Monday, May 13, 2013

Discovery of the electromagnetic Lagrangian

Lumo writes:
Schwarzschild is most famously associated with the first nontrivial exact solution to Einstein's equations of general relativity. But he would also study optics, photographic materials, celestial mechanics, quantum theory, stellar structure and statistics, Halley's comet, and spectroscopy. According to Wolfgang Pauli, Schwarzschild was the first man who wrote the correct form of the action for the electromagnetic field coupled to charges and currents.
I was surprised to learn that last Schwarzschild discovery a couple of years ago.

In my my book, I argue that the heart of special relativity was the 1905 discovery of the Lorentz covariance of Maxwell's equations.

Lorentz had a cruder concept in 1895 that he called the theorem of corresponding states. Einstein's 1905 paper postulated what Lorentz proved, but did not have the covariance concept. Poincare's 1905 paper presented the concept, and everyone else got it from him. It was not independently rediscovered by anyone else.

For one proof, Poincare presented a Lorentz invariant Lagrangian density that implies Maxwell's equations. The covariance follows from the invariance of the Lagrangian. I was going to credit Poincare with discovering the Lagrangian himself, but in researching the point for my book, I discovered that Karl Schwarzschild published it in 1903. Schwarzschild did not know that it was Lorentz invariant, or figure out the significance for special relativity, as Poincare did. Minkowski was one of the few people who grasped the significance of what Poincare did, and popularized the 4-dimensional geometrical view in 1908. Einstein was slow to understand Minkowski, but eventually caught to to what relativity was about in 1909.

Einstein's famous 1905 paper is one of the most widely praised scientific papers ever written, but it did not have the Lorentz group, spacetime, 4-dimensional geometry, relativistic Lagraangian, electromagnetic covariance, or gravitational implications. We got all those things from Poincare, and Poincare announced his results in 1905 before Einstein submitted his famous paper.

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