Einstein's field equations of general relativity
Gαβ= k T αβ

as k = -8 π G then Gαβ= -8 π G T αβ
analogous to Newton-Poisson 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 stress-energy tensor, is the source of the gravitational field in the Einstein Theory
Δ = 2 = δ2
——
δ x2
+ δ2
——
δ y2
+ δ2
——
δ z2


Gαβ = Rαβ -1

2
gαβ R
is the Einstein tensor constructed from curvature

Einstein's equations for the gravitational field
Rαβ -1

2
gαβ R = - 8 π G Tαβ

where the covariant metric tensor is given by:   gαβ =

g00 g01 g02 g03
g10 g11 g12 g13
g20 g21 g22 g23
g30 g31 g32 g33

(second-rank symmetric non-degerate tensor field)

The infinitesimal interval is given by:   d s 2 = g αβ d xα d xβ
We use the Einstein notation that implied summation over repeated indices thus:
d s 2 = g αβ d xα d xβ = d s 2 = Σ g αβ d xα d xβ
where (α = 0, 1, 2, 3 and β = 0, 1, 2, 3 ) Space-time is four-dimensional manifold
g αβ = g βα (is symmetric)

In General Relativity the Space-time 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 torsion-free.(In the Einstein-Cartan theory, the torsion Tαβγ is not vanish)

-using Christoffel symbols-
The Levi-Civita connection is given by:    Γ τμν = 1

2
gτα ( δμgνανgμααgμν )
where    gμν gντ = δμτ (Kronecker delta) and    δμ = δ / δ xμ

The Riemann tensor is given by:    Rαβγδ = δβ Γγαδ- δγ Γβαδ + Γβεδ Γγαε - Γγεδ Γβαε
The Ricci tensor is given by:    Rαβ = Rβα (for a Riemannian geometry is symmetric)
Rαβ = Rαδβδ
Rαβ = Rαδβγ g δγ
The Ricci scalar is given by:    R

Einstein's vacuum field equations
Rαβ= 0