得到过各方协助，但整体框架几乎就是个人心智的直接产物。他在纸上写下理论后，虽历经一次耗资不菲的日食观测远征才彻底获得验证并跻身全球名人，但它依然是极少数仅靠极度专注的思维与思想实验就推导出正确物理现实的典范。物理学界在此后的一百年里一直在“追逐这种快感”（chasing that high），每个学者都渴望拥有这种仅凭少量实证线索就能洞穿宇宙规律的浪漫体验。然而，这种路径在其他人身上鲜有成功，甚至爱因斯坦自己在后半生也没能再复制这一奇迹。 构建广义相对论所依赖的经验基石其实极其单薄： - **光速有限性与对称性**：不仅需要承认光速是有限常数，还需认识到狭义相对论中保护光速不变的背后对称性。 - **等效原理（Equivalence Principle）**：即惯性质量与引力质量完全相同的经验事实。 仅凭这两点，理论走向的选项就已经大幅收窄。Dwarkesh 提出，如果未来拥有成千上万个大语言模型（LLM），在可能的分支有限的情况下，它们完全可以遍历整个理论探索树（例如放弃同时性、深挖等效原理等）。但 Adam 提醒，爱因斯坦之所以能成功，在很大程度上是因为广义相对论的物理情境极端幸运，并非所有科学前沿都能如此轻易被纯思维推演。 --- ### `01:32:41` 弦理论的赌注与科学探索的分支因子 在被问及今天若有数百万个 AI 爱因斯坦同时自主运行，是否能在物理学前沿取得颠覆性突破时，Adam 表达了谨慎的乐观。他认为科学的不同领域具有完全不同的“分支因子”（branching fractions），这直接决定了你究竟有多需要依靠实验来砍掉死胡同。 弦理论在某种程度上就是“追逐爱因斯坦快感”的集大成者。为了将不包含任何量子力学机制的广义相对论与量子力学相统一，物理学家面临着巨大的观测鸿沟：若做量纲分析（dimensional analysis），想要在实验中直接探测到量子引力效应，需要建造银河系规模（galactic-sized）的巨型粒子对撞机。在完全无法获得新实验数据的前提下，弦理论学家只能孤注一掷地押注于类似爱因斯坦当年的逻辑——仅靠极度深入的数学思维与极少的实验输入来摸索正确道路。 ```mermaid flowchart TD accTitle: Theoretical Physics Branching Paths accDescr: Flowchart comparing the feasibility of pure theoretical exploration in string theory versus condensed matter physics. subgraph StringTheory["弦理论路线：纯理论探索"] A["经验输入匮乏<br/>(需银河级对撞机)"] --> B["核心工具:<br/>数学自洽性与极限回归"] B --> C{"可能自洽理论的数量?"} C -- "极少数 / 唯一" --> D["成功摸索出<br/>正确理论"] C -- "无限多可能" --> E["无法仅凭自洽性<br/>收敛至真实物理"] end subgraph CondensedMatter["凝聚态与实验主导学科"] F["分支因子极高"] --> G["存在海量候选假设"] G --> H["必须依赖实验<br/>硬性剪枝排除"] end ``` 这种做法能奏效的前提极其苛刻：理论物理学家手中仅有的工具是**数学自洽性**（mathematical consistency）以及**在已知极限下能否退化为现有物理学**。 - 如果自然界中在数学上自洽的引力理论只有唯一一个或极少数几个，那么只要做足够完备的自洽性校验，人类（或 AI）确实可能仅凭纯思维找到终极真理。 - 但如果自洽的理论存在无限多个，物理学家就永远不可能只凭自洽性与某种“美学直觉”摸索到真实世界的那一个。 相比之下，在凝聚态物理等其他学科中，候选方案实在太多，分支极度发散，除了直接做实验检验，没有任何捷径可以确定哪个分支符合现实。 --- ### `01:35:00` AI 数学时代的“消化不良”与超级解释者 随着未来机器建立起更深邃的物理学与数学理论，人类是否还有能力理解机器所理解的宇宙？Adam 的态度比很多悲观的预测更加积极。 在数学界，陶哲轩（Terry Tao）曾用“消化不良”（indigestion）这一生动的词汇来形容一种令人沮丧的前景：许多数学家担心大语言模型最终会沦为单纯的“做题证明机器”，吐出数十亿行晦涩难懂的 Lean 形式化代码。这些代码虽然能作为证明定理成立的无可辩驳的凭证，却无法为人类提供任何关于“为什么成立”的数学洞见。但 Adam 认为这种未来并不会真正发生，因为大语言模型不仅能成为**超级证明者**（superhuman provers），还将成为**超级解释者**（superhuman explainers）。 面对晦涩复杂的证明，AI 能以惊人的毅力不断尝试，将其重塑、提炼为人类易于消化的形式。早期实证已经在支持这种积极愿景： - **埃尔德什问题（Erdős problem）的突破**：几前个月被证明的一个埃尔德什猜想并未采用令人望而生畏的形式化 Lean 代码，而是以非形式化的直观证明呈现。随后人类数学家发表的跟进论文，成功吸收了机器提出的直观新思想，并将这些全新概念直接迁移、应用到了其他未知定理的证明之中。 - **单位距离猜想（Unit Distance Conjecture）的反例发现**：AI 找到的反例构造对人类而言完全清晰易懂。人类长期未能证伪该猜想，很大程度上是因为人类数学家先入为主地错误认为它是真的；而 LLM 具有无限的耐心，愿意在人类看来“完全是在浪费时间”的方向上死磕到底，最终攻破了被普遍视为低概率成功的猜想。 最后，Dwarkesh 感谢 Adam 在负责构建超级智能之余，抽空深入浅出地讲解这些诞生已逾百年的经典物理学奥秘；Adam 亦欣然回应，分享如此优美有趣的理论本身就是一件令人无比快乐的事。 ## Notable quotes > `00:00` 广义相对论是人类大脑构想或见证过的最美丽的产物。 — Dwarkesh Patel > `00:00` 连引力也不例外——光速不可逾越。 — Adam Brown > `30:00` 物质告诉时空如何弯曲，时空告诉物质如何运动。 — John Wheeler > `22:20` 在进行任何计算之前，你必须先对答案有一个很好的猜测。 — Jed Thompson > `52:15` 在事件视界处，你尚未死亡，但已注定毁灭（You are doomed, but you are not dead）。 — Adam Brown > `01:00:00` 你会看到我像在按快进键一样生活。 — Adam Brown > `01:05:00` 广义相对论的远距离极限必须还原为最初发现的牛顿物理学，事实也确实如此。 — Adam Brown > `01:10:00` 量子引力不尊重任何全局对称性（quantum gravity doesn't respect any global symmetries）。 — Adam Brown > `01:23:45` 坦白讲，我们的宇宙能被这样一个优美的理论所描述，本身就是宇宙的荣幸。 — Dwarkesh Patel > `01:30:00` 物理学界在此后的一百年里一直在追逐这种快感（chasing that high）。 — Adam Brown ## Mentioned Adam Brown, Dwarkesh Patel, Google DeepMind, Blueshift, 阿尔伯特·爱因斯坦 (Albert Einstein), 艾萨克·牛顿 (Isaac Newton), 詹姆斯·克拉克·麦克斯韦 (James Clerk Maxwell), 查尔斯·奥古斯丁·德·库仑 (Charles-Augustin de Coulomb), 约翰·米歇尔 (John Michell), 皮埃尔-西蒙·拉普拉斯 (Pierre-Simon Laplace), 卡尔·史瓦西 (Karl Schwarzschild), 约翰·惠勒 (John Wheeler), 罗杰·彭罗斯 (Roger Penrose), 斯蒂芬·霍金 (Stephen Hawking), 雅各布·贝肯斯坦 (Jacob Bekenstein), 阿瑟·爱丁顿 (Arthur Eddington), 陶哲轩 (Terry Tao), 安·兰德 (Ayn Rand), 保罗·埃尔德什 (Paul Erdős), Jed Thompson, Jane Street, Crusoe, Cursor, LIGO (激光干涉引力波天文台), 事件视界望远镜 (EHT), 人马座 A* (Sagittarius A*), M87, 庞德-雷布卡实验 (Pound-Rebka experiment), Lean, NanoGPT ## Video description <p>Adam Brown is back!</p><p>General relativity is said to be the most beautiful idea the human mind has ever produced. Most of us will never get to fully appreciate its elegance by taking the 20-lecture graduate course Adam taught on it at Stanford. But in this episode, Adam distills the key idea at its heart so clearly and compellingly that even I could keep up lol.</p><p>At the core of general relativity, Einstein is trying to figure out the principle behind a particular coincidence: that the mass that resists acceleration and the mass that gravity pulls on just happen to be exactly the same. Adam then leads us through the path of insight which Einstein called his “happiest thought.”</p><p>Then Adam lectures on black holes. First, by showing how even under special relativity you could create a perpetual motion machine if black holes weren’t truly black. And then, by explaining why the observations of an infalling observer and a distant bystander to the black hole would be so radically different</p><p>Adam leads Blueshift, the team at Google DeepMind cracking science and reasoning, which gave us the opportunity to discuss at the very end how close we are to AIs that could rediscover general relativity from scratch. Stay till the close for some philosophy of science.</p><p>Watch on <a target="_blank" href="https://youtu.be/QbdbAhaJoCQ">YouTube</a>; read the <a target="_blank" href="https://www.dwarkesh.com/p/adam-brown-gr">transcript</a>.