# Off-shell duality invariance of
Schwarzschild perturbation theory

Adam R. Solomon

Department of Physics and Astronomy, McMaster University,
1280 Main Street West, Hamilton ON, Canada

Perimeter Institute for Theoretical Physics,
31 Caroline Street North, Waterloo ON, Canada<EMAIL_ADDRESS>
###### Abstract

We explore the duality invariance of the Maxwell and linearized Einstein-
Hilbert actions on a non-rotating black hole background. On shell these
symmetries are electric-magnetic duality and Chandrasekhar duality,
respectively. Off shell they lead to conserved quantities; we demonstrate that
one of the consequences of these conservation laws is that even- and odd-
parity metric perturbations have equal Love numbers. Along the way we derive
an action principle for the Fackerell-Ipser equation and Teukolsky-Starobinsky
identities in electromagnetism.

###### Contents

  1. 1 Introduction
  2. 2 Schwarzschild background in the $2+2$ and GHP formalisms
    1. 2.1 $2+2$ decomposition
    2. 2.2 Geroch-Held-Penrose (GHP) formalism
  3. 3 Massless scalar
  4. 4 Electromagnetism
    1. 4.1 Electric-magnetic duality
    2. 4.2 Maxwell in GHP
  5. 5 Gravity
    1. 5.1 Odd sector
    2. 5.2 Even sector
  6. 6 Chandrasekhar duality
    1. 6.1 A complex master variable
    2. 6.2 Flat-space limit: linearized gravitational duality
    3. 6.3 Chandrasekhar duality off-shell
  7. 7 Physical implications: Love numbers
    1. 7.1 Equality of Love numbers from gravitational duality
  8. 8 Discussion
  9. A $2+2$ Ricci tensor components
    1. A.1 Odd perturbations
    2. A.2 Even perturbations

##  1 Introduction

> The black holes of nature are the most perfect macroscopic objects there are
> in the universe: the only elements in their construction are our concepts of
> space and time.
>
> Chandrasekhar [1]

The advent in the past decade of gravitational-wave astronomy and black hole
imaging have spurred a renewed observational interest in the foundational and
endlessly fascinating black hole solutions of general relativity (GR). The
Schwarzschild metric describing non-rotating black holes is in a sense
gravity’s analog of the hydrogen atom in quantum mechanics: it was the first
exact solution of Einstein’s equations to be discovered,111The history is
remarkable. Einstein published his field equations and an approximate solution
accounting for Mercury’s observed perihelion advance in November 1915.
Schwarzschild read this work while serving on the Russian front, and by
December 1915 had obtained his exact solution. Half a year later he died of an
autoimmune disease acquired at the front. and is still often the first
solution taught to students of GR.

The humble Schwarzschild metric is, of course, far from sufficient for
modelling gravitational-wave events: astrophysical black holes rotate and so
are more accurately described by the significantly more complicated Kerr
metric, and the two-body problem in general relativity is highly non-linear
and requires numerical techniques to solve near the merger. But some progress
can be made analytically, particularly during the inspiral and ringdown
phases, through a variety of perturbative schemes. Among the simplest is
_black hole perturbation theory_ , in which the metric is a small perturbation
around a black hole background, analogous to the flat-space perturbation
theory which is itself an essential topic in introductory GR courses.

Black hole perturbation theory, in other words, is a fundamental problem in GR
with significant relevance to modern experiments. In this paper we explore
some of the symmetries of this theory, particularly the _Chandrasekhar
duality_ between even- and odd-parity modes (which arrive to Earth as $+$ and
$\times$ polarizations), which most famously manifests itself in the fact that
the quasinormal mode spectra of both sectors are identical [1, 2].

We will have a particular emphasis on symmetries which hold _off shell_ , that
is, symmetries of the action rather than just of the equations of motion. Our
principal motivation for this is the role played by the action in Noether’s
theorem; it is also relevant for the quantum theory, e.g., [3, 4, 5, 6]. For
linear theories, which we consider in this work, it is always possible to
construct an action from the equations of motion, so the distinction between
on- and off-shell symmetries may seem somewhat artificial. Nevertheless there
are interesting differences, as is illustrated by the classical example of
electric-magnetic duality in Maxwell’s theory.

The electromagnetic field is described by the vector potential
$A=A_{\mu}\mathrm{d}x^{\mu}$. In terms of the field strength $F=\mathrm{d}A$,
the field equations in vacuum are222Here $\star$ is the Hodge star, which in
coordinates is $\star
F_{\mu\nu}=\frac{1}{2}\epsilon_{{\mu\nu}\alpha\beta}F^{\alpha\beta}$.

$\mathrm{d}\star F=0,\qquad\mathrm{d}F=0.$ (1.1)

The former is Maxwell’s equation, and the latter is the Bianchi identity,
which is satisfied for all field configurations since $\mathrm{d}^{2}=0$. If
we perform a _duality transformation_ , by sending $F\to\star F$ and $\star
F\to-F$,333In terms of the electric and magnetic fields this is
$(E,B)\to(B,-E)$. Note that $\star^{2}=-1$ on 2-forms in $3+1$ dimensions.
then the Maxwell equation becomes the Bianchi identity and vice versa, leaving
the full set of equations invariant. This is a particular case
($\theta=-\pi/2$) of an $SO(2)$ duality invariance of Maxwell’s equations,

$\begin{pmatrix}F\\\ \star
F\end{pmatrix}\to\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}F\\\ \star F\end{pmatrix}.$
(1.2)

Since electric-magnetic duality is a continuous symmetry, Noether’s theorem
tells us there must be an associated conservation law. To find this, one
varies the action under a duality transformation with a spacetime-dependent
parameter. However this does _not_ mean simply varying the Maxwell action
$S=\frac{1}{4}\int\mathrm{d}^{4}x\sqrt{-g}F_{\mu\nu}^{2}$ and setting $\delta
F_{\mu\nu}=\epsilon(x)\star F_{\mu\nu}$, because $A_{\mu}$ rather than
$F_{\mu\nu}$ is the dynamical variable which we vary in the action to obtain
Maxwell’s equations.

The Noether procedure requires us to vary $A$ by a functional $\delta A[A]$
implementing the duality symmetry, but it is impossible to construct a $\delta
A[A]$ such that $\mathrm{d}\delta A=\star F$. If there were, we could take an
exterior derivative to find $\mathrm{d}\star F=0$, i.e., Maxwell’s equation
for $A$, which is precisely what we do not want to assume.444Equivalently,
note that the Maxwell Lagrangian $E^{2}-B^{2}$ naively does not appear to be
invariant under rotations of $E$ and $B$, which are indeed a symmetry of the
action, but does appear to be invariant under _hyperbolic_ $(E,B)$ rotations,
which are _not_ a genuine symmetry. The best we can do is construct a symmetry
operator $\delta A[A]$ which is only a duality transformation (in the sense
that $\mathrm{d}\delta A=\star\mathrm{d}A$) on shell; the full expression
contains additional terms which vanish when the Maxwell equations are
satisfied [7, 8, 9].

Interestingly the off-shell duality transformation is typically _non-local_.
To see this we note that we could flip the roles of the Maxwell equation
$\mathrm{d}\star F=0$ and the Bianchi identity $\mathrm{d}F=0$ by taking the
former to define a potential, $\star F=\mathrm{d}\tilde{A}$, and the latter to
be the field equation for this “dual potential” $\tilde{A}_{\mu}$. This dual
potential is precisely the symmetry transformation,

$\delta A[A]=\tilde{A},$ (1.3)

where $\tilde{A}$ is a solution to the first-order equation
$\mathrm{d}\tilde{A}=\star\mathrm{d}A$. Since solving this equation requires
integration, in general $\tilde{A}$ will depend non-locally on $A$. For
instance, in a gauge where $\delta A_{0}=0$, the off-shell duality
transformation of $A_{i}$ is [8]

$\delta A^{i}=\nabla^{-2}(\epsilon^{ijk}\partial_{j}F_{0k}),$ (1.4)

with $\nabla^{-2}$ the inverse spatial Laplacian. This is a genuine symmetry
of the Maxwell action, which can be used to derive conserved quantities, and
which coincides with duality transformations $\delta F=\star F$ on shell,
i.e., when the Maxwell equations are satisfied.

The goal of this work is to discuss a similar story for the Chandrasekhar
duality in black hole perturbation theory. Along the way we will investigate
the dynamics of scalar, electromagnetic, and gravitational fields on the
Schwarzschild background in two covariant languages designed to exploit its
symmetries, the $2+2$ and Geroch-Held-Penrose (GHP) formalisms. These
approaches are complementary: the $2+2$ formulation is more intuitive but
specifically adapted to a non-rotating black hole, while GHP generalizes
straightforwardly to the full Kerr solution and is in a sense “more
fundamental” in that it is based on the algebraically-special structure of
black hole spacetimes. We will further see that objects arising naturally when
studying dynamics in the $2+2$ formulation have simple interpretations in GHP
language.

The rest of this paper is organized as follows. In section 2 we review the
Schwarzschild solution and introduce the $2+2$ and GHP formalisms. We study
the dynamics of a massless scalar field on Schwarzschild in section 3, the
electromagnetic field in section 4, and linearized gravity in section 5. In
section 6 we discuss the off-shell Chandrasekhar duality and in section 7
explore its physical consequences for tidal Love numbers, before concluding in
section 8.

Conventions: We work with vacuum general relativity in $3+1$ spacetime
dimensions with metric signature $(-,+,+,+)$ and choose an orientation such
that $\epsilon_{0123}=\sqrt{-g}$. We will use Greek letters $\mu,\nu,\cdots$
for four-dimensional spacetime indices, lower-case Latin letters $a,b,...$ for
the $(t,r)$ subspace $\mathcal{M}_{2}$, and upper-case letters $A,B,...$ for
the 2-sphere $S^{2}$.

##  2 Schwarzschild background in the $2+2$ and GHP formalisms

The black hole solutions in vacuum four-dimensional general relativity are
highly symmetrical. In this section we will review the Schwarzschild metric,
on which we will place various field theories, in two formalisms designed to
exploit these symmetries in a coordinate-independent manner. The first is the
$2+2$ formalism, which treats objects covariantly on the two-sphere and on the
$(t,r)$ plane. The second is the GHP formalism, which takes advantage of the
algebraically-special (type D) structure of black hole spacetimes in general
relativity.

The Schwarzschild metric in Boyer-Lindquist (or Schwarzschild) coordinates is

$g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=-f(r)\mathrm{d}t^{2}+\frac{1}{f(r)}\mathrm{d}r^{2}+r^{2}\underbrace{\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right)}_{\mathrm{d}\Omega_{S^{2}}^{2}},\qquad
f(r)\equiv 1-\frac{r_{\mathrm{s}}}{r}$ (2.1)

with $r_{\mathrm{s}}=2GM$ the Schwarzschild radius and
$\mathrm{d}\Omega^{2}_{S^{2}}$ the line element on the unit 2-sphere. As we
will see, kinetic terms for fields on a Schwarzschild background are often
more conveniently phrased in terms of a “tortoise coordinate” $r_{\star}$
defined by

$\mathrm{d}r_{\star}=\frac{\mathrm{d}r}{f(r)}.$ (2.2)

The horizon $r=r_{\mathrm{s}}$ is located at $r_{\star}=-\infty$ and spatial
infinity $r=\infty$ at $r_{\star}=\infty$.

###  2.1 $2+2$ decomposition

The Schwarzschild spacetime factorizes naturally into two submanifolds: the
$(t,r)$ plane $\mathcal{M}_{2}$ and the 2-sphere $S^{2}$. This is the basis of
the $2+2$ decomposition [10, 11, 12]. Let us write the four-dimensional
coordinates as $x^{\mu}=(x^{a},\theta^{A})$, where lower-case Latin letters
$a,b,...$ run over $(t,r)$ and upper-case letters $A,B,...$ run over
$(\theta,\phi)$. The metric factorizes into

$g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=g_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}+r^{2}\Omega_{AB}\mathrm{d}\theta^{A}\mathrm{d}\theta^{B},$
(2.3)

with

$g_{ab}=\begin{pmatrix}-f&0\\\
0&\frac{1}{f}\end{pmatrix},\qquad\Omega_{AB}=\begin{pmatrix}1&0\\\
0&\sin^{2}\theta\end{pmatrix}.$ (2.4)

To avoid a clutter of notation, we will use $\nabla_{\mu}$, $\nabla_{a}$, and
$D_{A}$ for the covariant derivatives with respect to $g_{\mu\nu}$, $g_{ab}$,
and $\Omega_{AB}$, respectively, and raise and lower indices with these
metrics. We also use the same symbol for $g_{\mu\nu}$ and $g_{ab}$; which one
is meant should be clear from context.555In particular, $\sqrt{-g}$ represents
the square root of the determinant of $g_{\mu\nu}$ in
$\int\mathrm{d}^{4}x\sqrt{-g}$ and of $g_{ab}$ in
$\int\mathrm{d}^{2}x\sqrt{-g}$.

The $r$ appearing in eq. 2.3 is a spacetime scalar on ${\mathcal{M}_{2}}$ and
need not be aligned with one of the coordinate directions, though it is in
Boyer-Lindquist coordinates. It and the 2-metric $g_{ab}$ obey the background
Einstein equations,

$rR=2\Box r,\qquad\nabla^{a}(r\nabla_{a}r)=r\Box r+(\partial
r)^{2}=1,\qquad\nabla_{a}\nabla_{b}r=\frac{1}{2}\Box rg_{ab},$ (2.5)

where $\Box=g^{ab}\nabla_{a}\nabla_{b}$ and $(\partial
r)^{2}=g^{ab}\partial_{a}r\partial_{b}r$. In coordinates, the Ricci scalar and
the norm of $\partial_{a}r$ are

$R=\frac{2r_{\mathrm{s}}}{r^{3}},\qquad(\partial r)^{2}=f.$ (2.6)

Note in particular that the latter of these allows us to use $f(r)$ in
coordinate-invariant expressions. We will find it convenient at times to use
the shorthand

$r_{a}=\partial_{a}r.$ (2.7)

As a consequence of its high degree of symmetry, equations of motion on the
Schwarzschild background admit fully separable solutions [13]. For a field of
integer spin $s$, the general solution for the field variable or an observable
constructed from it can be written in the schematic form (e.g., omitting
indices)

$\phi(x^{\mu})=\displaystyle\sum_{\ell=|s|}^{\infty}\displaystyle\sum_{m=-\ell}^{\ell}\underbrace{\int\mathrm{d}\omega
e^{-i\omega t}R_{\ell\omega}(r)}_{\phi_{\ell
m}(x^{a})}\underbrace{\vphantom{\int}\Theta_{\ell
m}(\theta)e^{im\phi}}_{S_{\ell m}(\theta^{A})}.$ (2.8)

A further consequence of symmetry is that the radial and angular functions
$R_{\ell\omega}(r)$ and $\Theta_{\ell m}(\theta)$ obey remarkably similar
equations. The main difference is that the periodic boundary conditions on the
angular coordinates constrain $S_{\ell m}(\theta,\phi)$ to the class of
spherical harmonic functions, which are eigenfunctions of the Laplacian on
$S^{2}$, while $R_{\ell\omega}(r)$ obeys a Schrödinger-like equation
(typically in terms of the tortoise coordinate $r_{\star}$ rather than $r$).

