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\begin{centering}

  \vspace{0cm}

\textbf{\Large{
On Late Time Tails in an Extreme  Reissner-Nordstr\"om \\  \vspace{0.2cm} Black Hole:   Frequency domain analysis}}

 \vspace{0.8cm}

  {\large Srijit Bhattacharjee${}^1$, Bidisha Chakrabarty${}^{2}$, \\ Partha Paul${}^{3,4}$, and Amitabh Virmani$^{3,4,5}$}

  \vspace{0.5cm}

\begin{minipage}{.9\textwidth}\small  \begin{center}
 ${}^{1}$
 Indian Institute of Information Technology, Allahabad \\
Devghat, Jhalwa, Uttar Pradesh 211015, India\\
  \vspace{0.5cm}
$^2$International Centre for Theoretical Sciences (ICTS), \\ Tata Institute of Fundamental Research, \\  Shivakote, 
Bengaluru 560 089, India \\
  \vspace{0.5cm}  
   ${}^{3}$Institute of Physics, Sachivalaya Marg, \\ Bhubaneswar, Odisha 751005, India  \\
  \vspace{0.5cm}
  ${}^{4}${Homi Bhabha National Institute, Training School Complex, \\ Anushakti Nagar, Mumbai 400085, India}\\
\vspace{4mm}
 ${}^{5}$Chennai Mathematical Institute, 
H1, SIPCOT IT Park, Siruseri  \\ Kelambakkam, Tamil Nadu 603103, India   \\


  \vspace{0.5cm}
{\tt srijitb@iiita.ac.in, bidisha.chakrabarty@icts.res.in, \\ pl.partha13@iopb.res.in, avirmani@cmi.ac.in}
\\ $ \, $ \\

\end{center}
\end{minipage}

\end{centering}


\begin{abstract}
In this brief note, we revisit the study of the leading order late-time decay tails of massless scalar perturbations outside an extreme Reissner-Nordstr\"om  black hole. Previous authors have analysed this problem in the time domain, we analyse the problem in the frequency domain.  
We first consider initial perturbations with generic regular behaviour across the horizon on characteristic surfaces. We reproduce some of the previous results of Sela [arXiv:1510.06169] using Fourier methods. Next we consider related initial data on $t=\mbox{const}$ hypersurfaces, and present decay results at timelike infinity, near future null infinity, and near future horizon. Along the way, using the $r_* \to -r_*$ inversion symmetry of an extreme Reissner-Nordstr\"om spacetime we relate higher multipole Aretakis and Newman-Penrose constants for massless scalar in this background.  

\end{abstract}

\newpage
\tableofcontents

\setcounter{equation}{0}
\section{Introduction}



More than 45 years ago, Price \cite{Price:1971fb} in his seminal analysis showed that when a Schwarzschild black hole is perturbed by a massless scalar field, at late time the perturbation typically decays as an inverse power in the Schwarzschild coordinate $t$. Price's law has been rigorously proved in the mathematical general relativity literature by Dafermos and Rodnianski \cite{Dafermos, DafermosReview}.
The problem of late time decay of scalar perturbation in 
 four-dimensional extreme Reissner-Nordstr\"om black hole has been analysed by several authors over the years, starting with the work of Bicak \cite{Bicak}. 
 
 Bicak observed that the effective potential for the massless scalar in an extreme Reissner-Nordstr\"om black hole has the same asymptotic form near the horizon as near infinity. Couch and Torrence \cite{CT} later showed that not only the effective potential has the same asymptotic form, it is in fact symmetric under $r_*$ going to $-r_*$, where $r_*$ is the tortoise coordinate for the extreme Reissner-Nordstr\"om  metric. 
 This surprising symmetry allows one to relate scattering dynamics near the horizon to the asymptotic region.   This symmetry adds several novel features to the late time dynamics of a massless scalar field in 
an extreme Reissner-Nordstr\"om black hole background 
 compared to a Schwarzschild black hole. This richness is one of the reasons that several authors have studied this problem \cite{Blaksley:2007ak, Lucietti:2012xr, Ori, Sela}. 
 
 Another reason the problem has attracted attention in the last few years is that Aretakis \cite{Aretakis:2011ha, Aretakis:2011hc, Aretakis:2012bm} has shown that a massless scalar has an instability at the future horizon of an extreme Reissner-Nordstr\"om black hole. More precisely, Aretakis showed that a massless scalar field decays at late time on and outside the future horizon, however, generically on the horizon its first radial derivative does not decay. This implies an instability. Since first radial derivative of the scalar decays away from the horizon but not on the horizon, it follows that the second-derivative must blow up  at late times \emph{on the horizon.} Aretakis instability is studied numerically in detail in \cite{Lucietti:2012xr}. They found excellent agreement with Aretakis' results. Using the Couch-Torrence symmetry, 
 Aretakis instability has been related to the similar growth in the behaviour of the derivatives of the massless scalar field at null infinity \cite{Bizon:2012we, Lucietti:2012xr}. Motivated by these developments,  more recently Ori and Sela  \cite{Ori, Sela} have re-analysed analytically the problem of late time decay of scalar perturbations outside an extreme Reissner-Nordstr\"om black hole along the lines of Price's \cite{Price:1971fb} analysis.  
 
In this note we revisit this problem. While the previous authors \cite{Blaksley:2007ak, Aretakis:2011ha, Lucietti:2012xr, Ori, Sela} have analysed the problem in the time domain, we analyse the problem in the frequency domain. Our analysis brings a different perspective. We reproduce and extend some of the previous results.  Using the Couch-Torrence symmetry, an initial data with regular behaviour across the horizon on the $v=0$ surface can be mapped to an initial data on the $u=0$ surface. Analysing this inverted initial data,
Sela \cite{Sela} has argued that there is a contribution to the late time tail in an extreme Reissner-Nordstr\"om background that is \emph{not} due to the curvature of the spacetime. This contribution  can be obtained from a flat space analysis of the inverted initial data. We first reproduce these results, including the exact coefficients, using rather simple Fourier methods.


