Riemann Tensor

General Relativity · Part 2

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Table of Contents

  1. 1.Definition and Properties
  2. 2.Curvature via Parallel Transport
  3. 3.Definition via Commutator
  4. 4.Component Expression
  5. 5.Symmetry Properties
  6. 6.Number of Independent Components
  7. 7.Special Case: 2 Dimensions
  8. 8.Special Case: 3 Dimensions
  9. 9.1. Antisymmetry in First Pair
  10. 10.2. Antisymmetry in Second Pair

Key Equations

$$\delta V^\rho = R^\rho_{\ \sigma\mu\nu}\, V^\sigma\, \delta x^\mu\, \delta x^\nu$$
$$[\nabla_\mu, \nabla_\nu] V^\rho = \left(\partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\right) V^\sigma$$
$$R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}$$
$$R_{\rho[\sigma\mu\nu]} = 0 \quad \Longleftrightarrow \quad R_{\rho\sigma\mu\nu} + R_{\rho\mu\nu\sigma} + R_{\rho\nu\sigma\mu} = 0$$

Equations are rendered with MathJax in the PDF with professional LaTeX typesetting.

Course Context

This PDF is part of the General Relativity course on CoursesHub.World. Explore Einstein's theory of gravity as curved spacetime. Covers differential geometry, curvature tensors, Einstein field equations, classic solutions including Schwarzschild and Kerr black holes, gra...

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