I have been observing the connections of differential geometry, representation theory and lie theory in general relativity, and was wondering what's essential in the view of a course on general relativity.
IMHO it is highly unlikely that RT would be an essential part of a graduate course in GR. Solid background in classical differential geometry ((pseudo)Riemannian manifolds, affine connections, tensor calculus, geodesics, etc.) and some familiarity with the calculus of variations (the Lagrangian and the Euler--Lagrange equations) are far more important.
Imtiyaz, do you speak about 'presentation'? Anyways, your thought seems not to deal with the matter under consideration.
consider space-time models without any causal restrictions you will get connections to representationtheories. (E.C.Zeeman, Hawking, etc)
As a long time researcher in this area (at the MPI in Potsdam), I can only support Juerg's somewhat short answer, in particular, the "!". First , some basic knowledge of pseudo-riemannian differential geometry is essential for the comprehension of the theory. Usually, providing such basis takes a considerable amount of time. Second, the focus in general relativity shifted from the finding and study of exact solutions of Einstein's field equations, like the Schwarzschild metric and the Kerr metric, to the solution of the initial-boundary-value problem of the equations. As a consequence, the needed mathematical methods shifted from differential geometric methods to methods from partial differential equations/functional analaysis. Teaching of the latter would be useful. After that, there should not be any time left for teaching representation theory.
Yes. Representation theory is a vital part of understanding the tools many physicists use. For example, Fourier analysis is a consequence of the theory.
http://en.wikipedia.org/wiki/Representation_theory#Harmonic_analysis
To get a better understanding of how these mathematical tools work, and what they are actually doing, a representation theory course is a vital tool.
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