</p><p><strong>Sponsors</strong></p><p>* <a target="_blank" href="https://janestreet.com/dwarkesh">Jane Street</a> has traders from all sorts of different backgrounds. For example, I recently got to speak with Jed Thompson, a trader who started his career in particle physics. Jed told me how the habits he built as a physicist (like never running a calculation without first having a good guess at the answer) helped him build good trading intuition. So no matter what field you’re working in right now, your experience may be more applicable than you think. Check out open positions at<a target="_blank" href="https://janestreet.com/dwarkesh"> janestreet.com/dwarkesh</a></p><p>* <a target="_blank" href="https://crusoe.ai/dwarkesh">Crusoe</a> gave me early access to their new serverless fine-tuning product, so I decided to try fine-tuning a Dwarkesh-style question generator. Crusoe made this really easy: I just turned my interview transcripts into training data and then kicked off a run – I never had to touch infra or tweak hyperparameters. After training was done, I ran a blind eval with my team: they preferred the fine-tuned model’s proposed questions over my own suggestions about 30% of the time. Serverless fine-tuning goes live next week. Learn more at<a target="_blank" href="https://crusoe.ai/dwarkesh"> crusoe.ai/dwarkesh</a></p><p>* <a target="_blank" href="https://cursor.com/dwarkesh">Cursor</a>’s iOS app lets me kick off real work no matter where I am. For example, recently I was at dinner with friends when I had an idea about how to investigate the past few years of progress in sample efficiency. I pulled out the Cursor app, dumped my thoughts into a voice note, and 15 minutes later, Cursor had cloned the relevant repo, done the necessary analysis, and written up its findings. And now I’m expanding that work into a full write-up. Without the Cursor app, the idea would’ve floated away. Check out the app now at<a target="_blank" href="https://cursor.com/dwarkesh"> cursor.com/dwarkesh</a></p><p>Timestamps</p><p>(00:00:00) – The coincidence that led Einstein to general relativity</p><p>(00:16:42) – Gravity is a consequence of curved spacetime, not a force</p><p>(00:31:46) – Why black holes prevent unlimited energy extraction</p><p>(00:47:12) – Black holes are the ultimate power plants</p><p>(01:13:50) – What falling into a black hole would actually feel like</p><p>(01:18:51) – The three ways we know black holes are real</p><p>(01:24:21) – The first time we saw gravity bend light</p><p>(01:29:33) – How far can AI get without experimental evidence?</p> <br/><br/>Get full access to Dwarkesh Podcast at <a href="https://www.dwarkesh.com/subscribe?utm_medium=podcast&#38;utm_campaign=CTA_4">www.dwarkesh.com/subscribe</a> [Source](https://api.substack.com/feed/podcast/206445844/59f976809738bf5ad158b7617346dda3.mp3)

[0:00] Dwarkesh Patel: 广义相对论是人类大脑构想或见证过的最美丽的产物。 [0:00] Adam Brown: 连引力也不例外——光速不可逾越。 [30:00] John Wheeler: 物质告诉时空如何弯曲，时空告诉物质如何运动。 [22:20] Jed Thompson: 在进行任何计算之前，你必须先对答案有一个很好的猜测。 [52:15] Adam Brown: 在事件视界处，你尚未死亡，但已注定毁灭（You are doomed, but you are not dead）。 [60:00] Adam Brown: 你会看到我像在按快进键一样生活。 [65:00] Adam Brown: 广义相对论的远距离极限必须还原为最初发现的牛顿物理学，事实也确实如此。 [70:00] Adam Brown: 量子引力不尊重任何全局对称性（quantum gravity doesn't respect any global symmetries）。 [83:45] Dwarkesh Patel: 坦白讲，我们的宇宙能被这样一个优美的理论所描述，本身就是宇宙的荣幸。 [90:00] Adam Brown: 物理学界在此后的一百年里一直在追逐这种快感（chasing that high）。

[0:00] Adam Brown: nothing can travel faster than light [0:00] Adam Brown: Not even gravity—nothing can go faster than light [0:00] Dwarkesh Patel: the most beautiful thing the human mind has ever conceived or seen [20:00] Adam Brown: Happiest thought [20:00] Adam Brown: Could gravity itself simply be an inertial force? [25:00] John Wheeler: Matter tells spacetime how to curve, and spacetime tells matter how to move. [65:00] Adam Brown: As R approaches the black hole event horizon... I have extracted exactly all of the energy from the brick... The brick falls into the black hole and I have extracted the entire mc^2 that used to be in the brick in my pulley system out there. [75:00] Adam Brown: fade to black [78:51] Adam Brown: mathematical monstrosity [95:00] Terence Tao: indigestion

[0:00] I'm back with Adam Brown. You currently need Blue Shift at Google Deep Mind, 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 vintage 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. 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, I think, as you say, 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. And it took Einstein one of the most famous minds in history about a decade to figure it out But when know when I teach it I do a 10 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've 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 and not make many of the same mistakes that were made by our forebearers. I think in 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. Okay, let's go. 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, two of 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 much more to it than that but it going to complete this arc of the centrality of nothing being able to go faster than the speed of light. 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 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 a object resists being accelerated. [5:00] That's 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 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 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 law of gravity says that the force caused by the gravitational interaction of two bodies is what called Newton constants 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, his famous inverse square law. And it points, it's 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 force law with this principle One option of course could be that this is true for non forces but not true 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 and so einstein and in fact many 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 okay so um you know 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 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. And again, for exactly the same reason, the electrostatics looks to be inconsistent with special relativity. But ultimately, it's not, or ultimately, this is not the full story. Electrostatic 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. In fact, the direction, 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 that, [10:00] hey, Maxwell's equations actually are fully consistent with nothing can go faster than the speed of light. And that consistency is reflected in a symmetry called the Lorentz symmetry of the Maxwell's field equations only noticed later after they were written down that eventually led Einstein to formulate his special theory of relativity. 