## 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 2.20/2.23)
• Tensor transformation equation (Carroll 2.30)
• Basis vectors (Physics Forums)
• Covariant derivative / Christoffel symbol (Carroll section 3.2)
• Torsion Tensor (C Eq 3.22)
• The Christoffel connection Γ  (C Eq 3.27)
• The geodesic equation (C Eq 3.44)
• Directional covariant derivative (C Eq 3.38)
• The parallel transport equation (C Eq 3.39, 3.40)
• Riemann tensor (c Eq 3.112/3.113)
• Bianchi identity (c Eq 3.140)
• Ricci tensor and scalar, Weyl tensor (c Eq 3.144-3.147)
• Einstein tensor and 'contravariant' derivative (c Eq 3.1452,2)
• Killing's equation, Killing vectors (c Eq3.174)
• Geodesic deviation equation (c Eq3.208)

#### Tensor tricks

• Multi-dimensional Chain Rule
• Partial derivative of components gives Kronecker delta (C Eq 1.152)
• Partial derivatives commute
• Metric is always symmetric (C section 2.5)
• Contracting with metric lowers / raises index
• Swap indices with metric or any similar tensor
• A relationship for the derivative of the metric determinant
• Fully contracted symmetric × antisymmetric tensor vanishes
• Symmetrising a tensor equation
• Two formulas involving four-velocity
• The projection tensor
• Contra / co-variant tensor transformation matrices
• Tensor contractions using matrices

#### 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 and 1.114)
• Energy Momentum tensor for dust in SR (C Eq 1.110)
• Energy-momentum tensor from action for matter (C Eq 4.75)
• Energy-momentum conservation equation (C Eq 3.92 & 4.8)
• Einstein's equation x 3 for general relativity (C Eq2.44-4.46)

#### More Maths

• Differential and Integration on Web
• 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)
• Stokes's theorem (C Eq 3.35)
• Euler-Lagrange Equation

All in Commentary Important Equations.pdf along with references and some notes.
Image from Wikipedi

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.