GR Lagrangian Part 2 Homework Statement Solution

[Malamala]

Homework Statement

This is a continuation of this problem. I will rewrite it here too:
The Lagrangian density for the $h=h^{00}$ term of the Einstein gravity tensor can be simplified to: $$L=-\frac{1}{2}h\Box h + (M_p)^ah^2\Box h - (M_p)^b h T$$ The equations of motion following from this Lagrangian looks roughly like (I didn't calculate this, they are given in the problem): $$\Box h = (M_p)^{a}\Box(h^2)-(M_p)^bT$$ For a point source $T=m\delta^3(x)$, solve the equation for h to second order in the source T, with $M_p=\frac{1}{\sqrt{G_N}}$ and calculate the correction to the Mercury angular frequency orbit around the sun compared to the first order approximation.

Homework Equations

The Attempt at a Solution

Continuing the same logic as before I write $$h=h_0+h_1+h_2...$$ where $h_0$ is first order in T, $h_1$ is second order and so on. Before, I got for the first order $$h_0=-\frac{M_p^b m }{4 \pi r}$$ Now if we go to the second order, the equation we need to solve is: $$\Box h_1 = (M_p)^a\Box(h_0^2)$$ which is equivalent to $$\Box (h_1 - (M_p)^a h_0^2) = 0$$ $$h_1 = (M_p)^a h_0^2 + f(x)$$ where $f(x)$ is such that $\Box f = \nabla f = 0$ (I replaced $\Box$ with $\nabla$ as the source is time independent). Is it ok up to now? Now first thing I am confused about, can I discard this $f$? in the equation of motion it seems like $h$ appear with $\Box$ so I think f will not affect the equations, but I am not 100% sure this is true. Now for the orbit, to first order I used $$m_{Mercury}\omega^2 r = \frac{m_{Mercury}}{M_p}(-\nabla h_0)$$ where that $M_p$ comes from the normalization (as I was told in the first post). And from here I got the first order approximation for $\omega$. For the second order, I tried the same thing (this is QFT class, so I don't think I am expected to use GR, and they ask for a rough approximation, anyway) $$m_{Mercury}\omega^2 r = \frac{m_{Mercury}}{M_p}(-\nabla (h_0+h_1))$$ However something is wrong as the correction should be very small, however, $h_1$ going like $h_0^2$, contains $m^2$ which here m is the mass of the sun so overall, the value of $h_1$ is bigger than $h_0$ which doesn't make sense for a second order correction. What is wrong with my calculations? Thank you!

 

[Orodruin]

Since $h_0 \ll 1$, clearly $h_0^2 \ll h_0$.