Application of the  frequency domain Green's function technique requires knowing initial data on a $t=\mbox{const}$ surface. 
Unfortunately, obtaining a precise relationship between  characteristic initial data specified on $u=0$ and $v=0$ null surfaces and initial data specified on a $t=\mbox{const}$ Cauchy surface is a difficult problem. However, to the extent the above mentioned flat space analysis is valid, it can be easily done. We use solution of flat space wave equation to obtain the correct fall off on $t=0$ surface near spatial infinity.  To this we add a sub-leading term (slower fall-off)  proportional to the ``initial static moment'' and compute its contribution to the  late time tail. This contribution arises due to backscattering from the weakly curved asymptotic region of the spacetime. 

Along the way, using the Couch-Torrence \cite{CT} symmetry we also relate higher multipole Aretakis and Newman-Penrose constants \cite{Newman:1968uj} for massless scalar in extreme Reissner-Nordstr\"om black hole background.  
 
 The rest of the paper is organised as follows. In section \ref{prelims} we review various interesting features that this problem has, namely, the Couch-Torrence symmetry and the construction of Aretakis and Newman-Penrose constants.
 In section \ref{late_time} we analyse the late time dynamics massless scalar in the frequency domain. We end with a brief summary and a discussion of open problems in section \ref{disc}. 

 
\section{Massless scalar in 4d extreme Reissner-Nordstr\"om (ERN) spacetime}
\label{prelims}
Massless scalar wave eqaution in 4d extreme Reissner-Nordstr\"om spacetime has a number of rich features. In this section we review some of these features. Along the way, we relate higher multipole Aretakis and Newman-Penrose constants.

\subsection{Couch-Torrence discrete conformal isometry}
The extreme Reissner-Nordstr\"om solution has a discrete conformal isometry \cite{CT}. A similar discrete isometry also exist for extreme D1-D5 string and for extreme D3 brane; see comments and references in \cite{Aharony:1999ti} and for recent discussions see \cite{Chow:2017hqe, Godazgar:2017igz}. We will use this symmetry in an important way in later sections, so we start with a brief review of this symmetry following \cite{Bizon:2012we, Lucietti:2012xr}. 
In static coordinates the extreme Reissner-Nordstr\"om metric takes the form,
\be
ds^2 = - \left( 1 - \frac{M}{r}\right)^2 dt^2 + \left( 1 - \frac{M}{r}\right)^{-2} dr^2 + r^2 d\Omega^2,
\ee
where $r$ is the area radial coordinate and $d\Omega^2$ is the line element of the unit 2-sphere. The Couch-Torrence symmetry is 
\be
\mathcal{T}: (t, r, \theta, \varphi) \to \left(t, M + \frac{M^2}{r-M}, \theta, \varphi \right). \label{CT}
\ee
It has number of interesting properties. 
It squares to identity
$
\mathcal{T}^2 = 1.
$
Its pull-back  on the Reissner-Nordstr\"om metric acts by a conformal transformation
\be
\mathcal{T}_*(g) =  \Omega^2 g, \qquad \mbox{where} \qquad \Omega = \frac{M}{r-M}.
\ee
On tortoise coordinate $r_*$ defined by
$
\frac{dr_*}{dr} = \left( 1 - \frac{M}{r}\right)^2,
$ that is, 
\be
r_* = r - M + 2M \log \left( \frac{|r-M|}{M} \right) - \frac{M^2}{r-M},
\ee
it acts as 
$
\mathcal{T}: r_* \to - r_*.
$
This last property implies that it interchanges the  ingoing and the outgoing Eddington-Finkelstein coordinates:
\begin{align}
\mbox{ingoing:}  \quad v&=t+r_*, & \mbox{outgoing:} \quad u&=t-r_*,&
\mathcal{T} :&  \quad u \to v. &
\end{align}

Since Ricci scalar of the extreme Reissner-Nordstr\"om metric vanishes, the conformally covariant operator is simply the box operator:
\be
L_g = \Box_g - \frac{1}{6} R = \Box_g.
\ee
Recall that under conformal transformation $\tilde g_{ab} = \omega^2 g_{ab}$, (see e.g., Wald's Appendix D, discussion around equation (D.13) \cite{Wald}),
\be
 L_{\omega^2 g} (\omega^{-1} \Phi) = \omega^{-3} L_g (\Phi) = \omega^{-3} \Box_g\Phi.
\ee
Moreover, from tensor transformation properties it follows that
\be
L_{\mathcal{T}_*(g)} (\mathcal{T}_*(\Phi))= \mathcal{T}_*(L_g (\Phi)).
\ee
Combining the two in the following way,
it follows that if $\Box_g   \Phi = 0,$ then 
\be
0 =  \Box_g   \Phi = \mathcal{T}_*(L_g (\Phi))
= L_{\mathcal{T}_*(g)} (\mathcal{T}_*(\Phi)) = L_{\Omega^2 g} (\Omega^{-1} \Omega (\mathcal{T}_*(\Phi)) = \Omega^{-3} \Box_g (\Omega \mathcal{T}_* (\Phi)).
\ee
That is,  if $\Phi$ is a solution then,  
\be
\tilde \Phi = \Omega\mathcal{T}_*(\Phi) \label{scalar_map} 
\ee
is also a solution. We will use mapping \eqref{scalar_map} to map solutions near the horizon to solutions near future null infinity and vice versa. 

\subsection{Aretakis constants, Newman-Penrose constants, and initial static moments}

We now briefly review the construction of Aretakis and Newman-Penrose  constants in an extreme Reissner-Nordstr\"om background, and relate them via the 
Couch-Torrence symmetry \eqref{scalar_map}. 

Previous studies have related Aretakis and Newman-Penrose constants for $l=0$ modes \cite{Bizon:2012we, Lucietti:2012xr, Godazgar:2017igz}. To the best of our knowledge, details for the $l\neq0$ have not been written out. In this subsection we write out those details explicitly.

