Question on asymptotic flatness What is the theoretical argument for the asymptotical flatness of the four-potential? Can one assume asymptotical flatness for the scalar dilaton field as well?
 A: To derive the $\textit{Peeling Theorem}$, let us invoke Penrose's Conformal treatment of infinity (https://link.springer.com/article/10.1007/s10714-010-1110-5). Given any physical space-time ($\mathcal{M},g_{ab}$) along with a positive definite scalar field $\Omega$, we can define un-physical space-time ($\hat{\mathcal{M}},\hat{g}_{ab}$), where $\hat{g}_{ab}=\Omega^2 g_{ab}$. Consider $\Omega$ and $\hat{g}_{ab}$ to be $\mathscr{C}^k$ smooth throughout $\hat{\mathcal{M}}$. The physical space-time $\mathcal{M}=$int$\hat{\mathcal{M}}$, while the boundary $\mathscr{I}=\partial \hat{\mathcal{M}}$. On $\mathscr{I}$, $\Omega = 0$ while $\hat{\nabla}_a\Omega \neq 0$. Let's assume $\mathscr{I}$ to be null hypersurface, which means the generator $\hat{N}_a=-\hat{\nabla}_a\Omega$ is null on $\mathscr{I}$.
Let $\gamma$ be any null ray in $\mathcal{M}$, with it's end point $P$ on $\mathscr{I}$. Choose $(o_A,\iota^A)$ to be spin frame in $\mathcal{M}$ with $r$ as the affine parameter on $\gamma$ (i.e. $Dr=l^a\nabla_a r=1$, $l^a=o^Ao^{A'}$ is the geodetic tangent vector of $\gamma$, with $Dl^a=0$). Let $\hat{o}_A=o_A$ and $\hat{\iota}^A=\iota^A$ constitute the corresponding spin-frame in $\hat{\mathcal{M}}$ and $\hat{r}$ be the associated affine parameter (with $\hat{D}\hat{r}=1$, $\hat{D}\hat{l}^a=0$). Fact that $\Omega$ is $\mathscr{C}^k$ smooth in $\hat{\mathcal{M}}$, we can write:
$$\Omega=A_0+A_1\hat{r}+A_2\hat{r}^2+\cdots+A_k\hat{r}^k+o(\hat{r}^k)$$ where $A_0,A_1,\cdots ,A_k$ are constants along $\gamma$. Let $\hat{r}=0$ on $\mathscr{I}$, then $\Omega |_{\mathscr{I}}=0$ implies $A_0=0$. Now, $\textit{asymptotic Einstein condition}$ corresponds to the shear-free condition on $\hat{N}_a|_{\mathscr{I}}$:
$$(\hat{\nabla}_a\hat{\nabla}_b\Omega-\frac{1}{4}\hat{g}_{ab}\hat{\nabla}^c\hat{\nabla}_c\Omega)|_{\mathscr{I}}=0$$
It follows that $\hat{l}^b\hat{l}^a\hat{\nabla}_a\hat{\nabla}_b\Omega|_{\mathscr{I}}=\hat{D}^2\Omega|_{\mathscr{I}}=A_2=0$. Also $\hat{l}_a=l_a\implies \hat{l}^a=\Omega^{-2}l^a$, then $\hat{D}r=\frac{\partial r}{\partial \hat{r}}=\Omega^{-2}$. One can now express $\hat{r}$ in terms of $r$. Function $A_2=0$ ensures that there are no logarithmic dependence on $r$, and that $\hat{r}$ and therefore, $\Omega$ can be written as polynomials in $1/r$.
If $\hat{\theta}_{AB\cdots L}$ be any arbitrary spinor field which is $\mathscr{C}^h$ smooth throughout $\hat{\mathcal{M}}$, one can expand $\hat{\theta}$ in powers of $\hat{r}$, and therefore, in polynomials in $1/r$. We can also assume $\hat{\theta}$ to be a conformal density of weight $w$, i.e. $\hat{\theta}_{\cdots}=\Omega^{-w}\theta_{\cdots}$, which will allow us to write $\theta_{\cdots}$ as polynomial in $1/r$.
As a particular example, consider $\theta_{AB\cdots L}$ to be any z.r.m. free spinor field in $\mathcal{M}$, which is a conformal density of weight $-1$. Now, retarded time coordinate $u$ can be treated as a parameter varying on null geodesics on $\mathscr{I}^{+}$, while advanced time coordinate is fixed throughout $\mathscr{I}^{+}$. Thus, if we start our analysis from $\mathscr{I}^{+}$, the coefficients appearing in $\theta_{\cdots}$ will turn out to be functions in $u,\zeta,\bar{\zeta}$ ($\zeta$ is the stereographic coordinate), and thus $\theta_{\cdots}$ will correspond to a purely outgoing field. As a special case, if $\theta_{AB}$ satisfies the peeling property, it implies that the Maxwell field $F_{ab}=\theta_{AB}\epsilon_{A'B'}+c.c.$ and therefore, the four potential $A_a$ will satisfy the Peeling property. This is also true for non-linear fields like Yang-Mills field (see https://journals.aps.org/prd/abstract/10.1103/PhysRevD.18.2901) and gravity.
