Is My Derivation of the Dominant Non-Linear Terms in MTW Exercise 19.3 Correct?

[zn5252]

hello
In part 8) of this Ex, MTW mentions that the dominant non linear terms must be proportional to the square (M/r)². The problem is that since I got the value :
h₀₀ = A⁰/r + 6Q^ijnⁱn^j/r³ (Q^ij is the quadrupole moment) and following the translation of coordinates suggested by MTW which is found in linearized theory , eq 4a of MTW chapter 18-Box18.2 , x^j_new= x^j_old - B^j/A⁰ where the B^j = 6Q^ijnⁱ/r which leads to the new metric perturbation in the new coordinate frame : h_00new = h_00old - ε_j,j - ε_j,j
where the ε^j = - B^j / A⁰
now I need to derive the ε^j with respect to j which leads to ε^j,j = 6Qⁱⁱ/A⁰r² - 12 Q^ijxⁱx^j/A⁰r⁴ (I discard this last term)
And the metric would then be :
g⁰⁰ = -1 + A⁰/2r - 6Qⁱⁱ/A⁰r²

I'm I right up to this point ? Did I miss anything perhaps ?
the poblem is that I would not see where the M² would come out here ? I have the r² and also the first linear term but the second term is proportional to the quadrupole moment divided by r²A⁰ which means no M squared ? and the A⁰ is linear in M

 

[zn5252]

I may have found it after a long struggle :
we have according to Equation 7.4 of T'hooft GR lectures ( https://docs.google.com/viewer?a=v&q=cache:V5lDvcnD2QwJ:www.phys.uu.nl/~thooft/lectures/genrel_2010.pdf+t+hooft+general+relativity&hl=en&pid=bl&srcid=ADGEESgNUwhdo8vmavylgtliXo2GgEqSIlKAuBBIU9gdx9kgs_HIoxG9ZDHOjR2kMEElCJ3e7lrWRCeM2TuprBETDoglG36U-kfA42lfRvLEbGzQr3P_EU6p7zRxHJbdduMTrHvWy-MP&sig=AHIEtbTwYwDR-VuoErk0SoftoaCX_8HEBQ ), a second term which is proportional to the square of h. I hope I got it right.

 

[zn5252]

Full solution could be found in Landau and Lifgarbagez page 359 for those interested...