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

T00 T01 T02 T03
T10 T11 T12 T13
T20 T21 T22 T23
T30 T31 T32 T33

T00 energy density
T01 T02 T03 T10 T20 T30 momentum density
T11 T22 T33 pressure
T12 T13 T23 T21 T31 T32 shear stress

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 equations for the gravitational field with cosmological constant
Rαβ -1

2
gαβ R + Λgαβ = 8 π G Tαβ

Λ = The cosmological constant is the energy density of space, or vacuum energy ( ≈ dark energy)

Einstein's vacuum field equations
Rαβ= 0