 \tau^{\mu\nu}= 16\pi (-g) \frac{\varphi}{\phi_0} T^{\mu\nu}  + \Lambda^{\mu\nu} + \Lambda_S^{\mu\nu} \,,
\label{heq}
\end{equation}
where
\begin{subequations}
\begin{eqnarray}
\Lambda^{\mu\nu} &\equiv& 16\pi \left [ (-g) t_{LL}^{\mu\nu} \right ](\tilde{\mathfrak{g}}^{\mu\nu})
+ {\tilde{h}^{\mu\alpha}}_{,\beta} {\tilde{h}^{\nu\beta}}_{,\alpha}
-\tilde{h}^{\alpha\beta} {\tilde{h}^{\mu\nu}}_{,\alpha\beta} \,,
\\
\Lambda_S^{\mu\nu} &\equiv&  \frac{(3+ 2\omega)}{\varphi^2} \varphi_{,\alpha} \varphi_{,\beta} \left ( \tilde{\mathfrak{g}}^{\mu\alpha}\tilde{\mathfrak{g}}^{\nu\beta} - \frac{1}{2} \tilde{\mathfrak{g}}^{\mu\nu}\tilde{\mathfrak{g}}^{\alpha\beta} \right ) \,,
\end{eqnarray}
\label{Lambdadef}
\end{subequations}
where the notation $[(-g) t_{LL}^{\mu\nu}](\tilde{\mathfrak{g}}^{\mu\nu})
$ denotes that the Landau-Lifshitz piece should be calculated using only $\tilde{\mathfrak{g}}$, in other words, exactly as in general relativity, except using the conformal metric, rather than the physical metric.   
The scalar field equation can also be rewritten in terms of a flat-spacetime wave equation, of the form
\begin{equation}
\Box_\eta \varphi = -8\pi \tau_s \,,
\label{phieq}
\end{equation}
where
\begin{eqnarray}
 \tau_s &=& -\frac{1}{3+2\omega} \sqrt{-g} \frac{\varphi}{\phi_0} \left ( T -2 \varphi \frac{\partial T}{\partial \varphi} \right )
- \frac{1}{8\pi}  \tilde{h}^{\alpha\beta} \varphi_{,\alpha\beta}
\nonumber \\
&&
+\frac{1}{16\pi} \frac{d}{d\varphi} \left [ \ln \left ( \frac{3+2\omega}{\varphi^2} \right ) \right ] \varphi_{,\alpha} \varphi_{,\beta} \tilde{\mathfrak{g}}^{\alpha\beta} \,.
\label{taustar}
\end{eqnarray}
In principle, Eqs.\ (\ref{gothg}) and (\ref{hdef}) can be combined to give $g_{\mu\nu}$ in terms of $\varphi$ and $\tilde{h}^{\mu\nu}$, although in practice, we will express it as a PN expansion.  The final result will be the relaxed field equations (\ref{heq0}) - (\ref{taustar}) expressed entirely in terms of $\tilde{h}^{\mu\nu}$, $\varphi$, and the matter variables.   The next task will be to solve these equations iteratively in a post-Newtonian expansion in the near-zone. Formally the solutions of these wave equations can be expressed using the standard retarded Green function, in the form
\begin{eqnarray}
\tilde{h}^{\mu\nu} (t,{\bf x}) &=& 4 \int \frac{\tau^{\mu\nu} (t-\mid {\bf x}-{\bf x}'\mid,{\bf x}')}{\mid {\bf x}-{\bf x}'\mid } d^3x' \,,
\nonumber \\
\varphi (t,{\bf x}) &=& 2 \int \frac{\tau_s (t-\mid {\bf x}-{\bf x}'\mid,{\bf x}')}{\mid {\bf x}-{\bf x}'\mid } d^3x' \,,
\end{eqnarray}
where the integration is over the past flat spacetime null cone of the field point $(t,{\bf x})$.   We will expand these integrals in the near-zone, and incorporate a slow-motion, weak-field expansion in terms of a small parameter $\epsilon \sim v^2 \sim m/r$; the strong-field internal gravity effects will be encoded in the functions $M_A(\phi)$.





