Field of point mass - MTW Solution to Ex 7.3C

1. Jan 30, 2013

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,

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2. Jan 30, 2013

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.