In ingoing Eddington-Finkelstein coordinates the extreme Reissner-Nordstr\"om metric is
\be
ds^2 = - \left( 1- \frac{M}{r} \right)^2 dv^2 + 2 dv dr + r^2 d\Omega^2.
\ee
Expanding the scalar in spherical harmonics in ingoing Eddington-Finkelstein coordinates as
\be
\Phi(v,r,\theta, \varphi) = \sum_{lm} \phi_{l}(v,r) Y_{lm}(\theta, \varphi),
\ee
we get equations for the mode functions $\phi_{l}(v,r)$
\be
2 r \partial_v \partial_r (r \phi_{l}) + \partial_r ((r-M)^2 \partial_r \phi_{l}) - l(l+1) \phi_{l}= 0.
\ee
Applying $\partial_r^l$ on this equation we see that 
\be
A_l [\psi] = \partial_r^l [r \partial_r(r \phi_{l})] \bigg{|}_{r=M}
\ee
is conserved, i.e., independent of $v$ along the horizon. These constants are called Aretakis constants.
For a solution of the wave equation of the form near the horizon,
\be
\phi_{l} (v, r) = \frac{1}{r}\sum_{k=0}^{\infty}c_k(v) \left(\frac{r}{M}-1\right)^k \label{near_horizon}
\ee
Aretakis constants are \cite{Ori, Sela}
\be
A_l = \frac{ (l+1)!}{M^l} \left( c_{l+1}  + \frac{l}{l+1}c_{l}\right). \label{Aretakis}
\ee
Note the factor of $\frac{1}{r}$ in equation \eqref{near_horizon}.

In outgoing Eddington-Finkelstein coordinates the extreme Reissner-Nordstr\"om metric is
\be
ds^2 = - \left( 1- \frac{M}{r} \right)^2 du^2 - 2 du dr + r^2 d\Omega^2.
\ee
Expanding the scalar in spherical harmonics in these coordinates as
\be
\Phi(u,r,\theta, \varphi) = \sum_{lm} \phi_{l}(u,r) Y_{lm}(\theta, \varphi),
\ee
we get equations for the mode functions
\be
-2 r \partial_u \partial_r (r \phi_{l}) + \partial_r ((r-M)^2 \partial_r \phi_{l}) - l(l+1) \phi_{l}= 0. \label{mode_eq}
\ee

We now illustrate a construction of Newman-Penrose (NP) constant for the special case of $l=2$. An analysis can indeed be written for general value of $l$, but it requires a complicated combinatorics, which is not illuminating. Later we will write a general expression. Setting $l=2$ for now, consider the solution of wave equation near infinity of the form,
\be
\phi_{2}  (u,r)
= \frac{1}{r}\sum_{k=0}^{\infty}  d_k(u) \left( \frac{R}{r} \right)^{k}, \label{inverse_power}
\ee
where $R$ is an arbitrary scale. 
Inserting this expansion in equation \eqref{mode_eq} and looking at inverse powers of $r$ we find successive equations, starting at $1/r$,
\bea
E_1 &\equiv& -3 d_0 +  R \dot d_1 = 0, \\
E_2 &\equiv&  - M d_0 - 2 R d_1 + 2 R^2 \dot d_2 = 0, \\
E_3 &\equiv&  M^2  d_0 - 4 M R d_1 + 3 R^3 \dot d_3 =0,
\eea
where over-dots denote $u$-derivatives.
The combination
$M^2 E_1 -2 M E_2 + E_3 
$
implies a conservation of 
\be
N_2 :=  d_3 -  \frac{4}{3} \left(\frac{M}{R} \right) d_2 +  \frac{1}{3}\left(\frac{M}{R} \right)^2 d_1, \label{N2}
\ee
at null infinity, i.e., $\partial_u N_2 =0$. This is an example of Newman-Penrose constants. 


How is this constant related to Aretakis constants?  Recall that applying the mapping \eqref{scalar_map} we can construct a solution near null infinity from a given solution near the horizon. Let us apply this mapping on the solution of the form
 \eqref{near_horizon} to get
 \bea
 \phi_{2} &=& \frac{M}{r-M} \left(M + \frac{M^2}{r-M}\right)^{-1} \left( c_0(u) + c_1(u) \left(\frac{M}{r-M}\right) + c_2 (u) \left(\frac{M}{r-M} \right)^2 + \ldots \right) \\ 
 &=& \frac{1}{r} \left( c_0 + c_1 \frac{M}{r}  + (c_1 + c_2) \left(\frac{M}{r}\right)^2 + 
 (c_1+ 2 c_2  + c_3) \left(\frac{M}{r}\right)^3+
 \ldots \right)
 \eea
Expanding this solution in inverse powers of $r$, we find coefficients in \eqref{inverse_power} to be
\begin{align}
d_1 &=\left( \frac{M}{R}\right) c_1,&
d_2 &= \left( \frac{M}{R}\right)^2 (c_1 + c_2),&
d_3 &= \left( \frac{M}{R}\right)^3 (c_1 +2 c_2 + c_3).
\end{align}
Substituting this in the  Newman-Penrose constant $N_2$, cf.~\eqref{N2}, we get
\be
N_2 =  \left(\frac{M}{R}\right)^3 \left(c_3 + \frac{2}{3}c_2 \right),
\ee
which is nothing but a factor times the Aretakis constant $A_2$, cf.~\eqref{Aretakis}. Similar construction can be written for higher Newman-Penrose constants. 

Taking the general solution of the form
\be
\phi_{l} (u,r) = \frac{1}{r}\sum_{k=0}^{\infty} \hat d_k(u) \left( \frac{M }{r -M} \right)^{k}, \label{inverse_power2}
\ee
and analysing the successive equations in inverse powers of $r$ one can  show that 
the combination
\be
N_l = \frac{ (l+1)!}{M^l} \left( \hat d_{l+1}  + \frac{l}{l+1}\hat d_{l}\right),
\ee
is constant. Applying the mapping \eqref{scalar_map} to the form of the solution \eqref{inverse_power2} near null infinity we get a solution near the horizon. The  Aretakis constant of that solution is proportional to $N_l$. Note the factor of $\frac{1}{r}$ in equation \eqref{inverse_power2}.


