# A toolbox for black hole scattering

Nava Gaddam and  Nico Groenenboom

###### Abstract

Hawking’s free field theory is expected to break down after Page time. In
previous work, we have shown that a primary dynamical reason for this
breakdown is the dominance of graviton fluctuations of the horizon that
mediate scattering processes. In this article, we present a toolbox for such
‘black hole scattering’ computations. The toolbox comprises of explicit
expressions for the graviton propagator near the horizon in an angular
momentum basis for all angular momentum modes of either parity, the leading
interaction rules, and most importantly a rewriting of the theory in terms of
a scalar theory with an interesting four-vertex. We demonstrate how this
rewriting drastically reduces the number of diagrams to be calculated in the
original formulation. Finally and perhaps most remarkably, we observe that the
black hole entropy appears to emerge from the multiplicity of external legs of
the dominant $2\rightarrow 2N$ amplitudes in this theory.

##  1 Introduction

Hawking argued that black holes violate predictability in the laws of physics
when free scalar fields propagate on a black hole background [1, 2]. A natural
question to ask is whether this conclusion would change if the scalar fields
were to interact with the inevitably (quantum) fluctuating black hole
spacetime. To answer such a question, it is imperative that calculations in
the presence of a fluctuating black hole horizon are doable. We have shown in
recent years that several relevant calculations can be done [3, 4, 5] that
teach us interesting (near-horizon) physics. It is the aim of the present
article to summarise these developments, extend them, and present a
comprehensive toolbox for further calculations.

The calculations of interest are those of scattering amplitudes. The
predominant question of interest is, can we calculate the amplitude of $N$
particles emerging from a Schwarzschild black hole when $M$ particles are
thrown into it? The interactions governing the scattering processes are
mediated by graviton fluctuations of the black hole horizon. To leading order,
these gravitons couple to the stress tensor of the minimally coupled scalar
field resulting in three-vertices of interest. Extension to higher order
interactions is both desirable and straightforward with the tools we develop,
although explicit calculations are likely to be hard to do in complete
generality.

Several questions may arise when scattering amplitudes in curved spacetimes
are considered. We list some of them that are rather natural in the context of
black holes here and provide conceptual answers:

  * •

Black holes break translational invariance. How could we then possibly compute
Feynman diagrams in momentum space? While four-momenta are indeed hard to
define, exploiting the spherical symmetry of the background, they can be
traded for two-momenta (along the longitudinal directions) and angular
momentum labels (after integrating the sphere out). Of course, the
longitudinal part of the metric is not flat, but its conformal flatness allows
us to trade the curvature for potentials. And the longitudinal momenta are
defined after this Weyl rescaling, without loss of generality.

  * •

How are kinematic variables defined if the spacetime is curved? After the Weyl
rescaling described above to exploit the conformal flatness of the
longitudinal part of the Schwarzschild metric, kinematic variables are defined
with the help of the longitudinal two-momenta arising from the flat part of
the metric. The price we pay is of course that the curvature effects have to
be traded for potentials.

  * •

The black hole contains a large number of degrees of freedom. But perturbation
theory is likely to breakdown when scattering processes involving many
external legs are considered. So, are we likely to learn much at all from
black hole scattering? While perturbation theory with many external legs does
breakdown in the vacuum [6], in the presence of a horizon, the size of the
black hole results in an emergent coupling constant
$\gamma\coloneqq\kappa/R=\sqrt{8\pi G}/R$, where $R$ is the Schwarzschild
radius [7, 3, 4]. For large black holes, the range of validity of perturbation
theory in the black hole background is much wider [3, 4], and considering many
external particles in fact results in the emergence of important time scales
(like Page time) [5].

  * •

Isn’t scattering on a black hole background governed by the Dray-’t Hooft
shockwave? Just as in its flat space counterpart [8, 9], the on-shell part of
the elastic $2\rightarrow 2$ amplitude contains the Dray-’t Hooft shockwave
[10, 11, 12, 13, 14, 15]. However, unlike in the case of flat space, there are
off-shell graviton perturbations that do not satisfy the classical equations
of motion but do contribute to the eikonal amplitudes [4]. Perhaps more
interestingly, particle production in amplitudes has no known counterpart as a
classical solution to Einstein’s equations. Therefore it is fair to say that
these scattering amplitudes capture far more information than is contained in
the shockwave.

  * •

Isn’t gravity (perturbatively) non-renormalisable? How far can computing
amplitudes take us? While it is true that ultraviolet physics will kick in at
some stage, one of the remarkable features of black hole scattering is that as
long as centre of mass energies of scattering satisfy $E\gg\gamma M_{Pl}$,
ultraviolet effects are heavily suppressed. For large black holes, $\gamma$ is
very small and this relation has a wide range of validity. In fact, we expect
the scattering to be of high energy near the horizon due to the large
redshift. Therefore, there is a lot more to be understood in the infrared than
one might have expected. These are all effects that can be explicitly
calculated as deviations from Hawking’s free field theory.

With those questions answered, we now list a few results of black hole
scattering that may be considered as significant successes.

##### Successes of black hole scattering to date

  * •

Black hole eikonal: As alluded to above, the emergence of a new eikonal phase
near the horizon that is different from, but somewhat analogous to, the flat
space eikonal is noteworthy. In flat space, the eikonal phases emerges at
large impact parameters when scattering energies are trans-Planckian $E\gg
M_{Pl}$. In the black hole eikonal, however, the range of validity is much
wider owing to a suppression by the size of the black hole: $E\gg\gamma
M_{Pl}$.

  * •

Black hole scattering $>$ shockwave: The Aichelburg-Sexl shockwave [16] in
flat space is equivalent to the flat space eikonal ladder which contains no
off-shell fluctuations [17]. The black hole eikonal ladder, on the other hand,
contains the Dray-’t Hooft shockwave [18] but also contains off-shell virtual
graviton fluctuations that carry more information than the on-shell classical
solution [4]. While the significance of this difference is not entirely
understood, it is certainly evident that black hole scattering carries more
information than the shockwave. Moreover, inelastic processes in black hole
scattering [5] have no known shockwave counterparts.

  * •

Emergence of time scales: Black holes have interesting time scales associated
with them, scrambling time and Page time are universal examples. One of the
significant successes of the black hole scattering program is the natural
emergence of these time scales from explicit calculations of amplitudes. Given
the scattering matrix associated with a certain process, there is a canonical
time scale (the Eisenbud-Wigner time delay [19, 20]) that estimates the time
that ingoing particles ‘spent’ in the scattering region. For elastic
$2\rightarrow 2$ black hole scattering, this time delay turns out to be
scrambling time [5]. Particle production on the other hand, via a computation
of $2\rightarrow 2N$ scattering, requires Page time [5]. This statement in
particular implies that Hawking’s free field theory on a fixed background is
explicitly invalid after Page time because interactions mediate information
return.

  * •

Emergence of black hole entropy: In a scattering matrix approach to quantum
black hole physics, it has not been clear how a coarse-grained entropy may
emerge. In the present article, we observe that the black hole entropy appears
to naturally arise from the multiplicity of the external legs of the dominant
$2\rightarrow 2N$ scattering.

  * •

Observational consequences: The inspiral phase of the observed gravitational
wave signals from compact binary mergers are well approximated by scattering
in flat space at large impact parameters [21, 22, 23].111See [24, 25] for
recent reviews. On the other hand, black hole scattering implies that modes
that are ingoing near the horizon create virtual gravitons that then release
outgoing modes that leak to the outside of the classical gravitation potential
in the form of gravitational echoes [26]. These modify the familiar quasi-
normal mode spectrum and provide a model independent prediction for echoes
from black holes. Gravitational echoes were previously thought to emerge from
exotic compact objects (ECOs) [27]. However, gravitational echoes from black
hole scattering will reflect on the black hole memory effect. In a broader
sense, it is fair to expect that black hole scattering will contribute to the
gravitational wave signals of the post-merger phase of the compact binary
mergers observed in nature. An estimation of the significance of this
contribution would go a long way towards an understanding of observational
signatures of black holes in nature. Finally, in analogy to the Post-Newtonian
and Post-Minkowskian expansions associated with the inspiral phase, a natural
Post-BH expansion possibly governs the post-merger phase that improves upon
the classical ringdown prediction.

The success of black hole scattering is not all-encompassing. There certainly
are shortcomings, some of which may be more important and/or harder to rectify
than others.

##### Some shortcomings so far

  * •

In its strict sense, it is not clear how one must properly define the S-matrix
in the presence of a black hole. In particular, how Hawking radiation may be
incorporated into the asymptotic out states is an interesting concern. We
essentially turn a blind eye to this issue, and compute scattering amplitudes
as if asymptotic states were well-defined. A consequence of this is that while
the corrections to Hawking’s picture are directly calculable in this approach,
incorporating Hawking’s leading order Bogoliubov transformations require
additional care.

  * •

An analytic expression for the graviton propagator in a black hole background
is unavailable. Therefore, we resort to an approximate version in the near-
horizon region. For several observables, this sufficiently captures the
dominant physics at low energies. Nevertheless, it would be of particular
interest to understand the limitations of this approximation with care.

  * •

Analyses of scattering processes have been limited to the leading three-vertex
arising from the graviton coupled to the minimally coupled scalar stress-
tensor. Vertex corrections, classical corrections from graviton self-
interactions, higher order interactions, other gauge and matter fields have
all yet to be studied.

  * •

Another obvious shortcoming of the black hole scattering program is a
technical one. All calculations so far have only been done in the presence of
a large black hole. Calculations in the presence of a time dependent
background are analytically intractable. While the scale at which these
effects become is not entirely clear, it is certainly of interest to
exhaustively understand what questions necessitate these effects to be
studied.

##### What is new in the present article?

In [3, 4], we have derived the graviton propagator for the even parity modes
in the Regge-Wheeler gauge [28] in the near horizon region and computed
elastic $2\rightarrow 2$ scattering amplitudes in a black hole eikonal limit
alluded to above. It is known that there are additional gauge redundancies in
the low angular momentum modes [3, 4, 29, 30]. Fixing this additional gauge
redundancy in the even parity mode of the s-wave, we have computed
$2\rightarrow 2N$ tree-level amplitudes and an infinite class of loop
corrections thereof in [5]. In the present article, we arrive at the following
results:

  * •

We derive the graviton propagator in the near horizon region for all modes of
the graviton (even and odd parity and all angular momentum modes with both
even and odd parity). This is done in Section 2.

  * •

Considering three-vertex interactions arising from the graviton coupling to
the stress tensor of a minimally coupled scalar field, we show that certain
modes of the graviton do not contribute to the physical S-matrix at this
level. This is presented in Section 3.

  * •

In Section 4, we show that the non-interacting mode can be decoupled from the
theory and integrated out. Consequently, we arrive a rewriting of the theory
in terms of an effective scalar theory with a four-vertex (without loss of
generality) that captures $2\rightarrow 2$ amplitudes involving the original
graviton exchange.

  * •

In Section 5, we show why the new rewriting of the theory in terms of a scalar
four-vertex is particularly efficient for computations. For instance, 972
diagrams that contribute to a three-loop $2\rightarrow 4$ amplitude are
captured by all of only four topologically distinct diagrams in the rewritten
theory.

  * •

Finally, in Section 6, we observe that the entropy of the black hole naturally
seems to arise from the multiplicity of the dominant $2\rightarrow 2N$
scattering.

##  2 Graviton perturbations in partial waves

The quadratic action for graviton perturbations (say $h_{\mu\nu}$) about a
spherically symmetric vacuum solution (say $g_{\mu\nu}$) to Einstein’s
equations, with $\bar{g}_{\mu\nu}=g_{\mu\nu}+\kappa h_{\mu\nu}$ and
$h=g^{\mu\nu}h_{\mu\nu}$, can be written as [31, 4]

$\displaystyle S~{}$
$\displaystyle=~{}-\dfrac{1}{2}\int\text{d}^{4}x\sqrt{-g}h^{\mu\nu}\left[\dfrac{1}{2}\left(2\nabla^{\rho}\nabla_{(\mu}h_{\nu)\rho}-\Box
h_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}h\right)-~{}\dfrac{1}{2}g_{\mu\nu}\biggr{(}\nabla^{\rho}\nabla^{\sigma}h_{\rho\sigma}-\Box
h\biggr{)}\right]$
$\displaystyle=~{}-\dfrac{1}{4}\int\text{d}^{4}x\sqrt{-g}\left(h^{\mu\nu}-\dfrac{1}{2}g^{\mu\nu}h\right)\left(2\nabla^{\rho}\nabla_{(\mu}h_{\nu)\rho}-\Box
h_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}h\right)+\dfrac{\kappa}{2}\int\text{d}^{4}x\text{
}h^{\mu\nu}T_{\mu\nu}\,,$ (2.1)

where we wrote the action in terms of the first order variation of the
Einstein tensor (square parenthesis in the first line) and in terms of the
first order variation of the Ricci tensor (the second parentheses in the
second equality) in the second line. The background metric of interest is

$\displaystyle\text{d}s^{2}~{}=~{}-2A(r)\text{d}x\text{d}y+r^{2}\left(x,y\right)\text{d}\Omega^{2}_{2}\quad\text{with}\quad
A(r)~{}=~{}\dfrac{R}{r}e^{1-\tfrac{r}{R}}\,.$ (2.2)

We discuss the choice of coordinates, and the non-vanishing Riemann tensor
components for this Schwarzschild metric in Appendix A. To exploit the
spherical symmetry of the background, we now expand the graviton in spherical
harmonics as in [28]:

$h_{\mu\nu}~{}=~{}\sum\limits_{\ell,m}h^{-}_{\ell
m,\mu\nu}+\sum\limits_{\ell,m}h^{+}_{\ell m,\mu\nu}\,,$ (2.3)

where $h^{-}_{\ell m,\mu\nu}$ are the so-called odd graviton modes and
$h^{+}_{\ell m,\mu\nu}$ the even modes. It can be checked that these even and
odd modes decouple in the quadratic action allowing us to consider them
separately [28, 4].

####  2.0.1 The odd harmonics

The odd harmonics are parametrised as

$h^{-}_{\ell
m,\mu\nu}=\begin{pmatrix}0&0&-h^{-}_{x}\csc\theta\partial_{\phi}&h^{-}_{x}\sin\theta\partial_{\theta}\\\
&0&-h^{-}_{y}\csc\theta\partial_{\phi}&h^{-}_{y}\sin\theta\partial_{\theta}\\\
&&h_{\Omega}\csc\theta\left(\partial_{\theta}\partial_{\phi}-\cot\theta\partial_{\phi}\right)&\tfrac{1}{2}h_{\Omega}\left(\csc\theta\partial_{\phi}^{2}+\cos\theta\partial_{\theta}-\sin\theta\partial_{\theta}^{2}\right)\\\
&&&-h_{\Omega}\sin\theta\left(\partial_{\theta}\partial_{\phi}-\cot\theta\partial_{\phi}\right)\end{pmatrix}Y_{\ell}^{m}.$
(2.4)

Using index notation, this can be written as

$\displaystyle h^{-}_{ab}$ $\displaystyle=0$ $\displaystyle
h^{-}_{aA}=h^{-}_{Aa}$ $\displaystyle=h_{a}\eta_{A}$ (2.5) $\displaystyle
h^{-}_{AB}$ $\displaystyle=h_{\Omega}\hat{\nabla}_{(A}\eta_{B)}\,,$

where the indices $(a,b)$ run over the longitudinal coordinates and $(A,B)$
take values in the transverse spherical directions. Of course, as this is a
partial wave expansion, all functions are only functions of the longitudinal
coordinates with the angular dependence explicitly extracted out. Furthermore,
we defined $\hat{\nabla}_{A}$ to be the covariant derivative on the unit two-
sphere and pseudo-tensor

$\displaystyle\eta_{A}~{}=~{}-{\epsilon_{A}}^{B}\partial_{B}Y_{\ell}^{m}$
(2.6)

on the two-sphere that is characteristic for the odd modes. As can be readily
checked, $h_{\Omega}$ falls away for the $\ell=0,1$ modes and in what follows,
we will remove it by gauge choice (for the $\ell>1$ modes), leaving us with
only $h^{-}_{aA}=h_{a}\eta_{A}$ to consider. Therefore, the quadratic action
for the odd harmonics $h_{a}^{-}$ can be written as

$\displaystyle
S~{}=~{}\dfrac{\lambda-1}{2}\sum\limits_{\ell,m}\int\text{d}^{2}x~{}h^{a}\left(\eta_{ab}\left(\partial^{2}-\mu^{2}\left(\lambda-\dfrac{3}{2}\right)\right)-\partial_{a}\partial_{b}+3\mu^{2}x_{[a}\partial_{b]}\right)h^{b}\,,$
(2.7)

with $\mu=R^{-1}$ and $\lambda=\ell^{2}+\ell+1$. Several manipulations had to
be performed to arrive at this action, all detailed in Appendix C. We first
plug in the odd graviton $h^{-}_{aA}=h_{a}\eta_{A}$ into the quadratic action
(2), then write the covariant derivatives out before integrating over the
sphere to reduce the theory to two dimensions. Then, we make a field
redefinition $h_{a}\rightarrow\sqrt{A\left(x,y\right)}\,\mathfrak{h}_{a}$ to
exploit the conformal flatness of the two dimensional metric
$g_{ab}=A\left(x,y\right)\eta_{ab}$ which allows us to trade all covariant
derivatives for partial derivatives and potentials. Furthermore, since the
quadratic operator is still not easily invertible notwithstanding these
manipulations, we make a near-horizon approximation where $r\sim R$ implying
$x,y\ll R$ and $A\left(x,y\right)\sim 1$. This yields the above action. As we
will show in Section 2.2, this action is reliable only for the multipole modes
$\ell\geq 2$. For the dipole $\ell=1$, there is additional gauge redundancy
that needs care.

Finally, it is worth noting that the action is proportional to $\lambda-1$ and
therefore vanishing for the $\ell=0$ or $\lambda=1$ mode, since there is no
odd harmonic in the monopole mode. Moreover, the action grows rapidly for
larger multipole moments rendering them to be subleading.

####  2.0.2 The even harmonics

The even harmonics are parametrised by

$h^{+}_{\ell
m,\mu\nu}=\begin{pmatrix}H_{xx}&H_{xy}&h^{+}_{x}\partial_{\theta}&h^{+}_{x}\partial_{\phi}\\\
&H_{yy}&h^{+}_{y}\partial_{\theta}&h^{+}_{y}\partial_{\phi}\\\
&&r^{2}(K+G\partial_{\theta}^{2})&r^{2}G(\partial_{\theta}\partial_{\phi}-\cot\theta\partial_{\phi})\\\
&&&r^{2}(K\sin^{2}\theta+G(\partial_{\phi}^{2}+\sin\theta\cos\theta\partial_{\theta}))\end{pmatrix}Y_{\ell}^{m}.$
(2.8)

Using index notation this can be written as

$\displaystyle h^{+}_{ab}~{}$ $\displaystyle=~{}H_{ab}Y_{\ell}^{m}$ (2.9)
$\displaystyle h^{+}_{aA}~{}$
$\displaystyle=~{}h^{+}_{Aa}~{}=~{}h_{a}^{+}\partial_{A}Y_{\ell}^{m}$ (2.10)
$\displaystyle h^{+}_{AB}~{}$
$\displaystyle=~{}Kg_{AB}Y_{\ell}^{m}+G\bigr{(}\partial_{A}\partial_{B}-\Gamma_{AB}^{C}\partial_{C}\bigr{)}Y_{\ell}^{m}\,.$
(2.11)

The $G$ and $h^{+}_{a}$ fields do not contribute to the monopole ($\ell=0$)
and we will remove them by a choice of gauge for the dipole and multipole
modes ($\ell\geq 1$). Just as in the case of the odd harmonics, this allows us
to only consider $h^{+}_{ab}=H_{ab}Y_{\ell}^{m}$ and
$h^{+}_{AB}=Kg_{AB}Y_{\ell}^{m}$ without any loss of generality. Following
steps identical to what we did for the odd harmonics, first integrating out
the two sphere, redefining the fields as $H_{ab}\to\mathfrak{h}_{ab}$ and
$K\to\mathcal{K}$, and making the near horizon approximation, the following
quadratic action for the even harmonics can be derived

$\displaystyle
S~{}=~{}\dfrac{1}{4}\int\text{d}^{2}x\biggr{(}\mathfrak{h}^{ab}\Delta^{-1}_{abcd}\mathfrak{h}^{cd}+\mathfrak{h}^{ab}\Delta^{-1}_{ab}\mathcal{K}+\mathcal{K}\Delta^{-1}_{ab}\mathfrak{h}^{ab}+\mathcal{K}\Delta^{-1}\mathcal{K}\biggr{)}\,,$
(2.12)

with

$\displaystyle\Delta^{-1}~{}$ $\displaystyle=~{}-\partial^{2}+\mu^{2}\,,$
(2.13a) $\displaystyle\Delta^{-1}_{ab}~{}$
$\displaystyle=~{}-\eta_{ab}\left(\partial^{2}-\dfrac{1}{2}\mu^{2}\left(\lambda-1\right)\right)+\partial_{a}\partial_{b}\,,$
(2.13b) $\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{2}\mu^{2}\left(\eta_{ac}x_{[b}\partial_{d]}+\eta_{bd}x_{[a}\partial_{c]}+\eta_{ab}x_{(c}\partial_{d)}-\eta_{cd}x_{(a}\partial_{b)}\right)$
$\displaystyle\qquad+\dfrac{\mu^{2}\left(\lambda+1\right)}{2}\bigr{(}\eta_{ab}\eta_{cd}-\eta_{a(c}\eta_{d)b}\bigr{)}\,.$
(2.13c)

The detailed calculation can be found in [4]. Just as in the case of the odd
modes, the conformal flatness of the longitudinal metric
$g_{ab}=A\left(x,y\right)\eta_{ab}$ has been exploited to write the above
action. Therefore all indices are raised and lowered with the flat metric.

####  2.0.3 Action of gauge transformations on the graviton

In order to understand the gauge fixing we employ, we first write down the
effect of a gauge transformation on the different graviton modes $H_{ab}$,
$h^{+}_{a}$, $h^{-}_{a}$, $G$, $K$, $h_{\Omega}$. These will also come to use
in defining ghosts for each partial wave. Let us define the diffeomorphisms
$\bar{\xi}^{\mu}$ along the longtidunal and transverse directions as

$\displaystyle\bar{\xi}_{a}~{}=~{}\sum_{\ell,m}\xi_{a}Y_{\ell}^{m}\quad\text{and}\quad\bar{\xi}_{A}~{}=~{}\sum_{\ell,m}\xi^{+}\partial_{A}Y_{\ell}^{m}+\sum_{\ell,m}\xi^{-}{\epsilon_{A}}^{B}\partial_{B}Y_{\ell}^{m}\,,$
(2.14)

where we split the transverse part into even ($\xi^{+}$) and odd modes
($\xi^{-}$). In what follows, $\delta^{(i)}A$ denotes the i-th order variation
of the tensor $A$

$\displaystyle g_{\sigma\rho}\delta^{(1)}\Gamma_{\mu\nu}^{\rho}~{}$
$\displaystyle=~{}-h_{\sigma\rho}\Gamma^{\rho}_{\mu\nu}+\dfrac{1}{2}\left(\partial_{\mu}h_{\sigma\nu}+\partial_{\nu}h_{\sigma\mu}-\partial_{\sigma}h_{\mu\nu}\right)$
$\displaystyle=~{}\dfrac{1}{2}\left(\nabla_{\mu}h_{\sigma\nu}+\nabla_{\nu}h_{\sigma\mu}-\nabla_{\sigma}h_{\mu\nu}\right)\,.$
(2.15)

##### The longitudinal modes

The action of the diffeomorphisms on the longitudinal modes work out to

$\displaystyle\delta
h_{ab}~{}=~{}\biggr{[}\partial_{a}\xi_{b}+\partial_{b}\xi_{a}-2\Gamma_{ab}^{c}\xi_{c}\biggr{]}Y_{\ell}^{m}-2\delta\Gamma_{ab}^{\mu}\bar{\xi}_{\mu}\,.$
(2.16)

This shows that

$\displaystyle\delta^{(1)}H_{ab}~{}$
$\displaystyle=~{}\tilde{\nabla}_{a}\xi_{b}+\tilde{\nabla}_{b}\xi_{a}\,,$
(2.17) $\displaystyle\delta^{(2)}H_{ab}~{}$
$\displaystyle=~{}\bar{\xi}^{c}\left(\tilde{\nabla}_{c}h_{ab}-\tilde{\nabla}_{a}h_{cb}-\tilde{\nabla}_{b}h_{ac}\right)-2\bar{\xi}^{A}\tilde{\nabla}_{(a}h_{b)A}+\bar{\xi}^{A}\partial_{A}h_{ab}~{}\eqqcolon~{}F_{ab}\left[h,\bar{\xi}\right]\,,$
(2.18)

where the tilde stands for operators that are only defined on the longitudinal
directions.222For instance, this would imply that
$\tilde{\nabla}_{a}h_{bA}=\partial_{a}h_{bA}-\Gamma^{c}_{ab}h_{cA}$ as
$h_{bA}$ is a vector along the longitudinal directions. The second order term
is sub-leading and will be irrelevant for our analysis. Furthermore, since the
gauge parameter $\bar{\xi}$ and the graviton are in different partial waves in
general, we refrain from further simplification of this term.

##### The transverse modes

The variation of the transverse modes under the said gauge transformations can
be written as

$\displaystyle\delta h_{AB}~{}$
$\displaystyle=~{}\partial_{A}\bar{\xi}_{B}+\partial_{B}\bar{\xi}_{A}-2\Gamma_{AB}^{\mu}\bar{\xi}_{\mu}-2\delta\Gamma_{AB}^{\mu}\bar{\xi}_{\mu}$
$\displaystyle=~{}2\xi^{+}\left(\partial_{A}\partial_{B}-\Gamma_{AB}^{C}\partial_{C}\right)Y_{\ell}^{m}+2\xi^{-}\left(\partial_{(A}{\epsilon_{B)}}^{C}\partial_{C}-\Gamma_{AB}^{C}{\epsilon_{C}}^{D}\partial_{D}\right)Y_{\ell}^{m}$
$\displaystyle\qquad+2g_{AB}V^{a}\xi_{a}Y_{\ell}^{m}-2\delta\Gamma_{AB}^{\mu}\xi_{\mu}\,.$
(2.19)

Comparing with the components appearing in $h_{AB}$, we see that

$\displaystyle\delta^{(1)}G~{}=~{}2\xi^{+}\,,\quad\delta^{(1)}K~{}=~{}2V^{a}\xi_{a}\quad\text{and}\quad\delta^{(1)}h_{\Omega}~{}=~{}2\xi^{-}\,,$
(2.20)

and

$\displaystyle\delta^{(2)}h_{AB}~{}$
$\displaystyle=~{}\bar{\xi}^{b}\biggr{(}-2\tilde{\nabla}_{(A}h_{B)b}-2V^{a}h_{ab}g_{AB}+\partial_{b}h_{AB}\biggr{)}$
$\displaystyle\qquad-\bar{\xi}^{C}\biggr{(}\tilde{\nabla}_{A}h_{CB}+\tilde{\nabla}_{B}h_{CA}-\tilde{\nabla}_{C}h_{AB}+2V^{a}h_{aC}g_{AB}\biggr{)}\,.$

##### The mixed modes

The diffeomorphisms also act on the modes that mix the longitudinal and
transverse ones in the following way

$\displaystyle\delta
h_{aA}~{}=~{}\biggr{[}\left(\partial_{a}-2V_{a}\right)\xi^{+}+\xi_{a}\biggr{]}\partial_{A}Y_{\ell}^{m}+\left(\partial_{a}-2V_{a}\right)\xi^{-}{\epsilon_{A}}^{B}\partial_{B}Y_{\ell}^{m}-2\delta\Gamma_{aA}^{\mu}\xi_{\mu}\,.$
(2.21)

This implies that

$\displaystyle\delta^{(1)}h_{a}^{+}~{}=~{}\left(\partial_{a}-2V_{a}\right)\xi^{+}+\xi_{a}\quad\text{and}\quad\delta^{(1)}h_{a}^{-}~{}=~{}\left(\partial_{a}-2V_{a}\right)\xi^{-}\,,$
(2.22)

and

$\displaystyle\delta^{(2)}h_{aA}~{}=~{}\bar{\xi}^{b}\biggr{(}-\partial_{A}h_{ab}+2\tilde{\nabla}_{[b}h_{a]A}+2V_{a}h_{bA}\biggr{)}+\bar{\xi}^{B}\biggr{(}2\tilde{\nabla}_{[B}h_{A]a}+2V_{a}h_{AB}-\partial_{a}h_{AB}\biggr{)}\,.$

###  2.1 Propagator for the monopole mode: $\ell=0$

The odd harmonic components, all of which contain angular derivatives, vanish
identically for the monopole since the corresponding spherical harmonic is
constant. For the same reason, so do the angular diffeomorphisms in (2.14).
The non-vanishing even harmonic mode is now

$h^{+}_{00,\mu\nu}~{}=~{}\begin{pmatrix}H_{xx}&H_{xy}&0&0\\\ &H_{yy}&0&0\\\
&&r^{2}K&0\\\ &&&r^{2}K\sin^{2}\theta\end{pmatrix}Y_{0}\,,$ (2.23)

and we have the two longitudinal diffeomorphisms in (2.14) to avail. One
convenient choice is to set the transverse scalar $K$ to zero. The
diffeomorphism that keeps it fixed to zero can be worked out from (2.20) to be
one that satisfies

$\displaystyle 2V^{a}\bar{\xi}_{a}~{}=~{}-KY_{0}\,.$ (2.24)

This implies that $\delta K=-K$ ensuring that there are no transverse degrees
of freedom left.

The last redundant degree of freedom that is left to be fixed can be chosen to
be $h_{xy}=0$. We will call this the traceless gauge because it is
equivalent333Note that the flat metric $\eta_{ab}$ is off-diagonal in the
lightcone coordinates we are working in. to $\eta^{ab}h_{ab}=0$. An
alternative choice was proposed in [29, 30] $x_{a}\epsilon_{bc}x^{c}h^{ab}=0$,
and was called the generalised Regge-Wheeler gauge. As we will find in Section
4, the traceless gauge has great calculational utility, while it was observed
in [29, 30] that the generalised Regge-Wheeler gauge makes the working out of
ghost actions simpler. The diffeomorphism that fixes the last gauge degree of
freedom should not change the transverse mode we already fixed and therefore,
it must satisfy the linearly independent relation $V^{a}\xi_{a}=0$. So, a
natural guess for the longitudinal diffeomorphism would be

$\displaystyle\bar{\xi}_{a}~{}=~{}-\dfrac{V_{a}}{2V^{2}}KY_{0}+f\left(x,y\right)\epsilon_{ab}V^{b}Y_{0}\,.$
(2.25)

To ensure that the second term indeed fixes the trace degree of freedom, we
demand that $\delta h_{ab}=-\eta^{ab}\mathfrak{h}_{ab}$ which can be worked
out to be

$\displaystyle\left(V\cdot\partial\right)f+f\left(\nabla\cdot
V\right)~{}=~{}-\eta^{ab}\mathfrak{h}_{ab}~{}\eqqcolon~{}-\mathfrak{h}\,.$
(2.26)

In Schwarzschild coordinates, this equation can be rewritten as

$\displaystyle\partial_{r}f+\dfrac{\left(\nabla\cdot
V\right)}{V^{r}}f~{}=~{}-\dfrac{\mathfrak{h}}{V^{r}}\,,$ (2.27)

which has a solution with the following integral representation

$\displaystyle
f~{}=~{}-I\left(r\right)\int\limits_{R}^{r}\dfrac{\mathfrak{h}\left(r^{\prime},t\right)}{I\left(r^{\prime}\right)V^{r}\left(r^{\prime}\right)}\text{d}r^{\prime}\quad\text{with}\quad
I\left(r\right)~{}=~{}\text{Exp}\left[-\int\limits_{R}^{r}\dfrac{\left(\nabla\cdot
V\right)\left(r^{\prime}\right)}{V^{r}\left(r^{\prime}\right)}\text{d}r^{\prime}\right]\,.$
(2.28)

####  2.1.1 Traceless gauge

As we have demonstrated above, in the traceless gauge, the only remaining off-
shell degrees of freedom are the diagonal longitudinal modes

$\displaystyle\mathfrak{h}_{0}^{ab}~{}=~{}\begin{pmatrix}\mathfrak{h}_{0}^{xx}&0\\\
0&\mathfrak{h}_{0}^{yy}\end{pmatrix}\,.$ (2.29)

The quadratic action is

$\displaystyle
S~{}=~{}\dfrac{1}{4}\int\text{d}^{2}x~{}\mathfrak{h}_{0}^{ab}\Delta^{-1}_{abcd}\mathfrak{h}_{0}^{cd}\,,$
(2.30)

where the quadratic operator is given by

$\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{2}\mu^{2}\left(\eta_{ac}x_{[b}\partial_{d]}+\eta_{bd}x_{[a}\partial_{c]}\right)+\dfrac{1}{2}\mu^{2}\bigr{(}\eta_{ab}\eta_{cd}-\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}\bigr{)}\,.$
(2.31)

