Friday, March 30, 2018

How Einstein learned about general covariance

Quanta mag reports:
Albert Einstein released his general theory of relativity at the end of 1915. He should have finished it two years earlier. When scholars look at his notebooks from the period, they see the completed equations, minus just a detail or two. “That really should have been the final theory,” said John Norton, an Einstein expert and a historian of science at the University of Pittsburgh.

But Einstein made a critical last-second error that set him on an odyssey of doubt and discovery — one that nearly cost him his greatest scientific achievement. The consequences of his decision continue to reverberate in math and physics today.

Here’s the error. General relativity was meant to supplant Newtonian gravity. This meant it had to explain all the same physical phenomena Newton’s equations could, plus other phenomena that Newton’s equations couldn’t. Yet in mid-1913, Einstein convinced himself, incorrectly, that his new theory couldn’t account for scenarios where the force of gravity was weak — scenarios that Newtonian gravity handled well. “In retrospect, this is just a bizarre mistake,” said Norton.

To correct this perceived flaw, Einstein thought he had to abandon what had been one of the central features of his emerging theory. ...

Einstein initially wanted his equations to be coordinate-independent (a property he called “general covariance”), meaning they’d produce correct, consistent descriptions of the universe regardless of which coordinate system you happened to be using. But Einstein convinced himself that in order to fix the error he thought he’d made, he had to abandon general covariance.

Not only did he fail at this, he doubled down on his error: He tried to show that coordinate independence was not a property that his theory could have, even in principle, because it would violate the laws of cause and effect. As one study of Einstein put it, “Nothing is easier for a first-rate mind than to form plausible arguments that what it cannot do cannot be done.”

Einstein pulled out of this dive just in time. By late 1915 he knew that the influential German mathematician David Hilbert was close to finalizing a theory of general relativity himself. In a few feverish weeks in November 1915, Einstein reverted to the equations of general relativity he’d had in hand for more than two years and applied the finishing touches.
Norton is an Einstein idolizer who makes this all about Einstein.

The problem was that Einstein did not understand general covariance. He only ever settled on it because of persuasion from Levi-Civita, Grossmann, and Hilbert. It was Grossmann who had the correct equations in 1913, that the Ricci tensor is zero. Einstein did not even know about the Ricci tensor.

The problem stems from Einstein not properly understanding special relativity in the first place. The core of the theory, and Poincare explained in 1905 and Minkowski in 1907, was that Maxwell's equations were covariant under Lorentz transformations. Einstein's 1905 paper only had the weaker principle of corresponding states that Lorentz published in 1895. Even as Einstein wrote later review papers on relativity, he never showed that he understood that Poincare and Minkowski proved covariance, or even the definition or importance of covariance.

The article makes it sound as if Einstein was competing with Hilbert, but actually they were collaborating.

The "completed equations", as applied to the solar system, were just Ricci = 0. Ricci is the covariant curvature tensor of the appropriate rank. I would credit the guys who figured out covariance and the Ricci tensor. With the discovery of dark energy, this equation is modified to say that Ricci is a small multiple of the metric tensor.

For details, see a scholarly account of the history of general relativity. One can debate which are the crucial ideas, but general covariance was not Einstein’s.

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