The spherical harmonics can be categorized by their transformation properties
under rotations. In four dimensions, there are two such classes: scalars and
vectors.666Degrees of freedom transforming under the tensor representation are
non-dynamical in $D=4$ but are present in higher dimensions. The scalar
harmonics are the familiar spherical harmonics,

$S_{\ell m}=Y_{\ell m}(\theta,\phi)\propto
P^{m}_{\ell}(\cos\theta)e^{im\phi},$ (2.9)

with $P^{m}_{\ell}(x)$ the associated Legendre polynomials. The vector
harmonics decompose into longitudinal and transverse, or electric and
magnetic, pieces, which are related to the scalar harmonics by

$\displaystyle E_{A,\ell m}$ $\displaystyle=D_{A}Y_{\ell m},$ (2.10a)
$\displaystyle B_{A,\ell m}$ $\displaystyle=-\epsilon_{AB}D^{B}Y_{\ell m},$
(2.10b)

with $\epsilon_{AB}$ the Levi-Civita tensor on the 2-sphere,
$\epsilon_{\theta\phi}=\sin\theta$. In coordinates these are

$\displaystyle E_{A,\ell m}\mathrm{d}\theta^{A}$
$\displaystyle=\partial_{\theta}Y_{\ell
m}\mathrm{d}\theta+\partial_{\phi}Y_{\ell m}\mathrm{d}\phi,$ (2.11a)
$\displaystyle B_{A,\ell m}\mathrm{d}\theta^{A}$
$\displaystyle=-\csc\theta\partial_{\phi}Y_{\ell
m}\mathrm{d}\theta+\sin\theta\partial_{\theta}Y_{\ell m}\mathrm{d}\phi.$
(2.11b)

The scalar harmonics obey the Laplace equation on the 2-sphere with eigenvalue
$-\ell(\ell+1)$,

$\displaystyle D^{2}Y_{\ell m}$
$\displaystyle=\frac{1}{\sqrt{\Omega}}\partial_{A}\left(\sqrt{\Omega}\Omega^{AB}\partial_{B}Y_{\ell
m}\right)$ $\displaystyle=-\ell(\ell+1)Y_{\ell m},$ (2.12)

where $\Omega\equiv\det(\Omega_{AB})=\sin^{2}\theta$, while the vector
harmonics $V_{A}=(E_{A},B_{A})$ are eigenfunctions with eigenvalue
$1-\ell(\ell+1)$,

$D^{2}V_{A}^{\ell m}=-\left[\ell(\ell+1)-1\right]V_{A}^{\ell m}.$ (2.13)

The spacetime integration measure appearing in a four-dimensional action
contains the 2-sphere integration measure $\mathrm{d}\Omega$,777We remind the
reader that in our notation,
$\int\mathrm{d}^{4}x\sqrt{-g}=\int\mathrm{d}^{4}x\sqrt{-\det g_{\mu\nu}}$
while $\int\mathrm{d}^{2}x\sqrt{-g}=\int\mathrm{d}^{2}x\sqrt{-\det g_{ab}}$.

$\int\mathrm{d}^{4}x\sqrt{-g}=\int\mathrm{d}^{2}x\sqrt{-g}r^{2}\mathrm{d}\Omega,\qquad\int\mathrm{d}\Omega\equiv\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}\sin\theta\mathrm{d}\theta\mathrm{d}\phi.$
(2.14)

We will be able to integrate over $S^{2}$ in actions on Schwarzschild using
the orthonormality relations of the spherical harmonics,

$\displaystyle\int\mathrm{d}\Omega Y_{\ell m}Y_{\ell^{\prime}m^{\prime}}$
$\displaystyle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}},$ (2.15a)
$\displaystyle\int\mathrm{d}\Omega V_{A,\ell
m}V^{A}_{\ell^{\prime}m^{\prime}}$
$\displaystyle=\ell(\ell+1)\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}},$
(2.15b) $\displaystyle\int\mathrm{d}\Omega E_{A,\ell
m}B^{A}_{\ell^{\prime}m^{\prime}}$ $\displaystyle=0.$ (2.15c)

###  2.2 Geroch-Held-Penrose (GHP) formalism

In this subsection we describe an alternative formalism for leveraging the
symmetry of black hole backgrounds: the Geroch-Held-Penrose (GHP) formalism,
which is itself built on the famous Newman-Penrose (NP) approach. While this
approach is somewhat more arcane than the $2+2$ formalism,888Due at least in
part to its heavy use of Icelandic runes. it more directly makes use of the
fundamental property underpinning the “magic” of the Schwarzschild and Kerr
spacetimes, namely the fact that they are _algebraically special_.

####  2.2.1 Newman-Penrose

Recall that the Weyl tensor $C_{{\mu\nu}\alpha\beta}$ of a generic spacetime
has four principal null directions;999Principal null directions are null
vectors $l^{\mu}$ satisfying
$l^{\nu}l_{[\rho}C_{\mu]\nu\alpha[\beta}l_{\sigma]}l^{\alpha}=0$ [14].
algebraically-special spacetimes are those where one or more of the four are
degenerate. The Kerr black hole is of algebraic _type D_ , with two singly-
degenerate principal null directions. These special vectors, $l^{\mu}$ and
$n^{\mu}$, point along outgoing and ingoing null rays, respectively. In the
Schwarzschild case they live on ${\mathcal{M}_{2}}$,

$l_{\mu}\mathrm{d}x^{\mu}=l_{a}\mathrm{d}x^{a},\quad
n_{\mu}\mathrm{d}x^{\mu}=n_{a}\mathrm{d}x^{a},$ (2.16)

and in fact can be thought of as zweibeins for the 2-metric,

$g_{ab}=-l_{a}n_{b}-n_{a}l_{b}.$ (2.17)

To complete the picture, we include null vectors parametrizing $S^{2}$: a
complex vector $m^{\mu}$ and its complex conjugate ${\bar{m}}^{\mu}$, with
$m_{\mu}\mathrm{d}x^{\mu}=m_{A}\mathrm{d}\theta^{A}$. These four vectors
together comprise a complex null tetrad
$e^{\mathbf{a}}_{\mu}=(l_{\mu},n_{\mu},m_{\mu},\bar{m}_{\mu})$, in the sense
that101010This is the usual vielbein relation
$g_{\mu\nu}=\eta_{\mathbf{{ab}}}e^{\mathbf{a}}_{\mu}e^{\mathbf{b}}_{\nu}$ with
the internal Minkowski metric written in the form
$\eta_{\mathbf{{ab}}}=\begin{pmatrix}0&-1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}.$ Here bold lowercase Latin letters represent 4D internal
Lorentz indices.

$g_{\mu\nu}=-2l_{(\mu}n_{\nu)}+2m_{(\mu}{\bar{m}}_{\nu)}.$ (2.18)

The vielbeins are normalized so that all of their inner products vanish except
for

$l_{\mu}n^{\mu}=-1,\quad m_{\mu}{\bar{m}}^{\mu}=1.$ (2.19)

This setup does not completely fix
$(l^{\mu},n^{\mu},m^{\mu},{\bar{m}}^{\mu})$, as there is some residual Lorentz
invariance. Insisting that $\ell^{\mu}$ and $n^{\mu}$ remain principal null
directions leaves a two-parameter symmetry comprising boosts of $l$ and $n$,

$l^{\mu}\to\alpha l^{\mu},\quad n^{\mu}\to\alpha^{-1}n^{\mu},$ (2.20)

and rotations of $m$ and ${\bar{m}}$,

$m^{\mu}\to e^{i\beta}m^{\mu},\quad{\bar{m}}^{\mu}\to
e^{i\beta}{\bar{m}}^{\mu},$ (2.21)

with $\alpha$ and $\beta$ real functions. We will choose the _Carter tetrad_
[15],

$\displaystyle l_{\mu}\mathrm{d}x^{\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(-\sqrt{f}\mathrm{d}t+\frac{1}{\sqrt{f}}\mathrm{d}r\right),$
(2.22a) $\displaystyle n_{\mu}\mathrm{d}x^{\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(-\sqrt{f}\mathrm{d}t-\frac{1}{\sqrt{f}}\mathrm{d}r\right),$
(2.22b) $\displaystyle m_{\mu}\mathrm{d}x^{\mu}$
$\displaystyle=\frac{r}{\sqrt{2}}\left(\mathrm{d}\theta+i\sin\theta\mathrm{d}\phi\right),$
(2.22c) $\displaystyle{\bar{m}}_{\mu}\mathrm{d}x^{\mu}$
$\displaystyle=\frac{r}{\sqrt{2}}\left(\mathrm{d}\theta-i\sin\theta\mathrm{d}\phi\right).$
(2.22d)

The frequently-used Kinnersley tetrad [16] is related by a rescaling (2.20)
with $\alpha=\sqrt{f/2}$. The Carter tetrad is particularly useful for our
purposes as it maintains symmetries of the background which can be obscured in
other bases [17].

In the Newman-Penrose formalism one works with spacetime scalars obtained by
projection along the null directions. For instance the Weyl tensor
$C_{{\mu\nu}\alpha\beta}$ is efficiently encoded in five complex Weyl scalars,
which are the “components” of the Weyl tensor in the complex null basis,

$\displaystyle\Psi_{0}=C_{lmlm},\quad\Psi_{1}=C_{lnlm},\quad\Psi_{2}=C_{lm{\bar{m}}n},\quad\Psi_{3}=C_{ln{\bar{m}}n},\quad\Psi_{4}=C_{n{\bar{m}}n{\bar{m}}},$
(2.23)

where
$C_{lm{\bar{m}}n}=C_{{\mu\nu}\alpha\beta}l^{\mu}m^{\nu}{\bar{m}}^{\alpha}n^{\beta}$
and so on. (In general we will use the notation $V_{\mu}l^{\mu}=V_{l}$, etc.)
For type-D spacetimes the only non-vanishing Weyl scalar is $\Psi_{2}$,
providing a remarkably compact characterization of the full Riemann tensor. In
the Schwarzschild case, the value of $\Psi_{2}$ in coordinates is111111The
resemblance to the Ricci scalar on ${\mathcal{M}_{2}}$, cf. eq. 2.6, is not
accidental. Using $R_{aAbB}=-r\nabla_{a}\nabla_{b}r\Omega_{AB}$ [12], we find
$\Psi_{2}=R_{aABb}l^{a}n^{b}m^{A}{\bar{m}}^{B}=\frac{1}{r}\nabla_{a}\nabla_{b}rl^{a}n^{b}=-\frac{1}{4}R$.

$\Psi_{2}=-\frac{r_{\mathrm{s}}}{2r^{3}}.$ (2.24)

####  2.2.2 Geroch-Held-Penrose

The GHP formalism soups up the NP formalism by working only with quantities
and operators which have simple transformation properties under the residual
Lorentz invariance (2.20)–(2.21). Defining $\lambda^{2}=\alpha e^{i\beta}$, we
will insist on working with tensors $\Phi$ that transform under eqs. 2.20 and
2.21 as

$\Phi\to\lambda^{p}\bar{\lambda}^{q}\Phi.$ (2.25)

Such a quantity is said to have _GHP type_ $\\{p,q\\}$. They are also called
spin- and/or boost-weighted, where the spin weight is $s=(p-q)/2$ and the
boost weight is $b=(p+q)/2$.

The residual Lorentz transformations (2.20)–(2.21) do not exhaust the symmetry
in choosing a tetrad, which is invariant under several discrete tetrad
interchanges: _complex conjugation_ , which swaps $m^{\mu}$ and
${\bar{m}}^{\mu}$; the _prime_ (′) operation, which interchanges both
$l\leftrightarrow n$ and $m\leftrightarrow{\bar{m}}$; and, less obviously, the
_star_ ($\star$) operation, $(l,n,m,{\bar{m}})\to(m,-{\bar{m}},-l,n)$, which
we will not use. These discrete invariances allow for a particularly
economical description of field equations, since one equation implies its
prime, conjugate, and prime conjugate versions.

Scalars with well-defined GHP type include the Weyl scalars, which inherit
their GHP types from the various factors of $l^{\mu}$, etc., in their
definitions (2.23),121212Tensors like $C_{\mu\nu\alpha\beta}$ are a priori
unweighted. as well as the spin coefficient $\rho$,131313And by extension
$\rho^{\prime}$, $\bar{\rho}$, and $\bar{\rho}^{\prime}$, although for
Schwarzschild $\rho$ and $\rho^{\prime}$ are real.

$\rho=-{\bar{m}}^{\mu}m^{\nu}\nabla_{\mu}l_{\nu},$ (2.26)

which is of GHP type $\\{1,1\\}$. Examples of scalars _without_ a well-defined
GHP type include the spin coefficients $\beta$ and $\epsilon$ (and their
primes and conjugates),

$\displaystyle\beta$
$\displaystyle=\frac{1}{2}\left(m^{\mu}{\bar{m}}^{\nu}\nabla_{\mu}m_{\nu}-m^{\mu}n^{\nu}\nabla_{\mu}l_{\nu}\right),$
(2.27) $\displaystyle\epsilon$
$\displaystyle=\frac{1}{2}\left(l^{\mu}{\bar{m}}^{\nu}\nabla_{\mu}m_{\nu}-l^{\mu}n^{\nu}\nabla_{\mu}l_{\nu}\right).$
(2.28)

These are the only non-zero spin coefficients for Schwarzschild and completely
describe the spin connection. In the Carter tetrad they take the coordinate
values

$\rho=-\rho^{\prime}=-\frac{\sqrt{f}}{\sqrt{2}r},\quad\beta=\beta^{\prime}=\frac{\cot\theta}{2\sqrt{2}r},\quad\epsilon=-\epsilon^{\prime}=\frac{r_{\mathrm{s}}}{4\sqrt{2f}r^{2}}.$
(2.29)

Analogously to the non-coordinate-invariant Christoffel symbols, $\beta$ and
$\epsilon$ can be used to construct covariant derivative operators with well-
defined GHP type. Unfortunately, the use of Icelandic runes for these
operators is firmly embedded in the literature:

$\displaystyle\operatorname{\text{\TH}}$
$\displaystyle=l^{\mu}\nabla_{\mu}-p\epsilon-q\bar{\epsilon},$
$\displaystyle\eth$
$\displaystyle=m^{\mu}\nabla_{\mu}-p\beta+q\bar{\beta}^{\prime},$ (2.30)
$\displaystyle\operatorname{\text{\TH}}^{\prime}$
$\displaystyle=n^{\mu}\nabla_{\mu}+p\epsilon^{\prime}+q\bar{\epsilon}^{\prime}$
$\displaystyle\eth^{\prime}$
$\displaystyle={\bar{m}}^{\mu}\nabla_{\mu}+p\beta^{\prime}-q\bar{\beta}$

The operator $\operatorname{\text{\TH}}$ sends a GHP type $\\{p,q\\}$ object
to one with type $\\{p+1,q+1\\}$, $\operatorname{\text{\TH}}^{\prime}$ to
$\\{p-1,q-1\\}$, $\eth$ to $\\{p+1,q-1\\}$, and $\eth^{\prime}$ to
$\\{p-1,q+1\\}$. Note that $\operatorname{\text{\TH}}$ and
$\operatorname{\text{\TH}}^{\prime}$ raise and lower the boost weight, while
$\eth$ and $\eth^{\prime}$ raise and lower the spin weight. For the Carter
tetrad in Schwarzschild, the GHP derivatives take the coordinate form [17]

$\displaystyle\operatorname{\text{\TH}}$
$\displaystyle=\frac{1}{\sqrt{2f}}\left(\partial_{t}+f\partial_{r}-\frac{br_{\mathrm{s}}}{2r^{2}}\right),$
$\displaystyle\eth$
$\displaystyle=\frac{1}{\sqrt{2}r}\left(\partial_{\theta}+i\csc\theta\partial_{\phi}-s\cot\theta\right)$
(2.31) $\displaystyle\operatorname{\text{\TH}}^{\prime}$
$\displaystyle=\frac{1}{\sqrt{2f}}\left(\partial_{t}-f\partial_{r}-\frac{br_{\mathrm{s}}}{2r^{2}}\right),$
$\displaystyle\eth^{\prime}$
$\displaystyle=\frac{1}{\sqrt{2}r}\left(\partial_{\theta}-i\csc\theta\partial_{\phi}+s\cot\theta\right).$

Note also that these derivatives have non-trivial commutators,

$[\operatorname{\text{\TH}},\operatorname{\text{\TH}}^{\prime}]=-2b\Psi_{2},\quad[\operatorname{\text{\TH}},\eth]=\rho\eth,\quad[\eth,\eth^{\prime}]=2s(\Psi_{2}+\rho\rho^{\prime})=-\frac{s}{r^{2}},$
(2.32)

along with their primes and complex conjugates.

