## Notes on Einstein Field Equation

Metric tensor:

(D: dimension of the spacetime)

Christoffel symbol:

Riemann tensor:

Ricci tensor:

Ricci scalar (curvature scalar):

Trace of energy-momentum tensor:

Einstein field equation:

or

Torsion tensor (have nothing to do with energy-momentum tensor ):

Properties of the Riemann tensor ():

(antisymmetric in first two indices)

(antisymmetric in last two indices)

(invariant under interchange of the first and last pair of indices)

or

Properties of the Ricci tensor:

Relation between R and T:

Einstein tensor:

so Einstein field equation can be rewritten as:

Geodesic equation:

Covariant derivative:

Energy-momentum tensor: