Thursday, 28 February 2019

Important Equations for General Relativity

Here are some important equations for General Relativity.
  • 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) 
  • Multi dimensional Chain Rule 
  • 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)
  • The most important formula (C Eq 3.27)
  • Stokes's theorem (C Eq 3.35)
  • The geodesic equation (C Eq 3.44)
They are in Commentary Important Equations.pdf along with references and some notes.

Image from WikipediBabylonian equations.

So far these are the basic maths that I have learnt in Carroll. I have done and not listed the Lorentz transformation. I hope I will add to the list as time goes on and include Lorentz and others.

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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