This is the operator that was found in [4] for the even modes, only made
traceless (as defined in Appendix B) to fix the redundant gauge degree of
freedom. In momentum space the operator reads

$\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{2}\mu^{2}\left(\eta_{ac}k_{[b}\partial^{k}_{d]}+\eta_{bd}k_{[a}\partial^{k}_{c]}\right)+\dfrac{1}{2}\mu^{2}\bigr{(}\eta_{ab}\eta_{cd}-\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}\bigr{)}\,$
(2.32)

where $\partial^{k}_{a}$ is the derivative with respect to $k_{a}$. In order
to invert this operator, we begin by observing that time-translational
invariance (in Schwarzschild coordinates) or dilation invariance (under
$x\rightarrow\lambda x$ and $y\rightarrow\lambda^{-1}y$ in Kruskal-Szekeres
coordinates) implies that the most general form of the inverse is symmetric in
the first and last two indices, in addition to being symmetric under exchange
of the first two indices with the last two:

$\displaystyle\Delta_{abcd}~{}$
$\displaystyle=~{}A\left(k^{2}\right)\eta_{ab}\eta_{cd}+B\left(k^{2}\right)\eta_{a(c}\eta_{d)b}+C\left(k^{2}\right)\left(\eta_{ab}k_{c}k_{d}+\eta_{cd}k_{a}k_{b}\right)$
$\displaystyle\qquad+D\left(k^{2}\right)\left(\eta_{a(c}k_{d)}k_{b}+\eta_{b(c}k_{d)}k_{a}\right)+E\left(k^{2}\right)k_{a}k_{b}k_{c}k_{d}\,.$
(2.33)

In addition, to respect the traceless gauge, we require the inverse to be
traceless (as discussed in Appendix B) such that

$\displaystyle\Delta^{-1}_{abcd}\Delta^{cdef}~{}$
$\displaystyle=~{}\delta^{(e}_{a}\delta^{f)}_{b}-\dfrac{1}{2}\eta_{ab}\eta^{ef}$
$\displaystyle=~{}\dfrac{B\mu^{2}}{2}\eta_{ab}\eta^{ef}-\left(B\mu^{2}+\dfrac{\mu^{2}Dk^{2}}{4}\right)\delta_{a}^{(e)}\delta_{b}^{f)}+\dfrac{\mu^{2}D}{4}\left(\eta_{ab}k^{e}k^{f}+\eta^{ef}k_{a}k_{b}\right)$
$\displaystyle\qquad-\left(\dfrac{\mu^{2}D}{2}+\mu^{2}Ek^{2}\right)k_{(a}\delta_{b)}^{(e}k^{f)}+\mu^{2}Ek_{a}k_{b}k^{e}k^{f}\,.$
(2.34)

where in the second equality, we plugged in444Notice that
$\partial^{k}_{a}F\left(k^{2}\right)=2F^{\prime}k_{a}$ and so on for other
similar terms. This implies that
$k_{[a}\partial^{k}_{b]}F\left(k^{2}\right)=2F^{\prime}k_{[a}k_{b]}=0$
allowing us to drop all derivative terms acting on scalar functions. the
general form (2.1.1). The solution to this equation is given by

$B\left(k^{2}\right)~{}=~{}-\dfrac{1}{\mu^{2}}\quad\text{and}\quad
D\left(k^{2}\right)~{}=~{}0~{}=~{}E\left(k^{2}\right)\,,$ (2.35)

with arbitrary functions $A\left(k^{2}\right)$ and $C\left(k^{2}\right)$.
These are determined by demanding the inverse $\Delta_{abcd}$ to be traceless
and are given by

$A\left(k^{2}\right)~{}=~{}\dfrac{1}{2\mu^{2}}\quad\text{and}\quad
C\left(k^{2}\right)~{}=~{}0\,.$ (2.36)

Therefore, the propagator is given by

$\displaystyle\mathcal{P}_{\ell=0}^{abcd}~{}=~{}\dfrac{1}{2\mu^{2}}\left(\eta^{ab}\eta^{cd}-\eta^{ac}\eta^{bd}-\eta^{ad}\eta^{bc}\right)\,.$
(2.37)

Evidently, the propagator has no momentum dependence. This is owed to the fact
that there are only two off-shell degrees of freedom in the monopole and
therefore, imposing equations of motion, we are left with no dynamical modes.

####  2.1.2 ‘Generalised Regge-Wheeler’ gauge

Instead of the traceless gauge, had we used the generalised Regge-Wheeler
gauge $x_{a}x^{b}\epsilon_{bc}h^{ac}=0$ proposed in [29], the quadratic
operator turn out to be the same as those found in [4]

$\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{2}\mu^{2}\left(\eta_{ac}x_{[b}\partial_{d]}+\eta_{bd}x_{[a}\partial_{c]}+\eta_{ab}x_{(c}\partial_{d)}-\eta_{cd}x_{(a}\partial_{b)}\right)$
(2.38)
$\displaystyle\qquad+\mu^{2}\bigr{(}\eta_{ab}\eta_{cd}-\eta_{a(c}\eta_{d)b}\bigr{)}\,,$
(2.39)

and the propagator is readily found to be the same as the one derived in [4]

$\displaystyle\mathcal{P}_{\ell=0}^{abcd}~{}=~{}\dfrac{1}{2\mu^{2}}\left(2\eta^{ab}\eta^{cd}-\eta^{ac}\eta^{bd}-\eta^{ad}\eta^{bc}\right)\,.$
(2.40)

While the propagator is still non-dynamical, it does contain the trace mode.
This trace can be separated out as
$\mathfrak{h}_{ab}=\hat{\mathfrak{h}}_{ab}+\frac{1}{2}\eta_{ab}\mathfrak{h}$.
As we will see in Section 4, depending on the form of the interactions allowed
in the theory, the trace mode can sometimes be integrated out and the two
gauge choices we discussed coincide.

###  2.2 Propagator for the dipole mode: $\ell=1$

Before fixing any gauge, the dipole mode $\ell=1$ contains the following
graviton degrees of freedom, as can be checked by explicit calculation for
each of the allowed values foor $m=-1,0,1$.

$\displaystyle h^{-}_{1m,\mu\nu}~{}$
$\displaystyle=~{}\begin{pmatrix}0&0&-h^{-}_{x}\csc\theta\partial_{\phi}&h^{-}_{x}\sin\theta\partial_{\theta}\\\
&0&-h^{-}_{y}\csc\theta\partial_{\phi}&h^{-}_{y}\sin\theta\partial_{\theta}\\\
&&0&0\\\ &&&0\end{pmatrix}Y_{1}^{m}\,,$ (2.41) $\displaystyle
h^{+}_{1m,\mu\nu}~{}$
$\displaystyle=~{}\begin{pmatrix}H_{xx}&H_{xy}&h^{+}_{x}\partial_{\theta}&h^{+}_{x}\partial_{\phi}\\\
&H_{yy}&h^{+}_{y}\partial_{\theta}&h^{+}_{y}\partial_{\phi}\\\
&&r^{2}(K-G)&0\\\ &&&r^{2}(K-G)\sin^{2}\theta\end{pmatrix}Y_{1}^{m}\,.$ (2.42)

####  2.2.1 Propagator for the even harmonics

We see from (2.42) that the only transverse scalar mode is $K-G$ which we may
conveniently just call $K$. Therefore, we have

$h^{+}_{1m,\mu\nu}~{}=~{}\begin{pmatrix}H_{xx}&H_{xy}&h^{+}_{x}\partial_{\theta}&h^{+}_{x}\partial_{\phi}\\\
&H_{yy}&h^{+}_{y}\partial_{\theta}&h^{+}_{y}\partial_{\phi}\\\ &&r^{2}K&0\\\
&&&r^{2}K\sin^{2}\theta\end{pmatrix}Y_{1}^{m}\,.$ (2.43)

Of the four gauge degrees of freedom to be fixed, three are even modes. The
convenient choice would be to gauge fix the transverse scalar and $h^{+}_{a}$
which we do by the following choice

$\xi_{a}~{}=~{}\left(\dfrac{1}{2}r^{2}\partial_{a}K-h^{+}_{a}\right)Y_{1}^{m}\quad\text{and}\quad\xi_{A}~{}=~{}-\dfrac{1}{2}r^{2}K\partial_{A}Y_{1}^{m}\,,$
(2.44)

to reduce the even mode to

$h^{+}_{1m,\mu\nu}~{}=~{}\begin{pmatrix}H_{xx}&H_{xy}&0&0\\\ &H_{yy}&0&0\\\
&&0&0\\\ &&&0\end{pmatrix}Y_{1}^{m}\,.$ (2.45)

The remaining degree of freedom must be removed from the odd mode. The
quadratic action for the even wave is therefore

$\displaystyle
S~{}=~{}\sum_{m=-1}^{1}\dfrac{1}{4}\int\text{d}^{2}x~{}\mathfrak{h}_{1m}^{ab}\Delta^{-1}_{abcd}\mathfrak{h}_{1m}^{cd}\,,$
(2.46)

where

$\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{\mu^{2}}{2}\left(\eta_{ac}x_{[b}\partial_{d]}+\eta_{bd}x_{[a}\partial_{c]}+\eta_{ab}x_{(c}\partial_{d)}-\eta_{cd}x_{(a}\partial_{b)}\right)+2\mu^{2}\bigr{(}\eta_{ab}\eta_{cd}-\eta_{a(c}\eta_{d)b}\bigr{)}\,.$
(2.47)

To find the propagator, we first observe that the even mode carries three
offshell degrees of freedom and the odd mode comes with one (after gauge
fixing the other). This implies that upon imposing equations of motion, there
are no dynamical degrees of freedom left. So, we expect that the derivative
terms in the quadratic operator will not contribute to the propagator. So, we
may use the general ansatz (2.1.1) but without requiring tracelessness.
Therefore, we find

$\displaystyle\Delta^{-1}_{abcd}\Delta^{cdef}~{}$
$\displaystyle=~{}\delta^{(e}_{a}\delta^{f)}_{b}$
$\displaystyle=~{}2\mu^{2}\left(A+B\right)\eta_{ab}\eta^{ef}-2\mu^{2}B\delta_{a}^{(e}\delta^{f)}_{b}+\text{derivative
terms}\,.$ (2.48)

where we used the momentum space representation ($x\to i\partial^{k}$ and
$\partial\to-ik$) in the second equality. The solution is found to be

$\displaystyle A\left(k^{2}\right)~{}$
$\displaystyle=~{}\dfrac{1}{2\mu^{2}}\,,\quad
B\left(k^{2}\right)~{}=~{}-\dfrac{1}{2\mu^{2}}\quad\text{and}\quad
C\left(k^{2}\right)~{}=~{}D\left(k^{2}\right)~{}=~{}E\left(k^{2}\right)~{}=~{}0\,.$
(2.49)

So, the propagator is given by

$\displaystyle\mathcal{P}_{1m}^{abcd}~{}=~{}\dfrac{1}{4\mu^{2}}\bigr{(}2\eta^{ab}\eta^{cd}-\eta^{ac}\eta^{bd}-\eta^{ad}\eta^{bc}\bigr{)}\,.$
(2.50)

####  2.2.2 Propagator for the odd harmonics

The action for the odd dipole mode can be read off from the general near-
horizon action (C.24)

$\displaystyle
S_{1,m}~{}=~{}-\int\text{d}^{2}x~{}\mathfrak{h}^{a}\left(\partial_{a}\partial_{b}-\eta_{ab}\partial^{2}-\dfrac{3}{2}\mu^{2}x_{a}\partial_{b}+\dfrac{3}{2}\mu^{2}x_{b}\partial_{a}+\dfrac{3}{2}\mu^{2}\eta_{ab}\right)\mathfrak{h}^{b}\,.$
(2.51)

A convenient choice of gauge to fix the redundant degree of freedom in the odd
mode $h^{a}$ is a Lorenz-like gauge since it is a spin one field in question.
We choose $\partial_{a}h^{a}=0$ whose corresponding diffeomorphism is given by
$\xi_{A}=-F\left(x\right){\epsilon_{A}}^{B}\partial_{B}$, where $F$ is given
by a solution to

$\displaystyle\partial^{2}F\left(x\right)-2V^{a}\partial_{a}F\left(x\right)-2F\left(x\right)\partial_{a}V^{a}~{}=~{}-\partial_{a}\mathfrak{h}^{a}\,.$
(2.52)

Owing to the gauge choice, the first and third terms in the quadratic operator
vanish identically. The fourth term can be integrated by parts as

$\displaystyle\int\text{d}^{2}x~{}\mathfrak{h}^{a}x_{b}\partial_{a}\mathfrak{h}^{b}~{}$
$\displaystyle=~{}\int\text{d}^{2}x~{}\mathfrak{h}^{a}\partial_{a}\left(x_{b}\mathfrak{h}^{b}\right)-\int\text{d}^{2}x~{}\mathfrak{h}^{a}\eta_{ab}\mathfrak{h}^{b}$
$\displaystyle=~{}-\int\text{d}^{2}x~{}\mathfrak{h}^{b}x_{b}\partial_{a}\mathfrak{h}^{a}-\int\text{d}^{2}x~{}\mathfrak{h}^{a}\eta_{ab}\mathfrak{h}^{b}\,,$
(2.53)

where the first integral drops due to the gauge choice and the second term
cancels the last term in (2.51), leaving us with

$\displaystyle
S_{1,m}~{}=~{}\int\text{d}^{2}x~{}\mathfrak{h}^{a}\eta_{ab}\partial^{2}\mathfrak{h}^{b}\,.$
(2.54)

The propagator is therefore easily found to be the familiar one for a massless
spin one field

$\displaystyle\mathcal{P}_{1m}^{ab}~{}=~{}\dfrac{\eta^{ab}}{k^{2}-i\epsilon}\,.$
(2.55)

In the generalised Regge-Wheeler gauge of [29, 30], $V^{a}h_{a}=0$, the
quadratic operator takes the following non-invertible form:
$\partial_{a}\partial_{b}-\partial^{2}\eta_{ab}$. However, defining
$h^{a}=h_{-}\epsilon^{ab}V_{b}$ results in a propagator for the field $h_{-}$.
It would be interesting to understand the apparent gauge-dependent emergent
dynamics for what is arguably a non-propagating mode [29, 30].

