An extremely useful relation between the Kronecker delta and the Levi-Civita symbol.
The brackets mean that we completely anti-symmetrize in the indices m,npic.twitter.com/Hp8tK8qZ3D
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Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.
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An extremely useful relation between the Kronecker delta and the Levi-Civita symbol.
The brackets mean that we completely anti-symmetrize in the indices m,npic.twitter.com/Hp8tK8qZ3D
that's what I get for working in SL(2,C) for too long
Look also for the SL(2,C) gauge theory of gravity by Moshe Carmeli. He considers gravity as an SL(2,C) Yang-Mills theory in which he recovers the Newman-Penrose (NP) equations (also used to study e.g. gravtitational waves). He formulates GTR in concise elegant 2-spinor formalism.
Alas, these things never work. They always come to grief over soldering forms. GR is not a gauge theory in the usual sense. Its symmetries are not internal. The soldering forms, and the torsion-free requirement which inevitably alludes to them, are the canaries in the coal mine.
Mr Laurens Gunnarsen, could you please kindly clarify what you said a bit more? Do you have more lead, or? Thank you.
The fundamental dynamical variables of gauge theories are connections on principal fiber bundles over spacetime. Attempts to make GR fit into this framework tend to focus on the frame bundle. But the frame bundle is no ordinary fiber bundle over spacetime. It's extremely special.
In particular, the frame bundle is "soldered" to the base manifold (i.e., spacetime) in a way that simply has no counterpart for other principal fiber bundles. It is exactly this feature that makes it possible to speak of torsion-free connections -- the only kind relevant to GR.
This soldering of the frame bundle to spacetime has a thousand profound ramifications, none of which can be naturally accommodated into the normal framework of gauge theory. And it doesn't help to replace the frame bundle by its double-cover, the spin bundle. It too is soldered.
Yep, It doesn't help much indeed.: to replace the frame bundle and its double cover spin bundle. (Like simply connected Spin(n) and O(n,R)) and the Clifford Algebras and also: SL(2,C) and SO(3,1).
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