Einstein Killed General Covariance Before He Saved It

“The laws of physics look the same in every coordinate system” sounds like a platitude, yet Einstein fought over it for eight years and, halfway through, published a proof that it could not be done.

Author: Koutian Wu; GitHub: ktwu01

The principle is called general covariance. Textbooks give it one line: write your equations as tensors, and they keep their form when you change coordinates. Hold the word “tensor” for a moment. The story of how the idea arrived is better than the one-liner. It has a flash of insight, a desperate call to an old classmate, a dead end Einstein built for himself, and a race against David Hilbert at the finish.

Start in 1905. Einstein is 26, reviewing patents in Bern, and publishes special relativity on the side. Its principle of relativity says that if you sit in a train moving at constant speed with the blinds down, no experiment can tell you whether the train is moving. Every frame in uniform straight-line motion gets the same laws. A frame, here, is just the set of rulers and clocks you use to measure where and when things happen.

The catch is the word “uniform.” Hit the brakes and your coffee sloshes forward, so you know at once that you are slowing down. Accelerating frames look second-class. Newtonian mechanics has to patch them with “fictitious forces” before the equations work. That bothered Einstein. Why should uniform motion be aristocracy and acceleration the peasantry?

In 1907, sitting at the patent office, he had what he later called the happiest thought of his life. A man falling freely from a roof does not feel his own weight. If he lets go of an apple, it floats beside him and falls with him.

That was the whole picture.

Today everyone has seen it. Astronauts drifting around a space station are falling around the Earth without pause. In 1907, Einstein pulled a startling conclusion out of it: within a small enough region, gravity and acceleration cannot be told apart. Locked in a windowless box, you feel the floor pressing on your feet. The box might be sitting on Earth, or a rocket might be pushing it through empty space. Let the box fall freely and gravity vanishes inside it. That is the equivalence principle.

It opened a door. If switching to an accelerating frame can make gravity disappear, then accelerating frames are not second-class, and the problem of gravity and the problem of accelerating frames are one problem. Einstein set himself a goal: a theory in which every frame is equal, uniform, accelerating, or spinning, with the same laws in all of them. He called this the general principle of relativity.

Then he was stuck for five years.

Stuck on what? Mathematics. “Every frame is equal” turns into a concrete demand on paper: change to any coordinates you like, and the equations must keep their form. Coordinates are a way of labeling events. You can use latitude and longitude, a neat square grid, or a crumpled fishing net whose cells are all different sizes and shapes. Special relativity only lets you switch between a few tidy grids. Einstein now wanted any crumpled net at all.

How do you write equations like that? He had no idea.

In 1912 he returned to Zurich and went to Marcel Grossmann, a mathematician and his old classmate, whose lecture notes Einstein had used to pass exams after skipping class. As the story was later told, Einstein opened with: “Grossmann, you have to help me, or I’ll go crazy.”

Grossmann went to the literature and came back with news: the mathematics already existed. Riemann had described the geometry of curved spaces in 1854, and around 1900 Ricci and Levi-Civita had built a toolkit they called the absolute differential calculus. We call it tensor analysis.

Now the word “tensor” can be unpacked. Think of a tensor as a physical object that exists apart from any coordinates. Coordinates are just the angle you photograph it from. Change the angle and every number in the photo changes, but by strict rules, with all components shifting together. So if a tensor equation holds in one coordinate system, it holds in all of them. That was exactly the “same form everywhere” Einstein wanted.

By then Einstein suspected that gravity was better described as the curvature of spacetime itself than as a force, and Grossmann’s tools could describe that curvature. The object that does it is the metric, which you can picture as a ruler placed at every point, telling you how far apart two nearby points really are. In flat spacetime the ruler is the same everywhere. Bend spacetime and the ruler changes from place to place.

Tools in hand, direction set. It should have been smooth from there.

Instead, in 1913 Einstein and Grossmann published a paper now known as the Entwurf, German for “outline.” Its field equations were not generally covariant.

The absurd part is that in his notebook from the winter of 1912 to 1913, Einstein had already written down equations very close to the final answer, built on the Ricci tensor. He decided they failed to reduce to Newtonian gravity when gravity is weak, worried they clashed with energy conservation, and threw them out. In hindsight, the trouble lay in how he handled coordinates. The equations were fine. He had the answer in his hands for almost three years.