###  2.3 Propagator for the multipole modes: $\ell>1$

All ten components of the graviton are non-trivial for the multipole modes.
So, the gauge choice we employed in [4] to fix four of those was

$\displaystyle\xi_{a}~{}=~{}\left(\dfrac{1}{2}r^{2}\partial_{a}G-h^{+}_{a}\right)Y_{\ell}^{m}\quad\text{and}\quad\xi_{A}~{}=~{}-\dfrac{1}{2}r^{2}G\partial_{A}Y_{\ell}^{m}-\dfrac{1}{2}h_{\Omega}{\epsilon_{A}}^{B}\partial_{B}Y_{\ell}^{m}\,.$
(2.56)

This diffeomorphism sets $h^{+}_{a}=G=h_{\Omega}=0$, resulting in the
following graviton modes

$\displaystyle h^{-}_{\ell m,\mu\nu}~{}$
$\displaystyle=~{}\begin{pmatrix}0&0&-h_{x}\csc\theta\partial_{\phi}&h_{x}\sin\theta\partial_{\theta}\\\
0&0&-h_{y}\csc\theta\partial_{\phi}&h_{y}\sin\theta\partial_{\theta}\\\
-h_{x}\csc\theta\partial_{\phi}&-h_{y}\csc\theta\partial_{\phi}&0&0\\\
h_{x}\sin\theta\partial_{\theta}&h_{y}\sin\theta\partial_{\theta}&0&0\end{pmatrix}Y_{\ell}^{m}\,$
(2.57) $\displaystyle h^{+}_{\ell m,\mu\nu}~{}$
$\displaystyle=~{}\begin{pmatrix}H_{xx}&H_{xy}&0&0\\\ H_{xy}&H_{yy}&0&0\\\
0&0&r^{2}K&0\\\ 0&0&0&r^{2}K\sin^{2}\theta.\end{pmatrix}Y_{\ell}^{m}\,.$
(2.58)

####  2.3.1 Propagator for the even harmonics

The near-horizon propagator for the even multipole modes was derived in full
generality in [3, 4], to which we refer for details. Here, we collect the
results from those references

$S~{}=~{}\sum_{\ell
m}\dfrac{1}{4}\int\text{d}^{2}x\biggr{(}\mathfrak{h}^{ab}\Delta^{-1}_{abcd}\mathfrak{h}^{cd}+\mathfrak{h}^{ab}\Delta^{-1}_{ab}\mathcal{K}+\mathcal{K}\Delta^{-1}_{ab}\mathfrak{h}^{ab}+\mathcal{K}\Delta^{-1}\mathcal{K}\biggr{)}\,,$
(2.59)

with

$\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{\mu^{2}}{2}\left(\eta_{ac}x_{[b}\partial_{d]}+\eta_{bd}x_{[a}\partial_{c]}+\eta_{ab}x_{(c}\partial_{d)}-\eta_{cd}x_{(a}\partial_{b)}\right)+\dfrac{\mu^{2}\left(\lambda+1\right)}{2}\bigr{(}\eta_{ab}\eta_{cd}-\eta_{a(c}\eta_{d)b}\bigr{)}$
$\displaystyle\Delta^{-1}_{ab}~{}$
$\displaystyle=~{}-\eta_{ab}\left(\partial^{2}-\dfrac{1}{2}\mu^{2}\left(\lambda-1\right)\right)+\partial_{a}\partial_{b}\,,$
$\displaystyle\Delta^{-1}~{}$ $\displaystyle=~{}-\partial^{2}+\mu^{2}\,.$
(2.60)

The propagators are given by compositions of the inverses of the above
operators:

$\displaystyle\mathcal{P}_{abcd}~{}$
$\displaystyle=~{}\Delta_{abcd}+\mathcal{P}_{\mathcal{K}}\text{p}_{ab}\text{p}_{cd}$
(2.61) $\displaystyle\mathcal{P}_{ab}~{}$
$\displaystyle=~{}-\mathcal{P}_{\mathcal{K}}\text{p}_{ab}$ (2.62)
$\displaystyle\mathcal{P}_{\mathcal{K}}~{}$
$\displaystyle=~{}\dfrac{\lambda+1}{\lambda-3}\dfrac{1}{k^{2}+\mu^{2}\lambda}\,,$
(2.63)

where

$\displaystyle\Delta_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{\mu^{2}\left(\lambda+1\right)}\bigr{(}2\eta_{ab}\eta_{cd}-\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}\bigr{)}$
(2.64) $\displaystyle\text{p}_{ab}~{}$
$\displaystyle=~{}\dfrac{\lambda-1}{\lambda+1}\eta_{ab}+\dfrac{2k_{a}k_{b}}{\mu^{2}\left(\lambda+1\right)}\,.$
(2.65)

Asymptotically, the propagator scales as $k^{2}$, a result of the effective
mass of the two-dimensional graviton. As was also pointed out in [4], the
shape of the propagators resembles that of massive gravity, with mass
$\mu^{2}\left(\lambda+1\right)$. Moreover, there is a change in sign for the
$\ell=0$ monopole and a pole in the $\ell=1$ dipole mode, indicating
additional redundant gauge degrees of freedom. These were resolved in the
previous subsections 2.1 and 2.2 to find the corresponding propagators.
Therefore, the above propagators are valid strictly for the $\ell>1$ multipole
modes.

####  2.3.2 Propagator for the odd harmonics

The near-horizon action for the odd harmonics of the multipole modes is
derived in detail in Appendix C. The result is (C.24) which we repeat here:

$\displaystyle\sum_{\ell,m}S_{\ell,m}~{}=~{}-\sum_{\ell,m}\dfrac{\lambda-1}{2}\int\text{d}^{2}x~{}\mathfrak{h}^{a}\left(\partial_{a}\partial_{b}-\eta_{ab}\partial^{2}-3\mu^{2}x_{[a}\partial_{b]}+\mu^{2}\eta_{ab}\left(\lambda-\dfrac{3}{2}\right)\right)\mathfrak{h}^{b}\,.$
(2.66)

The propagator is then found to be (C.34):

$\displaystyle\mathcal{G}^{ab}~{}=~{}\dfrac{\lambda-\frac{9}{2}}{\lambda-1}\dfrac{1}{k^{2}\left(\lambda-3\right)+\mu^{2}\left(\lambda-\frac{3}{2}\right)\left(\lambda-\frac{9}{2}\right)}\left(\eta^{ab}+\dfrac{k^{a}k^{b}}{\mu^{2}\left(\lambda-\frac{9}{2}\right)}\right)\,.$
(2.67)

This shape of this propagator resembles that of a massive spin-1 particle
whose mass diverges when $\lambda=3$ or $\ell=1$. This is not a physical pole
in momentum space and suggests additional gauge redundancy in that dipole mode
which warranted a separate study of its propagator in Section 2.2 where we
resolved this issue. Therefore, the propagator above is only valid and
reliable for the multipole modes $\ell\geq 2$.

##  3 Interactions with matter

In this section, we will proceed to (minimally) couple the theory we have
considered so far, to matter (with an action we label $S_{M}$). The leading
interaction term takes the form

$\displaystyle S_{int}~{}$
$\displaystyle=~{}\dfrac{1}{2}\kappa\int\text{d}^{4}x\sqrt{-g}~{}h^{\mu\nu}T_{\mu\nu}\quad\text{with}\quad
T_{\mu\nu}~{}\coloneqq~{}\dfrac{-2}{\sqrt{-g}}\dfrac{\delta S_{M}}{\delta
g^{\mu\nu}}\,.$ (3.1)

For the even modes, using (2.58), we have

$\displaystyle S^{+}_{int}~{}$ $\displaystyle=~{}\sum_{\begin{subarray}{c}\ell
m\end{subarray}}\dfrac{1}{2}\kappa\int\text{d}^{2}x\left(r^{2}A\left(x,y\right)\right)\text{d}\Omega_{2}~{}\left(g^{ac}g^{bd}H_{cd}T_{ab}+Kg^{AB}T_{AB}\right)Y_{\ell}^{m}$
$\displaystyle=~{}\sum_{\begin{subarray}{c}\ell
m\end{subarray}}\dfrac{1}{2}\kappa\int\text{d}^{2}x\left(r^{2}A\left(x,y\right)\right)\text{d}\Omega_{2}~{}\left(\eta^{ac}\eta^{bd}\dfrac{A\left(x,y\right)}{rA\left(x,y\right)^{2}}\mathfrak{h}_{cd}T_{ab}+\dfrac{\mathcal{K}}{r}g^{AB}T_{AB}\right)Y_{\ell}^{m}$
$\displaystyle=~{}\sum_{\begin{subarray}{c}\ell
m\end{subarray}}\dfrac{1}{2}\kappa\int\text{d}^{2}x~{}r~{}\text{d}\Omega_{2}~{}\left(\mathfrak{h}^{ab}T_{ab}+\mathcal{K}g^{AB}T_{AB}\right)Y_{\ell}^{m}\,,$
(3.2)

where in the second line, we used the field definitions
$rH_{ab}=A\left(x,y\right)\mathfrak{h}_{ab},~{}rK=\mathcal{K}$ and the fact
that $A\left(x,y\right)g^{ab}=\eta^{ab}$. Considering a minimally coupled
scalar field, we have

$\displaystyle T_{\mu\nu}~{}$
$\displaystyle=~{}\partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi}-\dfrac{1}{2}g_{\mu\nu}\partial_{\rho}\tilde{\phi}\partial^{\rho}\tilde{\phi}\,.$
(3.3)

Therefore, we have

$\displaystyle T_{ab}~{}$
$\displaystyle=~{}\partial_{a}\tilde{\phi}\partial_{b}\tilde{\phi}-\dfrac{1}{2}g_{ab}\partial_{\rho}\tilde{\phi}\partial^{\rho}\tilde{\phi}$
$\displaystyle=~{}\left(\dfrac{1}{r}\partial_{a}\phi+\phi\partial_{a}\dfrac{1}{r}\right)\left(\dfrac{1}{r}\partial_{b}\phi+\phi\partial_{b}\dfrac{1}{r}\right)-\dfrac{1}{2}g_{ab}\left(\dfrac{1}{r}\partial_{\rho}\phi+\phi\partial_{\rho}\dfrac{1}{r}\right)\left(\dfrac{1}{r}\partial^{\rho}\phi+\phi\partial^{\rho}\dfrac{1}{r}\right)$
$\displaystyle=~{}\dfrac{1}{r^{2}}\left(\partial_{a}\phi\partial_{b}\phi-\dfrac{1}{2}\eta_{ab}\eta^{cd}\partial_{c}\phi\partial_{d}\phi-\dfrac{\eta_{ab}}{2A\left(x,y\right)r^{2}}\gamma^{CD}\partial_{C}\phi\partial_{D}\phi\right)+\dots$
$\displaystyle\sim~{}\dfrac{1}{R^{2}}\left(\partial_{a}\phi\partial_{b}\phi-\dfrac{1}{2}\eta_{ab}\eta^{cd}\partial_{c}\phi\partial_{d}\phi-\dfrac{1}{2}\eta_{ab}\gamma^{CD}\hat{\partial}_{C}\phi\hat{\partial}_{D}\phi\right)$
$\displaystyle\sim~{}\dfrac{1}{R^{2}}\left(\partial_{a}\phi\partial_{b}\phi-\dfrac{1}{2}\eta_{ab}\eta^{cd}\partial_{c}\phi\partial_{d}\phi\right)~{}\eqqcolon~{}\dfrac{1}{R^{2}}\tilde{T}_{ab}\,,$
(3.4)

where we redefined the scalar field as $\tilde{\phi}\to\frac{1}{r}\phi$ in the
second line. The terms represented by the dots evidently drop out in the near-
horizon limit $r\sim R$, $A\left(x,y\right)\sim 1$. In the third line, we also
used that $g^{CD}=\frac{1}{r^{2}}\gamma^{CD}$ where $\gamma^{CD}$ is the
inverse of the round metric on the unit two-sphere. In the fourth line, we
defined the rescaled transverse derivatives
$\hat{\partial}_{A}\coloneqq\frac{1}{R}\partial_{A}$ in the near horizon
limit. The transverse effects are, therefore, naturally suppressed in the
near-horizon limit by an additional $\frac{1}{R}$ factor in comparison to the
longitudinal momenta. Therefore, in the fifth line, we ignore the last term of
the fourth line and arrive at the energy momentum tensor purely along the
longitudinal directions.

Similarly, for the transverse and mixed components of the stress tensor, we
have

$\displaystyle g^{AB}T_{AB}~{}$
$\displaystyle=~{}g^{AB}\partial_{A}\tilde{\phi}\partial_{B}\tilde{\phi}-g^{ab}\partial_{a}\tilde{\phi}\partial_{b}\tilde{\phi}-g^{CD}\partial_{C}\tilde{\phi}\partial_{D}\tilde{\phi}$
$\displaystyle=~{}-\dfrac{1}{A\left(x,y\right)}\eta^{ab}\left(\dfrac{1}{r}\partial_{a}\phi+\phi\partial_{a}\dfrac{1}{r}\right)\left(\dfrac{1}{r}\partial_{b}\phi+\phi\partial_{b}\dfrac{1}{r}\right)$
$\displaystyle\sim~{}-\dfrac{1}{R^{2}}\eta^{ab}\partial_{a}\phi\partial_{b}\phi$
(3.5) $\displaystyle T_{aA}~{}$
$\displaystyle=~{}\partial_{a}\tilde{\phi}\partial_{A}\tilde{\phi}~{}\sim~{}\dfrac{1}{R}\partial_{a}\phi\hat{\partial}_{A}\phi\,.$
(3.6)

Therefore, in the near-horizon limit, the interaction term for the even
graviton takes the following form:

$\displaystyle S^{+}_{int}~{}$
$\displaystyle=~{}\dfrac{\kappa}{2R}\sum_{\begin{subarray}{c}\ell m\\\
\ell_{1}m\\\
\ell_{2}m\end{subarray}}\int\text{d}^{2}x~{}\text{d}\Omega_{2}~{}\Biggr{[}\mathfrak{h}^{ab}\left(\partial_{a}\phi_{\ell_{1},m_{1}}\partial_{b}\phi_{\ell_{2},m_{2}}-\dfrac{1}{2}\eta_{ab}\partial_{c}\phi_{\ell_{1},m_{1}}\partial^{c}\phi_{\ell_{2},m_{2}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\mathcal{K}\eta^{ab}\partial_{a}\phi_{\ell_{1},m_{1}}\partial_{b}\phi_{\ell_{2},m_{2}}\Biggr{]}Y_{\ell}^{m}Y_{\ell_{1}}^{m_{1}}Y_{\ell_{2}}^{m_{2}}$
$\displaystyle=~{}\dfrac{\gamma}{2}\sum_{\begin{subarray}{c}\ell m\\\
\ell_{1}m\\\
\ell_{2}m\end{subarray}}\int\text{d}^{2}x~{}\Biggr{[}\left(\mathfrak{h}^{ab}-\dfrac{1}{2}\mathfrak{h}\eta^{ab}-\mathcal{K}\eta^{ab}\right)\partial_{a}\phi_{\ell_{1},m_{1}}\partial_{b}\phi_{\ell_{2},m_{2}}\Biggr{]}C\left[\ell
m;\ell_{1}m_{1};\ell_{2}m_{2}\right]\,,$ (3.7)

where we defined the coupling constant $\gamma\coloneqq\kappa/R$ and

$\displaystyle C\left[\ell
m;\ell_{1}m_{1};\ell_{2}m_{2}\right]~{}\coloneqq~{}\int\text{d}\Omega_{2}~{}Y_{\ell}^{m}Y_{\ell_{1}}^{m_{1}}Y_{\ell_{2}}^{m_{2}}\,.$
(3.8)

An interesting observation to be made from the form of the longitudinal
components is that $\eta^{ab}\tilde{T}_{ab}=0$ provided transverse momenta are
small. This implies that the trace mode in the graviton $\mathfrak{h}_{ab}$
does not couple to the matter fields. This allows us to combine the traceless
longitudinal graviton and the transverse scalar into a single tensorial mode

$\displaystyle\hat{\mathfrak{h}}_{ab}~{}\coloneqq~{}\mathfrak{h}_{ab}-\dfrac{1}{2}\eta_{ab}\mathfrak{h}-\mathcal{K}\eta^{ab}\,,$
(3.9)

that couples to the scalar fields as

$\displaystyle S^{+}_{int}~{}$
$\displaystyle=~{}\dfrac{\gamma}{2}\sum_{\begin{subarray}{c}\ell m\\\
\ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}\int\text{d}^{2}x~{}\Biggr{[}\hat{\mathfrak{h}}^{ab}\partial_{a}\phi_{\ell_{1},m_{1}}\partial_{b}\phi_{\ell_{2},m_{2}}\Biggr{]}C\left[\ell
m;\ell_{1}m_{1};\ell_{2}m_{2}\right]\,,$ (3.10)

where we suppressed the $\ell,m$ indices on the graviton mode. Owing to
residual gauge redundancy that needs to be fixed, $\mathfrak{h}$ and
$\mathcal{K}$ vanish identically in the monopole ($\ell=0$) mode whereas only
the latter vanishes for the dipole ($\ell=1$) mode as discussed in Section 2.1
and Section 2.2, respectively. For the multipole modes ($\ell>1$), the trace
of (3.9) shows that $\mathcal{K}=\frac{1}{2}\hat{\mathfrak{h}}$. In momentum
space, the above interaction action can be written as