\chapter{Formal Structure and Expansion of The Near-Zone Fields}

\label{chapter9}
\thispagestyle{myplain}
\lhead[\thepage]{Chapter 8. \emph{Formal Structure and Expansion of The Near-Zone Fields}}     
\rhead[Chapter 8. \emph{Formal Structure and Expansion of The Near-Zone Fields}]{\thepage}

\section{Formal Structure of The Near-Zone Fields}
\label{sec:nearzonefields}

Following Eq.~(\ref{Ncomponents}), we reintroduce a simplified notation for the field
$\tilde{h}^{\mu\nu}$ and the scalar field $\varphi$:
\begin{eqnarray}
N &\equiv& \tilde{h}^{00} \sim O(\epsilon) \,, \nonumber \\
K^i &\equiv& \tilde{h}^{0i} \sim O(\epsilon^{3/2}) \,, \nonumber \\
B^{ij} &\equiv& \tilde{h}^{ij} \sim O(\epsilon^2) \,, \nonumber \\
B &\equiv& \tilde{h}^{ii} \equiv \sum_i h^{ii} \sim O(\epsilon^2) \,,
\nonumber \\
\Psi &\equiv& \varphi - 1 \sim O(\epsilon) \,,
\end{eqnarray}
where we show the leading order dependence on $\epsilon$ in the near
zone.  To obtain
the equations of motion to 2.5PN order, we need to determine
the components of the physical metric and $\varphi$ to the following orders:
$g_{00}$ to $O(\epsilon^{7/2})$,
$g_{0i}$ to $O(\epsilon^{3})$,
$g_{ij}$ to $O(\epsilon^{5/2})$, and 
$\varphi$ to $O(\epsilon^{7/2})$.
From the Eqs.~(\ref{gothg}, \ref{hdef}), one can invert to find
$g_{\mu\nu}$ in terms of $\tilde{h}^{\mu\nu}$ and $\varphi$ to the appropriate order in $\epsilon$, as in PWI, Eq.\ (4.2).  Expanding to the required order, we find (compare to \eref{gr-metric})
\begin{subequations}
\label{STmetricexpand}
\begin{eqnarray}
g_{00} &=& -1 +  \left ( \frac{1}{2} N + \Psi \right) {\color {red} \epsilon}
+  \left ( \frac{1}{2} B - \frac{3}{8} N^2 - \frac{1}{2} N \Psi
- \Psi^2 \right ) {\color {red} \epsilon^{2}}\\
&& +  \left (\frac{5}{16} N^3 - \frac{1}{4} NB + \frac{1}{2} K^j K^j +\frac{3}{8} N^2 \Psi - \frac{1}{2} B \Psi +\frac{1}{2} N \Psi^2 + \Psi^3 \right ) {\color {red} \epsilon^{3}}
\nonumber +O(\epsilon^4) \,, 
\label{metric00}
\\
g_{0i} &=& -  K^i  {\color {red} \epsilon^{3/2}}+  \left ( \frac{1}{2} N + \Psi \right ) K^i  {\color {red} \epsilon^{5/2}}+O(\epsilon^{7/2}) \,, 
\label{metric0i}
\\
g_{ij} &=& \delta^{ij} \left \{ 1+  \left (\frac{1}{2} N - \Psi \right ) {\color {red} \epsilon}-  \left ( \frac{1}{8} N^2 + \frac{1}{2} B + \frac{1}{2} N \Psi - \Psi^2 \right ) {\color {red} \epsilon^{2}} \right \} +  B^{ij} {\color {red} \epsilon^{2}}
+O(\epsilon^3) \,, 
\label{metricij}
\\
(-g) &=& 1+  (N - 4\Psi)\; {\color {red} \epsilon}-  (B +4N \Psi - 10 \Psi^2 )\; {\color {red} \epsilon^{2}}+ O(\epsilon^3) \,.
\label{metricdet}
\end{eqnarray}
\label{metric}
\end{subequations}

In Eqs.~Eqs.~((\ref{STmetricexpand