Important Equations for General Relativity

Here are some important equations for General Relativity.

Babylonian equations

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

• What tensor rank?
• Multi-dimensional Chain Rule
• Partial derivative of components gives Kronecker delta
• Coordinates
• Vectors (C1.152)
• Tensors
• Partial derivatives commute
• Metric is always symmetric (C section 2.5)
• Contracting with metric lowers / raises index
• You can lower or raise indices on a tensor equation
• Swap indices with metric or any similar tensor
• Inverses and determinants
• Inverse of a matrix
• The determinant of the inverse is reciprocal of the determinant
• Determinant of a tensor in terms of Levi-Civita symbol (C Eq 2.66)
• Inverse tensor
• A relationship for the derivative of the determinant
• Fully contracted symmetric × antisymmetric tensor vanishes
• Symmetrising a tensor equation
• Two formulas involving four-velocity
• Second formula
• The projection tensor on four-velocity (C Eq 1.21)

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)
• Friedmann equations (C Eq 8.67)

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 Wikipedia Babylonian equations.

The rest of the fab four are

Commentary Tensor Tricks.pdf 

Commentary 2.5 The Metric.pdf 

Commentary Constants and conversion factors.pdf 

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.