$\displaystyle S^{+}_{int}~{}$
$\displaystyle=~{}\dfrac{\gamma}{2\left(2\pi\right)^{6}}\sum_{\begin{subarray}{c}\ell
m\\\ \ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}\int\text{d}^{2}x\int\mathrm{d}^{2}p\int\prod_{i=1}^{2}\mathrm{d}^{2}p_{i}~{}\Biggr{[}\hat{\mathfrak{h}}^{ab}\left(p\right)p_{1,a}p_{2,b}\Biggr{]}C\left[\ell
m;\ell_{1}m_{1};\ell_{2}m_{2}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\phi_{\ell_{1},m_{1}}\left(p_{1}\right)\phi_{\ell_{2},m_{2}}\left(p_{2}\right)e^{i\left(p+p_{1}+p_{2}\right)x}$
$\displaystyle=~{}\dfrac{\gamma}{2\left(2\pi\right)^{4}}\sum_{\begin{subarray}{c}\ell
m\\\ \ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}\int\mathrm{d}^{2}p\int\prod_{i=1}^{2}\mathrm{d}^{2}p_{i}~{}\Biggr{[}\hat{\mathfrak{h}}^{ab}\left(p\right)p_{1,a}p_{2,b}\Biggr{]}C\left[\ell
m;\ell_{1}m_{1};\ell_{2}m_{2}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\phi_{\ell_{1},m_{1}}\left(p_{1}\right)\phi_{\ell_{2},m_{2}}\left(p_{2}\right)\delta^{(2)}\left(p+p_{1}+p_{2}\right)$
$\displaystyle=~{}\dfrac{\gamma}{2}\sum_{\begin{subarray}{c}\ell m\\\
\ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}\int\dfrac{\mathrm{d}^{2}p_{1}}{\left(2\pi\right)^{2}}~{}\Biggr{[}\hat{\mathfrak{h}}^{ab}\left(-p\right)p_{1,a}\left(p-p_{1}\right)_{b}\Biggr{]}C\left[\ell
m;\ell_{1}m_{1};\ell_{2}m_{2}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\phi_{\ell_{1},m_{1}}\left(p_{1}\right)\phi_{\ell_{2},m_{2}}\left(p-p_{1}\right)$
$\displaystyle\eqqcolon~{}\dfrac{\gamma}{2}\sum_{\ell
m}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}~{}\hat{\mathfrak{h}}^{ab}_{\ell
m}\left(-p\right)\tilde{T}^{\ell m}_{ab}\left(p\right)\,,$ (3.11)

where we defined

$\displaystyle\tilde{T}^{\ell m}_{ab}\left(p\right)~{}$
$\displaystyle=~{}\sum_{\begin{subarray}{c}\ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}\int\dfrac{\mathrm{d}^{2}p_{1}}{\left(2\pi\right)^{2}}~{}p_{1,a}\left(p-p_{1}\right)_{b}~{}C\left[\ell
m;\ell_{1}m_{1};\ell_{2}m_{2}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\phi_{\ell_{1},m_{1}}\left(p_{1}\right)\phi_{\ell_{2},m_{2}}\left(p-p_{1}\right)\,.$
(3.12)

##  4 Decoupling the non-interacting graviton modes

The interacting dynamical graviton is a linear combination of the traceless
longitudinal mode and the transverse scalar, as can be seen from (3.9) and
(3.10). Therefore, it is desirable to rewrite the quadratic action (2.59) for
the traceless mode $\hat{\mathfrak{h}}_{ab}$. To this end, let us first write
the field redefinition as

$\displaystyle\begin{pmatrix}\hat{\mathfrak{h}}_{ab}\\\
\hat{\mathcal{K}}\end{pmatrix}~{}$
$\displaystyle\coloneqq~{}\hat{\Lambda}\begin{pmatrix}\mathfrak{h}_{cd}\\\
\mathcal{K}\end{pmatrix}~{}=~{}\begin{pmatrix}\delta^{cd}_{ab}-\frac{1}{2}\eta_{ab}\eta^{cd}&-\eta_{ab}\\\
\eta^{cd}&\Lambda\end{pmatrix}\begin{pmatrix}\mathfrak{h}_{cd}\\\
\mathcal{K}\end{pmatrix}~{}=~{}\begin{pmatrix}\mathfrak{h}_{ab}-\left(\frac{1}{2}\mathfrak{h}+\mathcal{K}\right)\eta_{ab}\\\
\mathfrak{h}+\Lambda\mathcal{K}\end{pmatrix}\,.$ (4.1)

For the rest of this section, working in momentum space turns out to be more
convenient where the above field redefinition remains and the momentum space
representation of $\Lambda$ is given by

$\Lambda~{}=~{}\dfrac{2}{\mu^{2}\left(\lambda+1\right)}\left(k^{2}+\mu^{2}\left(\lambda-1\right)\right)\,.$
(4.2)

The inverse of the redefinition matrix $\hat{\Lambda}$ can easily be found

$\hat{\Lambda}^{-1}~{}=~{}\begin{pmatrix}\delta^{cd}_{ab}-\frac{1}{2}\eta_{ab}\eta^{cd}+\frac{1}{4}\Lambda\eta_{ab}\eta^{cd}&\frac{1}{2}\eta_{ab}\\\
-\frac{1}{2}\eta^{cd}&0\end{pmatrix}\,.$ (4.3)

In matrix form, the Lagrangian in (2.59) can be written as

$\displaystyle\mathcal{L}~{}=~{}\dfrac{1}{4}\begin{pmatrix}\mathfrak{h}_{ab}&\mathcal{K}\end{pmatrix}\begin{pmatrix}(\Delta^{-1})^{abcd}&(\Delta^{-1})^{ab}\\\
(\Delta^{-1})^{cd}&\Delta^{-1}\end{pmatrix}\begin{pmatrix}\mathfrak{h}_{cd}\\\
\mathcal{K}\end{pmatrix}~{}\eqqcolon~{}\dfrac{1}{4}\begin{pmatrix}\mathfrak{h}_{ab}&\mathcal{K}\end{pmatrix}\mathbf{\Delta}^{-1}\begin{pmatrix}\mathfrak{h}_{cd}\\\
\mathcal{K}\end{pmatrix}\,.$ (4.4)

The propagators of the theory are defined by the inverse $\mathbf{\Delta}$.
The action of the field redefinition (4.1) can be absorbed into a redefinition
of the quadratic operator and its inverse as

$\displaystyle\begin{pmatrix}\left(\hat{\mathcal{P}}^{-1}\right)^{abcd}&0\\\
0&\hat{\mathcal{P}}^{-1}\end{pmatrix}~{}$
$\displaystyle=~{}\left(\hat{\Lambda}^{-1}\right)^{T}\mathbf{\Delta}^{-1}\hat{\Lambda}^{-1}$
(4.5) $\displaystyle\begin{pmatrix}\hat{\mathcal{P}}^{abcd}&0\\\
0&\hat{\mathcal{P}}\end{pmatrix}~{}$
$\displaystyle=~{}\hat{\Lambda}\mathbf{\Delta}\hat{\Lambda}^{T}\,.$ (4.6)

This reduces the action (2.59) to its diagonal degrees of freedom. In momentum
space, it reads

$\displaystyle S~{}=~{}\sum_{\ell
m}\dfrac{1}{4}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}\biggr{(}\hat{\mathfrak{h}}_{ab}\left(\hat{\mathcal{P}}^{-1}\right)^{abcd}\hat{\mathfrak{h}}_{cd}+\hat{\mathcal{K}}\hat{\mathcal{P}}^{-1}\hat{\mathcal{K}}\biggr{)}\,,$
(4.7)

where the propagators arising from the inverses of the above quadratic
operators are

$\displaystyle\hat{\mathcal{P}}^{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{\mu^{2}\left(\lambda+1\right)}\left(\eta_{ab}\eta_{cd}-\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}\right)-\dfrac{\lambda+1}{\lambda-3}\dfrac{1}{k^{2}+\mu^{2}\lambda}\left(\eta_{ab}+\tilde{\text{p}}_{ab}\right)\left(\eta_{cd}+\tilde{\text{p}}_{cd}\right)$
(4.8) $\displaystyle\hat{\mathcal{P}}~{}$
$\displaystyle=~{}\dfrac{4}{\mu^{2}\left(\lambda+1\right)}\,,$ (4.9)

where

$\tilde{\text{p}}_{ab}~{}=~{}\dfrac{2}{\mu^{2}\left(\lambda+1\right)}\left(k_{a}k_{b}-\dfrac{1}{2}k^{2}\eta_{ab}\right)\,.$
(4.10)

These expressions are strictly valid only for the multipole modes $\ell>1$.
However, by integrating out the non-interacting graviton modes, in what
follows, we will find a scalar rewriting of the theory that is valid for all
$\ell$. To this end, as we saw in (3.10), the field $\hat{\mathcal{K}}$ is
non-interacting and can therefore be integrated out. The position space field
redefinition for this freely propagating mode is

$\hat{\mathcal{K}}~{}=~{}\mathfrak{h}-\dfrac{2}{\mu^{2}\left(\lambda+1\right)}\left(-\partial^{2}+\mu^{2}\left(\lambda-1\right)\right)\mathcal{K}\,,$
(4.11)

and contains derivatives. Therefore, the Gaussian integral appears to contain
higher derivatives. However, given that the original theory we began with was
a two-derivative Einstein-Hilbert action, this is merely an artefact of the
field redefinitions and does not contribute to the physical S-matrix. It is
worth emphasising that this decoupling is only a feature of the cubic order of
interactions we consider in this article. Henceforth, we will ignore this free
scalar mode and focus on the interacting mode $\hat{\mathfrak{h}}_{ab}$. The
complete action in momentum space is now

$\displaystyle\sum_{\ell m}S_{\ell m}~{}$ $\displaystyle=~{}\sum_{\ell
m}\dfrac{1}{4}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}\biggr{(}\hat{\mathfrak{h}}^{\ell
m}_{ab}\left(-p\right)\left(\hat{\mathcal{P}}^{-1}\right)^{abcd}\left(p\right)\hat{\mathfrak{h}}^{\ell
m}_{cd}\left(p\right)\biggr{)}$ $\displaystyle\quad-\sum_{\ell
m}\dfrac{1}{2}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}~{}\phi_{\ell,m}\left(p^{2}+\mu^{2}\lambda\right)\phi_{\ell,m}+\sum_{\ell
m}\dfrac{\gamma}{2}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}~{}\Biggr{[}\hat{\mathfrak{h}}^{ab}_{\ell
m}\left(-p\right)\tilde{T}^{\ell m}_{ab}\left(p\right)\Biggr{]}\,,$ (4.12)

where we took the matter action to be that of a minimally coupled scalar field
as in [4].555See Section 4.3 of [4]. As before, $\lambda=\ell^{2}+\ell+1$. The
stress tensor is a convolution in momentum space

$\displaystyle\tilde{T}^{\ell m}_{ab}\left(p\right)~{}$
$\displaystyle\coloneqq~{}\sum_{\begin{subarray}{c}\ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}\int\dfrac{\mathrm{d}^{2}p_{1}}{\left(2\pi\right)^{2}}\left(p_{1,a}\right)\left(p-p_{1}\right)_{b}\phi_{\ell_{1},m_{1}}\left(p_{1}\right)\phi_{\ell_{2},m_{2}}\left(p-p_{1}\right)C\left[\ell
m;\ell_{1}m_{1};\ell_{2}m_{2}\right]$
$\displaystyle=~{}\sum_{\begin{subarray}{c}\ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}C\left[\ell
m;\ell_{1}m_{1};\ell_{2}m_{2}\right]\left(\partial_{a}\phi_{\ell_{1}m_{1}}\star\partial_{b}\phi_{\ell_{2}m_{2}}\right)\,.$
(4.13)

To simplify this action, we first transform the graviton mode as

$\hat{\mathfrak{h}}^{\ell
m}_{ab}\left(-p\right)\rightarrow\hat{\mathfrak{h}}^{\ell
m}_{ab}\left(-p\right)-\gamma\hat{\mathcal{P}}_{abcd}\left(p\right)\tilde{T}^{cd}_{\ell
m}\left(p\right)\,,$ (4.14)

to find the following action

$\displaystyle\sum_{\ell m}S_{\ell m}~{}$
$\displaystyle=~{}\dfrac{1}{4}\sum_{\ell
m}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}\biggr{(}\hat{\mathfrak{h}}^{\ell
m}_{ab}\left(-p\right)\hat{\mathcal{P}}^{abcd}\left(p\right)\hat{\mathfrak{h}}^{\ell
m}_{cd}\left(p\right)\biggr{)}-\dfrac{1}{2}\sum_{\ell
m}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}~{}\phi_{\ell,m}\left(p^{2}+\mu^{2}\lambda\right)\phi_{\ell,m}$
$\displaystyle\qquad\qquad-\sum_{\ell
m}\dfrac{\gamma^{2}}{4}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}~{}\Biggr{[}\tilde{T}^{\ell
m}_{ab}\left(-p\right)\hat{\mathcal{P}}^{abcd}\left(p\right)\tilde{T}^{\ell
m}_{cd}\left(p\right)\Biggr{]}\,.$ (4.15)

To arrive at this expression, we use the fact that
$\left(\hat{\mathcal{P}}^{-1}\right)^{abcd}\hat{\mathcal{P}}_{cdef}=\delta^{ab}_{ef}$.
We see that the new graviton mode has also been decoupled with this field
redefinition and can be integrated out, leaving us with

$\displaystyle\sum_{\ell m}S_{\ell m}~{}$
$\displaystyle=~{}-\dfrac{1}{2}\sum_{\ell
m}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}~{}\phi_{\ell,m}\left(p^{2}+\mu^{2}\lambda\right)\phi_{\ell,m}$
$\displaystyle\qquad-\sum_{\ell
m}\dfrac{\gamma^{2}}{4}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}~{}\Biggr{[}\tilde{T}^{\ell
m}_{ab}\left(-p\right)\hat{\mathcal{P}}^{abcd}\left(p\right)\tilde{T}^{\ell
m}_{cd}\left(p\right)\Biggr{]}\,.$ (4.16)

Now, using the form of the stress tensor in (3), we write

$\displaystyle\dfrac{-\gamma^{2}}{4}\sum_{\ell
m}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}~{}\tilde{T}^{\ell
m}_{ab}\left(-p\right)\hat{\mathcal{P}}^{abcd}\left(p\right)\tilde{T}^{\ell
m}_{cd}\left(p\right)~{}=~{}$
$\displaystyle\qquad\qquad\qquad=~{}\dfrac{-\gamma^{2}}{4}\sum_{\begin{subarray}{c}\ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\ell_{3}m_{3}\\\
\ell_{4}m_{4}\end{subarray}}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}\int\left(\prod_{i=1}^{4}\dfrac{\mathrm{d}^{2}p_{i}}{\left(2\pi\right)^{2}}~{}\phi_{\ell_{i}m_{i}}\left(p_{i}\right)\right)~{}p_{1,a}p_{2,b}\hat{\mathcal{P}}^{abcd}\left(p\right)p_{3,c}p_{4,d}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
C\left[1;2;3;4\right]\left(2\pi\right)^{2}\delta^{(2)}\left(p_{1}+p_{2}-p\right)\left(2\pi\right)^{2}\delta^{(2)}\left(p_{3}+p_{4}+p\right)$
$\displaystyle\qquad\qquad\qquad=~{}\dfrac{-\gamma^{2}}{4}\sum_{\begin{subarray}{c}\ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\ell_{3}m_{3}\\\
\ell_{4}m_{4}\end{subarray}}\int\left(\prod_{i=1}^{4}\dfrac{\mathrm{d}^{2}p_{i}}{\left(2\pi\right)^{2}}~{}\phi_{\ell_{i}m_{i}}\left(p_{i}\right)\right)~{}p_{1,a}p_{2,b}\hat{\mathcal{P}}^{abcd}\left(p_{1}+p_{2}\right)p_{3,c}p_{4,d}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
C\left[1;2;3;4\right]\left(2\pi\right)^{2}\delta^{(2)}\left(\sum_{i=1}^{4}p_{i}\right)\,,$
(4.17)

where we defined

$\displaystyle C\left[i;j;k;l\right]~{}\coloneqq~{}\sum_{\ell m}C\left[\ell
m;\ell_{i}m_{i};\ell_{j}m_{j}\right]C\left[\ell
m;\ell_{k}m_{k};\ell_{l}m_{l}\right]\,.$ (4.18)

Therefore, the effective matter action (4) can be written as

$\displaystyle\sum_{\ell m}S_{\ell m}~{}$
$\displaystyle=~{}-\dfrac{1}{2}\sum_{\ell
m}\int\dfrac{\mathrm{d}^{2}p}{\left(2\pi\right)^{2}}~{}\phi_{\ell,m}\left(p^{2}+\mu^{2}\lambda\right)\phi_{\ell,m}$
$\displaystyle\qquad-\dfrac{1}{4!}\sum_{\begin{subarray}{c}\ell_{1}m_{1}\\\
\ell_{2}m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\ell_{3}m_{3}\\\
\ell_{4}m_{4}\end{subarray}}\int\left(\prod_{i=1}^{4}\dfrac{\mathrm{d}^{2}p_{i}}{\left(2\pi\right)^{2}}~{}~{}\phi_{\ell_{i}m_{i}}\left(p_{i}\right)\right)~{}\left(2\pi\right)^{2}\delta^{(2)}\left(\sum_{i=1}^{4}p_{i}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
3C\left[i;j;k;l\right]V\left[p_{i};p_{j};p_{k};p_{l}\right]\,.\qquad\qquad$
(4.19)

with

$\displaystyle V\left[p_{1};p_{2};p_{3};p_{4}\right]~{}$
$\displaystyle\coloneqq~{}2\gamma^{2}p_{1,a}p_{2,b}\hat{\mathcal{P}}^{abcd}\left(p_{1}+p_{2}\right)p_{3,c}p_{4,d}\,.$
(4.20)

Since the interaction term is symmetric under exchange of the different scalar
partial wave legs, the corresponding Feynman rule for the vertex reads

$\displaystyle
i\Biggr{(}C\left[1;2;3;4\right]V\left[p_{1};p_{2};p_{3};p_{4}\right]+C\left[1;3;2;4\right]V\left[p_{1};p_{3};p_{2};p_{4}\right]+C\left[1;4;2;3\right]V\left[p_{1};p_{4};p_{2};p_{3}\right]\Biggr{)}\,.$
(4.21)

From the perspective of the original theory with the graviton, the three terms
correspond to $s$, $t$, and $u$ channel scattering.

