Adam Brown – A deep but accessible introduction to general relativity

The premise of the episode is a promise: general relativity is the most beautiful thing a single mind has produced, and it is possible to see why without sitting through a twenty-lecture graduate course. Adam Brown — now leading BlueShift at Google DeepMind, formerly a Stanford physicist working on cosmology, string theory and relativity — agrees the answer is yes, and the lecture is the attempt.

His opening observation is about why this is even possible. Relativity is unusual among the great twentieth-century theories in being essentially one person's work: Einstein pursued the idea doggedly for a decade and then wrote down a theory that describes the motion of the planets and the origin and fate of the universe. It took one of the most famous minds in history about ten years.

A ten-week course gets further than Einstein did in ten years, and not because the students are smarter. It is because they inherit Einstein plus everyone since, who took ideas that were thought incomprehensible to anyone below Einstein's intelligence and boiled them to essentials, without repeating their predecessors' mistakes. That compression is the whole trick of pedagogy, and it is what the next twenty minutes attempt in miniature.

The word "special" means restricted — it does not apply everywhere. Special relativity came ten years earlier, in 1905, and its slogan starts from one hypothesis: nothing travels faster than light. The theory takes that observation, promotes it to a principle, and takes the principle with maximum seriousness as the organising fact about spacetime.

It covers electromagnetism, and — though Einstein did not know it at the time — it applies straightforwardly to the strong and weak nuclear forces. What it does not obviously cover is gravity. General relativity, ten years later, is the repair, and "general" means precisely that gravity has been included. Its slogan is a two-word extension of the first: not even gravity.

To see the problem you have to go back to the theory Einstein was replacing: Newton's laws, Principia, 1687. The second law says force produces acceleration in proportion to mass, ma = F. The mass here is inertia — how much an object resists being accelerated — and this law survives into general relativity, though "force" and "acceleration" will need better definitions.

The first law is the zero-force special case: absent force, no acceleration, and objects continue in straight lines. This also survives — with the same caveat that both "force" and "straight line" will be upgraded.

What does not survive is the law of gravity. Newton's gravity supplies the missing piece: the force between two bodies is Newton's constant times the two masses over the distance squared, directed along the separation and attractive, hence the minus sign.

And here the tension is immediate and fatal. Jiggle the sun, and the inverse-square law read literally says the force at the Earth changes at once — not eight minutes later, but immediately. That is a superluminal influence, and it contradicts the principle that nothing outruns light. Newton's force law is inconsistent with special relativity.

One escape would be to say the speed limit holds for the other forces but not for gravity, and that gravitational signalling could build a faster-than-light telephone. Einstein had spent years eliminating exactly that possibility, and he was not going to concede it here. He and many contemporaries concluded that Newton's law is what has to give — and that is what turned out to be true.

There is a precedent for an inverse-square law being modified into relativistic consistency, and it is the electric force. Coulomb's law, written down a century after Newton, has the same shape: a constant times the two charges, along the separation, over distance squared.

For exactly the same reason, electrostatics looks inconsistent with relativity. It is not, because it is only a limit. The true theory is Maxwell's, which includes magnetic forces as well as electric ones; the electrostatic form is what you see when nothing moves. Once things move, correction terms appear, and they conspire to make the whole thing consistent.

The history ran the other way, which is the interesting part. Maxwell wrote his equations in the middle of the nineteenth century; only afterwards did people notice they were already consistent with nothing outrunning light. That consistency shows up as a symmetry — Lorentz symmetry of the Maxwell field equations — noticed after the fact, and it is what eventually led Einstein to special relativity.

So the template exists: take an inverse-square law, dress it in a relativistically invariant theory, and recover Newton as an approximation. In a grand sense that is what Einstein ends up doing. But it is a far more radical departure than Maxwell's generalisation, and the same formula that suggests the template also contains two hints about why.

The first hint is the sign. Gravity carries a minus; electrostatics carries a plus. Two positive masses attract; two like charges repel. So you cannot perform literally the same mathematical trick, because the same trick produces the same mathematics — like masses would repel.

The reason, stated ahead of the derivation, is spin. Electrostatics is mediated by a spin-1 particle, the photon. Gravity will be mediated by a spin-2 particle, and that is what flips the sign.

So Einstein had to find another route to relativistic invariance. What he had was one clue, and part of his genius was to treat it as highly significant rather than incidental. He called the insight behind it his happiest thought; it is often described as his most beautiful.

The clue is the second difference between the two laws, and it concerns what plays the role of charge. In gravity, the thing sitting where charge sits is mass.

In mechanics, mass has exactly one job: inertia, the resistance to acceleration. Charge is entirely separate and unrelated — a neutron is heavy and neutral, an electron is light and charged. There is no necessary relation between the two quantities.

Gravity breaks that separation. The mass in Newton's second law — the inertial mass resisting the force — is exactly the mass in Newton's gravitational law that determines how hard you are pulled. Same number, two roles. And that coincidence, which Newtonian physics simply registers as a fact, is the door Einstein walks through.