The event horizon of a black hole marks the point of no return. Once someone has passed it, even though spacetime in its vicinity is quite regular, they can no longer escape. Einstein and a roster of his leading contemporaries in mathematics and physics, however, in direct contradiction with our modern understanding, regarded spacetime there as breaking down, as ‘singular’, where terms in the equations disappear to zero or blow up to infinity. As University of Pittsburgh historian and philosopher of science John Norton explains, Einstein's concern about these terms now appears to us as a puzzling novice error. But a closer look at Einstein and the mathematical methods used reveals a different story.

The moment

It is a moment in every introductory class on general relativity. The simplest, Schwarzschild black hole is introduced. Here is the singularity at its center. Here is the event horizon. Once it is passed, there is no escape. However, while passing it, an observer would not notice anything special in the vicinity of the event horizon. The spacetime geometry is quite regular there.

Then comes the warning. The commonly used mathematical description of spacetime has some anomalies at the event horizon, the “Schwarzschild radius.” One term in the formula falls to zero and, worse, another diverges to infinity.

Do not be misled, there is no corresponding irregularity in the spacetime geometry. Its curvature stays finite. It is a novice mistake to think otherwise. These badly behaved terms just result from a peculiarity of the spacetime coordinate system used in the mathematics. If we choose another coordinate system, these pathologies go away. To think otherwise would be like imagining that there is a mystery at the Earth’s north and south poles because our maps show all longitudes meet at them. But this is just an accident of how we chose to lay out lines of longitude on the Earth!

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So, unlike the singularity now recognized at the center of a black hole, this divergence at the Schwarzschild radius, the event horizon, is not a real pathology of the spacetime, but rather a mathematical artifact that can be removed by changing the coordinate system used.

Lines of longitude converge at the North Pole

The puzzle

By modern lights, it is a novice mistake. Do only novices make it? Who did make it? It was the very physicist who discovered general relativity, Albert Einstein! He was quite sure that something goes terribly amiss at the Schwarzschild radius. When, in April 1922, the mathematician Jacques Hadamard asked Einstein what would happen if the radius were realized in a spacetime, Einstein replied in horror. It would be “an unimaginable misfortune [malheur] for theory…” He called it the “Hadamard catastrophe.”

Einstein was quite determined in this assessment. George Lemaître showed in 1933 that the pathology could be eradicated merely by changing the coordinate system. Einstein was undeterred. In 1935, he coauthored a paper with Nathan Rosen in which they introduced the notion of the Einstein-Rosen bridge. We now draw it as the familiar funnel shape illustrated at the head of this article. They wrote in calamitous terms of the singularity they perceived to be at the Schwarzschild radius: “For a singularity brings so much arbitrariness into the theory that it actually nullifies its laws.” They were willing to alter Einstein’s celebrated gravitational field equations to escape the arbitrariness.

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We have a serious puzzle to solve. Einstein, Hilbert, Klein. This is no collection of novices, but a roster of greatness in physics and mathematics. What were they thinking?

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