###  4.1 Approximate spherical symmetry

As shown in [4, 5], an important simplification of the theory can be obtained
by fixing one of the scalars in every pair to be in the s-wave such that the
Clebsch-Gordon coefficients diagonalise as

$\displaystyle C\left[\ell
m;\ell_{i}m_{i};\ell_{j}m_{j}\right]~{}=~{}2\delta_{\ell\ell_{i}}\delta_{mm_{i}}\quad\text{with}\quad\ell_{j}~{}=~{}0~{}=~{}m_{j}\,,$
(4.22)

where the factor of two accounts for the fact that either of the scalar legs
may be put in the s-wave. For the same reason, the $u$-channel (where $p_{1}$
and $p_{3}$ carry angular momentum) is projected out in this approximation.
This results in the following vertex

$\displaystyle
2i\gamma^{2}\Biggr{(}p_{1,a}p_{2,b}\hat{\mathcal{P}}^{abcd}\left(p_{1}+p_{2}\right)p_{3,c}p_{4,d}+p_{1,a}p_{4,b}\hat{\mathcal{P}}^{abcd}\left(p_{1}+p_{4}\right)p_{2,c}p_{3,d}\Biggr{)}\,.$
(4.23)

The first term corresponds to an exchange of the transverse scalar
$\mathcal{K}$ which is subleading, while the second term arises from an
exchange of the longitudinal field $\mathfrak{h}_{ab}$. This result
generalises the one of [4] where some further simplifying assumptions were
made (for instance, external scalars were assumed to be null in that
reference, while they have a small effective mass in the two-dimensional
theory).

###  4.2 Scattering of s-waves

Instead of considering the approximately spherically symmetric case, another
interesting possibility is to consider pure s-wave scattering of all external
legs [5]. Plugging in the propagator for the graviton monopole (2.37) and
taking the additional symmetry factors of identical external legs into
account, the vertex now reduces to

$\displaystyle\dfrac{\gamma^{2}}{\mu^{2}}\Biggr{[}\left(p_{1}\cdot
p_{2}\right)\left(p_{3}\cdot p_{4}\right)+\left(p_{1}\cdot
p_{3}\right)\left(p_{2}\cdot p_{4}\right)+\left(p_{1}\cdot
p_{4}\right)\left(p_{2}\cdot p_{3}\right)\Biggr{]}\,.$ (4.24)

##### A simple proposal for the inelastic ladder of ladders

It was shown in [5] that a certain class of infinitely many loop corrections
to the tree-level $2\rightarrow 2N$ amplitudes can be computed explicitly.
These were called the ‘ladder of ladders’ diagrams and were computed
essentially by replacing the exchanged virtual gravitons by a corresponding
quantity that captures the $2\rightarrow 2$ ladder. Several such ladders were
then glued together to produce the $2\rightarrow 2N$ ladder of ladders.

In this and the previous subsections, we saw that tree-level $2\rightarrow 2$
amplitude is equivalent to the four-vertex interaction in the effective scalar
theory. This implies that this four-vertex can be promoted to a physically
relevant four-point function of interest. Therefore, replacing this tree-level
vertex by the corresponding ladder and gluing several such ladders together,
we immediately find a simple way to reproduce the $2\rightarrow 2N$ ladder of
ladders amplitude.

##  5 Computationally effective theory

In the previous section, we found that certain modes in the graviton decouple
from cubic interactions and therefore do not contribute to the ensuing
physical S-matrix. This allowed us to combine the interacting modes and the
external scalar modes into an effective scalar theory with a four-vertex that
captures the $2\rightarrow 2$ graviton exchange. Pictorially, this is
illustrated in Fig. 1. While this may seem to be an unnecessary rewriting of
the original theory at hand, we will now argue that it is particularly
efficient for computations.

Figure 1: The three channels of $2\rightarrow 2$ tree-level graviton exchange
diagrams can be rewritten as a four-vertex in an effective scalar theory. This
vertex is given in (4.21). As argued in Section 4.2, this vertex can be
promoted to capture the $2\rightarrow 2$ eikonal ladder of [3, 4] to capture
the inelastic ladder of ladders amplitudes of [5].

One of the important utilities of this rewriting is the drastic reduction in
the number of diagrams that need to be computed using the four-vertex to
capture a significantly larger number of graviton exchange diagrams. This is
illustrated in Fig. 2, where we consider tree-level $2\rightarrow 4$
scattering. In the theory with graviton exchanges, there are eighteen such
diagrams that contribute for fixed external legs, nine of which we draw.
Rewritten in terms of the four-vertex, all of these nine diagrams are
contained in the single tree-level diagram show on the right in Fig. 2. This
can easily be computed and contains two copies of the four-vertex (4.21) and
the scalar propagator from the quadratic term in (4). Therefore, the complete
list of eighteen diagrams is computed by two topologically distinct scalar
four-vertex diagrams.

Figure 2: Consider nine different diagrams that capture $2\rightarrow 4$
scattering mediated by graviton exchanges at tree-level shown on the left hand
side in this figure. All of these are computed in one go by the tree-level
amplitude mediated by the four-vertex (4.21) depicted pictorially on the right
hand side above.

Finally, it is of particular importance to note that the utility of the four-
vertex rewriting is not restricted to tree-level diagrams. It was noticed in
[5] that general inelastic loop diagrams are difficult to compute. Such
general diagrams were called the cobweb diagrams. As an example, consider a
certain three-loop $2\rightarrow 4$ cobweb diagram666An appropriate
resummation of such diagrams is of particular interest to explore the
potential existence of new chaos exponents in higher moments in the spirit of
[32]. shown in Fig. 3. There are 972 such diagrams, one of which we draw as a
representative on the left hand side of the said figure. Such diagrams are all
computed by four topologically distinct scalar three-loop diagrams of the kind
shown on the right hand side of the same figure. Each such scalar diagram
contains five four-vertices and each such vertex is a combination of three
$2\rightarrow 2$ diagrams as shown in Fig. 1. Therefore, the total number of
diagrams contained in the four topologically distinct scalar diagrams is
$4\times 3^{5}=972$. As in Fig. 2 several diagrams of the kind on the left are
captured by a single diagram of the kind on the right. At loop level, not only
are the number of diagrams to be computed reduced but also the topologies of
the contributing diagrams in the four-vertex theory are conceivably simpler
and more tractable.

Figure 3: A representative of the three-loop cobweb diagrams is show on the
left. Several such diagrams are captured by a single three-loop scalar diagram
mediated by the four-vertex (4.21) shown on the right.

##  6 Black hole entropy from multiplicity

How black hole entropy may emerge from the scattering approach to black hole
dynamics is rather mysterious. The only game in town appears to be to impose a
short-distance cut-off that results in the desired result [10, 13]. Here, we
make an intriguing observation that appears to emerge from the second
quantised approach of the present article. The entropy contained in the
multiplicity of the external particles in inelastic scattering yields almost
exactly the black hole entropy.

In [5], inelastic amplitudes with particle production were computed. In
particular, it was shown that in $2\rightarrow 2N$ tree-level amplitudes grow
exponentially. As mentioned in the introduction, the characteristic time delay
associated with this scattering process is Page time, suggesting a breakdown
of Hawking’s free field theory thereafter. Of the $2N$ external particles, two
sets of $N$ particles are identical. Therefore, the total multiplicity of the
external states is given by

$\displaystyle\Omega~{}=~{}\dfrac{(2N)!}{(N!)^{2}}~{}\sim~{}4^{N}\,.$ (6.1)

The corresponding entropy is then

$\displaystyle S_{\Omega}~{}\sim~{}N\log\left(4\right)\,.$ (6.2)

It was also shown in [5] that the $2\rightarrow 2N$ amplitude is sharply
peaked about a specific value determined by the centre of mass energy, $E$, of
the scattering process

$\displaystyle N_{max}~{}=~{}\dfrac{E^{2}\kappa^{2}}{e}\,,$ (6.3)

where $\kappa^{2}=8\pi G$. To find the entropy associated with the
multiplicity of the external states, we need knowledge of the relevant centre
of mass energy of the scattering process to be considered. Lacking a first
principle derivation, there are several arguments to be made in favour of the
choice $E=M_{BH}$:

  * •

The canonical energy scale of the system we would like to probe is indeed
$M_{BH}$.

  * •

The Eisenbud-Wigner time-delay associated with elastic $2\rightarrow 2$
scattering has been shown to be scrambling time, agreeing with the expectation
from [11], only if we set $E=M_{BH}$ [5]. This is perhaps the strongest
indication in favour of this choice.

  * •

In similar vein, the Eisenbud-Wigner time-delay associated with inelastic
$2\rightarrow 2N$ scattering has been shown to be Page time when $E=M_{BH}$
[5]. An interpretation of this result is that when $E=M_{BH}$ energy is thrown
into the black hole, it doubles the energy contained in the black hole. After
the scattering process, momentum conservation implies that all the energy that
went in has returned and therefore, the black hole has halved in size
returning to its original mass. It is therefore natural that the half-life
time of the black hole is how long one has to wait for information to return.

Therefore, with $E=M_{BH}$, we have

$\displaystyle N_{max}~{}=~{}\dfrac{2}{e}4\pi
GM_{\text{BH}}^{2}~{}=~{}\dfrac{2}{eG}\dfrac{4\pi
R^{2}}{4}~{}=~{}\dfrac{2}{eG}A~{}=~{}\dfrac{2}{e}S_{BH}\,.$ (6.4)

Inserting this into (6.2), we find

$\displaystyle S_{\Omega}~{}\sim~{}S_{BH}\dfrac{2\log\left(4\right)}{e}\,.$
(6.5)

The proportionality factor is evidently of order one. With very little input,
it is remarkable that such a result emerges!

##  7 Summary

In this article, we have developed a toolbox that can be used to compute
scattering amplitudes in the presence of a fluctuating black hole background.
In particular, we found the graviton propagator near the horizon of a
Schwarzschild black hole in a partial wave basis for all angular momentum
modes of either parity. We also showed that not all graviton modes are
interacting at leading (three-point) order and found the propagator for the
interacting modes. We then found that this theory can be rewritten (without
loss of generality) as a scalar theory with a particular four-vertex that
captures the elastic $2\rightarrow 2$ graviton exchange. We demonstrated that
this rewriting is exceptionally effective for computations, dramatically
reducing the number of computations in comparison to the original formulation.

This toolbox can be used for many interesting computations including all
scattering amplitudes of interest. In addition to addressing some of the
shortcomings mentioned in the introduction, it would also be interesting to
address the issue of antipodal identification that appears to be a natural
boundary condition to glue the future of the past horizon and the past of the
future horizon together, for consistency [14, 33, 34, 35, 36, 37]. Extensions
to incorporate other standard model fields and charged particles is another
straightforward application. In upcoming work, we hope to address several such
applications that rely on the the tools developed in this article.

## Acknowledgements

We are grateful to Gerard ’t Hooft for various helpful discussions on related
topics. We also thank Fabiano Feleppa and Guillermo Cortina for ongoing
collaborations on related ideas. This work is supported by the Delta-Institute
for Theoretical Physics (D-ITP) that is funded by the Dutch Ministry of
Education, Culture and Science (OCW).

##  Appendix A Background metric and choice of coordinates

In this paper, we work in Kruskal-Szekeres coordinates that are defined as

$\displaystyle xy~{}$ $\displaystyle=$
$\displaystyle~{}2R^{2}\left(1-\dfrac{r}{R}\right)e^{\tfrac{r}{R}-1}\,,$ (A.1)
$\displaystyle x/y~{}$ $\displaystyle=$ $\displaystyle~{}e^{2\tau}\,,$ (A.2)

where $r\left(x,y\right)$ is implicitly defined, $\tau=\tfrac{t}{2R}$ and
$R=2GM$ is the Schwarzschild radius (whose inverse we call $\mu=1/R$). We will
often write these coordinates as a two-vector $x^{a}=\\{x,y\\}$ with small
Latin letters denoting the longitudinal coordinates and capital Latin letters
denoting the transverse coordinates on the sphere. We will also work in
natural units $\hbar=c=1$. The Schwarzschild metric in these coordinates is
given by

$\displaystyle\text{d}s^{2}~{}$ $\displaystyle=$
$\displaystyle~{}-2A(r)\text{d}x\text{d}y+r^{2}\left(x,y\right)\text{d}\Omega^{2}_{2}\,,$
(A.3) $\displaystyle A(r)~{}$ $\displaystyle=$
$\displaystyle~{}\dfrac{R}{r}e^{1-\tfrac{r}{R}}\,.$ (A.4)

where we employ the mostly plus signature. To leading order in the near
horizon region, we have $r\sim R$ and therefore $A\sim 1$ resulting in the
product space

$\displaystyle\text{d}s^{2}~{}=~{}-2\text{d}x\text{d}y+R^{2}\text{d}\Omega^{2}_{2}\,.$
(A.5)

##### Christoffel symbols:

The non-vanishing Christoffel symbols of the Schwarzschild metric in Kruskal-
Szekeres coordinates are given by

$\displaystyle\Gamma^{c}_{ab}~{}$
$\displaystyle=~{}\delta^{c}_{(a}U_{b)}-\dfrac{1}{2}g_{ab}U^{c}$
$\displaystyle\Gamma^{C}_{AB}~{}=~{}\delta^{C}_{(A}W_{B)}-g_{AB}W^{C}$ (A.6)
$\displaystyle\Gamma^{A}_{Ba}~{}$ $\displaystyle=~{}V_{a}\delta^{A}_{B}$
$\displaystyle\Gamma^{a}_{AB}~{}=~{}-V^{a}g_{AB}\,,$ (A.7)

where $V_{a}=\partial_{a}\log r$, $U_{a}=\partial_{a}\log A$ and
$W_{A}=\partial_{A}\log\sin\theta$. The rest of the symbols
$\Gamma_{ab}^{A}=0=\Gamma_{aA}^{b}$ vanish identically.

##### The Riemann tensor:

Using the definition

$R^{\rho}_{\text{
}\mu\sigma\nu}~{}=~{}\partial_{\sigma}\Gamma^{\rho}_{\mu\nu}-\partial_{\nu}\Gamma^{\rho}_{\mu\sigma}+\Gamma^{\rho}_{\sigma\kappa}\Gamma^{\kappa}_{\mu\nu}-\Gamma^{\rho}_{\mu\kappa}\Gamma^{\kappa}_{\sigma\nu}$
(A.8)

the only non-vanishing components of the Riemann tensor are related to the
following components by symmetries

$\displaystyle R_{xyxy}~{}$ $\displaystyle=~{}\partial_{x}\partial_{y}\log
A\,,$ (A.9) $\displaystyle R_{\theta\phi\theta\phi}~{}$
$\displaystyle=~{}r^{2}\sin^{2}\theta\left(1+\dfrac{2\partial_{x}r\partial_{y}r}{A}\right)\,,$
(A.10) $\displaystyle R_{aAbB}~{}$ $\displaystyle=~{}g_{AB}S_{ab}\,,$ (A.11)

where in the last line we defined a new tensor

$\displaystyle S_{ab}~{}$
$\displaystyle\coloneqq-\nabla_{(a}V_{b)}-V_{a}V_{b}~{}=~{}-\tilde{\nabla}_{(a}V_{b)}-V_{a}V_{b}\,.$
(A.12)

In the second equality above, we defined $\tilde{\nabla}$ which is the
covariant derivative only along the longitudinal directions. This second
equality holds because $\Gamma_{ab}^{C}=0$.

