## 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 Wikipedi

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.