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 the 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. And there's 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 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 mass, 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. 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 one particle, the photon, and gravity is going to be mediated by a spin two particle. And that responsible for the change in that sign now Okay so that why you can 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 parts 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 is how he would describe it. 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, 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 from newtonian physics so mass in electrostatics force forces and accelerations plays exactly one role it's sitting here it's the it's the inertia of the object and it what is resisting being accelerated so this is sometimes called the inertial mass that sitting here 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 relation, 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. 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 one part in a thousand or so. [15:00] 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. 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, 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 Okay so Einstein genius was to hone in on this as a central clue for how he is going to end up replacing Newton law And a reason it a central clue is because there is in fact another class of forces not fundamental forces like electromagnetism and 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. So here's a bucket. Here's some water filling the bucket. You better know your physics, Adam. Otherwise, you'll destroy the studio. No tricks. Will you put your finger in that and confirm that it's wet? Okay. So 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 of force of gravity, as we'd 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 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. 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 exclamation 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 um and that force is is known as the centrifugal force and it's 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 going around 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 around a bend. And, okay, what do we notice? What we notice is that your charge 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. 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 that you're not moving along straight lines, 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, Thank you. [20:00] For any time you have one of these inertial forces just caused by your inertia, it is guaranteed to be the case that the charge under that force is given by the inertial mass. Okay, 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 will 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's laws, but a necessary fact about the world. So this was Einstein 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 is just sitting there and 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. 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 them with his work at Jane Street 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. So even if you don't have a finance background, or for that matter, a physics background, you should still consider applying. Go to jainestreet.com slash twerkash to learn more. 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 Dworkesh, just sitting here at constant height. because you are experiencing the force of gravity if gravity is an inertial force because you experiencing a force down that means that you have to be moving along a knot straight line So that's you. 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 you know 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 uh 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 at [25:00] in San Francisco. Here is Greenland coming in from the North Pole. And here is England. Here's London. I can tell you're a physicist because of the very idealized forms of the continent. 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 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. 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 space with general relativity where this parabolic arc of the chalk as it 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 space time. And in fact, you are in a curved space time. So in Einstein's theory, the effect of matter is going to be to curve space-time. And through curving space-time, 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 space-time is curved. 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 space 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. 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 space-time. This says how much space-time is curved. This is some tensor, and the tensor will be zero if space-time were flat. And it's non-zero when space-time is not flat, it's curved in a particular way. On the right-hand side is not space-time anymore. On the right-hand side is matter. There's some constants, just our old friend Newton, [30:00] 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. 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 space-time on the left-hand side, or in a slogan, matter tells space-time how to curve. And then once matter is told space-time how to curve, the curvature of space-time tells matter how to move in the second half of the slogan, where the curvature of the space-time tells matter to move along straight lines of the curved space, and so experience fictitious forces if you try and pretend that space-time is flat. And that is Einstein's general theory of relativity in a nutshell backing up like an amazing thing about newtonian gravity is that he invented it you know allegedly due to some thought experiment to do with 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 It like massive cross hit like that that it describes planetary motion and also an apple falling off a tree And this was like this amazing thing that Newton unified the heavens and the earth and had 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. It describes the motion of Mercury and the planets in the solar system. And it describes the expansion of the entire universe. that's like a crazy you know huge numbers of orders of magnitude hit that does that darts all of those you're saying a moment ago one of the beautiful things about this theory is that it has reach uh 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 um than the high school version of light falls into it and can't get out of why black holes work the way they do. 