 The term static moment is often used in the literature \cite{Price:1971fb} to discuss time independent solutions of the wave equations. The usage of this term can be source of confusion, so we define it here what we mean by it. The mode expansion in the static coordinates
\be
\Phi = \frac{1}{r} \sum_{lm} \psi_l(t,r) Y_{lm},
\ee
results in the equations
\be
[\partial_{r_*}^2 - \partial_t^2] \psi_{l} = V_l(r) \psi_l,
\ee
with potential $V_l(r)$
\be
V_l(r) = \left( 1- \frac{M}{r}\right)^2 \left[ \frac{2M}{r^3}\left( 1- \frac{M}{r}\right) + \frac{l(l+1)}{r^2}\right].
\ee
This equation has two time independent solutions
\be
\psi_l = r (r-M)^l, \label{growing}
\ee
and
\be
\psi_l = \frac{r}{(r-M)^{l+1}}. \label{decaying}
\ee
Under the mapping \eqref{scalar_map} one static solution goes to the other, upto normalisation:
\be
\mathcal{T}_* ((r-M)^l) = \frac{M}{r-M} \cdot  \left( \frac{M^2}{r-M}\right)^l  =\frac{M^{2l+1}}{(r-M)^{l+1}}.
\ee




\section{Late time behavior of scalar perturbation in 4d ERN spacetime}
\label{late_time}

We are now in position to analyse the problem of late time decay of scalar field outside the horizon in an extreme Reissner-Nordstr\"om background. 


\subsection{Late time tails for non-compact initial data in flat space} 
We first reproduce some of the key results of Ori \cite{Ori} and Sela \cite{Sela} from a relatively simple Fourier analysis. In the next subsection we look at the contributions due to backscattering from the weakly curved asymptotic region.


To begin with, we  are interested in the characteristic initial value of the field $\psi_l$ specified at two intersecting null rays, $u=0$ and $v=0$, for the equation
\be
[\partial_{r_*}^2 - \partial_t^2] \psi_{l} = V_l(r) \psi_l, \label{wave_eq}
\ee
with potential $V_l(r)$
\be
V_l(r) = \left( 1- \frac{M}{r}\right)^2 \left[ \frac{2M}{r^3}\left( 1- \frac{M}{r}\right) + \frac{l(l+1)}{r^2}\right].
\ee
 The initial data is thus composed of two functions
\begin{align}
\psi^v_l (v) &= \psi_l(u=0, v), &
\psi^u_l (u) &= \psi_l(u, v=0). &
\end{align} 
See figure \ref{initial_data}. Due to linearity of the problem, we can analyse the two functions $\psi^u(u)$ and $\psi^v_l (v)$ separately. More precisely, we can split the characteristic initial value problem in two part: (i) non-vanishing data on $u=0$ surface  $\psi^v_l (v) = \psi_l(u=0, v)$ along with vanishing data on $v=0$ surface $\psi^u_l (u) = 0 $, (ii) non-vanishing data on $v=0$ surface  $\psi^u_l (u) = \psi_l(u, v=0)$ along with vanishing data on $u=0$ surface $\psi^u_l (u) = 0 $.
We can analyse the two parts separately and add the late time behaviour to obtain the final answer. In the following, this is how we will think of the evolution problem. For extreme Reissner-Nordstr\"om this logic has been employed by several authors in the past \cite{Blaksley:2007ak, Ori, Sela}.
 

 \begin{figure}[t]
\centering
\begin{center}
\includegraphics[width=0.5\textwidth]{ERN_initial_data.pdf}
\caption{Initial data for the characteristic initial value problem for scalar field in an extreme Reissner-Nordstr\"om background. The initial data is composed of two functions $\psi^v_l (v)= \psi_l(u=0, v)$ and $\psi^u_l (u) = \psi_l(u, v=0)$. }
\label{initial_data}
\end{center}
\end{figure}


Sela \cite{Sela} considered initial data of ``compact support'' --- an initial data for which the function $\psi^v_l (v)$ vanishes beyond certain value of $v$. In his analysis, the function $\psi_l^u(u)$ is taken to be supported near the event horizon $r=M$. Furthermore, this function is taken to admit Taylor expansion near $r=M$ as,
\be
\psi_l^u(u)=  c_0 + c_1  \left(\frac{r}{M}-1\right)+ c_2  \left(\frac{r}{M}-1\right)^2 + \ldots, \label{v_surface}
\ee
where $r$ is to be thought of as function of $u$ on the $v=0$ surface. 
We also take our initial data of this form for the function $ \psi^u(u)$. 
For the function $\psi^v_l (v)$ we consider a slightly more general behaviour than considered in \cite{Sela}. We allow for the initial static moment, i.e., as $r\to \infty$ the function $\psi^v_l (v)$ is taken to behave as 
\be
\psi^v_l (v) \sim \mu_l \frac{R^l}{r^{l}} + \text{compactly supported data}, \label{u_surface}
\ee
where, now, $r$ is to be thought of as function of $v$ on the $u=0$ surface and $R$ is an arbitrary scale we have introduced.  The coefficient $\mu_{l}$ is called the static moment. 



 For the initial data on the $u=0$ surface, it is believed that the late time tail arises due to backscattering from the weakly curved asymptotic region \cite{Price:1971fb, Klauder:1972je, Gundlach:1993tp}. The tail does not depend on the exact nature of the central object. For compactly supported initial data it goes as $t^{-2l-3}$ and for initial data with initial static moment it goes as $\mu_l t^{-2l-2}$\cite{Price:1971fb}.  

