The Soldier Who Solved Einstein's 'Unsolvable' Equation From the Trenches

Source: Dwarkesh Patel | Published: 2026-07-10T16:34:06Z

A Prussian artillery officer calculating shell trajectories during WWI found an exact solution to Einstein's general relativity field equations—solving in months what Einstein himself doubted could ever be solved exactly.

Einstein called this his "most beautiful thought." It would take him another eight years to go from that insight to the complete field equations. Adam Brown leads the BlueShift team at Google DeepMind; before becoming an AI researcher he was a physics professor at Stanford, working on everything from cosmology to string theory. In a single lecture, he distilled the theory that consumed a decade of Einstein's life to its essence.

The Thorn Newton Left Behind

Before Einstein, physics was ruled by Newton's law of gravitation: the force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. This formula described apples falling and planets orbiting the sun — Newton had unified the heavens and the earth in a single equation.

But the equation had a fatal flaw: it was instantaneous.

Under Newton's law, if the sun suddenly moved, Earth would feel the change in gravitational pull immediately — not eight minutes later, but at once. That meant gravitational influence could propagate at infinite speed, or in other words, you could use gravitational effects to transmit signals that bypass the speed of light.

This deeply troubled Einstein. He had spent years building special relativity on the principle that nothing can exceed the speed of light. Gravity couldn't be an exception. Newton's law had to go.

A Strange Coincidence Hiding a Key

Overthrowing a law requires a replacement. Einstein's clue came from a fact that Newtonian physics had long treated as mere coincidence.

Newton's second law says force equals mass times acceleration. The mass here is inertial mass — a measure of how hard an object is to push. A rock is harder to accelerate than a feather because its inertial mass is greater.

Newton's law of gravitation says the strength of gravity depends on an object's mass. This mass is gravitational mass — how much an object "weighs" in a gravitational interaction.

These are conceptually two entirely different quantities. Inertia is an object's resistance to changes in motion. Gravitational mass governs how strongly an object participates in gravitational interactions. In electromagnetism, there's no connection between the two: protons and neutrons have similar masses but wildly different charges; you can have a massive, charge-free neutron or a near-massless electron with a significant charge.

But in gravity, the two quantities are always exactly equal. Newton noticed this and confirmed it experimentally to one part in a thousand. By Einstein's time, the precision had reached one part in a billion. We now know they match to one part in 10¹⁵.

Why? Newton had no explanation. He accepted it as coincidence.

This coincidence leads directly to the famous result: in a vacuum, a feather and a brick hit the ground at the same time. The brick experiences more gravitational force, but it also has more inertial mass — a greater resistance to acceleration — and the two cancel exactly. This is no accident. The "what makes you hard to push" quantity and the "what determines how much gravity pulls you" quantity are the same thing.

Einstein refused to believe this was a coincidence. Something deep about the nature of gravity was hiding here.

The Clue in a Spinning Bucket

To follow Einstein's reasoning, you first need to understand a class of phenomena called "inertial forces."

Brown demonstrates with a thought experiment: tie a bucket of water to a rope, spin it fast enough, and swing it overhead in a loop. When the bucket is upside down, the water doesn't spill out.

From a bystander's perspective, the explanation is simple: the water wants to fall, but the bucket is moving fast enough that it rotates down before the water can escape. But from inside the bucket, the experience is completely different — you feel a force pressing you toward the bottom. That force is called centrifugal force.

Centrifugal force is an "inertial force," also called a "fictitious force." It isn't a real physical interaction; it's the expression of your inertia — your natural tendency to travel in a straight line — as seen from a rotating reference frame.

Here's the key point: the push you feel from centrifugal force depends on your inertial mass. This requires no explanation, because centrifugal force is inertia made manifest. The "charge" of an inertial force is, by construction, inertial mass.

Now connect this to the previous section: the "charge" of gravity is also inertial mass.

This led Einstein to a radical question: what if gravity is itself an inertial force?

You're Sitting in a Chair, But You're the One Curving

If gravity is an inertial force, it follows that people who feel no gravity are the ones traveling in straight lines.

An astronaut floating in orbit, weightless and feeling nothing — they're going straight.

You, sitting in your chair, feeling gravity press you into the seat — you're the one curving.

This sounds absurd. You're not moving at all.

But Brown offers a perfect analogy: the flight path map on an airplane's seatback screen.

Flying from San Francisco to London, the route on the map arcs way up toward Greenland, which looks like a massive detour. You know the shortest path between two points is a straight line — and yet the curved arc on the map is exactly what the plane flies.

