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I'm studying the book Gravity by Poisson & Will. Specifically, I'm interested in the post-Newtonian and post-Minkowskian approximations showed in chapters 6-10. The problem I'm having is conceptual, I understand the calculations but I don't get the names used. Let me try to explain what I think I understood and then someone can correct me.

Post-Minkowskian approximation: This consists on using the Landau-Lifshitz formalism to get to a wave equation for a new field $h^{\alpha \beta}$ that has a new source $t^{\alpha \beta}$ involving both matter fields and some terms involving the metric

\begin{equation} \Box h^{\alpha \beta}=-\frac{16 \pi G}{c^4}t^{\alpha \beta} \end{equation}

One then does an expansion in orders of $G$ and then solves iteratively until the desired order.

Post-Newtonian approximation: This involves the assumption of slow motion $\frac{v}{c}<<1$ in order to expand the metric (the actual metric $g^{\alpha \beta}$, not the new constructed field $h^{\alpha \beta}$) in powers of $c^{-1}$. Solving Einstein's equations order to order gives you the post-newtonian metric up to the desired order in powers of $c^{-1}$.

My problem: What I don't get is that the book I'm following embeds the post-Newtonian approximation within the Post-Minkowskian expansion in powers of $G$. That's where I get confused. It seems like to simplify the calculations of the post-Minkowskian iteration in powers of $G$ they also expand the fields in powers of $c^{-1}$, thus mixing both approximations. I'd like to know more precise definitions of each approximation and, also, the reason why this is presented in such a confusing way, conceptually speaking. I'm sure that historically it was done differently but this approach has more benefits.

P. C. Spaniel
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1 Answers1

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After a bit more than a year of studying the topic, I think I can answer my own question.

The Post-Minkowskian approximation (PMA) starts with Minkowski spacetime at zeroth order and uses a weak gravity approximation in powers of $G$. This means that particles are alowed to move at relativistc speed and even have zero rest mass. However, since the zeroth order solution to the equations of motion will give straight lines, the following corrections are best suited for describing scattering events and deviation angles, but not bound states. Interestingly, there has been some research on "continuing" the solutions from scattering events to bound systems and, surprisingly, the PMA can give information about bound states too[1].

The Post-Newtonian approximation (PNA) starts with Newtonian gravity at zeroth order and uses a slow-motion approximation in powers of $c^{-2}$. This endeavor was started by Chandrasekhar in the 60's [2]. However, there's a problem with this approximation: The slow-motion approximation creates instantaneous potentials (As oposed to the retarded potentials we would expect in Einstein's theory of gravity). These instantaneous potentials start giving ambigueties and divergencies as one goes up in the expansion. The solution is to embed the PNA inside a PMA , use the PNA for the metric tensor close to the matter distribution (where the slow-motion approximation and the instantaneous potentials make sense), use the PMA approximation for the far away zone where spacetime is approximately Minkowski and we need to deal with retarded gravitational waves, and then match both approximations in some buffer zone in between.

The confusion comes from the fact that the modern understanding of the PNA includes a PMA, so people sometimes refer to it as a PMA. However, the inclusion of a PMA is just a formal tool to avoid ill-defined differential equations and divergent integrals over instantaneous potentials; physically, it's still a PNA. The actual PMA stands on its own, as an approximation in powers of $G$ that works at all powers of $c^{-2}$, as can be seen in [1].

[1] https://arxiv.org/abs/1609.00354

[2] S. Chandrasekhar, The Post-Newtonian equations of Hydrodynamics in General Relativity, 1965.

P. C. Spaniel
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