###  A.1 The antisymmetric Levi-Civita tensor

Here we define the antisymmetric tensor on the transverse two-sphere as

$\displaystyle\epsilon_{AB}=r^{2}\sin\theta\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\,,$ (A.13)

with $\epsilon_{\theta\phi}=r^{2}\sin\theta=-\epsilon_{\phi\theta}$. Indices
are raised and lowered with the usual round metric on the two-sphere.
Therefore we have

$\displaystyle\epsilon_{A}^{\text{ }\text{ }B}=\begin{pmatrix}0&\sin\theta\\\
-\csc\theta&0\end{pmatrix}\quad\text{and}\quad\epsilon^{AB}=\frac{1}{r^{2}\sin\theta}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\,.$ (A.14)

Similarly, in the near horizon region $r\sim R$, we have the antisymmetric
tensor in the longitudinal directions

$\displaystyle\epsilon^{ab}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\quad\text{and}\quad\epsilon_{ab}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\,.$ (A.15)

##  Appendix B The group of traceless tensor operators

In this section, we consider the group $\bar{G}$ consisting of all invertible
rank four tensors that are (symmetric) traceless in their first and last pair
of indices. For $M,N\in\bar{G}$ the elements in the group are defined by

$\displaystyle\eta^{ab}M_{abcd}~{}$
$\displaystyle=~{}0~{}=~{}M_{abcd}\eta^{cd}\,.$ (B.1)

The group operation is given by

$\displaystyle\left(M\cdot N\right)_{abef}~{}\equiv~{}M_{abcd}N^{cd}_{\text{
}\text{ }\text{ }\text{ }ef}\,,$ (B.2)

where closure ($M\cdot N\in\bar{G}$) is guaranteed by the observation that
$M\cdot N$ is still traceless over $ab$ and $ef$. Of course, $\bar{G}$ is a
subgroup of the group $G$ that consists of all invertible rank four tensors.
The question of interest now is what the identity element $\bar{I}\in\bar{G}$
is, since this might differ from the identity element $I\in G$ given by
$I_{ab}^{cd}=\delta_{(a}^{c}\delta_{b)}^{d}$. Indeed we require $\bar{I}$ to
be traceless, but $I$ clearly isn’t. The obvious correction would be to make
it traceless

$\displaystyle\bar{I}_{ab}^{cd}~{}=~{}\delta_{(a}^{c}\delta_{b)}^{d}-\dfrac{1}{2}\eta_{ab}\eta^{cd}\,.$
(B.3)

It can now be checked that

$\displaystyle\eta^{ab}\bar{I}_{ab}^{cd}~{}=0~{}=~{}\bar{I}_{ab}^{cd}\eta_{cd}\quad\text{and}\quad\left(\bar{I}\cdot
M\right)_{abef}~{}=~{}\bar{I}_{ab}^{cd}M_{cdef}~{}=~{}M_{abef}$ (B.4)

hold, since $M$ is traceless. Therefore, we have

$\displaystyle M^{-1}\cdot M~{}=~{}M^{-1}_{abcd}M^{cd}_{\text{ }\text{ }\text{
}\text{ }ef}~{}=~{}\bar{I}_{abef}~{}=~{}\bar{I}\,.$ (B.5)

where we used that $\bar{I}_{abef}=\bar{I}_{efab}$ which ensures that there is
no ambiguity between using covariant or contravariant indices.

##  Appendix C Calculation of the propagator for the odd harmonics

We begin with the first line of the action (2):

$\displaystyle S~{}$
$\displaystyle=~{}-\dfrac{1}{2}\int\text{d}^{4}x\sqrt{-g}h^{\mu\nu}\left[\dfrac{1}{2}\left(2\nabla^{\rho}\nabla_{(\mu}h_{\nu)\rho}-\Box
h_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}h\right)-~{}\dfrac{1}{2}g_{\mu\nu}\biggr{(}\nabla^{\rho}\nabla^{\sigma}h_{\rho\sigma}-\Box
h\biggr{)}\right]$
$\displaystyle=~{}-\dfrac{1}{2}\int\text{d}^{4}x\sqrt{-g}h^{\mu\nu}\left[G^{(1)}_{\mu\nu}\right]\,,$
(C.1)

where $G^{(1)}_{\mu\nu}$ is the first order variation of the Einstein tensor.
Into this action, we will now plug in the odd harmonics

$h^{-}_{aA}~{}\eqqcolon~{}h_{a}\eta_{A}\quad\text{with}\quad\eta_{A}~{}\coloneqq~{}-{\epsilon_{A}}^{B}\partial_{B}Y_{\ell}^{m}\,.$
(C.2)

First, we notice that $\eta_{A}$ does not depend on the longitudinal
coordinates while $h_{a}$ is independent of the transverse spherical
coordinates. Since $g_{aA}=0$ and $h^{-}=g^{\mu\nu}h^{-}_{\mu\nu}=0$, we have

$\displaystyle G^{(1),-}_{aA}~{}$
$\displaystyle=~{}\dfrac{1}{2}g^{\rho\sigma}\bigr{(}\nabla_{\rho}\nabla_{a}h^{-}_{A\sigma}+\nabla_{\rho}\nabla_{A}h^{-}_{a\sigma}\bigr{)}-\dfrac{1}{2}\Box
h^{-}_{aA}\,.$ (C.3)

We now use the familiar property
$[\nabla_{\rho},\nabla_{\sigma}]h_{\mu\nu}=-R^{\tau}_{\text{ }\text{
}\mu\rho\sigma}h_{\tau\nu}-R^{\tau}_{\text{ }\text{
}\nu\rho\sigma}h_{\tau\mu}$, to rewrite the above as

$\displaystyle G^{(1),-}_{aA}~{}$
$\displaystyle=~{}\nabla_{(a}L_{A)}-\dfrac{1}{2}g^{\rho\sigma}\bigr{(}{R^{\tau}}_{A\rho
a}h^{-}_{\tau\sigma}+{R^{\tau}}_{\sigma\rho a}h^{-}_{\tau A}+{R^{\tau}}_{a\rho
A}h^{-}_{\tau\sigma}+{R^{\tau}}_{\sigma\rho A}h^{-}_{\tau
a}\bigr{)}-\dfrac{1}{2}\Box h^{-}_{aA}\,,$ (C.4)

where we defined $L_{\mu}\coloneqq\nabla^{\nu}h^{-}_{\mu\nu}$. Using the
Riemann tensor components written out in Appendix A and writing out the
covariant derivatives, we find

$\displaystyle
G^{(1),-}_{aA}~{}=~{}\dfrac{1}{2}\left(\partial_{a}-2V_{a}\right)L_{A}+\dfrac{1}{2}\partial_{A}L_{a}+S_{ab}h^{b}\eta_{A}-\dfrac{1}{2}\Box
h^{-}_{aA}\,,$ (C.5)

where we used $R_{AaBb}h_{-}^{bB}~{}=~{}S_{ab}h^{b}\eta_{A}$ from the
definition of the quantity $S_{ab}$ in Appendix A. We will now calculate the
quantities $L_{\mu}$ and the $\Box$ term. The first quantity to calculate is

$\displaystyle L_{A}~{}$ $\displaystyle=~{}g^{\mu\nu}\nabla_{\mu}h^{-}_{A\nu}$
$\displaystyle=~{}g^{\mu\nu}\partial_{\mu}h^{-}_{A\nu}-g^{\mu\nu}\Gamma_{\mu
A}^{\rho}h^{-}_{\rho\nu}-g^{\mu\nu}\Gamma_{\mu\nu}^{\rho}h^{-}_{A\rho}$
$\displaystyle=~{}\eta_{A}g^{ab}\left(\partial_{a}+2V_{a}\right)h_{b}\,,$
(C.6)

where we inserted $h_{-}^{aA}=h^{a}\eta^{A}$ and the relevant Christoffel
symbols to arrive at the last line. Similarly,

$\displaystyle L_{a}~{}$ $\displaystyle=~{}g^{\mu\nu}\nabla_{\mu}h^{-}_{a\nu}$
$\displaystyle=~{}g^{\mu\nu}\partial_{\mu}h^{-}_{a\nu}-g^{\mu\nu}\Gamma_{\mu
a}^{\rho}h^{-}_{\rho\nu}-g^{\mu\nu}\Gamma_{\mu\nu}^{\rho}h^{-}_{a\rho}$
$\displaystyle=~{}h_{a}\dfrac{1}{\sin\theta}g^{AB}\partial_{A}\left(\sin\theta\eta_{B}\right)$
$\displaystyle=~{}h_{a}\hat{\nabla}^{A}\eta_{A}$
$\displaystyle=~{}-h_{a}\hat{\nabla}^{A}\epsilon_{BC}\hat{\nabla}^{C}Y_{\ell
m}$ $\displaystyle=~{}0\,,$ (C.7)

where in the second last line, we identified the covariant derivative on the
two-sphere and to arrive at the last line, we observe that the covariant
derivatives acting on $Y_{\ell m}$ are in competition with the antisymmetry of
$\epsilon_{BC}$.

The term containing the d’Alembertian can be written as

$\displaystyle\Box h^{-}_{aA}~{}$ $\displaystyle=~{}\Box
h^{-}_{aA}-\partial^{c}\left(\Gamma_{ca}^{b}h^{-}_{bA}\right)-\partial^{\nu}\left(\Gamma_{\nu
A}^{B}h^{-}_{Ba}\right)-g^{\mu\nu}\Gamma^{\tau}_{\mu\nu}\left(\partial_{\tau}h^{-}_{aA}-\Gamma_{\tau
a}^{b}h^{-}_{bA}-\Gamma_{\tau A}^{B}h^{-}_{Ba}\right)$
$\displaystyle\qquad-g^{\mu\nu}\Gamma_{\mu
a}^{b}\partial_{\nu}h^{-}_{bA}+g^{\mu\nu}\Gamma_{\mu
a}^{\tau}\Gamma_{\nu\tau}^{b}h^{-}_{bA}-g^{\mu\nu}\Gamma_{\mu
A}^{B}\partial_{\nu}h^{-}_{Ba}$ $\displaystyle\qquad+g^{\mu\nu}\Gamma_{\mu
A}^{\tau}\Gamma_{\nu\tau}^{B}h^{-}_{Ba}+2g^{\mu\nu}\Gamma_{\mu
a}^{\tau}\Gamma_{\nu A}^{\kappa}h^{-}_{\kappa\tau}\,.$ (C.8)

We now insert $h^{-}_{aA}=h^{-}_{Aa}=h_{a}\eta_{A}$ and separate the
longitudinal terms from the transverse ones. This gives

$\displaystyle\Box h^{-}_{aA}~{}$
$\displaystyle=~{}\eta_{A}\biggr{[}g^{cd}\partial_{c}\partial_{d}h_{a}-\partial^{c}\left(\Gamma_{ac}^{b}h_{b}\right)-\partial^{b}\left(V_{b}h_{a}\right)-g^{\mu\nu}\Gamma_{\mu\nu}^{b}\partial_{b}h_{a}+g^{\mu\nu}\Gamma_{\mu\nu}^{c}\Gamma_{ac}^{b}h_{b}+g^{\mu\nu}\Gamma_{\mu\nu}^{b}V_{b}h_{a}$
$\displaystyle\qquad\qquad-\Gamma_{ac}^{b}\partial^{c}h_{b}+g^{cd}\Gamma^{e}_{ca}\Gamma^{b}_{de}h_{b}-2V_{a}V^{b}h_{b}-V_{b}\partial^{b}h_{a}+2V^{c}\Gamma_{ca}^{b}h_{b}-2V_{a}V^{b}h_{b}\biggr{]}$
$\displaystyle\qquad+h_{a}\biggr{[}g^{BC}\partial_{B}\partial_{C}\eta_{A}-\partial^{C}\left(\Gamma^{B}_{AC}\eta_{B}\right)-g^{\mu\nu}\Gamma_{\mu\nu}^{B}\partial_{B}\eta_{A}+g^{\mu\nu}\Gamma_{\mu\nu}^{C}\Gamma^{B}_{AC}\eta_{B}$
$\displaystyle\qquad\qquad-\Gamma_{AC}^{B}\partial^{C}\eta_{B}+g^{CD}\Gamma_{CA}^{E}\Gamma^{B}_{DE}\eta_{B}\biggr{]}\,.$
(C.9)

Writing longitudinal covariant derivatives with a tilde and transverse ones
with a hat, this can be compactly written as

$\displaystyle\Box h^{-}_{aA}~{}$
$\displaystyle=~{}\eta_{A}\tilde{\Box}h_{a}-4\eta_{A}V_{a}V^{b}h_{b}-2\eta_{A}V^{b}V_{b}h_{a}-\eta_{A}\left(\partial^{b}V_{b}\right)h_{a}+h_{a}\hat{\Delta}\eta_{A}\,,$
(C.10)

where $\hat{\Delta}$ is the Laplacian on the two sphere.

###  C.1 Integrating the two-sphere out

Piecing the above terms together, we have:

$\displaystyle G^{(1),-}_{aA}~{}$
$\displaystyle=~{}\dfrac{1}{2}\eta_{A}\biggr{[}-\tilde{\Box}h_{a}+\left(\tilde{\nabla}_{a}-2V_{a}\right)\left(\tilde{\nabla}_{b}+2V_{b}\right)h^{b}+2S_{ab}h^{b}+4V_{a}V_{b}h^{b}+2V^{2}h_{a}$
$\displaystyle\qquad\qquad+\left(\tilde{\nabla}\cdot
V\right)h_{a}\biggr{]}+\dfrac{1}{2}h_{a}\hat{\Box}\eta_{A}\,.$ (C.11)

We notice that

$\displaystyle\hat{\Box}\eta_{A}~{}$
$\displaystyle=~{}-\epsilon^{AB}\eta^{CD}\hat{\nabla}_{C}\hat{\nabla}_{D}\hat{\nabla}_{B}Y_{\ell
m}$
$\displaystyle=~{}-\epsilon^{AB}\left(\eta^{CD}\hat{\nabla}_{C}\hat{\nabla}_{B}\hat{\nabla}_{D}Y_{\ell
m}\right)$
$\displaystyle=~{}-\epsilon^{AB}\left(\hat{\nabla}_{B}\hat{\Box}Y_{\ell
m}-\eta^{CD}{\hat{R}^{E}}_{~{}DCB}\hat{\nabla}_{E}Y_{\ell m}\right)$
$\displaystyle=~{}-\epsilon^{AB}\left(\hat{\nabla}_{B}\hat{\Box}Y_{\ell
m}+{\hat{R}^{E}}_{~{}B}\hat{\nabla}_{E}Y_{\ell m}\right)$
$\displaystyle=~{}-\epsilon^{AB}\left(\hat{\nabla}_{B}\hat{\Box}Y_{\ell
m}+\hat{R}\hat{\nabla}_{B}Y_{\ell m}\right)$
$\displaystyle=~{}-\dfrac{\lambda-2}{r^{2}}\eta_{A}\,,$ (C.12)

where the second equality is allowed because the spherical harmonics are
scalar functions on the sphere, and in the last equality, we used that the
Ricci tensor of the two sphere satisfies
$\hat{R}_{AB}=\frac{1}{2}\hat{R}g_{AB}=\frac{1}{r^{2}}g_{AB}$.

Therefore, the action for the odd harmonics can now be written as

$\displaystyle S~{}$
$\displaystyle=~{}-\sum_{\ell,m,\ell^{\prime},m^{\prime}}\int\text{d}^{4}x\sqrt{-g}h^{a}_{\ell
m}\eta^{A}_{\ell m}\left[G^{(1),-}_{aA,\ell^{\prime}m^{\prime}}\right]$
$\displaystyle=~{}-\sum_{\ell,m,\ell^{\prime},m^{\prime}}\dfrac{1}{2}\int\text{d}^{2}x~{}A\left(x,y\right)r\left(x,y\right)^{2}\int\text{d}\Omega_{2}~{}\eta^{A}_{\ell
m}\eta_{A,\ell^{\prime}m^{\prime}}$ $\displaystyle\qquad\qquad h^{a}_{\ell
m}\biggr{[}\left(-\tilde{\Box}+2V^{c}V_{c}+\left(\tilde{\nabla}\cdot
V\right)+\dfrac{\lambda-2}{r^{2}}\right)g_{ab}+\left(\tilde{\nabla}_{a}-2V_{a}\right)\left(\tilde{\nabla}_{b}+2V_{b}\right)$
$\displaystyle\qquad\qquad\qquad+2S_{ab}+4V_{a}V_{b}\biggr{]}h^{b}_{\ell^{\prime}m^{\prime}}$
$\displaystyle=~{}-\dfrac{\lambda-1}{2}\sum_{\ell,m}\int\text{d}^{2}x~{}A\left(x,y\right)h^{a}_{\ell
m}\biggr{[}\left(-\tilde{\Box}+2V^{c}V_{c}+\left(\tilde{\nabla}\cdot
V\right)+\dfrac{\lambda-2}{r^{2}}\right)g_{ab}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\left(\tilde{\nabla}_{a}-2V_{a}\right)\left(\tilde{\nabla}_{b}+2V_{b}\right)+2S_{ab}+4V_{a}V_{b}\biggr{]}h^{b}_{\ell
m}\,,$ (C.13)

where in the third equality, we inserted the definition of $\eta_{A}$ and
applied Stokes’ theorem before integrating over the sphere. Remarkably, the
action is proportional to $\lambda-1=\ell^{2}+\ell$. Therefore, the
contribution of these odd modes for large $\ell$ is heavily suppressed in
comparison to the even modes.