In this language, the scalar spherical harmonics are eigenfunctions of
$\eth\eth^{\prime}$,

$\eth\eth^{\prime}Y=-\frac{\ell(\ell+1)}{2r^{2}}Y,$ (2.33)

and are a special case of the _spin-weighted_ spherical harmonics,

$\frac{1}{2}\left(\eth^{\prime}\eth+\eth\eth^{\prime}\right)Y_{s}=-\frac{\ell(\ell+1)-s^{2}}{2r^{2}}Y_{s},$
(2.34)

which can be obtained from the scalar harmonics by raising and lowering the
spin weight with $\eth$ and $\eth^{\prime}$,

$\eth
Y_{s}=-\frac{\sqrt{\ell(\ell+1)-s(s+1)}}{\sqrt{2}r}Y_{s+1},\quad\eth^{\prime}Y_{s}=\frac{\sqrt{\ell(\ell+1)-s(s-1)}}{\sqrt{2}r}Y_{s-1}.$
(2.35)

The $|s|=1$ spin-weighted harmonics are related to the vector harmonics by
[17]

$\displaystyle E_{A}$
$\displaystyle=\frac{\sqrt{\ell(\ell+1)}}{2}\left(Y_{-1}\tilde{m}_{A}-Y_{1}\bar{\tilde{m}}_{A}\right),$
(2.36a) $\displaystyle B_{A}$
$\displaystyle=-i\frac{\sqrt{\ell(\ell+1)}}{2}\left(Y_{-1}\tilde{m}_{A}+Y_{1}\bar{\tilde{m}}_{A}\right),$
(2.36b)

where
$\tilde{m}_{A}\mathrm{d}\theta^{A}=\mathrm{d}\theta+i\sin\theta\mathrm{d}\phi$.

##  3 Massless scalar

We want to compute the action for linearized gravity on Schwarzschild,
performing separation of variables and utilizing the $2+2$ decomposition. Many
of the basic steps of the computation are present in the simpler cases of a
scalar and vector field, so we will work our way up to gravity one integer
step in spin at a time.

The action for a massless scalar is141414We remind the reader that to avoid a
clutter of notation we are using $\sqrt{-g}$ for both $\sqrt{-\det
g_{\mu\nu}}$ and $\sqrt{-\det g_{ab}}$, with the meaning clear depending
whether we are integrating over $\mathrm{d}^{4}x$ or $\mathrm{d}^{2}x$. Note
also that in Boyer-Lindquist coordinates, $\sqrt{-\det g_{ab}}=1$.

$S=-\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{-g}(\partial_{\mu}\phi)^{2}.$ (3.1)

The field $\phi$ admits a spherical harmonic expansion of the form (2.8),

$\phi(x^{\mu})=\displaystyle\sum_{\ell=0}^{\infty}\displaystyle\sum_{m=-\ell}^{\ell}\phi_{\ell
m}(x^{a})Y_{\ell m}(\theta^{A}).$ (3.2)

Inserting this into eq. 3.1 and integrating over $S^{2}$ we find a sum over
actions for each $(\ell,m)$ mode,

$\displaystyle S$
$\displaystyle=-\frac{1}{2}\int\mathrm{d}^{2}x\sqrt{-g}r^{2}\int\mathrm{d}\Omega\left[(\partial_{a}\phi)^{2}+r^{-2}(\partial_{A}\phi)^{2}\right]$
$\displaystyle=-\frac{1}{2}\int\mathrm{d}^{2}x\sqrt{-g}\displaystyle\sum_{\ell\ell^{\prime}mm^{\prime}}\int\mathrm{d}\Omega\left[r^{2}\partial_{a}\phi_{\ell
m}\partial^{a}\phi_{\ell^{\prime}m^{\prime}}+\ell(\ell+1)\phi_{\ell
m}\phi_{\ell^{\prime}m^{\prime}}\right]Y_{\ell m}Y_{\ell^{\prime}m^{\prime}}$
$\displaystyle=-\frac{1}{2}\int\mathrm{d}^{2}x\sqrt{-g}\displaystyle\sum_{\ell,m}\left[r^{2}(\partial\phi_{\ell
m})^{2}+\ell(\ell+1)\phi_{\ell m}^{2}\right]$
$\displaystyle\equiv\displaystyle\sum_{\ell,m}S_{\ell m}.$ (3.3)

To simplify notation, we will drop the $\ell m$ subscripts and focus on an
individual mode, with the summation over all modes implied. This is kosher
because in linear theories modes of different $(\ell,m)$ decouple.

The $2D$ field $\phi$ is not canonically normalized, as its kinetic term is
multiplied by a factor of $r^{2}$. We can remove this with a field
redefinition [12, 18],

$\psi\equiv r\phi,$ (3.4)

in terms of which the action is

$\boxed{S=\int\mathrm{d}^{2}x\sqrt{-g}\left[-\frac{1}{2}(\partial\psi)^{2}-\frac{1}{2r^{2}}\left(\ell(\ell+1)+\frac{r_{\mathrm{s}}}{r}\right)\psi^{2}\right].}$
(3.5)

We identify the usual scalar potential on a Schwarzschild background [2],

$V(r)=\frac{\ell(\ell+1)}{r^{2}}+\frac{r_{\mathrm{s}}}{r^{3}}.$ (3.6)

If we drop our insistence on covariance and write the action in terms of the
coordinates $(t,r)$,

$S=\int\mathrm{d}t\mathrm{d}r\left(\frac{1}{2}f^{-1}(\partial_{t}\psi)^{2}-\frac{1}{2}f(\partial_{r}\psi)^{2}-\frac{1}{2}V(r)\psi^{2}\right),$
(3.7)

we find that the kinetic and gradient terms again have nonstandard factors in
front. To canonically normalize we transform to the tortoise coordinate
$\mathrm{d}r=f\mathrm{d}r_{\star}$ [18],

$S=\int\mathrm{d}t\mathrm{d}r_{\star}\left(\frac{1}{2}(\partial_{t}\psi)^{2}-\frac{1}{2}(\partial_{r_{\star}}\psi)^{2}-\frac{1}{2}V(r)\psi^{2}\right).$
(3.8)

For completeness let us write the action (3.1) in GHP language. Writing the
metric in terms of the null vectors, cf. eq. 2.18, we have

$\displaystyle S$
$\displaystyle=-\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{-g}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$
$\displaystyle=\int\mathrm{d}^{4}x\sqrt{-g}\left(l^{\mu}n^{\nu}-m^{\mu}{\bar{m}}^{\nu}\right)\partial_{\mu}\phi\partial_{\nu}\phi$
$\displaystyle=\int\mathrm{d}^{4}x\sqrt{-g}\left(\operatorname{\text{\TH}}\phi\operatorname{\text{\TH}}^{\prime}\phi-\eth\phi\eth^{\prime}\phi\right).$
(3.9)

If we separate variables and integrate over the 2-sphere, then the action for
a single mode is

$S_{\ell
m}=\int\mathrm{d}^{2}x\sqrt{-g}\left(r^{2}\operatorname{\text{\TH}}\phi\operatorname{\text{\TH}}^{\prime}\phi-\frac{\ell(\ell+1)}{2}\phi^{2}\right).$
(3.10)

##  4 Electromagnetism

The next step on the road to gravity, which is the spin-2 case, is the spin-1
case, which is electromagnetism. The Maxwell action is

$S=-\frac{1}{4}\int\mathrm{d}^{4}x\sqrt{-g}F_{\mu\nu}^{2},\qquad
F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}.$ (4.1)

The vector potential is a superposition of separable solutions:

$A^{\mu}=\displaystyle\sum_{\ell=1}^{\infty}\displaystyle\sum_{m=-\ell}^{\ell}A^{\mu}_{\ell
m}.$ (4.2)

Herein we will focus on a single mode and drop $\ell m$ subscripts, with the
summation implied. Under a $2+2$ decomposition the vector potential is

$A_{\mu}\mathrm{d}x^{\mu}=A_{a}(x^{a})Y\mathrm{d}x^{a}+a(x^{a})B_{A}\mathrm{d}\theta^{A}.$
(4.3)

Here we have used our gauge freedom to remove the longitudinal mode, which is
proportional to $E_{A}\mathrm{d}\theta^{A}$.

Gauge invariance adds a wrinkle that was not present for the scalar: in order
to avoid losing information when fixing a gauge at the level of the action
rather than the equations of motion, one must make a _complete_ gauge fixing,
in the sense that there are no integration constants left when fixing a gauge
vector (rather than necessarily that all gauge freedom is exhausted, although
we will insist on this too) [19, 20]. Our gauge choice satisfies this
requirement [18].

Performing separation of variables and integrating over the 2-sphere, we
obtain

$S=\int\mathrm{d}^{2}x\sqrt{-g}\mathcal{L},$ (4.4)

where

$\boxed{\mathcal{L}=\underbrace{\vphantom{\bigg{[}}-\frac{1}{4}r^{2}F_{ab}^{2}-\frac{1}{2}\ell(\ell+1)A_{a}^{2}\vphantom{\bigg{]}}}_{\mathcal{L}_{\mathrm{even}}}\underbrace{\vphantom{\bigg{[}}-\frac{1}{2}\ell(\ell+1)\left[(\partial
a)^{2}+\frac{\ell(\ell+1)}{r^{2}}a^{2}\right]\vphantom{\bigg{]}}}_{\mathcal{L}_{\mathrm{odd}}}.}$
(4.5)

We see that the even-parity (or electric) field $A_{a}$ and the odd-parity (or
magnetic) field $a$ decouple.

The even sector has only one dynamical degree of freedom but depends on two
variables $A_{a}$. To isolate this dynamical field we integrate in an
auxiliary variable $\lambda(t,r)$:

$\displaystyle\mathcal{L}_{\mathrm{even,aux}}$
$\displaystyle=\mathcal{L}_{\mathrm{even}}+\frac{1}{4}r^{2}\left(F_{ab}+r^{-2}\lambda\epsilon_{ab}\right)^{2}$
$\displaystyle=\lambda\epsilon^{ab}\partial_{a}A_{b}-\frac{1}{2}\frac{\lambda^{2}}{r^{2}}-\frac{1}{2}\ell(\ell+1)A_{a}^{2}.$
(4.6)

The $\lambda$ equation of motion fixes it to be proportional to $F_{ab}$ on-
shell,

$\lambda=r^{2}\epsilon^{ab}\partial_{a}A_{b}=-r^{2}F_{tr}.$ (4.7)

Inserting this back into $\mathcal{L}_{\mathrm{even,aux}}$ we obtain
$\mathcal{L}_{\mathrm{even}}$, establishing their dynamical equivalence.
However we can also obtain an action for $\lambda$ alone by integrating out
$A_{a}$ using its equation of motion,

$A^{a}=\frac{1}{\ell(\ell+1)}\epsilon^{ab}\partial_{b}\lambda,$ (4.8)

and plugging back into the action,

$\mathcal{L}=-\frac{1}{2\ell(\ell+1)}(\partial\lambda)^{2}-\frac{1}{2}\frac{\lambda^{2}}{r^{2}}-\frac{1}{2}\ell(\ell+1)(\partial
a)^{2}-\frac{1}{2}\frac{\ell^{2}(\ell+1)^{2}}{r^{2}}a{}^{2}.$ (4.9)

We canonically normalize the fields by scaling out appropriate factors of
$\sqrt{\ell(\ell+1)}$,

$\psi_{+}\equiv\frac{\lambda}{\sqrt{\ell(\ell+1)}},\qquad\psi_{-}\equiv\sqrt{\ell(\ell+1)}a,$
(4.10)

so that

$\mathcal{L}=\displaystyle\sum_{\pm}\left[-\frac{1}{2}(\partial\psi_{\pm})^{2}-\frac{1}{2}\frac{\ell(\ell+1)}{r^{2}}\psi_{\pm}^{2}\right].$
(4.11)

We conclude that $\psi_{\pm}$ are the “master variables” for the electric
($+$) and magnetic ($-$) sectors (see also Ref. [18]), each satisfying a
Schrödinger equation with the usual vector potential [2].

###  4.1 Electric-magnetic duality

The Lagrangian (4.11) is manifestly invariant under electric-magnetic duality,
which acts as a rotation on the vector $(\psi_{+},\psi_{-})^{T}$. The
infinitesimal version is

$\delta(\psi_{+},\psi_{-})=(\psi_{-},-\psi_{+}),$ (4.12)

that is,

$\displaystyle\delta\lambda$ $\displaystyle=\ell(\ell+1)a,$ (4.13a)
$\displaystyle\delta a$ $\displaystyle=-\frac{\lambda}{\ell(\ell+1)}.$ (4.13b)

Since eq. 4.11 is dynamically equivalent to the original Maxwell action (4.5),
related by auxiliary variables, a symmetry of one is a symmetry of the other.
To construct the symmetry operators $\delta A_{a}$ and $\delta a$ for eq. 4.5
we need only use eqs. 4.8 and 4.7 relating $A_{a}$ and $\lambda$ on shell to
find

$\displaystyle\delta A_{a}$ $\displaystyle=\epsilon_{ab}\partial^{b}a,$
(4.14a) $\displaystyle\delta a$
$\displaystyle=-\frac{r^{2}}{\ell(\ell+1)}\epsilon^{ab}\partial_{a}A_{b}.$
(4.14b)

This is an _off-shell_ symmetry of the action (4.5). As discussed in the
introduction, this symmetry is non-local. This is reflected in the
transformation law for $a$, which contains the inverse spherical Laplacian in
the form $1/\ell(\ell+1)$.151515Recalling that $-\ell(\ell+1)$ is the
eigenvalue of the spherical Laplacian $D^{2}$ for scalar spherical harmonics,
we see that $D^{2}(\delta a\,Y)=r^{2}\epsilon^{ab}\partial_{a}A_{b}\,Y$.
Interestingly the symmetry transformation for $A_{a}$ is local.

The transformation law (4.14) has a natural interpretation in terms of Hodge
duality. Consider the dual field strength tensor,

$\star F_{\mu\nu}=\frac{1}{2}\epsilon_{{\mu\nu}\alpha\beta}F^{\alpha\beta}.$
(4.15)

The Maxwell equation is $\mathrm{d}\star F=0$, so that on-shell $\star
F=\mathrm{d}\tilde{A}$ can be expressed in terms of a dual potential
$\tilde{A}_{\mu}$. It turns out that the off-shell duality transformation
$\delta A_{\mu}$ is just such a dual potential, that is,

$\delta A_{\mu}=\tilde{A}_{\mu}$ (4.16)

where

$\tilde{A}_{\mu}\mathrm{d}x^{\mu}=\epsilon_{ab}\partial^{b}a(x)Y(\theta)\mathrm{d}x^{a}-\frac{r^{2}}{\ell(\ell+1)}\epsilon^{ab}\partial_{a}A_{b}(x)B_{A}(\theta)\mathrm{d}\theta^{A}$
(4.17)

solves

$\star F_{\mu\nu}=\partial_{\mu}\tilde{A}_{\nu}-\partial_{\nu}\tilde{A}_{\mu}$
(4.18)

on shell. The fact that $A_{\mu}$ and $\tilde{A}_{\mu}$ are related by
integration, $\star\mathrm{d}A=\mathrm{d}\tilde{A}$, underlies the non-local
nature of $\delta A_{\mu}$.

If we further package the electric and magnetic master variables into a
complex scalar,

$\psi\equiv\frac{\psi_{+}-i\psi_{-}}{\sqrt{2}},$ (4.19)

then the action (4.11) is simply

$S=\int\mathrm{d}^{2}x\sqrt{-g}\left[-\partial_{a}\psi\partial^{a}\bar{\psi}-\frac{\ell(\ell+1)}{r^{2}}\psi\bar{\psi}\right].$
(4.20)

Electric-magnetic duality acts as $\delta\psi=i\psi$, which is manifestly a
symmetry. It is straightforward to obtain the conserved current via the
standard Noether procedure,

$\displaystyle J_{a}$
$\displaystyle=i(\bar{\psi}\partial_{a}\psi-\psi\partial_{a}\bar{\psi})$
$\displaystyle=\psi_{+}\partial_{a}\psi_{-}-\psi_{-}\partial_{a}\psi_{+}.$
(4.21)

Intriguingly, the complex master field $\psi$, which we obtained by
integrating out non-dynamical fields and canonically normalizing, turns out to
be proportional to $(\ell,m)$ modes of the middle Newman-Penrose scalar
$\phi_{1}=(1/2)(F_{ln}-F_{m{\bar{m}}})$,

$\sqrt{\frac{\ell(\ell+1)}{2}}\psi_{\ell m}=r^{2}(\phi_{1})_{\ell m}.$ (4.22)

For this reason, it will be illuminating to recontextualize the foregoing
$2+2$ calculation in the GHP formalism.

