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?

 

[blue_raver22]

The Einstein field equations are a set of equations that describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. They are a cornerstone of general relativity, which is the theory of gravity proposed by Albert Einstein.

To understand how the Einstein field equations become the Poisson equation, we first need to understand the concept of curvature in general relativity. In general relativity, the curvature of spacetime is described by the Riemann curvature tensor, denoted by R^, which has 256 components. However, in order to simplify the equations, we can use the Bianchi identity to reduce the number of independent components to 20. These 20 components represent the tidal forces, or the stretching and squeezing of space, caused by the presence of matter and energy.

In Newtonian physics, the gravitational force is described by the Poisson equation, which relates the gravitational potential (ϕ) to the mass distribution (ρ) through the equation ∇^2ϕ = 4πGρ, where G is the gravitational constant. This equation is used to calculate the gravitational force between two objects in Newtonian gravity.

Now, in general relativity, the curvature tensor R^ can be related to the tidal force tensor, which is equivalent to the Newtonian gravitational force. This is shown in part (a) of the exercise, where R^ is given by - (∂ϕ/∂xμ∂xν)eμ ⊗ eν. This means that the tidal force tensor in general relativity is equivalent to the gradient of the gravitational potential. This is the key link between the Einstein field equations and the Poisson equation.

In part (b) of the exercise, it is shown that the Ricci tensor, denoted by Ric, is equivalent to ∇^2ϕ. The Ricci tensor is a contracted version of the Riemann curvature tensor, and it represents the curvature of space at a specific point. This means that the curvature of space at a point is related to the gravitational potential at that point, just like in the Poisson equation.

Finally, in part (c) of the exercise, it is shown that the Einstein tensor, denoted by G, is equivalent to 2∇^2ϕ. The Einstein tensor is a combination of the Ricci tensor and the scalar curvature, and it represents the curvature of spacetime as a whole. This means that the overall curvature