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Teleparallel Gravity [Part 2]

Teleparallel Gravity [Part 2]. 王元君 Ong Yen Chin LeCosPA Quantum Cosmology Weekly Report December 28 , 2010 . Recall. Teleparallelism uses a connection with vanishing curvature but nonzero torsion: . 2 of 15. Recall: What I don’t understand….

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Teleparallel Gravity [Part 2]

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  1. Teleparallel Gravity [Part 2] 王元君 Ong Yen Chin LeCosPA Quantum Cosmology Weekly Report December 28, 2010

  2. Recall Teleparallelism uses a connection with vanishing curvature but nonzero torsion: 2 of 15

  3. Recall: What I don’t understand… • In above derivation [and in Nakahara’s “Geometry, Topology and Physics”, Second Edition], Contortion tensor (upto overall sign difference depending on conventions in various papers): • In most teleparallel and f(T) papers: 3 of 15

  4. Minus Sign Problem Resolved 4of 15

  5. Geodesics Equation 5 of 15

  6. Geodesics Equation • Pereira et al. interpret this as: “This is a force equation, with torsion playing the role of gravitational force.” An Introduction to Teleparallel Gravity, R. Aldrovandi and J. G. Pereira 6 of 15

  7. Geodesics Equation 7of 15

  8. Math 464: Notes on Differential Geometry, Matt Visser 8 of 15

  9. Formulation as Gauge Theory • Instead of treating the tetrads as fundamental, we introduce a translational gauge on the Minkowski tangent space [now treated as internal space, c.f. U(1) in electromagnetism]. • A local translation of the Minkowski space coordinates goes like: 9of 15

  10. Formulation as Gauge Theory • The translational gauge potential is a 1-form assuming values in the Lie algebra of the translation group , and is the generator of infinitesimal translations. 10 of 15

  11. Formulation as Gauge Theory • The covariant derivative is so • If we require that then the covariant derivative is gauge invariant: 11 of 15

  12. Formulation as Gauge Theory • is then identified with the tetrad . • The commutatorsatisfies 12 of 15

  13. f(T) Gravity f(T) Gravity 13 of 15

  14. Flat FLRW tetrad: 14 of 15

  15. f(T) Gravity 15 of 15

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