# MTW Exercise 19.3 last part

1. Mar 30, 2013

### 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 + 6Qijninj/r3 (Qij 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 , xjnew= xjold - Bj/A0 where the Bj = 6Qijni/r which leads to the new metric perturbation in the new coordinate frame : h00new = h00old - εj,j - εj,j
where the εj = - Bj / A0
now I need to derive the εj with respect to j which leads to εj,j = 6Qii/A0r2 - 12 Qijxixj/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

2. Mar 31, 2013

3. Apr 1, 2013

### zn5252

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