###  4.2 Maxwell in GHP

Analogously to the Weyl tensor, the electromagnetic field strength tensor
$F_{\mu\nu}$ can be fully encoded in three complex Maxwell scalars,

$\phi_{0}=F_{lm},\quad\phi_{1}=\frac{1}{2}\left(F_{ln}-F_{m{\bar{m}}}\right),\quad\phi_{2}=F_{{\bar{m}}n},$
(4.23)

of GHP types $\\{2,0\\}$, $\\{0,0\\}$, and $\\{-2,0\\}$, respectively. We
remind the reader of the notation $F_{lm}=F_{\mu\nu}l^{\mu}m^{\nu}$, etc. The
Maxwell Lagrangian is

$\displaystyle\mathcal{L}$ $\displaystyle=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$
$\displaystyle=-(-l^{(\mu}n^{\nu)}+m^{(\mu}{\bar{m}}^{\nu)})(-l^{\alpha}n^{\beta}+m^{\alpha}{\bar{m}}^{\beta})F_{\mu\alpha}F_{\nu\beta}$
$\displaystyle=\phi_{1}^{2}-\phi_{0}\phi_{2}+\mathrm{c.c.}$ (4.24)

Now we introduce an auxiliary complex scalar $\lambda$ of GHP type
$\\{0,0\\}$, meant to equal $\phi_{1}$ on-shell, by sending
$\mathcal{L}\to\mathcal{L}-(\phi_{1}-\lambda)^{2}-(\bar{\phi}_{1}-\bar{\lambda})^{2}$,

$\boxed{\mathcal{L}=2\phi_{1}\lambda-\lambda^{2}-\phi_{0}\phi_{2}+\mathrm{c.c.}}$
(4.25)

Instead of decomposing $A_{\mu}$ into $\mathcal{M}^{2}$ tensors $A_{a}(t,r)$
and $a(t,r)$ as in the $2+2$ decomposition, in the GHP formalism we encode it
in the four scalars $(A_{l},A_{n},A_{m},A_{\bar{m}})$. The gauge choice we
made earlier can be written in a GHP-invariant manner as

$\eth^{\prime}A_{m}+\eth A_{\bar{m}}=0.$ (4.26)

In this gauge, the even modes live in $A_{l}$ and $A_{n}$ while the odd modes
live in $A_{m}$ and $A_{\bar{m}}$ through the combination

$\eth A_{\bar{m}}-\eth^{\prime}A_{m}=i\frac{\ell(\ell+1)}{r^{2}}aY.$ (4.27)

To work with the equations of motion coming from the Lagrangian (4.25), it is
helpful to establish just a bit more notation. First, we write the Maxwell
scalars in terms of operators $\mathcal{T}_{i}$ acting on $A$ [21],

$\displaystyle\phi_{0}=\mathcal{T}_{0}A$ $\displaystyle=-\eth
A_{l}+(\operatorname{\text{\TH}}-\rho)A_{m},$ (4.28a)
$\displaystyle\phi_{1}=\mathcal{T}_{1}A$
$\displaystyle=\frac{1}{2}\left(-\operatorname{\text{\TH}}^{\prime}A_{l}+\operatorname{\text{\TH}}A_{n}+\eth^{\prime}A_{m}-\eth
A_{\bar{m}}\right),$ (4.28b) $\displaystyle\phi_{2}=\mathcal{T}_{2}A$
$\displaystyle=\eth^{\prime}A_{n}-(\operatorname{\text{\TH}}^{\prime}-\rho^{\prime})A_{\bar{m}}.$
(4.28c)

Second, we introduce Wald’s notion of _adjoint operators_ [22]. The adjoint
$\mathcal{O}^{\dagger}$ of an operator $\mathcal{O}$ satisfies
$A\mathcal{O}B-B\mathcal{O}^{\dagger}A=\nabla_{\mu}v^{\mu}$ for some vector
$v^{\mu}$ and tensors (with indices suppressed) $A$ and $B$, so that under an
integral we obtain the adjoint when integrating by parts,

$\int\mathrm{d}^{4}x\sqrt{-g}A\mathcal{O}B=\int\mathrm{d}^{4}x\sqrt{-g}B\mathcal{O}^{\dagger}A.$
(4.29)

The adjoints of the GHP derivatives are

$\operatorname{\text{\TH}}^{\dagger}=-\operatorname{\text{\TH}}+2\rho,\quad\eth^{\dagger}=-\eth,$
(4.30)

along with their primes. The adjoints of $\mathcal{T}^{i}$ are [21]

$\displaystyle\mathcal{T}_{0}^{\dagger}$
$\displaystyle=l^{\mu}\eth-m^{\mu}(\operatorname{\text{\TH}}-\rho),$ (4.31a)
$\displaystyle\mathcal{T}_{1}^{\dagger}$
$\displaystyle=\frac{1}{2}\left[l^{\mu}(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})-n^{\mu}(\operatorname{\text{\TH}}-2\rho)-m^{\mu}\eth^{\prime}+{\bar{m}}^{\mu}\eth\right],$
(4.31b) $\displaystyle\mathcal{T}_{2}^{\dagger}$
$\displaystyle=-n^{\mu}\eth^{\prime}+{\bar{m}}^{\mu}(\operatorname{\text{\TH}}^{\prime}-\rho^{\prime}).$
(4.31c)

We now have the tools to vary the Maxwell Lagrangian (4.25) with respect to
$A$,

$\left(\mathcal{T}_{0}^{\dagger}\mathcal{T}_{2}+\mathcal{T}_{2}^{\dagger}\mathcal{T}_{0}\right)A+\mathrm{c.c.}=2\mathcal{T}_{1}^{\dagger}\lambda+\mathrm{c.c.}$
(4.32)

Note that this is a vector-valued equation, per the definitions of
$\mathcal{T}_{i}^{\dagger}$. The components along $l$ and $n$ determine
$A_{l}$ and $A_{n}$ in terms of $\lambda$ and its complex conjugate,

$\displaystyle A_{l}$
$\displaystyle=-\frac{1}{2\eth\eth^{\prime}}(\operatorname{\text{\TH}}-2\rho)(\lambda+\bar{\lambda}-\mathfrak{g})$
(4.33a) $\displaystyle A_{n}$
$\displaystyle=\frac{1}{2\eth\eth^{\prime}}(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})(\lambda+\bar{\lambda}+\mathfrak{g})$
(4.33b)

where

$\mathfrak{g}\equiv\eth^{\prime}A_{m}+\eth A_{\bar{m}}$ (4.34)

is zero in the gauge used in the previous subsection; we will fix
$\mathfrak{g}=0$ herein. We can also integrate out $A_{m}$ and $A_{\bar{m}}$
using the imaginary part of the $\lambda$ equation of motion,

$\displaystyle\lambda-\bar{\lambda}$ $\displaystyle=\phi_{1}-\bar{\phi}_{1}$
$\displaystyle=\eth^{\prime}A_{m}-\eth A_{\bar{m}},$ (4.35)

which implies

$A_{m}=\frac{1}{2\eth\eth^{\prime}}\eth(\lambda-\bar{\lambda}),\qquad
A_{\bar{m}}=-\frac{1}{2\eth\eth^{\prime}}\eth^{\prime}(\lambda-\bar{\lambda}).$
(4.36)

Now that we have solutions for each component of $A_{\mu}$ in terms of
$\lambda$, we can plug them into the Lagrangian (4.25) to find a theory for
$\lambda$ alone. However, to avoid the complications of dealing with the
inverse $\eth\eth^{\prime}$ operator, we first perform a simple field
redefinition,

$\lambda=\eth\eth^{\prime}\psi$ (4.37)

so that the solution for $A_{\mu}$ is

$\displaystyle A_{l}$
$\displaystyle=-\frac{1}{2}\operatorname{\text{\TH}}(\psi+\bar{\psi}),$
(4.38a) $\displaystyle A_{n}$
$\displaystyle=\frac{1}{2}\operatorname{\text{\TH}}^{\prime}(\psi+\bar{\psi}),$
(4.38b) $\displaystyle A_{m}$
$\displaystyle=\frac{1}{2}\eth(\psi-\bar{\psi}).$ (4.38c)

To integrate out $A_{\mu}$ we plug this solution into eq. 4.25. The Maxwell
scalars evaluated on this solution are

$\displaystyle\phi_{0}$ $\displaystyle=\eth\operatorname{\text{\TH}}\psi,$
(4.39a) $\displaystyle\phi_{1}$
$\displaystyle=\frac{1}{2}\left[\operatorname{\text{\TH}}\operatorname{\text{\TH}}^{\prime}\left(\psi+\bar{\psi}\right)+\eth\eth^{\prime}(\psi-\bar{\psi})\right],$
(4.39b) $\displaystyle\phi_{2}$
$\displaystyle=\eth^{\prime}\operatorname{\text{\TH}}^{\prime}\psi.$ (4.39c)

Putting these in the action we find, freely integrating by parts,161616It is
helpful to recall the GHP commutators (2.32), particularly
$[\operatorname{\text{\TH}},\eth]=\rho\eth$. On GHP type $\\{0,0\\}$ objects
such as $\psi$,
$[\operatorname{\text{\TH}},\operatorname{\text{\TH}}^{\prime}]=[\eth,\eth^{\prime}]=0$.
Together with the adjoints (4.30), these imply that up to total derivatives
$(\eth\operatorname{\text{\TH}}A)(\eth^{\prime}\operatorname{\text{\TH}}^{\prime}B)=(\operatorname{\text{\TH}}\operatorname{\text{\TH}}^{\prime}A)(\eth\eth^{\prime}B)=(\eth\eth^{\prime}A)(\operatorname{\text{\TH}}\operatorname{\text{\TH}}^{\prime}B)$.

$\displaystyle\mathcal{L}$
$\displaystyle=2\phi_{1}(\eth\eth^{\prime}\psi)-(\eth\eth^{\prime}\psi)^{2}-\phi_{0}\phi_{2}+\mathrm{c.c.}$
$\displaystyle=\left(\operatorname{\text{\TH}}\operatorname{\text{\TH}}^{\prime}\psi-\eth\eth^{\prime}\psi\right)\eth\eth^{\prime}\bar{\psi}+{\mathrm{c.c.}}$
$\displaystyle=2\left(\operatorname{\text{\TH}}\operatorname{\text{\TH}}^{\prime}\psi-\eth\eth^{\prime}\psi\right)\eth\eth^{\prime}\bar{\psi}.$
(4.40)

This is a remarkably simple result. To switch back to
$\lambda=\eth\eth^{\prime}\psi$, we integrate by parts and use the GHP
commutators,

$\eth\eth^{\prime}(\operatorname{\text{\TH}}\operatorname{\text{\TH}}^{\prime}-\eth\eth^{\prime})=\left[(\operatorname{\text{\TH}}-2\rho)(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})-\eth\eth^{\prime}\right]\eth\eth^{\prime},$
(4.41)

to write

$\boxed{\mathcal{L}=2\bar{\psi}\left[(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})(\operatorname{\text{\TH}}-2\rho)-\eth\eth^{\prime}\right]\lambda.}$
(4.42)

where $\bar{\psi}=(\eth\eth^{\prime})^{-1}\bar{\lambda}$. The equation of
motion obtained by varying with respect to $\bar{\psi}$ is

$\left[(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})(\operatorname{\text{\TH}}-2\rho)-\eth\eth^{\prime}\right]\lambda=0.$
(4.43)

On shell $\lambda=\phi_{1}$, for which this is the _Fackerell-Ipser equation_
[23] in GHP notation [24].

Electric-magnetic duality transformations act as complex rotations on the
Maxwell scalars, $\phi_{i}\to e^{i\theta}\phi_{i}$, essentially since they are
the components of the (anti-)self-dual parts of the Maxwell tensor. The action
(4.42) is indeed manifestly invariant under $\lambda\to e^{i\theta}\lambda$,
or infinitesimally $\delta\lambda=i\lambda$ (along with
$\delta\bar{\psi}=-i\bar{\psi}$).171717The Lagrangian (4.42) does not look
real, but it is up to a total derivative, as can be explicitly checked using
the commutators and adjoints of the GHP derivatives, and in particular the
identity
$\operatorname{\text{\TH}}^{\dagger}(\operatorname{\text{\TH}}^{\prime})^{\dagger}\eth\eth^{\prime}=\eth\eth^{\prime}\operatorname{\text{\TH}}\operatorname{\text{\TH}}^{\prime}.$

A natural extension of the setup with $\phi_{1}$ as an auxiliary field is to
introduce auxiliary fields for all three Maxwell scalars, that is, a triplet
$(\lambda_{0},\lambda_{1},\lambda_{2})$ which on-shell satisfy
$\lambda_{i}=\phi_{i}$.181818This is essentially the construction of, e.g.,
Ref. [25] for the chiral formulation of Maxwell theory. First let us note that
we can “chop off” the $+{\mathrm{c.c.}}$ in the real Maxwell Lagrangian (4.24)
by adding the total derivative $(i/4)F_{\mu\nu}(\star
F)^{\mu\nu}=\phi_{0}\phi_{2}-\phi_{1}^{2}-{\mathrm{c.c.}}$,

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{4}F_{\mu\nu}^{2}+\frac{i}{4}F_{\mu\nu}(\star
F)^{\mu\nu}$ $\displaystyle=2\left(\phi_{1}^{2}-\phi_{0}\phi_{2}\right).$
(4.44)

Now we add in the full triplet of auxiliary fields,

$\displaystyle\mathcal{L}$
$\displaystyle\to\mathcal{L}-2(\phi_{1}-\lambda_{1})^{2}+2(\phi_{0}-\lambda_{0})(\phi_{2}-\lambda_{2})$
$\displaystyle=2\left(2\phi_{1}\lambda_{1}-\lambda_{0}\phi_{2}-\lambda_{2}\phi_{0}-\lambda_{1}^{2}+\lambda_{0}\lambda_{2}\right).$
(4.45)

The $\lambda_{i}$ equations of motion set $\lambda_{i}=\phi_{i}$ as desired,
while the $A$ equation of motion is

$2\mathcal{T}_{1}^{\dagger}\lambda_{1}-\mathcal{T}^{\dagger}_{0}\lambda_{2}-\mathcal{T}^{\dagger}_{2}\lambda_{0}=0,$
(4.46)

or in vector notation,

$\displaystyle 0$
$\displaystyle=l^{\mu}\left[(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})\lambda_{1}-\eth\lambda_{2}\right]+m^{\mu}\left[(\operatorname{\text{\TH}}-\rho)\lambda_{2}-\eth^{\prime}\lambda_{1}\right]$
$\displaystyle-n^{\mu}\left[(\operatorname{\text{\TH}}-2\rho)\lambda_{1}-\eth^{\prime}\lambda_{0}\right]-{\bar{m}}^{\mu}\left[(\operatorname{\text{\TH}}^{\prime}-\rho^{\prime})\lambda_{0}-\eth\lambda_{1}\right]$
$\displaystyle\equiv\mathcal{E}_{l}l^{\mu}+\mathcal{E}_{n}n^{\mu}+\mathcal{E}_{m}m^{\mu}+\mathcal{E}_{\bar{m}}{\bar{m}}^{\mu}.$
(4.47)

This formulation yields first-order constraints among the $\phi_{i}$ on-shell.
These are equivalent to the _Teukolsky-Starobinsky identities_ , which are
second-order differential relations between $\phi_{0}$ and $\phi_{2}$, or
equivalently fourth-order relations for $\phi_{0}$ and $\phi_{2}$ separately.
To obtain the Teukolsky-Starobinsky identities we therefore need to take
combinations of derivatives of $\mathcal{E}_{a}$ to remove $\phi_{1}$. The
correct combinations are

$\displaystyle\eth^{\prime}\mathcal{E}_{n}-(\operatorname{\text{\TH}}-3\rho)\mathcal{E}_{m}$
$\displaystyle=0$ $\displaystyle\Longrightarrow$
$\displaystyle(\operatorname{\text{\TH}}-2\rho)(\operatorname{\text{\TH}}-2\rho)\phi_{2}$
$\displaystyle=\eth^{\prime 2}\phi_{0},$ (4.48a)
$\displaystyle\eth\mathcal{E}_{l}-(\operatorname{\text{\TH}}-3\rho)\mathcal{E}_{\bar{m}}$
$\displaystyle=0$ $\displaystyle\Longrightarrow$
$\displaystyle(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})\phi_{0}$
$\displaystyle=\eth^{2}\phi_{2},$ (4.48b)
$\displaystyle\eth\mathcal{E}_{m}+\eth^{\prime}\mathcal{E}_{\bar{m}}$
$\displaystyle=0$ $\displaystyle\Longrightarrow$
$\displaystyle(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})\eth^{\prime}\phi_{0}$
$\displaystyle=(\operatorname{\text{\TH}}-2\rho)\eth\phi_{2},$ (4.48c)

where the T-S identities following the arrows can be found in, e.g., eq. 43 of
Ref. [21]. The third identity can also be obtained from
$(\operatorname{\text{\TH}}-2\rho)\mathcal{E}_{l}+(\operatorname{\text{\TH}}^{\prime}-2\rho^{\prime})\mathcal{E}_{n}$.
Here we have used the background equation
$\operatorname{\text{\TH}}\rho^{\prime}=\operatorname{\text{\TH}}^{\prime}\rho=\rho\rho^{\prime}-\Psi_{2}$.

