How Einstein was once lost, almost losing the general theory of relativity

By 1913, Albert Einstein had almost completed the general theory of relativity. But one simple mistake led to the fact that for two years he painfully revised his theory. And today mathematicians are still struggling with the difficulties that stood in his way.

Albert Einstein released his general theory of relativity at the end of 1915. I should have finished it two years earlier. When the researchers studied his notes of that period, they saw practically complete equations in which only a couple of details were missing. “It was supposed to be a definitive theory,” said John Norton , an Einstein expert and a science historian from the University of Pittsburgh.

But at the last moment Einstein made a critical mistake, which sent him on the path of doubts and discoveries — so complicated that he almost cost him his greatest scientific achievement. The consequences of his decision continue to respond in the mathematics and physics of today.

And this error. GTR was supposed to oust Newtonian gravity. This means that she had to explain the same physical phenomena that Newton's equations coped with, as well as other phenomena that Newtonian theory could not explain. However, in mid-1913, Einstein convinced himself, and it was his mistake, that his new theory does not describe those cases in which gravity turns out to be weak — and Newton's theory described these cases well. “Looking back, this mistake seems very strange,” said Norton.
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Einstein believed that in order to correct the apparent shortcoming, it was necessary to abandon what was the basis of his new theory.

The Einstein gravitational field equations, the GRT equations, describe how the form of space-time responds to the presence of matter and energy. To describe these changes, you need to assign a coordinate system to space-time — something like lines of latitude and longitude — denoting which points are located.

The most important thing to understand about coordinate systems is that they were invented by people. In one system, a point can have coordinates (0, 0, 0), and in another - (1, 1, 1). Physical properties did not change, we just marked the point differently. “These tags are about us, not about the world,” said James Weatherr , a philosopher of science from the University of California, Irvine.

At first, Einstein wanted his equations not to depend on coordinates (he called this the principle of general covariance ), that is, that they give out correct and consistent descriptions of the Universe regardless of which coordinate system you use. But Einstein convinced himself that in order to correct the mistake which he thought he had made, it was necessary to abandon the general covariance.

He not only failed to do this, he also multiplied errors: he tried to show that his theory could not have independence from coordinates, even in principle, since this would violate the laws of cause and effect. As pointed out in one of Einstein's research papers , “for a first-class mind, there is nothing easier than to come up with plausible reasons why nobody can do what he cannot do.”

But Einstein got out of this situation just in time. By the end of 1915, he already knew that the influential German mathematician David Gilbert came very close to completing his general theory of relativity. For several turbulent weeks in November 1915, Einstein returned to the equations of GTR that he had originally had, and added a few final touches. In November 1915, at the first of four lectures at the Prussian Academy of Sciences, he announced his achievement. And since then, our views on the physical world have changed forever.

Today's Einstein equations obey the principle of general covariance. They give out the same physical truths about the universe - how space-time is bent in the presence of energy and matter - regardless of which coordinates you use for the markup.

However, mathematicians and physicists are still struggling with the difficulties associated with coordinate systems, which slowed down Einstein 100 years ago. For example, a monumental attempt to reconcile GTR with quantum theory stumbles, in particular, because of the difficulties associated with developing a theory of quantum gravity, which has the same covariance that Einstein's equations have reached. "In a sense, we can say that we do not have an adequate quantum theory of gravity because we do not know how to express solutions of the Einstein equations so that dependence on coordinates disappears," said Veserol.

In practice, difficulties usually arise in order to violate the covariance of the Einstein equations — that is, to choose a specific coordinate system, well suited for solving a particular problem. This difficulty is especially strongly hampered by mathematicians studying the hypothesis of the stability of a black hole. For each specific task, some one coordinate system is better suited to the others - and the choice of the coordinate system and its tweaks to changing the solution is in the field of mathematical art.

New evidence would have been obtained much simpler if there had been one, universal coordinate system, well suited for any task and any space-time configuration. But, as Einstein discovered during those years of burdensome wanderings, the Universe does not recognize any privileged choice of coordinates.

“It’s not just that we don’t have that choice,” said Veserol. “The fact is that one of the lessons given to us by Einstein is that it would be a mistake to expect such a choice.”