How Einstein field equation becomes the Poisson equation?

[lili 73]

I want to show that ∇²ϕ=ρ/2, which governs gravity in Newtonian physics?
I found solution of this question in [General Relativity for Mathematicians, R.K.Sachs and H.Wu, 1997, page 112&271].
Solution refer to optional exercise as follows:

Let R^ be the (0, 4) –tensor field physically equivalent to the curvature tensor of (ℝ⁴,g) .
(a) Show, that R^ corresponds to - (∂ϕ/∂xμ∂xν)eμ ⊗ eν. The tidal force tensor of Newtonian physics.
(b) Show Ric(∂4, ∂4)= ∇²ϕ.
(c) Show that , in the sense of the convections ,the Einstein tensor G becomes 2∇²ϕ .

Please help me to understand how Einstein field equation becomes the Poisson equation?

 

[conquest]

what exactly is $\phi$ and how is it related to g.

 

[lili 73]

in that book ,the author use a native spacetime (ℝ4,g) with g,

g=-(1+2 ϕ)du⁴⊗ du⁴+(1-2 ϕ)$\sum$du^α⊗ du^α ,where ϕ:ℝ3→ℝ will be interpreted as a time –independent Newtonian gravitational potential .
(sigma from 1 to 3).

 

[conquest]

Have you tried calculating the christoffel symbols and then just writing out local expressions for the curvature and Ricci curvature?