Thursday, 28 February 2019

Important Equations for General Relativity

Here are some important equations for General Relativity.
They are in Commentary Important Equations.pdf along with references and some notes.

Mathematics
  • Definition of ##{\partial }_{\mu }## (Carroll 1.54)
  • (Anti)symmetrisation operator (Carroll 1.79)
  • Vector as derivative (Carroll 2.16)
  • Commutator (Carroll 2.20)
  • Tensor transformation equation (Carroll 2.30)
  • Basis vectors (Physics Forums)
  • Differential equation rules (Wikipedia)
  • Pullback / Pushforward operators (Carroll A.9, A.10)
  • Levi-Civita symbol and tensor (Carroll section 2.8)
  • p-forms (Carroll section 2.9)
  • Exterior derivative (Carroll 2.76)
  • Wedge product (Carroll 2.73)
  • Hodge star operator (Carroll 2.82)
  • Covariant derivative / Christoffel symbol (Carroll 3.2)
  • Torsion Tensor (C Eq 3.22)
  • The Christoffel connection Γ  (C Eq 3.27)
  • Stokes's theorem (C Eq 3.35)
  • The geodesic equation (C Eq 3.44)
  • Euler-Lagrange Equation
  • Directional covariant derivative (C Eq 3.38)
  • The parallel transport equation (C Eq 3.39, 3.40) 
  • Riemann tensor
  • Ricci tensor and scalar, Weyl tensor (c Eq 3.144-3.147)
  • Einstein tensor and 'contravariant' derivative (c Eq 3.1452,2)

Tensor Tricks

  • Multi-dimensional Chain Rule
  • Partial derivative of components gives Kronecker delta
  • Partial derivatives commute
  • Metric is always symmetric (C section 2.5)
  • Contracting with metric lowers index
  • Swap indices with metric

Physics

  • Electro Magnetic Field Tensor (C Eq 1.69)
  • Maxwell's equations (C Eq 1.96-1.98)
  • Energy Momentum tensor for a perfect fluid (C Eq 3.93)
  • Energy-momentum conservation equation (C Eq 3.92)

Image from WikipediBabylonian equations.

Carroll's (1.68) where the Levi-Civita symbol is defined says "the Levi-Civita symbol is a ##(0,4)## tensor. It is NOT a tensor as he reminds us elsewhere.

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