We note that $\phi_{1}$ is special not just because it appeared naturally in
the dynamical construction of the previous subsection, but also because it is
closely related to the _Killing-Yano 2-form_ and its dual,

$\phi_{1}=\frac{i}{4r}F^{\mu\nu}\left(Y_{\mu\nu}-i\star Y_{\mu\nu}\right),$
(4.49)

where

$Y=r^{3}\sin\theta\mathrm{d}\theta\wedge\mathrm{d}\phi,\quad\star
Y=r\mathrm{d}t\wedge\mathrm{d}r.$ (4.50)

The Killing tensor, which underlies separability, is the square of the
Killing-Yano tensor, in coordinates,

$k_{AB}=-r^{4}\Omega_{AB},\quad k_{a\mu}=0.$ (4.51)

To connect explicitly to the $2+2$ formulation of the previous subsection, we
note the useful identities

$\displaystyle m_{A}\bar{m}_{B}$
$\displaystyle=\frac{r^{2}}{2}\left(\Omega_{AB}-i\epsilon_{AB}\right),$
(4.52a) $\displaystyle l_{a}n_{b}$
$\displaystyle=\frac{1}{2}\left(-g_{ab}+\epsilon_{ab}\right).$ (4.52b)

Using these we can calculate the Maxwell scalars in terms of $2+2$ quantities,

$\displaystyle\phi_{1}$
$\displaystyle=\frac{1}{2}\left(\epsilon^{ab}\nabla_{a}A_{b}-i\frac{\ell(\ell+1)}{r^{2}}a\right)Y,$
(4.53a) $\displaystyle\phi_{0}$
$\displaystyle=\left(\partial_{a}aB_{A}-A_{a}E_{A}\right)l^{a}m^{A},$ (4.53b)
$\displaystyle\phi_{2}$
$\displaystyle=-\left(\partial_{a}aB_{A}-A_{a}E_{A}\right)n^{a}\bar{m}^{A},$
(4.53c) $\displaystyle\phi_{0}\phi_{2}$
$\displaystyle=\frac{1}{4}\left(A_{a}^{2}E_{A}^{2}+(\partial
a)^{2}B_{A}^{2}\right).$ (4.53d)

We conclude with speculation about the structure discussed in this section and
its generalization to Kerr. There the Fackerell-Ipser equation is not
separable, which is why it is typical to work with the _Teukolsky equations_
[26] for the extreme-weight scalars $\phi_{0}$ and $\phi_{2}$, which are
separable due to the aforementioned Killing tensor structure [13]. It would be
very interesting to obtain an action principle for the Teukolsky equations
analogously to the one we have constructed for the Fackerell-Ipser equation
and Teukolsky-Starobinsky identities. We note that in Ref. [27] such an action
was constructed using the fact that the Teukolsky equations are linear, which
may provide a hint: the Teukolsky Lagrangian derived there is of the form
$\mathcal{L}\sim\rho^{-2}\phi_{2}\mathcal{O}\phi_{0}$, where $\mathcal{O}$ is
the Teukolsky operator for $\phi_{0}$. It would also be interesting to
understand how the Debye and Hertz potentials which appear in reconstruction
methods [28, 22, 29, 30] arise from the action formulation. We leave these
important open questions for future work.

##  5 Gravity

Consider linear perturbations around the Schwarzschild metric
$\bar{g}_{\mu\nu}$,191919For black hole perturbation theory in $2+2$ language
see, e.g., Refs. [10, 11, 12, 31]. The factor of $2/M_{\mathrm{Pl}}$ is to
canonically normalize the metric fluctuation.

$g_{\mu\nu}=\bar{g}_{\mu\nu}+\frac{2}{M_{\mathrm{Pl}}}h_{\mu\nu},$ (5.1)

and expand the Einstein-Hilbert action to quadratic order in $h_{\mu\nu}$,

$\displaystyle S$
$\displaystyle=\frac{M_{\mathrm{Pl}}^{2}}{2}\int\mathrm{d}^{4}x\sqrt{-g}R[g]$
$\displaystyle=\bar{S}+\delta_{1}S+\delta_{2}S+\mathcal{O}(h^{3}).$ (5.2)

The even- and odd-parity perturbations decouple at this order, so each is
described by a separate quadratic action:

$\delta_{2}S=\displaystyle\sum_{\ell=2}^{\infty}\displaystyle\sum_{m=-\ell}^{\ell}\left(S^{\ell
m}_{\mathrm{even}}+S_{\mathrm{odd}}^{\ell m}\right).$ (5.3)

Herein we will drop bars on background quantities, since we will only be
interested in $\delta_{2}S$.

Expanding the Ricci scalar to second order in perturbations is a non-trivial
task, and ultimately not necessary, since we can write the action in first-
order form. To see this, consider a metric variation $g\to g+\delta g$ and
Taylor expand the action,

$S[g+\delta g]=S[g]+\delta S+\frac{1}{2}\delta^{2}S+\cdots.$ (5.4)

Matching to eq. 5.2 we see that

$\delta_{2}S=\frac{1}{2}\delta^{2}S.$ (5.5)

It is a foundational result in GR that
$\delta\int\mathrm{d}^{4}x\sqrt{-g}R=\int\mathrm{d}^{4}x\sqrt{-g}G_{\mu\nu}\delta
g^{\mu\nu}$. Taking a second variation we obtain

$\displaystyle\delta_{2}S$
$\displaystyle=\frac{M_{\mathrm{Pl}}^{2}}{4}\int\mathrm{d}^{4}x\sqrt{-g}\delta
G_{\mu\nu}\delta g^{\mu\nu}$
$\displaystyle=-\int\mathrm{d}^{4}x\sqrt{-g}h^{\mu\nu}G[h]_{\mu\nu}$ (5.6)

where $G[h]_{\mu\nu}\equiv\delta G_{\mu\nu}[g+h]$ is the linear-in-$h$ part of
the Einstein tensor for $g_{\mu\nu}+h_{\mu\nu}$,

$G[h]_{\mu\nu}=\nabla_{\alpha}\nabla_{(\mu}h_{\nu)}^{\alpha}-\frac{1}{2}\Box
h_{\mu\nu}-\frac{1}{2}\nabla_{\mu}\nabla_{\nu}h-\frac{1}{2}\left(\nabla_{\mu}\nabla_{\nu}h^{\mu\nu}-\Box
h\right)g_{\mu\nu}.$ (5.7)

For simplicity (and to facilitate comparison to the literature) we will
continue to call this $\delta G_{\mu\nu}$, with the understanding that it is
evaluated on $g_{\mu\nu}+h_{\mu\nu}$ rather than
$g_{\mu\nu}+2M_{\mathrm{Pl}}^{-1}h_{\mu\nu}$. Integrating by parts we recover
the standard Fierz-Pauli Lagrangian for a spin-2 field,

$\delta_{2}S=\int\mathrm{d}^{4}x\sqrt{-g}\left(-\frac{1}{2}\nabla_{\alpha}h_{\mu\nu}\nabla^{\alpha}h^{\mu\nu}+\nabla_{\alpha}h_{\mu\nu}\nabla^{\nu}h^{\mu\alpha}-\nabla_{\mu}h\nabla_{\nu}h^{\mu\nu}+\frac{1}{2}\nabla_{\mu}h\nabla^{\mu}h\right).$
(5.8)

The $2+2$ components of $\delta G_{\mu\nu}[g+h]$ are standard and can be found
in, e.g., Refs. [10, 11, 12].202020We leave the analogous GHP analysis for
future work. We present relevant components in appendix A. The quadratic
action (5.6) is expanded as

$h^{\mu\nu}\delta G_{\mu\nu}=h^{ab}\delta G_{ab}+\frac{2}{r^{2}}h^{aA}\delta
G_{aA}+\frac{1}{r^{4}}h^{AB}\delta G_{AB}.$ (5.9)

We remind the reader that $\mathcal{M}_{2}$ indices are raised with $g^{ab}$
and $S^{2}$ indices with $\Omega^{AB}$.

There are at least two useful gauges which can be safely fixed at the level of
the action [20]. One is the standard Regge-Wheeler gauge, in which $h_{aA}$ is
purely odd and $h_{AB}=r^{2}K\Omega_{AB}$. Another is the “$\alpha$ gauge”
used in, e.g., Refs. [32, 33, 18], where $h_{aA}$ contains both even and odd
pieces and $h_{AB}=0$. The gauge choice affects the auxiliary structure of the
action. To see this, consider the gauge-invariant variables $\tilde{h}_{ab}$
and $\tilde{K}$ defined in Ref. [11], which correspond (by construction) to
$h_{ab}$ and $K$ in the Regge-Wheeler gauge, and in $\alpha$ gauge contain
derivatives,

$\displaystyle\tilde{h}_{ab}$
$\displaystyle=h_{ab}-2\nabla_{(a}\left(r^{2}r_{b)}\alpha\right),$ (5.10a)
$\displaystyle\tilde{K}$ $\displaystyle=-2fr\alpha.$ (5.10b)

We will remain agnostic about which of these two gauges to pick, and write
down expressions for both.

In these gauges, the components of $h_{\mu\nu}$ are

$\displaystyle h_{ab}$ $\displaystyle=\displaystyle\sum_{\ell,m}h^{\ell
m}_{ab}Y_{\ell m},$ (5.11a) $\displaystyle h_{aA}$
$\displaystyle=\displaystyle\sum_{\ell,m}r^{2}\left(\alpha_{\ell
m}r_{a}E_{A}^{\ell m}+h^{\ell m}_{a}B_{A}^{\ell m}\right),$ (5.11b)
$\displaystyle h_{AB}$ $\displaystyle=\displaystyle\sum_{\ell,m}r^{2}K_{\ell
m}Y_{\ell m}\Omega_{AB},$ (5.11c)

where we remind the reader that $r_{a}\equiv\partial_{a}r$. As usual we will
drop the summation and the subscripts and focus on a single $(\ell,m)$ mode.
In Regge-Wheeler gauge we set $\alpha=0$, and in $\alpha$ gauge we set $K=0$.
We will also find it convenient to decompose $h_{ab}$ into its trace and
tracefree parts,

$h_{ab}=\hat{h}_{ab}+\frac{1}{2}hg_{ab},\quad\hat{h}^{a}{}_{a}=0,$ (5.12)

and to work with the Ricci tensor rather than the Einstein tensor,

$\delta G_{\mu\nu}=\delta R_{\mu\nu}-\frac{1}{2}\delta Rg_{\mu\nu},\quad\delta
R=g^{ab}\delta R_{ab}+r^{-2}\Omega^{AB}\delta R_{AB}$ (5.13)

In terms of these variables, the even and odd actions are

$\displaystyle S_{\mathrm{even}}$
$\displaystyle=\int\mathrm{d}^{2}x\sqrt{-g}\mathrm{d}\Omega\left[r^{2}(Kg^{ab}-\hat{h}^{ab})\delta
R_{ab}+\frac{1}{2}h\Omega^{AB}\delta R_{AB}-2r^{2}r^{a}\alpha E^{A}\delta
R_{aA}\right],$ (5.14a) $\displaystyle S_{\mathrm{odd}}$
$\displaystyle=-2\int\mathrm{d}^{2}x\sqrt{-g}\mathrm{d}\Omega
r^{2}h^{a}B^{A}\delta R_{aA}.$ (5.14b)

To integrate over the 2-sphere, we note that the $S^{2}$ scalars $\delta
R_{ab}$ and $\Omega^{AB}\delta R_{AB}$ are expanded in $Y_{\ell m}$, while the
even and odd parts of $\delta R_{aA}$ can be written as

$\delta R_{aA}=\delta R^{E}_{a}E_{A}+\delta R^{B}_{a}B_{A}.$ (5.15)

Performing the integral over $S^{2}$ and writing the actions as
$S=\int\mathrm{d}^{2}x\sqrt{-g}\mathcal{L}$, the Lagrangians are

$\displaystyle\mathcal{L}_{\mathrm{even}}$
$\displaystyle=r^{2}(Kg^{ab}-\hat{h}^{ab})\delta
R_{ab}+\frac{1}{2}h\Omega^{AB}\delta
R_{AB}-2\ell(\ell+1)r^{2}r^{a}\alpha\delta R^{E}_{a},$ (5.16a)
$\displaystyle\mathcal{L}_{\mathrm{odd}}$
$\displaystyle=-2\ell(\ell+1)r^{2}h^{a}\delta R^{B}_{a},$ (5.16b)

where $h_{ab}$ denotes $h_{ab}^{\ell m}$, etc. Let us treat the odd and even
sectors separately.

###  5.1 Odd sector

The odd piece of the Ricci tensor is (see appendix A)

$\delta
R^{B}_{a}=\frac{1}{2r^{2}}\nabla^{b}\left(r^{4}F_{ab}\right)+\frac{(\ell+2)(\ell-1)}{2}h_{a},$
(5.17)

where

$F_{ab}=\partial_{a}h_{b}-\partial_{b}h_{a},$ (5.18)

so the Lagrangian (5.16b) is

$\displaystyle\mathcal{L}_{\mathrm{odd}}$
$\displaystyle=-2\ell(\ell+1)r^{2}h^{a}\delta R^{B}_{a}$
$\displaystyle=-\ell(\ell+1)\left(h^{a}\nabla^{b}\left(r^{4}F_{ab}\right)+(\ell+2)(\ell-1)r^{2}h_{a}^{2}\right)$
$\displaystyle=-\ell(\ell+1)\left(\frac{1}{2}r^{4}F_{ab}^{2}+(\ell+2)(\ell-1)r^{2}h_{a}^{2}\right),$
(5.19)

where in the last line we have integrated by parts. Note that
$(\ell+2)(\ell-1)=\ell(\ell+1)-2$. Finally we rescale

$h_{a}\to\frac{h_{a}}{\sqrt{2\ell(\ell+1)}}$ (5.20)

so the action takes the form

$\boxed{\mathcal{L}_{\mathrm{odd}}=-\frac{1}{4}r^{4}F_{ab}^{2}-\frac{(\ell+2)(\ell-1)}{2}r^{2}h_{a}^{2}.}$
(5.21)

In coordinates this is [18]

$\mathcal{L}_{\mathrm{odd}}=\frac{1}{2}r^{4}(\dot{h}_{1}-h_{0}^{\prime})^{2}+\frac{1}{2}(\ell+2)(\ell-1)r^{2}\left(\frac{1}{f}h_{0}^{2}-fh_{1}^{2}\right),$
(5.22)

where $h_{a}\mathrm{d}x^{a}=h_{0}\mathrm{d}t+h_{1}\mathrm{d}r$, and overdots
and primes denote $\partial_{t}$ and $\partial_{r}$, respectively.