The contradiction resolves: the map is flat, but Earth is spherical. When you unfold a spherical surface onto a plane, straight lines appear curved and curved-looking routes are actually the shortest paths. The great circle arc from San Francisco to London is a straight line on the sphere; it only looks like a detour on the flat projection.

Spacetime works the same way. We're used to thinking about "straight lines" in flat spacetime. But if spacetime itself is curved, all our intuitions about straight and bent are wrong.

Chalk thrown through the air, tracing a parabola, in free fall, subject to no other forces — that's a straight line in curved spacetime. You sitting motionless in a chair, with the ground constantly pushing up on you, deflecting you away from that natural path — you're the one taking a detour.

The core picture of general relativity snaps into focus: matter curves spacetime; curved spacetime determines how matter moves. If your path doesn't follow a geodesic — the shortest path in curved spacetime — you feel a "force." That force is gravity, and it is, in essence, an inertial force.

From that first insight in 1907 to the complete field equations in 1915, Einstein spent eight years.

A Prussian Artillery Officer in the Trenches Who Solved for Black Holes

When Einstein wrote down his equations, he believed they were so complex that exact solutions might never be found — only approximations.

He was wrong.

Karl Schwarzschild was a Prussian artillery officer serving in World War I. In the gaps between calculating shell trajectories, he read Einstein's newly published field equations. Within months, he had found an exact solution — now called the Schwarzschild solution — describing the curvature of spacetime around a spherical mass.

The solution contained a peculiarity: at a specific radius (now called the Schwarzschild radius, equal to 2GM/c²), the equations blow up. Schwarzschild himself didn't fully understand what this meant, and Einstein got it wrong too — he wrote extensively, and incorrectly, about objects being repelled at that radius. Physicists puzzled over this solution for half a century.

In retrospect, the meaning is clear. Brown offers an intuitive argument:

Imagine a pulley system slowly lowering a brick toward a massive object from a great distance. Energy can be extracted during this process. The fraction of the brick's rest-mass energy (mc²) you can extract depends on how deep you lower it. Near Earth's surface, this fraction is about 7×10⁻¹⁰ — tiny. Near denser objects, the fraction grows. But if you lower the brick to the Schwarzschild radius, 2GM/c², this fraction approaches 1: you can extract 100% of the brick's rest-mass energy.

What happens if you go deeper? You lose control. Gravity becomes infinitely strong; the brick is torn from your grasp and you can never retrieve it. That's a black hole.

The Most Efficient Power Plant in the Universe

This energy extraction calculation leads naturally to a striking comparison of energy efficiency.

Chemical combustion taps the energy in atomic bonds — about 10⁻¹⁰ of rest-mass energy. Nuclear fission does better, around 10⁻³. Nuclear fusion better still, around 10⁻². But neither fission nor fusion changes the total number of protons and neutrons; the vast majority of available energy — locked in their rest masses — remains completely inaccessible.

Black holes are the exception. Given a black hole and a pulley system, you can in principle extract 100% of the rest-mass energy from whatever you feed it. That's the theoretical maximum efficiency permitted by physics.

A curious footnote: Earth's escape velocity is about 11 km/s, and the ratio of gravitational binding energy to rest mass is roughly 7×10⁻¹⁰. The ratio of chemical energy to rest mass in hydrogen-oxygen rocket fuel is about 1.5×10⁻¹⁰ — the same order of magnitude, arising from entirely different physics. This is why chemical rockets can reach space, but just barely: most of the fuel goes to propelling the fuel itself, and the fraction that reaches orbit is tiny.

Falling Into a Black Hole: Two Perspectives

The event horizon — at the Schwarzschild radius — has a counterintuitive property: from the outside, someone falling into a black hole never seems to arrive; but from the perspective of the person falling, crossing the event horizon is completely unremarkable.

Brown explains how both can be true simultaneously.

From an outside observer's perspective: the closer the falling person gets to the event horizon, the stronger gravitational time dilation becomes. The observer watches the falling person's clock run slower and slower. Light from the faller becomes increasingly redshifted as the event horizon approaches, until only a gradually reddening, fading image remains, dissolving into darkness. The outside observer never sees the person cross.

From the falling person's perspective: the clock runs normally, nothing feels out of the ordinary, crossing the event horizon is as uneventful as passing through empty space. For a sufficiently massive black hole, tidal forces at the event horizon are negligible — you wouldn't feel any stretching at all. For a black hole with the mass of an entire galaxy, you could live inside it for a very long time — generations, even civilizations — without ever realizing you've crossed a point of no return, until the singularity finally closes in.

