Published May 7, 2020 | Version v1
Technical note Open

Notes on the Weyl tensor, decomposition of Riemann tensor, Ruse-Lanczos identity and duality of the curvature tensor

Authors/Creators

  • 1. University of Brasilia

Description

These notes have the pedagogical purpose of exposing mathematical accounts about of the Weyl tensor. The development of calculations is presented to obtain the curvature tensor from the conformal transformation, consequently resulting in the Weyl conformal tensor and the Riemann tensor reveals itself  decomposed into its parts. The Ruse-Lanczos identity is a useful mathematical tool to build duality identities in the two pairs of antisymmetric indices of the Riemann tensor components. 

Files

curvatura_Weyl_Tensor.pdf

Files (124.6 kB)

Name Size Download all
md5:a4308d12547f153f05ced630c4258acb
124.6 kB Preview Download

Additional details

References

  • W.C. Santos, Introduction to Einstein-Maxwell equations and the Rainich conditions, e-Print: arXiv:1606.08527 [gr-qc], (2016)
  • R.M. Wald, General Relativity, The University of Chicago Press (1984)
  • M. Nakahara, Geometry, topology and physics, Bristol, UK: Hilger, Graduate student series in physics, (1990)
  • G.S. Hall, Symmetries and Curvature Structure in General Relativity, World Scientific Publishing Co. Pte. Ltd., (2004)
  • H. Weyl, Eine neue Erweiterung der Relativitatstheorie, Ann. der Physik 59, 101-133, (1919)
  • J. D. Jackson and L. B. Okun, Historical roots of gauge invariance, Rev. Mod. Phys. 73, 663, (2001)
  • N. Straumann, Hermann Weyl and the early history of Gauge Theories, in 'Symmetries in Algebra and Number Theory', contributions to 'On the Legacy of Hermann Weyl', p.173, Universittsverlag Gttingen, (2009)
  • N. Straumann, Early History of Gauge Theories and Weak Interactions, Invited talk at the PSI Summer School on Physics with Neutrinos, Zuoz, Switzerland, August 4-10, 1996., e-Print: arXiv:hep-ph/9609230, (1996)
  • S.W. Hawking and G.F. Ellis, The large scale structure of space-time, Cambridge University Press, (1993)
  • N. Rosen, Weyl's Geomety and Physics, Foundations of Physics, Vol. 12, No. 03, (1982)
  • R. Penrose, Conformal treatment of infinity in Relativity, Groups and Topology, eds. B.S. DeWitt and C. DeWitt, Gordon and Breach Science Publishers Ltd., (1964)
  • R. D'Inverno, Introducing Einstein's Relativity, Oxford, UK: Clarendon (1992)
  • M.P. Hobson, G.P. Efstathiou, A.N. Lasenby, General Relativity, an introduction for physicists, Cambridge University Press (2006)
  • B. F. Schutz, A First Course in General Relativity, Second Edition, Cambridge University Press (2009)
  • S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, Inc. (1972)
  • P. Szekeres, The gravitational Compass, J. Math. Phys., 6 1387-1391, (1965)
  • F. De Felice and J. S. Clarke, Relativity on Curved Manifolds, Cambridge University Press, (1990)
  • H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of Einstein's Field Equations, 2nd Edition, Cambridge Monographs on Mathematical Physics, (2003)
  • J.Griffiths and J. Podolsk, Exact Space-Times in Einstein's General Relativity, Cambridge Monographs on Mathematical Physics, (2009)
  • M. Carmeli, Group Theory and General Relativity,Representations of the Lorentz group and their applications to the gravitational field (International series in pure and applied physics), McGraw-Hill, 1st edition, (1977)