How can I find the metric in weak field limit for specific theory? What is the general approach to finding a modified version of Poisson equation by means of the weak field limit of a specific gravitational theory? What is the first step? Can you introduce the main procedure? Is it different for each theory?
Thank you.
 A: The usual steps are:


*

*Derive the full Euler-Lagrange equations from the Lagrangian of your theory.  There will be an analog of the Einstein equation (from the variation of $\mathcal{L}$ with respect to the metric) and some equations of motion for the other fields $\Psi^\alpha$ in your theory.  Don't forget to vary the derivative operators when taking the variations with respect to the metric.

*Find a background solution.  For "weak field", we usually mean that the metric is Minkowski ($g_{ab} = \eta_{ab}$).  If there is any other field content in your theory, you will need to ensure that their equations of motion are also obeyed in the case of flat spacetime as well.  This will imply certain conditions on the background values $\bar{\Psi}^\alpha$ of the other fields.  Often you'll find that the "background values" of the fields are $\bar{\Psi}^\alpha = 0$, but in some cases (such as spontaneous symmetry breaking) these fields will not vanish in the background solution.

*Plug the following ansatz into the equations of motion (both metric and "matter"):
$$
g_{ab} = \eta_{ab} + \epsilon \eta_{ab} + \mathcal{O}(\epsilon^2), \qquad  \Psi^\alpha = \bar{\Psi}^\alpha + \epsilon \psi_\alpha + \mathcal{O}(\epsilon^2).
$$
Here, $\epsilon$ is a parameter that parametrizes our deviations from our flat "background" solution.  When $\epsilon = 0$, we recover the background solution;  and the $\mathcal{O}(\epsilon)$ terms in the equations of motion give us the linearized equations.

*Write out the linearized metric equation in terms of time & space derivatives of the components of the fields.  Assume that all time derivatives vanish.  If your modified gravity theory is not too avant-garde, the resulting system of equations will depend on the background fields $\eta_{ab}$ and $\bar{\Psi}^\alpha$ and the metric perturbations $h_{ab}$.  In particular, the analog of the Poisson equation will be whatever your equations imply about the component $h_{tt}$ of the metric perturbation.


For further reference, check out Wald's General Relativity for the mathematical details.  (Perturbation theory is covered in Section 7.5, and how to perform variations with respect to the metric is covered in Section E.1.)  Clifford Will's The Confrontation between General Relativity and Experiment, and his earlier book Theory and Experiment in Gravitational Physics, both have a nice exposition of how one goes from a "full" modified gravity theory to a quasi-Newtonian or post-Newtonian theory.