The event horizon's peculiarity is its teleological nature: it's not a boundary you can detect locally, but a global judgment about your future. Once crossed, the singularity is inevitable; no amount of rocket thrust can escape — but that inevitability may not arrive for a very long time.

Three Ways We Became Certain Black Holes Are Real

It took physicists a long time to accept that black holes actually exist. After Schwarzschild published his solution, Einstein and many others believed it was merely a pathological edge case — a quirk of the equations that no real astrophysical object would ever realize.

What changed their minds came first from theory, then from observation.

In the 1960s, Roger Penrose (who later won the Nobel Prize in Physics) proved that black hole formation is a generic feature of general relativity. You don't need finely tuned initial conditions; starting from generic states, black holes form naturally. This theoretically closed the back door to "black holes can't actually exist."

On the observational side, there have been three milestones.

The first: tracking stellar orbits at the center of the Milky Way. At the galactic center sits an invisible object called Sagittarius A*, around which surrounding stars trace peculiar elliptical orbits. Decades of observations allowed astronomers to calculate the mass and size of this invisible object — millions of times the mass of the sun, packed into an extraordinarily small volume. Only a black hole fits.

The second: LIGO. Shortly after it began operating in late 2015, LIGO detected vibrations in spacetime itself. Two laser interferometers at separate locations simultaneously measured identical ripples in spacetime, ruling out local vibration. Tracing the signal back to its source: 1.3 billion light-years away, two black holes each about 30 times the sun's mass had collided and merged — the collision occurred 1.3 billion years ago, the gravitational waves traveled across the cosmos, and arrived at Earth just weeks after LIGO switched on. Hundreds of similar events have been detected since.

The third: the Event Horizon Telescope. A planet-scale array of radio telescopes, working in concert, captured direct images of the glowing material surrounding Sagittarius A* and the black hole at the center of a neighboring galaxy — the superheated glow of matter being consumed just before it crosses the point of no return.

An Experiment Repeatedly Ruined by Weather and War

General relativity predicts that light bends when passing near a massive object — and that the bending is exactly twice what Newtonian theory predicts. This factor-of-two difference is the key to distinguishing the two theories.

Testing this requires observing starlight passing close to the sun. Under normal circumstances this is impossible: the sun is far too bright. There is one exception: a total solar eclipse, when the moon blocks the sun and stars become visible at noon.

Several expeditions set out in the 1910s. All failed. The first traveled to Argentina; everything was in place when clouds rolled in. The second was a German team, sponsored by the arms manufacturer Krupp, headed to Crimea to observe an eclipse — then World War I broke out, Russia and Germany went to war, and the entire team was arrested by Russian forces and held for the duration of the war.

For Einstein, this turned out to be good luck. During the war he recalculated, and discovered an error in his earlier prediction. Using only the equivalence principle, he had obtained a deflection equal to Newton's prediction; the correct answer, from the full theory, was double that.

In 1919, British physicist Arthur Eddington led expeditions to multiple locations to observe a solar eclipse and successfully measured the deflection of starlight — perfectly matching Einstein's revised prediction, twice the Newtonian value. A British experiment confirming a German theory, published shortly after the end of World War I: the result became something of a symbol of postwar reconciliation. Einstein became a global celebrity overnight, and general relativity became the consensus of physics.

How Far Pure Thought Can Carry You

There's one aspect of general relativity worth sitting with: how thin its empirical foundation actually is.

You need to know that the speed of light is finite. You need the symmetries of special relativity. You need the experimental fact that inertial mass equals gravitational mass. That's it. From these few inputs, one person derived a single set of equations that describes apples falling, planetary orbits, the expansion of the universe, and black holes.

Brown calls this an extreme outlier in the history of physics — a theory almost entirely the product of a single mind, one that traveled an extraordinary distance on thought alone.

His assessment: this path doesn't always work. In condensed matter physics, you often have no choice but to run experiments to know which theory is right. String theory has spent decades betting on pure thought — trusting that there is a unique internally consistent theory of quantum gravity, and that mathematical consistency alone can find it — but the verdict is still out. Whether this strategy succeeds depends on how many consistent theories exist. If there's only one, consistency might be enough to find it. If there are infinitely many consistent theories, consistency isn't selective enough, and only experiment can point you to the right answer.

On whether AI can drive future discoveries in physics, Brown is cautiously optimistic. He points to a recent AI proof of the Erdős conjecture — a proof that wasn't inscrutable machine output, but a genuinely new line of reasoning that human mathematicians could follow, and that they then used to prove further theorems. His view is that sufficiently powerful AI won't only be a proving machine — it will be an explaining machine, one that translates impenetrable proofs into knowledge that humans can actually absorb.