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 you know that those equations were so complicated, no one would ever come up with exact solutions to them. And we just always be having to do approximations. And that's not to be correct, that Schwarzschild, who was a Prussian artillery officer in the First World War, in between writing, you know, 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 had written them down within a matter of months, in fact 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 were not called a black hole at the time. In fact, people were extremely confused about what this 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 wrote all sorts of wrong things about how objects would maybe bounce off the event horizon. Just totally confused. 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, as we set it up, is 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 know 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 kilometres per second. It's the escape velocity. But for objects that are heavier or more compact, the escape velocity is larger. So for a scumple of Jupiter, it would be hundreds of kilometres a second. You need to shoot it off. [35:00] 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 idly 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 velocity is, and just putting the velocity equal to the speed of light, this gives a critical radius of 2 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. The connection you're pointing out is a Newtonian one. Did anybody make this connection before? 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 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 than that 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. 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 of force so it an inverse r distance law not an inverse square distance law that 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 just put it on the surface of the earth And so this is the amount of energy I've got out here a long way away and I've extracted from the brick. 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. And 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. 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 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 extracted a fraction 7 times 10 to the minus 10 of the original rest mass energy of the brick Doing useful work a long way away 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, the relativistic effects, where the first order term is just Newtonian, and then the next order terms will give you the GR corrections to the Newtonian answer. 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 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 squared of the oxygen hydrogen you're going to mix together is given by 1.5 times 10 to the minus [40:00] 10. First observation, these two are close to each other, even though they came from completely different calculations. This was a gravitational calculation that was 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 and 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. 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. In particular, this number is a few times bigger than this number which means that almost all of the the vast majority your payload fraction is quite small when 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. 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's heavier or more compact, the fraction of energy that you extract by lowering the object down to to the surface is going to be larger. So for example, if you lower it not down to the Earth's surface, but down to the Sun's surface, this would be larger. You know, a million times larger because the Sun is, you know, a few million times the mass of the Earth, but then it's also bigger, so that takes it away a little bit, and you end up with 2 times 10 to the minus 6, famous redshift from the sun surface and you can escape from there You can imagine cramming a sun mass into a Earth radius to make this formula even bigger In fact that pretty much exactly what so you know sun mass Earth radius which is about 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'll be extracting by lowering it down to the surface. But it really feels like something has to give before we make an object that is too mass, 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 can get more than 100 of the mass of your brick back by lowering it down to the surface of this object And that feels wrong That feels in fact sort of more wrong than what going on here because now you got all this energy a long way away and you could perhaps use it to make a whole new brick. 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. 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 you can imagine two ways that you could avoid this conclusion one would be somehow that gravity becomes very weak when you get close to a massive object, weaker than the Newtonian law would predict. 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 [45:00] 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. That's actually the opposite of how general relativity results this. General relativity resolved 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, a finite distance 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 formed black hole. 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 be cool to try fine tuning a question generator using the transcripts of my old interviews The models have 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 hyperparameters. 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-server 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 slash thwarkash. So far, everything we've written down on the board is Newtonian. It's just 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. They're going to give us intuition for what it's like outside and indeed inside a black hole. 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, 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 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 static what is the local force of gravity that you experience Or you can imagine that you sitting here The reason you static is you firing a rocket very hard And you know what is the how much acceleration do you do locally feel So by whatever mechanism you're remaining static, what is the local force of gravity that you feel? The local force of gravity 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 minus 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 1, R is very big, and you get Newton's force law back again. And for the Earth, this is very small, as we discussed, [50:00] 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 tailor 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 fourth 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. 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 a black hole, This is a very significant radius over here, 2 gm over c squared. It's called the event horizon. It 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. 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 constructive for the way gravitational attraction happens in general relativity. 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 a long way away from the black hole helps you stay away from the black hole stops you falling in 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. 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, it's not too hard to write down, if you wrote the version of this formula that applies when you have angular momentum, you would see that pushing you away from the black hole. 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 the 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. 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 black 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 convert yourself to light and try and shoot yourself out, not if you fire your rocket. [55:00] infinitely hard. The other place, of course, is r equals zero, which is that's 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 of race, if you try and resist the force of gravity. Okay, that's formula number one. 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. Asks about gravitational time dilation. So let's again imagine that you're sitting here. This is Dorker sitting here, some radius R away from the black hole. And I'm sitting out here way off at infinity, just watching you. And we're static relatively 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? Of course, as far as you're concerned, your watch is ticking at one second per second As far as I concerned my watch is ticking at one 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 as 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 wristwatch runs than mine? And it says that The time interval as measured by your wristwatch is given by the time interval as measured by my wristwatch a long way away times this exact same square root factor that's showing up all over the place, times the square root of 1 minus 2 gm over r c squared. And so this factor here is less than 1. So if I think one second has passed, you think less than one second has passed. In other words, if I slowly lure 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 I observe your wristwatch 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 slower. This is an effect that is now considerably within the precision of, for example, GPS. It 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 account for that difference and subtract it off in order to get an accurate read. This is known as gravitational time dilation, and notice it's quite different from the relativistic time dilation you see in special relativity which is caused by two objects being in motion relative to each other Here we not in motion relative to each other we both static we 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. 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... [60:00] the more relevant inertial frame than the other one. You're exactly right. Yeah, so in special relativity, if you and I are moving relatively 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 is 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. You really, we both agree that you are deeper in the gravitational well than I am. 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 sped up. If you are observing me, you see me living my life and fast forward. 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 going to think that it is lower frequency than it frequency than 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 redshift, gravitational redshift. It gets redshifted, 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 redshifted 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 a 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 and so the third formula says suppose that you Dorkesh 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 I have, 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 just Avogadro numbers of carbon atoms or let 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 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 minus 2 gm of r c squared out. Mass down here suffers this red shifting as it goes up and has less energy by the time it reaches infinity than it did to begin with. 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 the amount [65:00] 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 up from the black hole. And so if we know that formula, then we can in fact calculate exactly this formula. 