###  C.2 Weyl transformation and the near-horizon approximation

The quadratic operator in (C.1) reads

$\displaystyle\mathcal{D}_{ab}~{}$
$\displaystyle=~{}A\left(x,y\right)\biggr{[}\left(-\tilde{\Box}+2V^{c}V_{c}+\left(\tilde{\nabla}\cdot
V\right)+\dfrac{\lambda-2}{r^{2}}\right)g_{ab}+\left(\tilde{\nabla}_{a}-2V_{a}\right)\left(\tilde{\nabla}_{b}+2V_{b}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+2S_{ab}+4V_{a}V_{b}\biggr{]}$
$\displaystyle=~{}A\left(x,y\right)\biggr{[}\left(-\tilde{\Box}+2V^{c}V_{c}+\left(\tilde{\nabla}\cdot
V\right)+\dfrac{\lambda-2}{r^{2}}\right)g_{ab}+\tilde{\nabla}_{a}\tilde{\nabla}_{b}+2\left(\tilde{\nabla}_{a}V_{b}\right)+2V_{b}\tilde{\nabla}_{a}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-2V_{a}\tilde{\nabla}_{b}+2S_{ab}\biggr{]}$
$\displaystyle=~{}A\left(x,y\right)\biggr{[}\left(-\tilde{\Box}+2V^{c}V_{c}+\left(\tilde{\nabla}\cdot
V\right)+\dfrac{\lambda-2}{r^{2}}\right)g_{ab}+\tilde{\nabla}_{a}\tilde{\nabla}_{b}+2\left(\tilde{\nabla}_{a}V_{b}\right)+2V_{b}\tilde{\nabla}_{a}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-2V_{a}\tilde{\nabla}_{b}-2V_{a}V_{b}\biggr{]}$

where in the third equality, we plugged in the definition of $S_{ab}$ from
(A.12). Given the complicated background spacetime, this operator is not
invertible. However, we will now exploit the conformal flat nature of the
metric $g_{ab}=A\left(x,y\right)\eta_{ab}$ to make the following field
redefinition

$h_{a}~{}=~{}\sqrt{A\left(x,y\right)}\,\mathfrak{h}_{a}\,.$ (C.15)

This transforms the quadratic operator as

$\mathcal{D}_{ab}~{}\longrightarrow~{}\sqrt{A\left(x,y\right)}\mathcal{D}_{ab}\sqrt{A\left(x,y\right)}\,.$
(C.16)

In order to cycle the $\sqrt{A\left(x,y\right)}$ from the far right in the
above equation to the other side, we will make extensive use of the following
identity

$\partial_{a}\sqrt{A\left(x,y\right)}~{}=~{}\sqrt{A\left(x,y\right)}\left(\partial_{a}+\dfrac{1}{2}U_{a}\right)\,.$
(C.17)

We will now write all quantities appearing in the odd harmonic action (C.1) in
terms of the Weyl transformed field and the flat metric $\eta_{ab}$.
Therefore, in what follows, we will use the symbol “$\rightarrow$” to show the
step where the Weyl transformation and replacement of $g_{ab}$ by
$A\left(x,y\right)\eta_{ab}$ have been made.

The first term of interest is

$\displaystyle\tilde{\nabla}^{a}\tilde{\nabla}^{b}h_{b}~{}$
$\displaystyle=~{}\partial^{a}\partial^{b}h_{b}$
$\displaystyle\to~{}A\sqrt{A}\dfrac{1}{A}\partial^{a}\dfrac{1}{A}\partial^{b}\sqrt{A}~{}\mathfrak{h}_{b}$
$\displaystyle=~{}\left(\partial^{a}-\dfrac{1}{2}U^{a}\right)\left(\partial^{b}+\dfrac{1}{2}U^{b}\right)\mathfrak{h}_{b}\,.$
(C.18)

As desired there are only flat space derivatives and potential terms that are
artefacts of the curvature. Next, we consider

$\displaystyle\tilde{\nabla}_{a}\tilde{\nabla}^{a}h_{b}~{}$
$\displaystyle=~{}g^{ac}\tilde{\nabla}_{a}\tilde{\nabla}_{c}h_{b}$
$\displaystyle=~{}g^{cd}\partial_{c}\partial_{d}h_{b}-U_{c}\partial^{c}h_{b}-U_{b}\partial^{c}h_{c}+U^{a}\partial_{b}h_{a}-\dfrac{1}{2}\left(\partial^{c}U_{c}\right)h_{b}-\dfrac{1}{2}\left(\partial^{c}U_{b}\right)h_{c}+\dfrac{1}{2}\left(\partial_{b}U_{c}\right)h^{c}$
$\displaystyle\to~{}g^{cd}\left(\partial_{c}+\dfrac{1}{2}U_{c}\right)\left(\partial_{d}+\dfrac{1}{2}U_{d}\right)\mathfrak{h}_{b}-U_{c}\left(\partial^{c}+\dfrac{1}{2}U^{c}\right)\mathfrak{h}_{b}-U_{b}\left(\partial^{c}+\dfrac{1}{2}U^{c}\right)\mathfrak{h}_{c}$
$\displaystyle\qquad+U^{c}\left(\partial_{b}+\dfrac{1}{2}U_{b}\right)\mathfrak{h}_{c}-\dfrac{1}{2}\left(\partial^{c}U_{c}\right)\mathfrak{h}_{b}-\dfrac{1}{2}\left(\partial^{c}U_{b}\right)\mathfrak{h}_{c}+\dfrac{1}{2}\left(\partial_{b}U_{c}\right)\mathfrak{h}^{c}$
$\displaystyle=~{}\eta_{bc}\biggr{[}\partial^{2}-\dfrac{1}{4}U_{d}U^{d}\biggr{]}\mathfrak{h}^{c}+\biggr{[}-U_{b}\partial_{c}+U_{c}\partial_{b}\biggr{]}\mathfrak{h}^{c}\,,$
(C.19)

where in the second equality, we used that
$g^{cd}\Gamma^{e}_{cb}\Gamma^{a}_{de}=0$. Finally we consider the single
derivative terms:

$\displaystyle\tilde{\nabla}^{a}h_{b}~{}$
$\displaystyle=~{}\partial^{a}h_{b}-\dfrac{1}{2}U_{b}h^{a}-\dfrac{1}{2}U^{a}h_{b}+\dfrac{1}{2}\delta_{b}^{a}U^{c}h_{c}$
$\displaystyle\to~{}\partial^{a}\mathfrak{h}_{b}-\dfrac{1}{2}U_{b}\mathfrak{h}^{a}+\dfrac{1}{2}\delta_{b}^{a}U^{c}\mathfrak{h}_{c}\,.$
(C.20)

Putting these results together gives the following operator:

$\displaystyle\mathcal{D}_{ab}~{}$
$\displaystyle\to~{}\partial_{a}\partial_{b}+U_{[a}\partial_{b]}-4V_{[a}\partial_{b]}+\dfrac{1}{2}\left(\partial_{a}U_{b}\right)-\dfrac{1}{4}U_{a}U_{b}-2V_{a}V_{b}$
$\displaystyle\qquad+\eta_{ab}\left(-\partial^{2}+\dfrac{1}{4}U_{d}U^{d}-V_{c}U^{c}+2V_{c}V^{c}+\left(\partial_{c}V^{c}\right)+A\left(x,y\right)\dfrac{\lambda-2}{r^{2}}\right)\,.$
(C.21)

The biggest advantage of all the manipulations done so far is that the theory
is now entirely defined in two dimensional flat space with the Minkowski
metric $\eta_{ab}$. The remaining derivatives are all partial derivatives with
the curvature traded for complicated potential terms. This allows us the
luxury of defining familiar a Fourier transforms. So far all calculations were
exact. The problem still remains that the operator is not invertible
analytically. To remedy this problem, we will employ a near-horizon
approximation to find an inverse near the horizon.

### The near-horizon approximation

To leading order, near the horizon we have $r\sim R$. This implies that the
longitudinal coordinates satisfy $x,y\ll R$ or equivalently $\mu x_{a}\ll 1$.
The potentials now become

$\displaystyle V_{a}~{}\sim~{}\dfrac{1}{2}\mu^{2}x_{a}\quad\text{and}\quad
U_{b}~{}\sim~{}-\mu^{2}x_{b}\,.$ (C.22)

The quadratic operator therefore simplifies to

$\displaystyle\mathcal{D}^{ab}~{}=~{}\partial^{a}\partial^{b}-\eta^{ab}\partial^{2}-3\mu^{2}x^{[a}\partial^{b]}+\mu^{2}\eta^{ab}\left(\lambda-\dfrac{3}{2}\right)\,.$
(C.23)

Therefore, the action for the odd modes can be written as

$\displaystyle\sum_{\ell,m}S_{\ell,m}~{}=~{}\sum_{\ell,m}-\dfrac{\lambda-1}{2}\int\text{d}^{2}x~{}\mathfrak{h}^{a}\left(\partial_{a}\partial_{b}-\eta_{ab}\partial^{2}-3\mu^{2}x_{[a}\partial_{b]}+\mu^{2}\eta_{ab}\left(\lambda-\dfrac{3}{2}\right)\right)\mathfrak{h}^{b}\,.$
(C.24)

The quadratic operator is now easy to invert, using ideas developed in [4].

###  C.3 Propagator for the odd harmonics

To invert the operator, we first perform a Fourier transform, resulting in

$\displaystyle\mathcal{D}^{ab}~{}$
$\displaystyle=~{}\eta^{ab}\left(k^{2}+\mu^{2}\left(\lambda-\dfrac{3}{2}\right)\right)-k^{a}k^{b}-3\mu^{2}k^{[a}\partial_{k}^{b]}\,,$
(C.25)

where we used that

$\displaystyle\int\text{d}^{2}x\mathfrak{h}^{a}\left(-3\mu^{2}x_{[a}\partial_{b]}\right)\mathfrak{h}^{b}~{}$
$\displaystyle=~{}\int\text{d}^{2}x\int\text{d}^{2}k\int\text{d}^{2}k^{\prime}\mathfrak{h}^{a}\left(k\right)\mathfrak{h}^{b}\left(k^{\prime}\right)e^{ik\cdot
x}\left(-3\mu^{2}x_{[a}\partial_{b]}\right)e^{ik^{\prime}\cdot x}$
$\displaystyle=~{}\int\text{d}^{2}x\int\text{d}^{2}k\int\text{d}^{2}k^{\prime}\mathfrak{h}^{a}\left(k\right)\mathfrak{h}^{b}\left(k^{\prime}\right)e^{ik\cdot
x}\left(-3i\mu^{2}x_{[a}k^{\prime}_{b]}\right)e^{ik^{\prime}\cdot x}$
$\displaystyle=~{}\int\text{d}^{2}x\int\text{d}^{2}k\int\text{d}^{2}k^{\prime}\mathfrak{h}^{a}\left(k\right)\mathfrak{h}^{b}\left(k^{\prime}\right)e^{ik\cdot
x}\left(-3\mu^{2}k^{\prime}_{[b}\partial^{k^{\prime}}_{a]}\right)e^{ik^{\prime}\cdot
x}$
$\displaystyle=~{}\int\text{d}^{2}x\int\text{d}^{2}k\int\text{d}^{2}k^{\prime}e^{i(k+k^{\prime})\cdot
x}\mathfrak{h}^{a}\left(k\right)\left(3\mu^{2}\partial^{k^{\prime}}_{[a}k^{\prime}_{b]}\right)\mathfrak{h}^{b}\left(k^{\prime}\right)$
$\displaystyle=~{}\left(2\pi\right)^{2}\int\text{d}^{2}k^{\prime}\mathfrak{h}^{a}\left(-k^{\prime}\right)\left(-3\mu^{2}k^{\prime}_{[a}\partial^{k^{\prime}}_{b]}\right)\mathfrak{h}^{b}\left(k^{\prime}\right)\,,$
(C.26)

where in the second equality we rewrote the partial derivative, in the third,
we commuted $x$ and $k$ before rewriting $x$ as a partial derivative with
respect to $k$, integrated by parts in the fourth, and used antisymmetry in
the final equality.

We would now like to find the Green’s function $\mathcal{G}^{bc}$ that
satisfies

$\displaystyle\mathcal{D}_{ab}\,\mathcal{G}^{bc}~{}=~{}\delta_{a}^{c}\,.$
(C.27)

Dilation invariance of the background $x\to\lambda x,\,y\to\lambda^{-1}y$
corresponds to time translation invariance of the black hole background. This
suggests the following form for the Green’s function

$\displaystyle\mathcal{G}^{bc}~{}=~{}F\left(k^{2}\right)\left(\eta^{bc}+G\left(k^{2}\right)k^{b}k^{c}\right)\,,$
(C.28)

where $F,\,G$ are to be determined. Acting with the operator $\mathcal{D}$ on
$\mathcal{G}$ gives

$\displaystyle\mathcal{D}_{ab}\,\mathcal{G}^{bc}~{}$
$\displaystyle=~{}F\delta_{a}^{c}\left(k^{2}+\mu^{2}\left(\lambda-\dfrac{3}{2}\right)\right)+Fk_{a}k^{c}\left(G\mu^{2}\left(\lambda-\dfrac{3}{2}\right)-1\right)$
$\displaystyle\qquad-3\mu^{2}\biggr{(}k_{[a}\partial_{k}^{c]}F+k_{[a}\partial^{k}_{b]}\left(FGk^{b}k^{c}\right)\biggr{)}\,.$
(C.29)

Action of the the derivatives on $F,\,G$ can be worked out using the chain
rule resulting in $\partial_{a}F=2F^{\prime}k_{a}$ and
$\partial_{a}G=2G^{\prime}k_{a}$. However, since all derivatives contain an
antisymmetrisation, we see that derivatives on the scalar factors vanish:

$\displaystyle
k_{[a}\partial_{k}^{c]}F~{}=~{}2F^{\prime}k_{[a}k^{c]}~{}=~{}0\,.$ (C.30)

Therefore, the total contribution from the derivatives is given by

$\displaystyle
k_{[a}\partial_{k}^{c]}F+k_{[a}\partial^{k}_{b]}\left(k^{b}k^{c}FG\right)~{}$
$\displaystyle=~{}FGk_{[a}\delta^{b}_{b]}k^{c}+FGk_{[a}\delta^{c}_{b]}k^{b}$
$\displaystyle=~{}\dfrac{1}{2}FG\biggr{(}k_{a}k^{c}\delta^{b}_{b}-k_{b}k^{c}\delta_{a}^{b}+k_{a}k^{b}\delta^{c}_{b}-k^{2}\delta^{c}_{a}\biggr{)}$
$\displaystyle=~{}FGk_{a}k^{c}-\dfrac{1}{2}FGk^{2}\delta^{c}_{a}\,.$ (C.31)

Inserting this contribution into the defining equation gives

$\displaystyle\mathcal{D}_{ab}\,\mathcal{G}^{bc}~{}$
$\displaystyle=~{}F\delta_{a}^{c}\left(k^{2}+\mu^{2}\left(\lambda-\dfrac{3}{2}\right)+\dfrac{3}{2}\mu^{2}Gk^{2}\right)+Fk_{a}k^{c}\left(G\mu^{2}\left(\lambda-\dfrac{9}{2}\right)-1\right)~{}=~{}\delta_{a}^{c}\,.$
(C.32)

The solution to this equation is easily seen to be given by

$\displaystyle
F~{}=~{}\dfrac{\lambda-\frac{9}{2}}{\lambda-3}\dfrac{1}{k^{2}+\mu^{2}\frac{\left(\lambda-\frac{3}{2}\right)\left(\lambda-\frac{9}{2}\right)}{\lambda-3}}\quad\text{and}\quad
G~{}=~{}\dfrac{1}{\mu^{2}\left(\lambda-\frac{9}{2}\right)}\,.$ (C.33)

We now finally have the propagator for the odd harmonics which reads

$\displaystyle\mathcal{G}^{ab}~{}=~{}\dfrac{\lambda-\frac{9}{2}}{\lambda-1}\dfrac{1}{k^{2}\left(\lambda-3\right)+\mu^{2}\left(\lambda-\frac{3}{2}\right)\left(\lambda-\frac{9}{2}\right)}\left(\eta^{ab}+\dfrac{k^{a}k^{b}}{\mu^{2}\left(\lambda-\frac{9}{2}\right)}\right)\,.$
(C.34)

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