Physically we can think of eq. 5.21 as describing a two-dimensional vector
with an $r$-dependent mass,212121The physical intuition behind this is a bit
more apparent if we perform the $2+2$ decomposition before linearizing. In
Regge-Wheeler gauge the odd modes only contribute to the Ricci scalar via the
tangent and transverse extrinsic curvatures, in the form of a mass term and a
Maxwell term, respectively, for $g^{AB}B_{B}\delta g_{aA}$ [31]. The factor of
$r^{-2}$ in $g^{AB}=r^{-2}\Omega^{AB}$ motivates the compensating factor of
$r^{2}$ in our definition (5.11b) of $\delta g_{aA}$. where we remind the
reader that $r$ is a background scalar rather than necessarily a coordinate
direction.

Note the close resemblance to $\mathcal{L}_{\mathrm{even}}$ for the Maxwell
field (4.5). We can repeat the same trick to integrate out the two fields
$h_{a}$ in favor of a single dynamical field. We integrate in an auxiliary
variable $\lambda(x^{a})$ via a perfect square so as not to affect the
dynamics,

$\displaystyle\mathcal{L}_{\mathrm{odd,aux}}$
$\displaystyle=\mathcal{L}_{\mathrm{odd}}+\frac{1}{4}\left(r^{2}F_{ab}+\lambda\epsilon_{ab}\right)^{2}$
$\displaystyle=\frac{1}{2}\left(r^{2}\lambda\epsilon^{ab}F_{ab}-\lambda^{2}-(\ell+2)(\ell-1)r^{2}h^{2}\right).$
(5.23)

This is dynamically equivalent to $\mathcal{L}_{\mathrm{odd}}$, which is
recovered by plugging in the solution to the $\lambda$ equation of motion,
$\lambda=(1/2)r^{2}\epsilon^{ab}F_{ab}$, and we will write it as
$\mathcal{L}_{\mathrm{odd}}$ accordingly. The introduction of $\lambda$ gives
us the option to integrate out $h_{a}$ by solving its equation of motion,

$(\ell+2)(\ell+1)r^{2}h_{a}=\epsilon_{ab}\partial^{b}(r^{2}\lambda).$ (5.24)

Substituting this into the action we have

$\mathcal{L}_{\mathrm{odd}}=-\frac{1}{2}\frac{[\partial(r^{2}\lambda)]^{2}}{(\ell+2)(\ell-1)r^{2}}-\frac{1}{2}\lambda^{2}.$
(5.25)

We perform a further rescaling to canonically normalize the kinetic term,

$\lambda=\frac{\sqrt{(\ell+2)(\ell-1)}}{r}\Psi_{-},$ (5.26)

so that the action becomes, using the background equations of motion (2.5),

$\mathcal{L}_{\mathrm{odd}}=-\frac{1}{2}(\partial\Psi_{-})^{2}-\frac{1}{2}\left(\frac{\ell(\ell+1)}{r^{2}}-\frac{3}{2}R\right)\Psi_{-}^{2}.$
(5.27)

The mass term explicitly evaluates to

$\displaystyle\frac{\ell(\ell+1)}{r^{2}}-\frac{3}{2}R$
$\displaystyle=\frac{\ell(\ell+1)}{r^{2}}-\frac{3r_{\mathrm{s}}}{r}$
$\displaystyle=\frac{V_{-}(r)}{f(r)},$ (5.28)

where $V_{-}(r)$ is the _Regge-Wheeler potential_ [34]. Putting everything
together we obtain the odd-sector Regge-Wheeler action,

$\boxed{S_{\mathrm{odd}}=\int\mathrm{d}^{2}x\sqrt{-g}\left[-\frac{1}{2}(\partial\Psi_{-})^{2}-\frac{1}{2}\frac{V_{-}}{f}\Psi_{-}^{2}\right].}$
(5.29)

The equation of motion,

$\Box\Psi_{-}=\frac{V_{-}}{f}\Psi_{-},$ (5.30)

where

$\displaystyle\Box$ $\displaystyle=\partial_{a}(g^{ab}\partial_{b})$
$\displaystyle=\frac{1}{f}\left(-\partial_{t}^{2}+\partial_{r_{\star}}^{2}\right),$
(5.31)

is the usual Regge-Wheeler equation [34] for $\Psi_{-}$,

$\left(-\partial_{t}^{2}+\partial_{r_{\star}}^{2}\right)\Psi_{-}=V_{-}\Psi_{-}.$
(5.32)

This means that $\Psi_{-}$ must be proportional to the Regge-Wheeler variable
up to time derivatives. Indeed, recalling Martel and Poisson’s [11] gauge-
invariant definition of the Cunningham-Price-Moncrief variable [35], which is
itself a time integral of the original Regge-Wheeler variable [34], we find
agreement with $\Psi_{-}$ up to a numerical factor:

$\displaystyle\Psi_{\mathrm{CPM}}$
$\displaystyle=\frac{r^{3}}{(\ell+2)(\ell-1)}\epsilon^{ab}F_{ab}$
$\displaystyle=\frac{2r}{(\ell+2)(\ell-1)}\lambda$
$\displaystyle=\frac{2}{\sqrt{(\ell+2)(\ell-1)}}\Psi_{-}.$ (5.33)

We conclude the discussion of the odd sector by noting an interesting
alternative approach discussed in, e.g., Ref. [10]. Consider the
$\mathcal{M}^{2}$ 1-form $h=h_{a}\mathrm{d}x^{a}$. The action is

$S_{\mathrm{odd}}=-\frac{1}{2}\int\mathrm{d}^{2}x\left(r^{4}F\wedge\star
F+(\ell+2)(\ell-1)r^{2}h\wedge\star h\right),$ (5.34)

and the equation of motion is

$\mathcal{E}=\mathcal{E}_{a}\mathrm{d}x^{a}=-\star\mathrm{d}(r^{4}\star
F)-(\ell+2)(\ell-1)r^{2}h.$ (5.35)

Taking a divergence by applying $\mathrm{d}\star$, we find that the 1-form
$r^{2}\star h$ is closed,

$\mathrm{d}(r^{2}\star h)=0.$ (5.36)

By the Poincaré lemma we can write it in terms of a scalar potential $\phi$,

$r^{2}h=\star\mathrm{d}\phi,$ (5.37)

or in index notation,

$r^{2}h_{a}=-\epsilon_{ab}\partial^{b}\phi.$ (5.38)

Comparing to eq. 5.24 we see that this potential is related to our auxiliary
variable $\lambda$ by

$\phi=-\frac{r^{2}}{(\ell+2)(\ell-1)}\lambda.$ (5.39)

The auxiliary field method is a technique for consistently implementing eq.
5.37 at the level of the action. In particular, if we were to naïvely plug the
solution (5.37) directly into the original action (5.21), the resulting theory
would be of fourth order in derivatives of $\phi$, and could not describe the
same physics: it contains two degrees of freedom rather than one, and
possesses an Ostrogradski ghost instability [36].

###  5.2 Even sector

The action for the even sector is given by eq. 5.16a. Expressions for relevant
components of the perturbed Ricci tensor are in appendix A. The resulting
actions after many intergrations by parts are

$\displaystyle\mathcal{L}_{\mathrm{even}}^{\mathrm{RW}}$
$\displaystyle=-2rr^{c}\hat{h}^{ab}\nabla_{a}\hat{h}_{bc}-2r\hat{h}^{ab}r_{a}\partial_{b}h-\frac{r^{2}R+\ell(\ell+1)+1}{2}\hat{h}_{ab}^{2}+\frac{\ell(\ell+1)+2}{4}h^{2}$
$\displaystyle\hphantom{{}=}-2r^{2}\nabla_{a}\hat{h}^{ab}\nabla_{b}K+r^{2}\partial
h\cdot\partial K+2rr^{a}K\partial_{a}h+\ell(\ell+1)hK+r^{2}(\partial K)^{2},$
(5.40a) $\displaystyle\mathcal{L}_{\mathrm{even}}^{\alpha}$
$\displaystyle=-2rr^{c}\hat{h}^{ab}\nabla_{a}\hat{h}_{bc}-2r\hat{h}^{ab}r_{a}\partial_{b}h-\frac{r^{2}R+\ell(\ell+1)+1}{2}\hat{h}_{ab}^{2}+\frac{\ell(\ell+1)+2}{4}h^{2}$
$\displaystyle\hphantom{{}=}+\ell(\ell+1)r^{2}\left(r^{2}(\epsilon^{ab}r_{a}\partial_{b}\alpha)^{2}+2f\alpha^{2}-2r_{b}\alpha\nabla_{a}\hat{h}^{ab}-\frac{1}{r^{4}}h\nabla_{a}\left(r^{4}r^{a}\alpha\right)\right),$
(5.40b)

in Regge-Wheeler gauge and in $\alpha$ gauge, respectively. We begin by noting
the well-known fact that these expressions are significantly more complicated
than eq. 5.21.222222One is compelled to wonder who ordered this, especially
given that, as we will see, the two sectors have essentially the same
dynamics.

It is convenient to perform a coordinate-like decomposition on objects with
indices by projecting along $r_{a}$ and the timelike direction
$t_{a}=\epsilon_{ab}r^{b}=f\partial_{a}t$, in terms of which the metric is
[11]

$fg_{ab}=r_{a}r_{b}-t_{a}t_{b}.$ (5.41)

In particular, we do not lose any information by projecting the traceless
perturbation $\hat{h}$ once along $r_{a}$ [12],

$\hat{h}_{a}\equiv\hat{h}_{ab}r^{b},$ (5.42)

as we can reconstruct $\hat{h}_{ab}$ via232323To see this, consider all
contractions with $r_{a}$ and $t_{a}$.

$f\hat{h}_{ab}=2r_{\langle
a}\hat{h}_{b\rangle}=r_{a}\hat{h}_{b}+r_{b}\hat{h}_{a}-(r\cdot\hat{h})g_{ab},$
(5.43)

where angular brackets denote traceless symmetrization,
$T_{\langle{ab}\rangle}=T_{({ab})}-\frac{1}{2}Tg_{ab}$. This simplifies the
actions somewhat,

$\displaystyle\mathcal{L}_{\mathrm{even}}^{\mathrm{RW}}$
$\displaystyle=-2r\hat{h}^{ab}\nabla_{a}\hat{h}_{b}-\frac{\ell(\ell+1)+1}{f}\hat{h}_{a}^{2}-2r\hat{h}^{a}\partial_{a}h+\frac{\ell(\ell+1)+2}{4}h^{2}$
$\displaystyle\hphantom{{}=}-2r^{2}\nabla_{a}\hat{h}^{ab}\nabla_{b}K+r^{2}\partial
h\cdot\partial K+2rr^{a}K\partial_{a}h+\ell(\ell+1)hK+r^{2}(\partial K)^{2},$
(5.44a) $\displaystyle\mathcal{L}_{\mathrm{even}}^{\alpha}$
$\displaystyle=-2r\hat{h}^{ab}\nabla_{a}\hat{h}_{b}-\frac{\ell(\ell+1)+1}{f}\hat{h}_{a}^{2}-2r\hat{h}^{a}\partial_{a}h+\frac{\ell(\ell+1)+2}{4}h^{2}$
$\displaystyle\hphantom{{}=}+\ell(\ell+1)r^{2}\left(r^{2}(t^{a}\partial_{a}\alpha)^{2}+2f\alpha^{2}-2r_{b}\alpha\nabla_{a}\hat{h}^{ab}-\frac{1}{r^{4}}h\nabla_{a}\left(r^{4}r^{a}\alpha\right)\right).$
(5.44b)

For concreteness, let us fix $\alpha$ gauge. We will discuss Regge-Wheeler
gauge at the end of the section. After the gauge freedom has been used up,
there are four fields for one underlying dynamical degree of freedom. Two
auxiliary variables are apparent by inspection of the action (5.44):
$t^{a}\hat{h}_{a}\sim h_{tr}$ and $h$. Here we will essentially follow the
procedure of Ref. [18] and begin by integrating out the former. To isolate the
components of $\hat{h}_{a}$ we decompose it as

$\hat{h}_{a}=\hat{h}_{0}t_{a}+\hat{h}_{1}r_{a}.$ (5.45)

We will also need to perform some simple field redefinitions to demix fields.
We begin by shifting $h$,

$h=\tilde{h}-2\hat{h}_{1}.$ (5.46)

Note that $h$ contains both $h_{tt}$ and $h_{rr}$, whereas $\tilde{h}\sim
h_{rr}$. In this field basis the action is

$\displaystyle\mathcal{L}_{\mathrm{even}}^{\alpha}$
$\displaystyle=\ell(\ell+1)\hat{h}_{0}^{2}+\frac{\ell(\ell+1)+2}{4}\tilde{h}^{2}-\ell(\ell+1)\tilde{h}\hat{h}_{1}-2rt^{a}\hat{h}_{0}\partial_{a}\tilde{h}+2rr^{a}\tilde{h}\partial_{a}\hat{h}_{1}$
$\displaystyle\hphantom{{}=}-\frac{\ell(\ell+1)}{r^{2}}\tilde{h}\nabla_{a}\left(r^{4}r^{a}\alpha\right)+\ell(\ell+1)r^{2}\left[r^{2}(t^{a}\partial_{a}\alpha)^{2}+2f\alpha^{2}\right]$
$\displaystyle\hphantom{{}=}+2\ell(\ell+1)\left[\hat{h}_{0}t^{a}\partial_{a}(r^{2}\alpha)+2rr^{a}\hat{h}_{1}\partial_{a}(r\alpha)+(1+3f)r\hat{h}_{1}\alpha\right]$
(5.47)

We can integrate out $\hat{h}_{0}$ using its equation of motion,

$\ell(\ell+1)\hat{h}_{0}=rt^{a}\partial_{a}\left(\tilde{h}-\ell(\ell+1)r\alpha\right),$
(5.48)

to find

$\displaystyle\mathcal{L}_{\mathrm{even}}^{\alpha}$
$\displaystyle=-\frac{r^{2}}{\ell(\ell+1)}t^{a}t^{b}\partial_{a}\tilde{h}\partial_{b}(\tilde{h}-2\ell(\ell+1)r\alpha)+\frac{\ell(\ell+1)+2}{4}\tilde{h}^{2}-\ell(\ell+1)\tilde{h}\hat{h}_{1}$
$\displaystyle\hphantom{{}=}+2rr^{a}\tilde{h}\partial_{a}\hat{h}_{1}-\frac{\ell(\ell+1)}{r^{2}}\tilde{h}\nabla_{a}\left(r^{4}r^{a}\alpha\right)$
$\displaystyle\hphantom{{}=}+2\ell(\ell+1)\hat{h}_{1}\left(2rr^{a}\partial_{a}(r\alpha)+(1+3f)r\alpha\right)+2\ell(\ell+1)r^{2}f\alpha^{2}.$
(5.49)

Now we perform a second field redefinition,242424The reason for this
particular order of operations is that integrating out $\hat{h}_{0}$
simplifies the kinetic term for $\alpha$. comprising a shift to demix $\alpha$
and $\tilde{h}$ and an overall rescaling,

$\alpha=\frac{\Lambda}{r^{2}f}\psi+\frac{\tilde{h}}{2\ell(\ell+1)r},$ (5.50)

where we have introduced the function [11]

$\Lambda(r)\equiv\ell(\ell+1)+1-3f.$ (5.51)

The action becomes

$\displaystyle\mathcal{L}_{\mathrm{even}}^{\alpha}$
$\displaystyle=\left(\frac{\ell(\ell+1)+1}{4}+\left(\frac{1}{2\ell(\ell+1)}-1\right)f\right)\tilde{h}^{2}-\Lambda\tilde{h}\hat{h}_{1}-\frac{\ell(\ell+1)\Lambda}{f}r^{a}\tilde{h}\partial_{a}\psi$
$\displaystyle\hphantom{{}=}+\frac{2\Lambda
r}{f}t^{a}t^{b}\partial_{a}\tilde{h}\partial_{b}\psi-\frac{(\ell+2)(\ell-1)(\ell(\ell+1)+\Lambda)}{r}\tilde{h}\psi+\frac{2\ell(\ell+1)\Lambda^{2}}{r^{2}f}\psi^{2}$
$\displaystyle\hphantom{{}=}+\frac{2\ell(\ell+1)}{f}\hat{h}_{1}\left[2\Lambda
r^{a}\partial_{a}\psi+\frac{3f(\ell(\ell+1)-f)-\ell(\ell+1)-1}{r}\psi\right].$
(5.52)

Note that $\psi$ is precisely the gauge-invariant Zerilli-Moncrief function
defined in Ref. [11], multiplied by $-1/4$.