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 2gm 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 derive this using Newtonian physics. this is exactly correct not just in the newtonian limit but all the way uh to where the effects of general relativity are important now if you're a very very long way away from the black hole r is much much bigger than 2 gm of a c squared then you can tailor expand this formula and the first order term is just the old newtonian formula it better be it better be that the long distance limit of general relativity recovers the newtonian physics that we that we originally discovered. But as you get closer and closer to the black hole, this starts to deviate from the Newtonian answer and deviate in the Newtonian answer that exactly is going to end up resolving our original thought experiment to do with lowering a brick down towards a black hole. 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 still a long way from the black hole then I extracted 1 minus 1 equals 0 I haven extracted any energy from the black hole As I lower it closer and closer to the black hole, well, 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. So I start off with a brick a very, 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'd lower it 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 MC squared 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 mc squared from the brick? No. Is it possible to extract the full mc squared 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. so you know most power plants today operate by burning chemical energy that is not very efficient that gets you know if the pair factor of 10 to the minus 10 because chemical bonds are super weak compared to the rest masses of objects and you're really only extracting a tiny fraction of the rest mass of the fuel that you're you're considering you can level up from there by going to 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 3 for fission or 10 to the minus 2 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 [70:00] 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 M, I can extract, up to quantum corrections essentially, I can extract essentially 100% of the rest mass energy that I've gone in. It is the most efficient possible power plant, because by building an apparatus like this, in principle, I could extract 100% of the energy of whatever I started with. I intuitively get how energy equals mass, and then there's 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 mean for them to have 1 or 2 or 5 of their original mass Yeah that 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 are still conserved out there in the universe. It's just you need to assign a nuclear 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 another bit, ends up in protons and neutrons. So it is a very interesting fact, once you turn on quantum gravity, that black holes eat nuclear 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 we talking about quantum gravity now to a general principle that quantum gravity doesn respect any global symmetries It doesn respect nuclear number symmetry It doesn respect any of these symmetries And that's a whole other can of worms that we can open some other day. 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 speedrun 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 out my phone, I dumped this idea into a voice note in the cursor app, and I went back to dinner. And then I got a notification about 15 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 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 able to just kick off the investigation right then and there with the Cursor app If you want to try Cursor iOS app go to cursor slash Dwarkesh Okay, Adam, I'd like to think on this podcast we impart not only theoretical but practical knowledge as well. 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? 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 hard you need to fire your rocket to not fall into the black hole. 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 hole, start picking up corrections, start picking up general relativity corrections to the Newtonian inverse square law. And so what I will see as I watch you fall towards the black hole is first you'll go faster and faster. [75:00] faster towards the black hole as you fall down the gravitational potential of the black hole. 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 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. 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. The longer the wavelength of light, the harder it is even to really see you. You start getting delocalized by the wavelength of the light and eventually I just stop seeing you entirely There a final photon that I see that you emit and then you just fade to black fade through red to black I never see you cross the event horizon This was noticed by people in the early days of general relativity and greatly confused them and they started to think that you would experience something funny yourself as you fell across the event horizon. 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 one second per second. If you look back at me, there's some funny stuff going on to do with 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 normal. The event horizon is not a particularly violent place for you. You can calculate the tidal forces as you across the event horizon, or as you approach and then across the event horizon. And they're not particularly big. 