 
For the function $\psi^u(u)$, following \cite{Blaksley:2007ak, Ori, Sela}, we use the Counch-Torrence symmetry to map the problem from near the horizon to near infinity. The problem near infinity can be analysed again using the well developed techniques mentioned above. The map of the initial data is \eqref{CT}:
\be
\psi^v_l(v) \sim c_0 + c_1\left( \frac{M }{r -M} \right) + \ldots + c_l \left( \frac{M }{r -M} \right)^{l}   +
c_{l+1} \left( \frac{M }{r -M} \right)^{l+1} + \ldots
\ee
where now $r$ is to regarded as a function of $v$ along the  $u=0$ surface. Expanding in powers of $r$ results in an expansion 
\be
\psi^v_l(v) \sim \hat c_0  + \hat c_1 \frac{R}{r} +\hat c_2 \frac{R^2}{r^2} + \ldots +  \hat c_l \frac{R^l}{r^l} + \hat c_{l+1}\frac{R^{l+1}}{r^{l+1}} + \ldots.
\ee
where we have inserted an arbitrary scale $R$. In this expansion there is a term that goes as static moment. 
There are terms that are stronger than the static moment and there are also terms weaker than the static moment. The coefficient  $\hat c_{k}$ receives contributions from $c_{k}$, $k \le l$. 


Again using linearity of the problem, the effective problem that we need to analyse is therefore, 
\be
\psi^v_l(v) \sim \hat c_0  + \hat c_1 \frac{R}{r} +\hat c_2 \frac{R^2}{r^2} + \ldots +  (\hat c_l +\mu_l) \frac{R^l}{r^l} + \hat c_{l+1}\frac{R^{l+1}}{r^{l+1}} + \ldots + \text{compactly supported data}. \label{final_data_weak_field}
\ee
with $\psi^u_l(u)=0.$ Generically, if the Aretakis and the Newman-Penrose constants are non-zero, the coefficients of $\frac{1}{r^l}$ and $\frac{1}{r^{l+1}}$ would be non-zero in equation \eqref{final_data_weak_field}.


Sela \cite{Sela}, building upon the work of Barack \cite{Barack}, argued that for the initial data of the form \eqref{final_data_weak_field}, there is a contribution to the late time in an extreme Reissner-Nordstr\"om background that is not due to the curvature of the spacetime. The term
$
\hat c_{l+1}\frac{R^{l+1}}{r^{l+1}} 
$
in the expansion of the data \eqref{final_data_weak_field} results in a leading order tail as it disperses 
in flat space.\footnote{In some sense, this result is the ``Couch-Torrence dual'' of the results of section 4 of \cite{Lucietti:2012xr}, where they have obtained such tails from a purely $\text{AdS}_2 \times \text{S}^2$ analysis. $\text{AdS}_2 \times \text{S}^2$ is conformal to flat space, and the massless scalar wave equation is conformally invariant.} We reproduce Sela's results from a relatively simple Fourier analysis. 



The Fourier transform of the field $\psi_l(t,r)$ 
\be
\psi_l(\omega,r) = \int_{-\infty}^\infty e^{i\omega t} \psi_l(t,r) dt,
\ee  
satisfies the equation
\be
\label{radeq}
\left( -\omega^2 - \partial_r^2 + \frac{l(l+1)}{r^2} \right)\psi_l(\omega, r) = 0.
\ee  
General solution to this equation is 
\be
\label{soln1}
\psi_l(\omega,r) = A(\omega)\sqrt{r}J_{l+1/2}(\omega r) + B(\omega)\sqrt{r}Y_{l+1/2}(\omega r).
\ee
To obtain regular solutions at $ r=0 $ we must set $ B(\omega) = 0 $. Thus, we get 
\be
\label{soln2}
\psi_l(\omega,r) = A(\omega)\sqrt{r}J_{l+1/2}(\omega r) .
\ee
The solution in time domain is simply the inverse Fourier transform,
\be
\label{soln3_1}
\psi_l(t,r) =  \frac{1}{2\pi}\sqrt{r} \int_{-\infty}^{\infty} A(\omega)J_{l+1/2}(\omega r) e^{-i\omega t} d\omega.
\ee
If we know the function $A(\omega)$, we will be able to do this integral and would know the full solution for the field $\psi_l(t,r)$, in particular its late time behaviour. Due to linearity of the problem we can consider each term in the expansion \eqref{final_data_weak_field} separately. 


To determine $A(\omega)$ corresponding to $r^{-k}$ term, we use the fact that the initial data behaves as
$\hat c_{k}\frac{R^{k}}{r^{k}} $
on the $u=0$ surface. We make the ansatz $ A(\omega) = 2 \pi A_0 \ \omega^p $ to get from \eqref{soln3_1}
\begin{eqnarray}
\label{soln4}
 \psi_l(t,r) &=& A_0 \sqrt{r}\int_{-\infty}^{\infty} \omega^p J_{l+1/2}(\omega r) e^{-i\omega t} d\omega \\
 &=& 2 \ A_0 \ \sqrt{r} \ e^{i(p+l+1/2)\frac{\pi}{2}} \int_{0}^{\infty} \omega^p \ J_{l+1/2}( \omega r) \ \cos\left[ (p+l+1/2)\frac{\pi}{2} +\omega t \right] d\omega,
\end{eqnarray}
where we have use the appropriate symmetry property of $J_{l+1/2}( \omega r)$ under $\omega$ to $-\omega$ and have converted the integral along the positive $\omega$ axis.
This last integral can be easily done using the identity (6.699-1) or (6.699-2)  of Gradshteyn and Ryzhik \cite{gradshteyn2007}.