The upshot of all these field redefinitions is that two of the remaining three
fields are manifestly non-dynamical: $\hat{h}_{1}$ is a Lagrange multiplier
(it appears linearly) and $\tilde{h}$ is auxiliary (it appears quadratically
but without derivatives). The constraint obtained by varying with respect to
$\hat{h}_{1}$ fixes $\tilde{h}$ in terms of $\psi$,

$\tilde{h}=\frac{2\ell(\ell+1)}{f}\left[2r^{a}\partial_{a}+\frac{3f(\ell(\ell+1)-f)-\ell(\ell+1)-1}{r\Lambda}\right]\psi,$
(5.53)

while the equation of motion for $\tilde{h}$ is

$\displaystyle 0$
$\displaystyle=\Lambda\hat{h}_{1}+\frac{(\ell(\ell+1)+1-f)\Lambda}{f}r^{a}\partial_{a}\psi+\frac{(\ell+2)(\ell-1)(\ell(\ell+1)+\Lambda)}{r}\psi$
$\displaystyle\hphantom{{}=}+\frac{2r\Lambda}{f}t^{a}t^{b}\nabla_{a}\nabla_{b}\psi-\left(\frac{\ell(\ell+1)+1}{2}+\left(\frac{1}{\ell(\ell+1)}-2\right)f\right)\tilde{h}.$
(5.54)

This fixes $\hat{h}_{1}$ once we use eq. 5.53, although we do not need to know
$\hat{h}_{1}$ in order to integrate it out of the action, as it multiplies the
constraint (5.53) that it enforces. We will however need this equation in
order to construct off-shell duality operators for the metric perturbations.

Plugging eq. 5.53 into the action we finally obtain, after some integrations
by parts and algebra,

$\mathcal{L}_{\mathrm{even}}^{\alpha}=-4\ell(\ell+1)(\ell+2)(\ell-1)\left[(\partial\psi)^{2}+\frac{V_{+}}{f}\psi^{2}\right],$
(5.55)

where

$V_{+}=\frac{f}{3r^{2}}\left(\Lambda+\frac{2(\ell+2)^{2}(\ell-1)^{2}\left(1+\ell(\ell+1)\right)}{\Lambda^{2}}\right)$
(5.56)

is the _Zerilli potential_ [37]. Finally we canonically normalize,

$\psi=\frac{1}{2\sqrt{\ell(\ell+1)(\ell+2)(\ell-1)}}\Psi_{+},$ (5.57)

to obtain the Zerilli action for the even sector:

$\boxed{\mathcal{L}_{\mathrm{even}}=-\frac{1}{2}(\partial\Psi_{+})^{2}-\frac{1}{2}\frac{V_{+}}{f}\Psi^{2}_{+}.}$
(5.58)

The main benefit of working with $\alpha$ gauge is that the field
redefinitions we needed to perform did not involve derivatives, but a choice
of gauge is not a choice of physics, and indeed in Regge-Wheeler gauge we can
follow a similar procedure to reduce the action (5.40a) to the Zerilli action
(5.58). We begin again by integrating out $h_{tr}\sim\hat{h}_{0}$, while
$h_{tt}\sim h-2\hat{h}_{1}$ is a Lagrange multiplier that imposes a constraint
on $K$ and $h_{rr}\sim h+2\hat{h}_{1}$ (and in turn drops out of the action).
To demix the remaining two variables and canonically normalize we perform a
field redefinition,

$h+2\hat{h}_{1}=\sqrt{\frac{\ell(\ell+1)}{2(\ell+2)(\ell-1)}}\frac{\Lambda}{rf}\Psi_{+}+\frac{1}{f}\left(2rr^{a}\partial_{a}-\Lambda\right)K.$
(5.59)

The even sector is inordinately complicated, and the procedure we have done is
not unique, and may not be the simplest or clearest. Alternative approaches
would therefore be interesting to explore. An obvious alternative is to
integrate out $h$ first rather than $\hat{h}_{0}$. Furthermore, the
decomposition (5.45) can be swapped for a more elegant argument in terms of
differential forms analogously to the odd sector [12], which may therefore
admit an auxiliary variable formulation. And of course an approach eliding the
Regge-Wheeler and Zerilli equations altogether in favor of the Teukolsky
equation would be of exceptional interest.

##  6 Chandrasekhar duality

The linearized Einstein-Hilbert action is a complicated functional of the
metric perturbations (cf. eqs. 5.21 and 5.40), but by integrating out the non-
dynamical degrees of freedom we obtained a simple action in terms of the
Regge-Wheeler and Zerilli variables,

$\boxed{S=\displaystyle\sum_{\ell=2}^{\infty}\displaystyle\sum_{m=-\ell}^{\ell}\displaystyle\sum_{\pm}\int\mathrm{d}^{2}x\sqrt{-g}\left(-\frac{1}{2}(\partial\Psi_{\pm})^{2}-\frac{1}{2f}V_{\pm}\Psi^{2}_{\pm}\right),}$
(6.1)

where $V_{+}$ and $V_{-}$ are the usual Zerilli [37] and Regge-Wheeler [34]
potentials, respectively.

It is important to pause here to emphasize the difference between on-shell and
off-symmetries. We could have constructed eq. 6.1 directly from the Regge-
Wheeler and Zerilli equations, but it was a non-trivial exercise to get there
from the Einstein-Hilbert action using standard field theory tools. Having
done this exercise, we will be able to construct an off-shell duality symmetry
of the original action (5.2).

First let us demonstrate the duality invariance of the Regge-Wheeler/Zerilli
action (6.1). It is a remarkable fact that the two seemingly-disparate
potentials $V_{\pm}$ (cf. eqs. 5.28 and 5.56) can be written in a unified form
in terms of a single _superpotential_ [38, 39, 1, 40, 41, 2],252525Note also
the relation $V_{+}=V_{-}-2\partial_{r_{\star}}^{2}\Lambda$ [42]. For
electric-magnetic duality on charged black hole backgrounds, see Ref. [43].

$V_{\pm}=W^{2}\mp r^{a}\partial_{a}W+\beta,$ (6.2)

where the superpotential $W(r)$ and constant $\beta$ are given by

$W(r)=-\left(\frac{3}{2}\frac{rRf}{\Lambda}+\sqrt{-\beta}\right),\quad\beta\equiv-\left(\frac{\ell(\ell+1)(\ell+2)(\ell-1)}{6r_{\mathrm{s}}}\right)^{2}.$
(6.3)

It is straightforward to check that the action (6.1) is invariant under the
duality symmetry

$\boxed{\delta\Psi_{\pm}=\left(r^{a}\partial_{a}\mp W\right)\Psi_{\mp}.}$
(6.4)

The transformation (6.4) is an off-shell symmetry of the action, and coincides
on shell with the venerable _Chandrasekhar duality_ [38, 39, 1, 40].262626For
Chandrasekhar duality beautifully visualized, see Ref. [44]. This “hidden”
symmetry of the linearized Einstein equations relates a solution $\Psi_{\pm}$
to the Regge-Wheeler or Zerilli equation to a solution $\Psi_{\mp}$ to the
other equation, which can be constructed in frequency space via272727The
prefactor is not strictly necessary since any constant multiple of this will
also be a solution. However we include this prefactor to emphasize the
existence of _algebraically-special_ modes for which $\omega^{2}=\beta$, where
special care must be taken. We will not discuss algebraically-special modes in
this work.

$\Psi_{\pm}=\frac{1}{\beta-\omega^{2}}\left(\frac{\mathrm{d}}{\mathrm{d}r_{\star}}\Psi_{\mp}\mp
W\Psi_{\mp}\right).$ (6.5)

We note that, intriguingly, this symmetry structure also appears in
_supersymmetric quantum mechanics_ , the theory of $0+1$-dimensional
supersymmetry [45].282828Concretely, we can write eq. 6.4 in a manner
suggestive of supersymmetric quantum mechanics by defining raising and
lowering operators,  $A\equiv\frac{\partial}{\partial r_{\star}}+W,\qquad
A^{\dagger}=-\frac{\partial}{\partial r_{\star}}+W,$ (6.6) and writing the
symmetry as  $\delta\begin{pmatrix}\Psi_{+}\\\
\Psi_{-}\end{pmatrix}=\begin{pmatrix}0&-A^{\dagger}\\\
A&0\end{pmatrix}\begin{pmatrix}\Psi_{+}\\\ \Psi_{-}\end{pmatrix}.$ (6.7) The
Chandrasekhar duality is responsible for the crucial result that, for four-
dimensional black holes in GR, the even and odd sectors are _isospectral_ ,
meaning they share the same quasinormal mode spectrum.292929This is a
consequence of the fact that, if $\Psi_{\pm}$ satisfies the boundary
conditions which define a quasinormal mode, then the $\Psi_{\mp}$ generated by
eq. 6.5 does as well, so the Chandrasekhar transformation relates quasinormal
modes of even and odd parity without changing the frequency (excluding
algebraically-special modes). As we will see, this is also true for the
infalling boundary conditions used to calculate Love numbers.

With the off-shell symmetry (6.4) in hand, we can compute conserved quantities
using the Noether procedure. The conservation law, in coordinates, is

$\partial_{t}J^{t}+\partial_{r_{\star}}J^{r_{\star}}=0,$ (6.8)

with the current

$\displaystyle J^{t}$
$\displaystyle=\dot{\Psi}_{+}A^{\dagger}\Psi_{-}-\dot{\Psi}_{-}A\Psi_{+}$
$\displaystyle=-\Psi_{+}^{\prime}\dot{\Psi}_{-}-\dot{\Psi}_{+}\Psi_{-}^{\prime}+W\left(\Psi_{-}\dot{\Psi}_{+}-\Psi_{+}\dot{\Psi}_{-}\right),$
(6.9a) $\displaystyle J^{r_{\star}}$
$\displaystyle=\dot{\Psi}_{+}\dot{\Psi}_{-}-(A\Psi_{+})(A^{\dagger}\Psi_{-})-\beta\Psi_{+}\Psi_{-}$
$\displaystyle=\dot{\Psi}_{+}\dot{\Psi}_{-}+\Psi_{+}^{\prime}\Psi_{-}^{\prime}+W\left(\Psi_{+}\Psi_{-}^{\prime}-\Psi_{-}\Psi_{+}^{\prime}\right)-\left(W^{2}+\beta\right)\Psi_{+}\Psi_{-}.$
(6.9b)

Here overdots denote derivatives with respect to $t$, and primes denote
$\partial_{r_{\star}}$ derivatives.

###  6.1 A complex master variable

Similarly to the spin-1 case, we can combine the Regge-Wheeler and Zerilli
variables into a complex variable,

$\Psi\equiv\frac{\Psi_{+}+i\Psi_{-}}{\sqrt{2}},$ (6.10)

in terms of which the Lagrangian (6.1) takes a very simple form,

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{2}\displaystyle\sum_{\pm}\left((\partial\Psi_{\pm})^{2}+\frac{V_{\pm}}{f}\Psi_{\pm}^{2}\right)$
$\displaystyle=-\partial_{a}\Psi\partial^{a}\bar{\Psi}+\frac{1}{2f}r^{a}\partial_{a}W(\Psi^{2}+\bar{\Psi}^{2})-\frac{W^{2}+\beta}{f}\Psi\bar{\Psi}$
(6.11)

as does the duality transformation,

$\delta\Psi=i\left(r^{a}\partial_{a}\bar{\Psi}+W\Psi\right).$ (6.12)

Let us confirm this is a symmetry. Under a general variation, the Lagrangian
changes as

$\delta\mathcal{L}=\bar{\mathcal{E}}\delta\Psi+\mathcal{E}\delta\bar{\Psi},$
(6.13)

where the equation of motion $\mathcal{E}$ is

$\mathcal{E}\equiv\Box\Psi+\frac{1}{f}r^{a}\partial_{a}W\bar{\Psi}-\frac{W^{2}+\beta}{f}\Psi.$
(6.14)

In terms of the quantity

$\bar{Q}=r^{a}\partial_{a}\bar{\Psi}+W\Psi,$ (6.15)

the variation of the Lagrangian under $\delta\Psi=i\bar{Q}$ is

$\displaystyle\delta\mathcal{L}$
$\displaystyle=i\left(\bar{Q}\bar{\mathcal{E}}-Q\mathcal{E}\right)$
$\displaystyle=2\operatorname{Im}Q\mathcal{E}.$ (6.16)

Now we calculate $Q\mathcal{E}$ and freely integrate by parts,

$\displaystyle Q\mathcal{E}$
$\displaystyle=\left(r^{a}\partial_{a}\Psi+W\bar{\Psi}\right)\left(\Box\Psi+\frac{1}{f}r^{a}\partial_{a}W\bar{\Psi}-\frac{W^{2}+\beta}{f}\Psi\right)$
$\displaystyle=W\left(-\partial_{a}\Psi\partial^{a}\bar{\Psi}+\frac{r^{a}\partial_{a}W}{f}\left(\Psi^{2}+\bar{\Psi}^{2}\right)-\frac{W^{2}+\beta}{f}\Psi\bar{\Psi}\right).$
(6.17)

The last line is manifestly real, so that the variation of the Lagrangian
vanishes as expected,

$\delta\mathcal{L}=2\operatorname{Im}Q\mathcal{E}=0.$ (6.18)

Using similar manipulations we can also calculate the conserved current,

$J_{a}=v_{a}-\bar{v}_{a}+W\left(\bar{\Psi}\partial_{a}\Psi-\Psi\partial_{a}\bar{\Psi}\right)-\frac{W^{2}+\beta}{2f}r_{a}\left(\Psi^{2}-\bar{\Psi}^{2}\right)$
(6.19)

where we have defined

$v^{a}\partial_{a}=-(\partial_{r}\Psi)(\partial_{t}\Psi)\partial_{t}+\frac{f^{2}(\partial_{r}\Psi)^{2}+(\partial_{t}\Psi)^{2}}{2}\partial_{r}$
(6.20)

such that $r^{a}\partial_{a}\Psi\Box\Psi=\nabla_{a}v^{a}$.

Analogously to the spin-1 case, it is natural to wonder whether this complex
master variable is related to the middle-weight Weyl scalar, $\Psi_{2}$. A new
complication in the gravitational case is that $\Psi_{2}$ has a background
value, and accordingly its perturbation $\delta\Psi_{2}$ is not gauge-
invariant. Nevertheless one can construct a gauge-invariant version
$\widetilde{\delta\Psi_{2}}$ which contains the Regge-Wheeler and Zerilli
variables [46],

$\widetilde{\delta\Psi_{2}}=\frac{\sqrt{(\ell-1)\ell(\ell+1)(\ell+2)}}{2r^{3}}\left[\frac{\Lambda}{(\ell+2)(\ell-1)}\Psi_{+}+i\Psi_{-}\right]Y(\theta)$
(6.21)

This is not quite our master variable $\Psi$, as the real (even) piece is a
rescaling of the Zerilli variable. We leave a further exploration of this
question for future work.