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 because it's because it's because they're closer and you end up getting getting stretched but if i take a big enough black hole you wouldn notice anything funny happening whatsoever um the the bigger the black hole the smaller the tidal effects And if I took a black hole the mass of the galaxy you be basically fine as you cross the event horizon. If I took an even bigger black hole than that, you could live out your entire life having crossed the event horizon of the black hole before you hit the singularity, which is fatal. When you cross the event horizon, you are doomed. You are doomed because once you cross the event horizon, you must proceed to the singularity. There's no way you can fire a rocket to stop yourself hitting the singularity. You are doomed, but you are not dead. You are only for sure dead once you hit the singularity and get spaghettified, get mangled by the tidal forces. But for a large enough black hole, you can be doomed and not even know it. The event horizon 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. You could, in principle, for a large enough black hole, live out your entire life if you had a black hole that was you know 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 As you saying GR explains or predicts a lot of phenomenon, and some we think are correct, some we don't know are correct. Why do we think black holes are correct, but not wormholes? Yeah, that's a great question. And people did not believe it to begin with. Schwarzschild wrote down his solution almost immediately after Einstein Reuters 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 was 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, that 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. [80:00] 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's 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 star. 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 processing 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 is it, how massive is it. And what you find is that it's very massive indeed. And it's also very small indeed. You know it small because the stars get super close to it but don 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 star, the black hole at the center of our 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 space-time itself. And there's a famous event pretty much immediately after we turned it on in late 2015 where we felt space-time shaking. and you knew it was space-time shaking not just the earth shaking because we had a bunch of these detectors then two now now four at different points on the earth and they all shaped in exactly the same way so it couldn't just be explained by a truck passing one and not the other or a seismic event in 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 both of which weighed about 30 times as much as the Sun on the other side of the universe about 1 billion light years away That collision happened about 1 billion years ago and just happened to reach the Earth within weeks of us turning on the Lego 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. There is a... Later, we had what's called the Event Horizon Telescope, which is a ginormous conglomeration of radio telescopes all over the Earth, that were able to look very closely at the black hole at the centre of our galaxy, Sagittarius A star, and the even bigger black hole at the centre of our neighbouring galaxy, and see very, very faintly the radio emissions of matter falling into these black holes that shine super brightly as it does so, and we were able to see that in the radio emissions of these things. 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. It's so beautiful that not only can a single mind come up with this theory, but 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. It 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 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 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 oh um 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 that 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. And so that would be... [85:00] bending of light. Certainly, historically, that was the most influential. So, according to general relativity, all energy gravitates, and all energy is affected by gravity. And so, light, as it's passing a massive object like the sun, will get bent in the direction of the sun. Actually, the same will happen in Newtonian physics. If you just say, suppose you have a particle going along, and you know how much it gets bent as it passes. the sun depending on its impact parameter but also its velocity and the faster it's going the less it gets bent so you just take that newtonian formula and you plug in velocity equal to speed of light and see what the answer you get at you get a certain amount of bending through newtonian physics and general relativity you can do the same calculation and you actually get double the newtonian answer so this was a big i mean the 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, you know, based on his understanding of the equivalence principle for what this answer should be And so he wrote down the answer And then in response to him and a number of other people being interested in this there were people sending out expeditions to go and try and measure it And I think the very first thing that Newton did is he phoned up the observatory and said you know, can you measure, look at distant stars behind the sun and measure how light bends as it passes the sun? And this was, you know, this was the true theorist move because the, I think the director of the Mount Wilson Observatory said like, absolutely 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. So during the 1910s there was a whole bunch of expeditions sent to measure the deflection of light. They'd go you know 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. And I think the first one that they did was in 1911. They went to Argentina for an eclipse and then everything set up. I mean this is the problem with this thing And then you get there all the way to Argentina a very long way in those days and that just washed out by the by the clouds and you don don see anything and it very frustrating and then the next one that went along was there 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, and so that also fails. And 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. And so during the war, while everything shut down and no one is thinking about eclipse expeditions, he corrects this mistake and comes up with a new prediction that actually