Einstein's field equations of general relativity 

G_{αβ}= k T _{αβ} 
as  k = 8 π G  then  G_{αβ}= 8 π G T _{αβ} 
analogous to NewtonPoisson equation:  ^{2} Φ = 4 π G ρ  
where: Φ is the gravitational potencial, ρ is the matter density and G the gravitational constant.  
ρ , the matter density, is the source of the gravitational field in the Newtonian Theory and  
T _{αβ}, the stressenergy tensor, is the source of the gravitational field in the Einstein Theory 
Δ = ^{2} =  δ^{2} —— δ x^{2}  +  δ^{2} —— δ y^{2}  +  δ^{2} —— δ z^{2} 
 is the Einstein tensor constructed from curvature 
Einstein's equations for the gravitational field  


where the covariant metric tensor is given by:  g_{αβ} = 

The infinitesimal interval is given by: d s ^{2} = g _{αβ} d x^{α} d x^{β} 
We use the Einstein notation that implied summation over repeated indices thus: d s ^{2} = g _{αβ} d x^{α} d x^{β} = d s ^{2} = Σ g _{αβ} d x^{α} d x^{β} where (α = 0, 1, 2, 3 and β = 0, 1, 2, 3 ) Spacetime is fourdimensional manifold 
g _{αβ} = g _{βα} (is symmetric) 
In General Relativity the Spacetime is a Hausdorff topological space, covered by coordinate patches.
Thus there should exist a set of coordinate transition fuctions ( differentiable functions) in the overlap regions.
Thus we need tensors. Differentiation in tensor fields need make a connection.
The conection is symmetric. Thus torsionfree.(In the EinsteinCartan theory, the torsion T_{αβ}^{γ} is not vanish)
using Christoffel symbols
The LeviCivita connection is given by:  Γ ^{τ}_{μν} =  1 — 2  g^{τα} ( δ_{μ}g_{να}+δ_{ν}g_{μα}δ_{α}g_{μν} ) 
where g^{μν} g_{ντ} = δ^{μ}_{τ} (Kronecker delta)  and δ_{μ} = δ / δ x^{μ} 
The Riemann tensor is given by: R_{αβγ}^{δ} = δ_{β} Γ_{γα}^{δ} δ_{γ} Γ_{βα}^{δ} + Γ_{βε}^{δ} Γ_{γα}^{ε}  Γ_{γε}^{δ} Γ_{βα}^{ε} 
The Ricci tensor is given by: R_{αβ} = R_{βα} (for a Riemannian geometry is symmetric) R_{αβ} = R_{αδβ}^{δ} R_{αβ} = R_{αδβγ} g ^{δγ} 
The Ricci scalar is given by: R 
Einstein's vacuum field equations 

R_{αβ}= 0 