Matching the resulting answer at $u=0$ with 
\be
\label{incond}
\psi_l(u=0, r) = \hat c_{k}\frac{R^{k}}{r^{k}},
\ee
gives 
\be 
p=k-1/2,
\ee 
and fixes the constant $A_0$. Substituting the constant $A_0$ in terms of $\hat c_{k}$ gives a final answer 
\bea
\label{soln9}
\psi_l(t,r) &=& -\frac{\hat{c}_k R^k 2^{k+1}\Gamma(k+1)}{\pi (2l+1)!!}  \sin(k\pi) \ \Gamma(l-k+1)\nn \\ & & \qquad \ r^{l+1} \ t^{-(k+l+1)}  \ F \left( \frac{l+k+2}{2}, \frac{l+k+1}{2}; l+\frac{3}{2}; \frac{r^2}{t^2} 
\right),
\eea
where $F(a,b;c;z)$ is the standard Hypergeometric function. For $ k \leq l $ this expression vanishes due to the $\sin(k\pi)$ factor. However, for $ k \geq l+1 $, the $ \Gamma(l-k+1) $  factor develops a pole that exactly cancels with the zero of the $ \sin $ function and gives a finite result. Setting $k=l+1$, equation \eqref{soln9} simplifies to 
\be
\label{solution_simple}
\psi_l(t,r) = 2\hat{c}_{l+1}R^{l+1} \frac{[(l+1)!]^2}{(2l+2)!} (-1)^{l+1} (4r)^{l+1} \left( 1- \frac{r^2}{t^2} \right)^{-l-1} t^{-2l-2}.
\ee
At timelike infinity, i.e., in the limit $ t \gg r$ \eqref{soln9} becomes
\be
\psi_l(t,r) \sim -\frac{\hat{c}_k R^k 2^{k+1}\Gamma(k+1)}{\pi (2l+1)!!} \  \sin(k\pi) \ \Gamma(l-k+1) \ 
r^{l+1} \ t^{-(k+l+1)}.
\ee
The leading contribution to the late time tail comes from $k=l+1$. We get
\be
\label{solution}
\psi(t,r | t \gg r) \sim 2\hat{c}_{l+1} R^{l+1}(-1)^{l+1} (4r)^{l+1} \frac{[(l+1)!]^2}{ (2l+2)!}  t^{-(2l+2)}.
\ee
This expression precisely matches with Sela's equation (6.18) including the pre-factors. 

We can  use solution \eqref{solution_simple} to obtain the tail behaviour near future null infinity. In order to achieve the limit we must take
$r \to \infty$ together with $u := t-r$ finite, i.e., $u \ll r$. The leading contribution to the tail comes once again from $k=l+1$.
In this limit we find using equation (9.131-2) of Gradshteyn and Ryzhik \cite{gradshteyn2007},
\be
\psi_l(t,r | u \ll r ) \sim 2^{l+2} \hat{c}_{l+1} R^{l+1}(-1)^{l+1} \frac{[(l+1)!]^2}{ (2l+2)!}  u^{-l-1}.
\ee
This expression precisely matches with Sela's equation (6.11) including the pre-factors, provided we relate our $u$ to Sela's retarded time $u_s$: $u_s = u/2$. 

We can also use the solution \eqref{solution_simple} to obtain the fall off  behaviour at spatial infinity:
\be
\psi_l(t=0,r) \sim 2^{3l+4} \hat{c}_{l+1} R^{l+1}  \frac{[(l+1)!]^2}{ (2l+2)!}  r^{-l-1}, \label{spatial}
\ee
together with $\partial_t \psi_{l}(t=0,r) = 0.$


There are other contributions 
to the $t^{-(2l+2)} $ late time tail. They arise due to  backscattering from the curvature of spacetime.  
For initial data \eqref{final_data_weak_field}, these contributions come from terms $r^{-k}$ for $k < l+1$. It is  expected that they should go as 
\be
M^{l+1-k} \hat c_k t^{-(2l+2)}.
\ee
We do not address these contributions in this note. Though, it seems likely that the iterative scheme of \cite{Barack} can be adopted in frequency domain to compute such contributions.

 We look at a related problem for $k=l$ in the next subsection. 

\subsection{Contributions due to asymptotic curvature of spacetime} 

Equation \eqref{spatial} can be interpreted as initial data at $t=0$ surface in the extreme Reissner-Nordstr\"om background, see figure \ref{initial_data2}. We conclude that for an initial data at $t=0$ surface with $r_*^{-l-1}$ decay near spatial infinity, there is a  contribution  to $t^{-2l-2}$ tail at late times.   More precisely,  if 
\be
\psi_l(t=0,r) \sim \mu_{l+1}  R^{l+1} r_*^{-l-1}, \label{spatial}
\ee
then the late time tail is 
\be
\psi(t,r_* | t \gg r_* \gg M) \sim (-1)^{l+1}  2^{-l-1} \mu_{l+1}  R^{l+1}  (r_*)^{l+1} t^{-(2l+2)},
\ee
and
\be
\psi_l(t,r | u \ll r_* ) \sim (-1)^{l+1}  2^{-2 -2 l}  \mu_{l+1}  R^{l+1}   u^{-l-1}.
\ee
From our discussion above, it is clear that such an initial data generically will have a non-zero Newman-Penrose constant, and its 
Couch-Torrence reflection will have a non-zero Aretakis constant. 

 \begin{figure}[t]
\centering
\begin{center}
\includegraphics[width=0.5\textwidth]{ERN_initial_data2.pdf}
\caption{Time symmetric initial data for scalar field in the extreme Reissner-Nordstr\"om background. The initial data consists of a function specified on $t=0$ surface.  $\partial_t\psi_l(t=0,r)$ taken to be zero. }
\label{initial_data2}
\end{center}
\end{figure}



Now we address the question if in addition to \eqref{spatial} there is an $r_*^{-l}$ term present at $t=0$ surface near spatial infinity, then how it  contributes to the late time tail. The contribution arises due to  backscattering from the curvature of spacetime. If the initial data has non-zero static moment such a term would be present. 

 To compute this contributions, fortunately, we do not need to do much.  Since the late time tail arises due to backscattering from the weakly curved asymptotic region \cite{Price:1971fb, Klauder:1972je, Gundlach:1993tp}, the computation is exactly the same as in the Schwarschild background. 
 
 
We very briefly review the Green's function approach to late time tails for Schwarzschild spacetime following \cite{Andersson:1996cm} for compact sources, and supplement it with a discussion for extended source of the type:
\be
\psi_l(t=0,r) \sim c_{l}  R^{l}  r_*^{-l}. \label{spatial2}
\ee
 
The retarded Green's function for wave operator appearing in \eqref{wave_eq} satisfy
\be
\left[\partial_t^2 - \partial_{r_*}^2  + V(r_*)\right] G(r_*, r_*'; t ) = \delta (t) \ \delta (r_*-r_*') 
\ee
with the boundary conditions
\be
G(r_*, r_*'; t ) = 0, \qquad \mbox{for} \qquad t < 0. 
\ee
We are interested in analysing the Green's function in the frequency domain. Therefore,  we do a Fourier transform via
\be
\tilde G(r_*, r_*'; \omega )  = \int_0^\infty \, dt \, G(r_*, r_*'; t )  \, e^{i\omega t}.
\ee
The range of the $r_*$ coordinate for black hole spacetimes is $-\infty$ to $\infty$.  In frequency domain the solutions to the wave equations we are interested in should satisfy the outgoing boundary conditions at infinity,  and ingoing boundary conditions at the horizon. In terms of the $r_*$ coordinate, these become
\begin{align}
&\tilde \psi_l(r_*,\omega) \to e^{i\omega r_*} & & \mbox{as} & & r_* \to \infty,&\\
&\tilde \psi_l(r_*,\omega) \to e^{-i\omega r_*} & & \mbox{as} & &   r_* \to -\infty.&
\end{align}
The Fourier transform of the Green's function $\tilde G(r_*, r_*'; \omega )$ satisfy
\be
\left[ -\omega^2 - \partial_{r_*}^2 + V(r_*) \right] \tilde G(r_*, r_*'; \omega) = 0,
\ee
and is analytic in the upper half plane. Now recall that for a second order ODE with homogeneous boundary conditions,  Green's function can be uniquely constructed simply using two auxiliary functions
$f(r_*, \omega)$ and $g(r_*, \omega)$ where $f(r_*, \omega)$ satisfy the left boundary condition and $g(r_*, \omega)$ satisfy the right boundary condition (see e.g., \cite{MorseFeshbach}).  We adopt normalisations such that
\begin{align}
&g(r_*, \omega)  \to  e^{i\omega r_*}& & \mbox{as} & & r_* \to \infty,& \\
&f(r_*, \omega)  \to e^{-i\omega r_*}& & \mbox{as} & & r_* \to -\infty.&
\end{align}
Then the Green's function is given by
\be
\tilde G(r_*,r_*'; \omega) = \begin{cases} 
\frac{f(r_*, \omega) g(r_*', \omega)}{W(\omega)}, & \quad \text{if } \quad r_* < r_*' \\
\frac{f(r_*', \omega) g(r_*, \omega)}{W(\omega)}, & \quad \text{if } \quad r_* > r_*'.
  \end{cases}
\ee
where $W(\omega)$ is the Wronskian of the two solutions $f(r_*, \omega)$ and $g(r_*, \omega)$:
$
W(\omega) = g \partial_{r_*} f - f \partial_{r_*}g.
$
The Wronskian is independent of $r_*$. The late time tails comes from the branch cut along the negative imaginary axis of the 
Green's function $\tilde G(r_*,r_*'; \omega) $ in the complex $\omega$ place \cite{Ching:1995tj}.

Andersson \cite{Andersson:1996cm} has presented a very clear computation of the branch cut of the Green's function in the low-frequency asymptotic expansion using some results from \cite{Futterman}. Instead of reviewing those details here, we simply write equation (40) of that reference (which has a typo of an over-all minus sign)  \cite{Andersson:1996cm}:
\be
G^C(r_*,r_*',t) = - 2 \pi  i M \sqrt{r_* r_*'} \int_{0}^{-i \infty} \omega \ J_{l+1/2}(\omega r_*) \ J_{l+1/2}(\omega r_*') \ e^{-i \omega t}d\omega.
\label{cut}
\ee
The late time solution using this Green's function is simply \cite{MorseFeshbach, Ching:1995tj}
\be
\label{soln1}
\psi^C_l(r_*,t) = \int_0^{\infty} G^C(r_*,r_*^{\prime},t)  \ \partial_t\psi_0(r_*^{\prime},0) \ dr_*^{\prime} - \int_0^{\infty}  \partial_t G^C(r_*,r_*^{\prime},t) \ \psi_0(r_*^{\prime}), \, 
\ee
where we have implicitly used the fact that the leading contribution only comes from the asymptotic region, and our non-compact initial data has support only in the $r_* \gg M$ asymptotic region.

Next we consider the non-compact source with 
\eqref{spatial2} together with
$ \partial_t\psi_0(r_*,0) = 0 $. With this initial data we get from \eqref{soln1}
\be\label{soln3}
\psi_l(r_*,t) =  2 \pi \mu_l R^l M \sqrt{r_*} \int_0^{ -i \infty}  d\omega \, \omega^2 \ e^{-i \omega t} \ J_{l+1/2}(\omega  r_*) \int_{0}^\infty dr^\prime_* \, {r^\prime_*}^{-l+1/2} \ J_{l+1/2}( \omega r^\prime_*).
\ee
We can evaluate the first integral in equation \eqref{soln3} using identity (6.561-14) of Gradshteyn and Ryzhik \cite{gradshteyn2007} to get
\bea
\psi_l(r_*,t) &=&   \frac{2 \sqrt{2 \pi}}{(2l-1)!!}  \mu_l R^l M \sqrt{r_*} \int_0^{ -i \infty}  d\omega \, \omega^{l+1/2} \ e^{-i \omega t} \ J_{l+1/2}(\omega  r_*).
\eea
Now, to compute the tail at timelike infinity, we approximate $ \omega r_* \ll 1$ to get
\bea
\psi_l (t,r_* | t \gg r_* \gg M)   &\sim&  \frac{4 \mu_l R^l M r_*^{l+1}}{(2l-1)!!(2l+1)!!}  \int_0^{- i \infty} \ \omega^{2l+1} \ e^{-i\omega t} \ d\omega\\
& \sim&  (-1)^{l+1} 4 \mu_l R^l M  \frac{ (2l)!!}{(2l-1)!!} r_*^{l+1} t^{-2l-2}.
\eea
This expression matches with equation  (69) of reference \cite{Leaver:1986gd} and equations IV-1 and IV-2 of reference \cite{Moncrief}. In these papers, computations are done very differently, and in somewhat different contexts. 

