Field of Point Mass: MTW Ex 7.3C Solution

[zn5252]

hello,
Please see attached snapshot. My question is regarding the way MTW found the term h₀₀ or just for the hbar from equation 7.
Now since we are in the static case, Equation 7 becomes a Laplacian. But I could not figure out how the 1/r term came to be and where did the 16Pi go ?
Any help is much appreciated.
Thanks,

 

[elfmotat]

Write the Laplace equation as the following:

$$\nabla\cdot (\nabla\bar{h}_{\mu \nu })=-16\pi T_{\mu \nu }$$

Now apply Gauss' Law:

$$\int \nabla\cdot (\nabla\bar{h}_{\mu \nu })d^3 x=\int (\nabla\bar{h}_{\mu \nu })\cdot \mathbf{n}d^2x= -16\pi \int T_{\mu \nu }d^3x$$

$$4\pi r^2 |\nabla\bar{h}_{00}|=-16\pi M$$

I'll let you take it from there.