# Basic questions on the PPN formalism in General Relativity

I'm trying to learn about testing modified gravity using the PPN formalism. I have several textbooks that I am reading through (including Clifford Will's book), and have some basic questions on the topic. (My apologies if these questions are rather elementary.)

To my understanding, the PPN formalism sounds like it says that the goal is to find the metric $g_{\mu\nu}$ and then find the perturbation $h_{\mu\nu}$. Once we find $h_{\mu\nu}$, we can check how closely (or not) it matches to a known solution at lowest order (such as Schwarzschild, for example).

My understanding of using the PPN formalism is: first I solve $\delta S=0$ to get the equation(s) of motion for $g_{\mu\nu}$ (and possibly some scalar fields $\phi$ depending on the specific model). One thing that bother me is that having the equation of motion is not the same as having $g_{\mu\nu}$.

In a nutshell, my question is: once I get the equation of motion by varying my gravitational action, what is the next step (to test modified gravity via the PPN formalism)? (I have more specific questions below)

1. After finding an equation of motion from solving $\delta S=0,$ am I supposed to insert $g_{\mu\nu}=\eta_{\mu\nu}+\epsilon h_{\mu\nu} + ....$ into my equations of motion (and the corresponding $R_{\mu\nu}$ and $R$ found from approximating $g_{\mu\nu}\approx \eta_{\mu\nu}+\epsilon h_{\mu\nu}$) and then solve to various orders in $\epsilon$?

2. If I have a scalar field, do I need to go to higher orders in expansions of $\epsilon$ and $\Phi$ (e.g. I should write $\Phi\approx \Phi_o + \delta\varphi + \frac{\lambda}{2}\delta\varphi^2+....$? Should I also be expanding the Ricci Tensor and Ricci Scalar to higher orders also?

3. Do we always assume that the perturbations $\delta \varphi$ or $\delta X^\lambda$ are of order $\epsilon$? (If so, why?)

4. Suppose (for example) that I want to compare a theory of modified gravity against the predictions from GR (say, for light deflection around a Schwarzschild metric in GR, for example). Far from a gravitating source, in GR, the Schwarzschild background should be Minkowski. Do I assume this to be the case for the new theory of gravity also? (If yes, is the reason for this because "far away" in this context can mean only a few hundred lightyears where spacetime is relatively well described as Minkowski (e.g. the scales that I consider are small enough that I can just match to Minkowski spacetime instead of some expanding spacetime)?)

5. Let us assume that in our model, at lowest order gravity does not reduce to the usual EFE ($R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G T_{\mu\nu}$), but some other form of gravity. (Let us suppose that it reduces to something like trace-free gravity . In Trace Free Gravity, the field equations are $R_{\mu\nu}-\frac{1}{4}Rg_{\mu\nu}=8\pi G \left( T_{\mu\nu}-\frac{1}{4}Tg_{\mu\nu}\right)$. I realise that under most circumstances this would be bad, but Trace Free Gravity can be shown to reproduce the same predictions as GR). My question is: would this change anything in my steps to finding the PPN parameters? Should I substitute the fields (from question 3) into the trace-free form of the gravity equations instead of putting them into the usual EFE? (e.g. in step 4 described here)

6. As practice, I am trying to work through this paper. I arrived at Eq.(12) by plugging Eq. (10) and (11) into Eq. (9) and dropping all terms higher than first order in $h_{\mu\nu}$ and $\varphi$. However, in attempting to arrive at Eq. (13) I plugged in the same terms as above (plug in $\Phi\approx \Phi_o + \varphi$ and $g_{\mu\nu}\approx\eta_{\mu\nu}+h_{\mu\nu}$) and I also inserted $\frac{1}{\Phi_o}\partial_\nu\varphi=\partial_\sigma h^\sigma _\nu - \frac{1}{2}\partial_\nu h$. After dropping terms higher than 2nd order in perturbations, I find $$\partial_\mu\partial_\nu\varphi - \frac{1}{2}\eta_{\mu\nu}\partial_\alpha\partial^\alpha \varphi-\frac{\Phi_o}{2}\left[ \Box\left( h_{\mu\nu}-\eta_{\mu\nu}\frac{h}{2}\right)\right]=8\pi g T_{\mu\nu} + \partial_\mu\partial_\nu \varphi - \eta_{\mu\nu}\Box\varphi$$ (the first two terms on the left hand side are not in Eq. (13) of the paper). The fact that the first two terms are still around suggests that there is something that I am not doing correctly or understanding. (Or is there a justification that I can drop these terms that I am not seeing? I don't think this is the case since $\partial_\mu\partial_\nu\varphi$ is on the right hand side as also.)

1. When working with a gauge, do we essentially treat our gauge choice as another equation that we can plug into the equation of motion to help simplify it (so long as the gauge satisfies $\tilde{h}_{\mu\nu}=h_{\mu\nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu$)? (in the paper above the author uses $\frac{1}{\Phi_o}\partial_\nu\varphi=\partial_\sigma h^\sigma _\nu - \frac{1}{2}\partial_\nu h$.)

2. Is there exist a good instructive worked example (hopefully with lots of intermediate steps filled in) of applying the PPN formalism to a theory of modified gravity?

Any help with clearing up my confusion is greatly appreciated.

• It seems to me you are thinking about the post-Minkowski expansion where you expand in the perturbation of the flat background. For the post-Newtonian expansion one essentially uses a preferred frame and expands around Newtonian physics in it. The easiest way to get a first intuition is to use SI units, you then easily identify the terms that are $\mathcal{O}(\epsilon)$ as those that are $\mathcal{O}(c^{-2})$. – Void Jan 31 '18 at 15:46
• thanks @Void. Is there any reference that does this (keeping the SI units around) which you would recommend? – Bob Jan 31 '18 at 17:39