Thursday, December 19, 2013

Grossmann's contribution to relativity

Tilman Sauer just wrote a detailed paper on Marcel Grossmann and his contribution to the general theory of relativity.
Grossmann's contribution to the Entwurf theory consisted in the following.

* He clarified the mathematical foundation of the theory based on a general line element (8) and generalized the concept of a tensor to a structure of n th rank in m-dimensional space.

* He identified the absolute differential calculus by Ricci and Levi-Civita as the relevant mathematical toolbox for the problem of a relativistic theory of gravitation and transformed it into a tensor calculus both with respect to notation and by generalizing it to mixed tensor densities.

* He proved that the conservation law for matter (11) was a generally covariant equation by showing that it expresses the covariant divergence of Theta-mu-nu.

* He identified the Riemann tensor as a relevant and rich resource for the problem of constructing generally covariant gravitational field equations, and he showed Einstein several ways of extracting a second rank object from the Riemann tensor that would have the desired limiting form in the case of weak static fields.

* After the failure of the mathematical strategy of constructing a field equation from the Riemann tensor, he proved the central identity (39) from which the gravitational field equations of the Entwurf theory were read off.

* In joint work with Einstein, he showed how the Einstein-Grossmann theory can be formulated in terms of a variational principle and clarified its transformational properties in light of the hole argument
This is a substantial part of the theory, but Einstein wrote privately:
Grossmann will never lay claim to being co-discoverer. He only helped in guiding me through the mathematical literature but contributed nothing of substance to the results.
Grossmann ought to be considered a co-discoverer. After the collaboration, Einstein wrote nonsense papers for over a years, and did not get back on track until he collaborated with Hilbert in 1915.

It appears that Grossmann and Hilbert are not credited because of a prejudice against mathematicians. Their contributions are considered to be just math, not physics. But finding the equations of general relativity was largely a mathematical problem, given Newtonian gravity and special relativity.

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