To compute the tail near null infinity, we approximate $ \omega r_* \gg 1$. A similar calculation then  gives 
\be
 \psi_l (t,r_*) \sim (-1)^{l+1} 2 \mu_l R^l M \frac{ l!}{(2l-1)!!}  u^{-l-1}.
\ee
This expression also matches with  equation  (68) of reference \cite{Leaver:1986gd}.


\section{Discussion}
\label{disc}

In this note we have revisited the study of the leading order late-time decay tails of massless scalar perturbations outside an extreme Reissner-Nordstr\"om black hole. While previous studies have analysed this problem in the time domain, we analyse the problem in the frequency domain.   


A detailed analysis of this problem was reported by  Sela \cite{Sela} (for electromagnetic and gravitational perturbations see \cite{Sela:2016}). His analysis is quite involved. 
The merit of our work lies in its simplicity. 
We are able to obtain most of the key results of Sela's analysis, including all pre-factors, using rather straightforward Fourier methods.




We find that initial perturbations with generic regular behaviour across the horizon decays at late times decays as $t^{-2l-2}$ near timelike infinity $(t \gg r_*)$. It decays as $u^{-l-1}$ near null infinity. The inversion map \eqref{scalar_map} maps the decay behaviour near future null infinity to the decay behaviour $v^{-l-1}$ near the horizon. 


For initial data \eqref{final_data_weak_field} of the form, there are other contributions 
to the $t^{-(2l+2)} $ late time tail. They arise due to  backscattering from the curvature of spacetime, from  terms $r^{-k}$ for $k < l+1$ in 
\eqref{final_data_weak_field}. These contributions should go as 
\be
\mbox{(pre-factor)} \ M^{l+1-k} \hat c_k t^{-(2l+2)}.
\ee
We have not addressed these contributions in this note. Though, it seems likely that the iterative scheme of \cite{Barack} can be adopted in the frequency domain to compute such contributions.


We also considered initial data at $t=0$ surface. We showed that a term proportional to the static moment also contributes to the strength of the tail. From the Couch-Torrence symmetry, it follows that such a term
if present near the bifurcation surface, will also contributes to the $v^{-l-1}$ tail near the horizon. 




In section \ref{prelims} using the Couch-Torrence symmetry we also related higher multipole Aretakis and Newman-Penrose constants for massless scalar in extreme Reissner-Nordstr\"om black hole background.  


All of our  analysis is only valid in the asymptotic regions, either near infinity or near the horizon $|r_*| \gg M$. 
We have not attempted to compute the correct radial dependence of the coefficient of the tail in full generality. From general results in the literature we do expect the correct radial dependence of the tail at time-like infinfity to be the static solution to the extreme Reissner-Nordstr\"om potential \cite{Price:1971fb, Ori, Sela}, cf.~\eqref{decaying}
\be
\frac{r}{M} \left(\frac{r}{M}-1\right)^{-l-1}
\ee
with a constant pre-factor.  We expect the constant pre-factor gets contributions from the Newman-Penrose as well as from the  Aretakis constant. This has been observed in numerical simulations \cite{Lucietti:2012xr}. The proportionality to the Aretakis constant is briefly discussed in \cite{Ori, Sela}, but details have not been presented. 

Together with the suggestion of references \cite{Winicour, Lucietti:2012xr} that ``initial static moments'' are more precisely thought of as initial data with non-zero Newman-Penrose constants, it is natural to conjecture that the total tail coefficient will be proportional to the sum of (appropriately normalised) Aretakis and Newman-Penrose constants. It will be interesting to understand this circle of ideas better in the future. 

It will also be interesting to reproduce the late time tails from a microscopic CFT analysis for extremal black holes for which the CFT dual descriptions are known. 

In a recent paper \cite{Camps}, authors studied moduli space scattering of two extreme Reissner-Nordstr\"om black holes. They obtained the asymptotic gravitational radiation field wave-form at ``moderately'' late times, when the two black holes have not merged. They found that the asymptotic radiation field exhibits a quadrupolar late time tail of $t^{-2l-2}$ for $l=2$. It will be interesting to understand how their results relate to our analysis. 

We hope to report on some of these problems in the future. 



\subsection*{Acknowledgements} We thank Sayan Kar for discussions, and Shahar Hod and Orr Sela  for email correspondence.   Preliminary versions of these results were reported by AV at a ``Quantum Spacetime Seminar'' at TIFR, ``Indian Strings Meeting 2016'' at IISER Pune, and  ``IAGRG-2017'' conference at  IIT Guwahati. We thank Shiraz Minwalla and Julian Sonner for useful feedback. 
AV also thanks BHU Varanasi for warm hospitality towards the final stages of this work.
PP thanks CMI Chennai for warm hospitality towards the final stages of this work. 
The work of AV is supported in part by the DST-Max Planck Partner Group ``Quantum Black Holes'' between CMI, Chennai and AEI, Golm.
The work of SB is supported by the IIIT Allahabad seed grant on the project titled "Probing the interior of AdS Black Holes", and Department of Science and Technology, Government of India under the Early Career Research Award file no ECR/2017/002124. BC acknowledges hospitality at Kavli Asian Winter School ICTS/Prog-KAWS2018/01 and ICTP Trieste while this work was in progress. 

 
 
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\end{thebibliography}




\end{document}