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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 :

h_{00}= A^{0}/r + 6Q^{ij}n^{i}n^{j}/r^{3}(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}/A^{0}where the B^{j}= 6Q^{ij}n^{i}/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}/ A^{0}

now I need to derive the ε^{j}with respect to j which leads to ε^{j,j}= 6Q^{ii}/A^{0}r^{2}- 12 Q^{ij}x^{i}x^{j}/A^{0}r^{4}(I discard this last term)

And the metric would then be :

g^{00}= -1 + A^{0}/2r - 6Q^{ii}/A^{0}r^{2}

I'm I right up to this point ? Did I miss anything perhaps ?

the poblem is that I would not see where the M^{2}would come out here ? I have the r^{2}and also the first linear term but the second term is proportional to the quadrupole moment divided by r^{2}A^{0}which means no M squared ? and the A^{0}is linear in M

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# MTW Exercise 19.3 last part

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