Worse, he then found a reason why general covariance was impossible. This is the famous hole argument, or Lochbetrachtung.

It goes roughly like this. Picture a region of spacetime containing no matter, a “hole,” with the distribution of matter outside fixed completely. If the field equations are generally covariant, you can quietly change coordinates inside the hole while leaving everything outside untouched, and get a mathematically different solution. Same world outside, two gravitational fields inside. A theory like that cannot even give one answer for the field produced by a given arrangement of matter. Einstein saw this as fatal, and in 1913 and 1914 he argued in print that a good theory of gravity must give up general covariance.

He shot down his own principle, in public.

I find this stretch of the story wonderful. The thing he wanted most, he refuted himself with an elegant argument, and then spent two years building a theory on top of that refutation.

By 1915 the Entwurf theory was cracking. First, Mercury. Its closest point to the Sun creeps forward a little each century, and after subtracting the tug of the other planets, about 43 arcseconds per century remained unexplained. Astronomers had even proposed a hidden planet, Vulcan, to account for it. Einstein and his friend Michele Besso had run the Entwurf numbers in 1913 and got around 18. Second, rotation. Einstein found the theory could not accommodate a rotating frame, which contradicted the whole point of making every frame equal.

In the summer of 1915 he gave a week of lectures in Göttingen. In the audience sat David Hilbert, one of the strongest mathematicians alive. Hilbert understood, and started working on the problem himself. Einstein was no longer alone on this road.

In November 1915 he delivered a paper to the Prussian Academy every Thursday, four weeks running. On November 4 he dropped the Entwurf and went back to the Ricci tensor. On November 18 he computed Mercury with the new equations: 43 arcseconds. He later wrote to his friend Paul Ehrenfest that for several days he was beside himself with joy. The same paper put the bending of starlight grazing the Sun at 1.7 arcseconds, twice his 1911 figure, a number Eddington’s eclipse expedition would confirm in 1919. On November 25 he submitted the final field equations.

The left side describes how spacetime curves. The right side describes how matter and energy are distributed. Both sides are tensors, so the equation keeps this form on any crumpled fishing net.

Hilbert submitted his own paper on November 20, and who wrote down the final equations first was argued over for decades. In 1997 historians found the proofs of Hilbert’s paper. They do not show the final equations explicitly, and part of one page had been cut away, so the argument started again. The two men themselves made peace quickly. That December, Einstein wrote to Hilbert offering reconciliation.

And the hole argument? How did Einstein climb out of the pit he had dug?

He realized that coordinates are labels with no physical meaning of their own. Change coordinates inside the hole and the two “different” solutions describe the same physical world with the events renumbered. What you can actually measure are coincidences between events: two particles meeting at a point, a flash of light reaching your eye at a moment. Those coincidences are the same in every coordinate system. He laid this out in letters to Ehrenfest and Besso at the turn of 1916, and it became known as the point-coincidence argument.

This step cuts deeper than the equations. It says a point in spacetime has no identity apart from what happens there. Newton’s empty stage of absolute space, existing before anything else, was taken down here.

There is one more twist.

In 1917 the physicist Erich Kretschmann pointed out that, with enough mathematical effort, any physical theory can be rewritten in generally covariant form, Newtonian gravity included. So general covariance by itself has no physical content. It is a way of writing things down.

Einstein replied in 1918 and conceded the point. He added that general covariance still has heuristic value. Among all theories written in covariant form, general relativity is the simplest and most natural, while Newtonian gravity forced into covariant form becomes ugly and clumsy.

So the opening sentence, that the laws of physics look the same in every coordinate system, began as a physical principle and slowly settled into a mathematical discipline. The parts with real physical weight are the equivalence principle and the idea that spacetime has no fixed structure given in advance, its shape set entirely by matter and fields. That second idea is now called background independence. Many books blur the general principle of relativity and general covariance together, and the history explains why. In Einstein’s head they were tangled from the start.

A postscript. In 1918 Hermann Weyl, following the same line of thought, tried to make electromagnetism geometric through a kind of invariance under changing the scale of your rulers. The first version failed. Reworked in 1929, it became gauge symmetry, and the Standard Model of particle physics is built on gauge symmetry from the ground up.

A man sits in a chair at the patent office and imagines someone falling off a roof.

Eight years later he writes an equation that keeps its form on any crumpled fishing net, having proved along the way that it could not be done.