will be double the Newtonian prediction. And 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 it was double the Newtonian prediction And that really what launches Einstein as a global celebrity is that this British experiment confirming a sort of German origin theory was part of the post-war reconciliation and that Einstein had figured out anything. And that is, I'd say, the point at which general relativity became the consensus view. And people were super convinced of this very impressive test. nowadays we've done huge more tests than that you know very precise orbital dynamics you can see it in the orbit of mercury and indeed even in the other planets um you can just measure the red shifting as light as it as it goes the gravitational effect on the propagation of light the energy of light uh all over the place but historically that was the most impressive confirmation of general liberty one question you could 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 who's 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. [90:00] Not required. It's not required. Yeah, 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? These AI companies are really increasing the demand curve on the theoretical physicists, but yeah. That's right. Yeah, not so cheap anymore. But how far can that get you? 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, you know, and the product of a single mind. He got lots of help in various ways, but it really was just like a singular vision that he pursued for years. um and it was pretty you know 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 it it perhaps one of the most extreme examples of this where just somebody just sits down and thinks very hard and writes down a true theory And in some sense, physics has been chasing that high ever since. People love that romantic vision of themselves, just sitting down and thinking with very few empirical insights and 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. Yeah, how far you could get just by thinking, what do you need to do general relativity? You need the fineness 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, that Einstein came up with in special relativity, then you probably want the equivalence principle. It's an empirical fact that the inertial masses and the gravitational mass are the same for everything. But that's pretty sparse. And from just those two things, there's still a few options. But if you have lots and lots of large language models, 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 super significant And this other thing now okay now abandon simultaneity and see how far that takes you so still there only a finite number of things to explore there i don know how that how well i think we got very very lucky with general relativity that that's quite so powerful under those circumstances but um if you just have lots and lots of einsteins and you just give each of them various options you could presumably see them parallel 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 are we in a different era now and really that would there's like limited usefulness of that parallelism i think there is usefulness i do think that different parts of science have different branching fractions and how much you need uh experiment to cut off that branch yeah you know I talked about chasing the high of Einstein. Arguably, string theory has really been going all in on that. So Einstein's theory is just generativity. There's also quantum mechanics. I'm trying to marry those two in a consistent way that generativity doesn't do it all. There's no quantum mechanics in generativity. 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 particle colliders It just very hard to see any of that stuff So that didn't stop people. I mean, it stopped many people, but many people didn't stop and keep trying to do it. And so there you just have to kind of hope that it works out sort of like how I did with general relativity, where just by thinking very, 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, the tool you have at your disposal is mathematical consistency and doesn't reduce correctly in 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. 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 gone all in on that, I would say, is just trying to, with minimal experimental input, believing that there's only one consistent theory of gravity, just by doing sufficient consistency checks, you can find it. [95:00] for other examples it's you know much harder there's clearly many condensed metaphysics often you simply need to go and do an experiment to turn out which one is correct whenever our future 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 like once this sort of step is taken that we will actually be in a position to understand what our AI civilization understands? 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, some people are, many mathematicians are worried that these LLMs are just going to turn into proof machines. So 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 explains. 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 early days but it pretty supportive of that more positive vision So if you look at there was a ERDOS problem that was proved a few months ago now, and it wasn't just an incomprehensible set of lean. In fact, it wasn't even proved in lean and do it, it was 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 Erdl's 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 here. One is that it's totally comprehensible. the disproof of the unit distance conjecture that it came up with. I mean, it's a mathematician. 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 that 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 is that they just have extreme patience, even for doing things that perhaps look like low probability of success. Adam, thanks so much for coming back on and doing this while you could be building super intelligence. You're explaining hundred year old physics, but it was very interesting. It's a super fun subject. I'm so happy to share it.