###  6.2 Flat-space limit: linearized gravitational duality

We can gain some physical insight by looking at the flat-space limit,
$r_{\mathrm{s}}\to 0$. The expression (6.4) for $\delta\Psi_{\pm}$ diverges
due to the $1/r_{\mathrm{s}}$ scaling in $W(r)$, which can be remedied by
sending $\delta\Psi_{\pm}\to r_{\mathrm{s}}\delta\Psi_{\pm}$ before taking the
limit. In this limit we have an $SO(2)$ symmetry acting on
$(\Psi_{+},\Psi_{-})$ similar to the electromagnetic case,

$\displaystyle\delta\Psi_{+}$ $\displaystyle=-\Psi_{-},$ (6.22a)
$\displaystyle\delta\Psi_{-}$ $\displaystyle=\Psi_{+}.$ (6.22b)

Direct calculation shows that, on shell, this duality generates rotations
between the Riemann tensor and its dual,

$\displaystyle\delta R_{{\mu\nu}{ab}}$
$\displaystyle={\star}R_{{\mu\nu}{ab}},$ (6.23a)
$\displaystyle\delta{\star}R_{{\mu\nu}{ab}}$
$\displaystyle=-R_{{\mu\nu}{ab}},$ (6.23b)

where the dual Riemann tensor is defined as

${\star}R_{{\mu\nu}{ab}}=\frac{1}{2}\epsilon_{{\mu\nu}\rho\sigma}R^{\rho\sigma}{}_{ab}.$
(6.24)

This is the well-known gravitational “electric-magnetic” duality, lifted to an
off-shell symmetry for linear perturbations around flat space [47].

We conclude that the symmetry (6.4) is an extension of electromagnetic duality
to Schwarzschild backgrounds. An off-shell duality symmetry has also been
found to hold for Minkowski [47], de Sitter [48], and anti-de Sitter
backgrounds [49]. Adding to this list Schwarzschild, which is less symmetric
than the others, raises interesting questions: which other backgrounds possess
a linearized duality symmetry, and what physical mechanism underlies these
symmetries?

###  6.3 Chandrasekhar duality off-shell

The symmetry (6.4) can be lifted to a symmetry of the linearized Einstein-
Hilbert action in terms of the metric perturbations, eqs. 5.21 and 5.40,
analogously to electromagnetism. The calculation itself is cumbersome and not
especially enlightening, so we will outline the steps without presenting full
expressions. Let us begin with the transformation of the odd-sector variable
$h_{a}$. Using its solution (5.24) and undoing various rescalings, we have

$\delta
h_{a}=\frac{1}{\sqrt{2\ell(\ell+1)r^{2}}}\epsilon_{ab}\partial^{b}\left(r\delta\Psi_{-}\right),$
(6.25)

where $\delta\Psi_{-}$ is given by eq. 6.4. That expression is constructed
from $\Psi_{+}$, which we in turn write in terms of even-sector metric
perturbations by following the chain of field redefinitions. For the even
sector, we vary the expressions in terms of $\Psi_{+}$ for $h_{ab}$ and
$\alpha$ or $K$, use eq. 6.4, and relate $\Psi_{-}$ to $h_{a}$ via

$\Psi_{-}=\frac{r^{3}}{2\sqrt{(\ell+2)(\ell-1)}}\epsilon^{ab}F_{ab}.$ (6.26)

In this way we construct (rather complicated) expressions $\delta
h_{\mu\nu}[h]$ which one can verify by explicit calculation comprise an off-
shell symmetry of eqs. 5.21 and 5.40. Interestingly they can be simplified
somewhat using the equations of motion, in which case the expressions become
entirely local. A natural question for future investigation is whether the
$\delta h_{\mu\nu}$ constructed this way is equal to a dual potential
$\tilde{h}_{\mu\nu}$. Since only the electric part of the Weyl tensor has a
non-vanishing background value, the linearized duality transformations do not
simply rotate $C_{{\mu\nu}\alpha\beta}$ and $\tilde{C}_{{\mu\nu}\alpha\beta}$.

##  7 Physical implications: Love numbers

Another aspect of black hole perturbation theory in which symmetry has
recently been found to play a crucial role is in the computation of _tidal
Love numbers_. In particular, the puzzle over the unexpected vanishing of
black hole Love numbers [50, 51, 52, 53, 54] spurred the discovery of
underlying symmetry structures [55, 56, 57, 58]. It turns out that the duality
symmetry which is the focus of this paper also plays a role in the symmetry
story for Love numbers.

Consider the Regge-Wheeler action (5.22) in the static sector, i.e., setting
time derivatives to zero,

$\displaystyle\mathcal{L}_{\mathrm{odd}}^{\omega=0}$
$\displaystyle=\frac{1}{2}r^{4}h_{0}^{\prime
2}+\frac{(\ell+2)(\ell-1)}{2}r^{2}\left(\frac{1}{f}h_{0}^{2}-fh_{1}^{2}\right),$
(7.1)

where primes denote $r$ derivatives. In the static limit $h_{1}$ is auxiliary
and decouples from $h_{0}$, so can be consistently set to zero. The Regge-
Wheeler variable $\Psi_{-}$ is related to $h_{0}$ by

$\Psi_{-}=r^{3}h_{0}^{\prime},\quad
h_{0}=\frac{f}{r^{2}}\partial_{r}\left(r\Psi_{-}\right).$ (7.2)

In Ref. [55] it was shown that the static Regge-Wheeler equation is invariant
under _ladder symmetries_ which are responsible for the vanishing of tidal
Love numbers in the odd sector. These come in the form of raising and lowering
operators which relate solutions of the Regge-Wheeler equation to a solution
with $\ell$ raised or lowered by one,

$\displaystyle D^{+}_{\ell}$
$\displaystyle=-r^{2}f\partial_{r}+\frac{\ell^{2}+3}{2(\ell+1)}r_{\mathrm{s}}-\ell
r,$ (7.3a) $\displaystyle D^{-}_{\ell}$
$\displaystyle=r^{2}f\partial_{r}+\frac{\ell^{2}(r_{\mathrm{s}}-2r)-2\ell(r-r_{\mathrm{s}})+4r_{\mathrm{s}}}{2\ell}.$
(7.3b)

At the lowest rung of the ladder, $\ell=2$, there is a further symmetry given
by $\delta\Psi_{-}^{\ell=2}=Q_{2}\Psi_{-}^{\ell=2}$, with

$Q_{2}=r^{6}f\partial_{r}-3r^{5}f.$ (7.4)

It follows that any $\ell$ mode is symmetric under the “horizontal” ladder
symmetry

$\delta\Psi_{-}=Q_{\ell}\Psi_{-},$ (7.5)

where $Q_{\ell}$ is built recursively from $Q_{2}$,

$Q_{\ell}\equiv D^{+}_{\ell-1}Q_{\ell-1}D_{\ell}^{-}.$ (7.6)

Transforming from $\Psi_{-}$ to $h_{0}$, we see that the metric transforms
under the horizontal odd-sector ladder symmetry as

$h_{0}\to
h_{0}+\frac{f}{r^{2}}\partial_{r}\left[rQ_{\ell}(r^{3}h_{0}^{\prime})\right],\quad
h_{1}\to h_{1}.$ (7.7)

It is straightforward to check that eq. 7.7 is a symmetry of eq. 7.1. However,
such a symmetry of the Zerilli equation is not apparent. Indeed, the argument
for the vanishing of Love numbers for the Zerilli equation in Ref. [55] relied
on the fact, as we will show, that the duality invariance (6.4) implies that
the even and odd Love numbers are equal.303030This is to some extent an
artifact of our insistence on working with the Regge-Wheeler and Zerilli
master equations. The main result in Ref. [55] worked with the Teukolsky
equation, which, besides not being limited to the Schwarzschild case, contains
both even and odd modes.

Ladder operators for the Zerilli equation can be constructed straightforwardly
by sandwiching a Regge-Wheeler ladder operator between two applications of the
duality symmetry, e.g., for the horizontal operators,

$\delta\Psi_{+,\ell}=\left(\partial_{r_{\star}}-W\right)Q_{\ell}\left(\partial_{r_{\star}}+W\right)\Psi_{+,\ell}.$
(7.8)

It would be very interesting to know whether this symmetry is responsible for
universal relations such as I-Love-Q [59, 60].

###  7.1 Equality of Love numbers from gravitational duality

Let us finish by establishing that the vanishing of the duality Noether
current requires the tidal Love numbers in the even and odd sectors to be
equal. Following Ref. [61], we calculate the Love numbers for static solutions
by imposing regularity at the horizon and examining the behavior of the fields
at infinity,

$\Psi_{\pm}\to\bar{\Psi}_{\pm}\left(r^{\ell+1}+\hat{\lambda}_{\pm}r^{-\ell}\right),$
(7.9)

where $\hat{\lambda}_{\pm}$ are the Love numbers for the even ($+$) and odd
($-$) sectors and $\bar{\Psi}_{\pm}$ are constants.

Since we are looking at static solutions, conservation of the Noether current
(6.8) becomes the statement that the $r_{\star}$ component (6.9b) is constant.
First we need to ensure that the duality transformation (6.4) preserves the
boundary conditions, namely, if $\Psi_{\pm}$ is regular at the horizon, then
so is $\partial_{r_{\star}}\Psi_{\pm}\mp W(r)\Psi_{\pm}$. From eq. 6.3 we see
that $W(r_{\mathrm{s}})$ is finite, which leaves us to check that
$\partial_{r_{\star}}\Psi_{\pm}=f(r)\partial_{r}\Psi_{\pm}$ is regular at
$r=r_{\mathrm{s}}$. We can see this by solving the Regge-Wheeler and Zerilli
equations perturbatively near the horizon,

$f\partial_{r}(f\partial_{r}\Psi_{\pm})=V_{\pm}\Psi_{\pm}.$ (7.10)

It is convenient to use $f$ as our radial coordinate, so that we can simply
expand around $f=0$ to look at the horizon. Using the fact that the Regge-
Wheeler and Zerilli potentials both scale as $f$ near the horizon, and that
$\partial_{r}=f^{\prime}(r)\partial_{f}\approx\partial_{f}/r_{\mathrm{s}}$, we
have

$f\partial_{f}(f\partial_{f}\Psi_{\pm})\approx\frac{(\ell-1)\ell(\ell+1)(\ell+2)\pm
3}{\ell(\ell+1)+1}f\Psi_{\pm}.$ (7.11)

Near the horizon this is solved by

$\Psi_{\pm}=c_{1}^{\pm}\left(1+\mathcal{O}(f)\right)+c_{2}^{\pm}\ln
f\left(1+\mathcal{O}(f)\right).$ (7.12)

Regularity at the horizon demands $c_{2}^{+}=c_{2}^{-}=0$, so that
$f\partial_{r}\Psi_{\pm}\approx f\partial_{f}\Psi_{\pm}\to 0$ as $f\to
0$.313131We must also check that the subdominant terms do not blow up at the
horizon. Assuming that the subdominant terms on the log side go as $f^{n}\ln
f$, then these contribute harmlessly to $f\partial_{f}\Psi_{\pm}$ for $n>0$,
and moreover they vanish upon perturbatively solving the equation of motion.
So if $\Psi_{\pm}$ is a solution with boundary conditions suitable for
computing Love numbers, then
$\tilde{\Psi}_{\pm}\equiv\Psi_{\pm}+\delta\Psi_{\pm}$ is as well.

Now we simply need to compute $J^{r_{\star}}$ at the horizon and at infinity
and equate the two, where for static solutions

$J^{r_{\star}}=\Psi_{+}^{\prime}\Psi_{-}^{\prime}+W\left(\Psi_{+}\Psi_{-}^{\prime}-\Psi_{-}\Psi_{+}^{\prime}\right)-(W^{2}+\beta)\Psi_{+}\Psi_{-}.$
(7.13)

We begin by evaluating this at the horizon. Primes denote $r_{\star}$
derivatives, and we are assuming that $\Psi_{\pm}$ are regular at the horizon,
so $\Psi_{\pm}^{\prime}=(1-r_{\mathrm{s}}/r)\partial_{r}\Psi_{\pm}=0$ at
$r=r_{\mathrm{s}}$. From eq. 6.3 we see $W^{2}(r_{\mathrm{s}})+\beta=0$, so
that the current vanishes for static solutions with regular boundary
conditions,

$J^{r_{\star}}=0.$ (7.14)

At infinity, we again have $W^{2}(\infty)+\beta=0$, so the leading-order terms
will be those with only one derivative,

$\displaystyle J^{r_{\star}}$ $\displaystyle\to
W(\infty)\left(\Psi_{+}\Psi_{-}^{\prime}-\Psi_{-}\Psi_{+}^{\prime}\right)$
$\displaystyle=-\frac{\bar{\Psi}_{+}\bar{\Psi}_{-}}{6r_{\mathrm{s}}}\ell(\ell+1)(\ell-1)(\ell+2)(2\ell+1)\left(\hat{\lambda}_{+}-\hat{\lambda}_{-}\right).$
(7.15)

Since $J^{r_{\star}}=0$ everywhere, we conclude that

$\boxed{\hat{\lambda}_{+}=\hat{\lambda}_{-},}$ (7.16)

i.e., the even and odd sectors are forced to have equal Love numbers as a
consequence of symmetry.

It turns out that both of these Love numbers are strictly zero [50, 51, 52,
53, 54], which is a consequence of a different symmetry than the duality
considered in this paper [55, 56, 57, 58], but these conclusions are distinct
from each other, i.e., eq. 7.16 does not just say $0=0$. The equality of Love
numbers follows from the invariance under duality of the boundary conditions.
This is clearly the case for black holes, where regularity at the horizon
implies the duality-invariant $c_{2}^{+}=c_{2}^{-}$, but one could also in
principle imagine a horizonless compact object with non-zero but
(approximately) equal Love numbers, provided that whatever boundary conditions
are chosen at its surface are invariant under duality and that the object is
sufficiently compact that $J^{r_{\star}}\approx 0$.

##  8 Discussion

We have computed the actions for scalar, electromagnetic, and linearized
fields on a Schwarszchild background in the $2+2$ formalism. In each case we
focused on isolating and canonically normalizing the underlying dynamical
degrees of freedom. In the cases of electromagnetism and gravity, this
exercise revealed a manifest electric-magnetic duality symmetry, which holds
off shell and accordingly can be used to construct conserved quantities.

As a physical application of the Noether current associated to linearized
gravitational duality, we showed that duality forces the even- and odd-parity
perturbations to have identical tidal responses. Combining this duality with a
“ladder” symmetry [55] which causes the odd Love numbers to vanish therefore
extends that particular argument for vanishing Love numbers to even
perturbations. It would be interesting to explore whether these symmetries
play a role in universal relations for compact objects.

In the case of electromagnetism, we found a clear connection to objects
arising in the Newman-Penrose and Geroch-Held-Penrose formalisms: the
dynamical master variable is related to the middle-weight Maxwell scalar
$\phi_{1}$. This observation enabled us to derive actions for the Fackerell-
Ipser equation and Teukolsky-Starobinsky identities. It would be quite
interesting to extend these constructions to the Teukolsky equation for the
extreme-weight Maxwell scalars, to gravity, and to Kerr, which is the case of
prime astrophysical interest. We leave these questions for future work.

## Acknowledgements

I am grateful to Lam Hui, Austin Joyce, Riccardo Penco, and Luca Santoni for
collaboration, and many insightful discussions, on duality and other topics in
black hole perturbation theory. This work made substantial use of xAct and the
diffgeo Mathematica package by Matthew Headrick. My research is partially
supported by funds from the Natural Sciences and Engineering Research Council
(NSERC) of Canada. Research at the Perimeter Institute is supported in part by
the Government of Canada through NSERC and by the Province of Ontario through
MRI.

##  Appendix A $2+2$ Ricci tensor components

In this appendix we reproduce the components of the linearized Ricci tensor
$\delta R_{ab}$ in the $2+2$ split [10, 11, 12], using the partially gauge-
fixed metric perturbation (5.11). We do not present a complete list but focus
on components of $\delta R_{ab}$ necessary to compute the linearized Einstein-
Hilbert action, cf. eqs. 5.16a and 5.16b.

###  A.1 Odd perturbations

In the odd sector, the only non-zero component is

$\displaystyle\delta R^{B}_{aA}$
$\displaystyle=\left(\frac{1}{2r^{2}}\nabla^{b}\left(r^{4}F_{ab}\right)-h_{a}\right)B_{A}+h_{a}D^{B}D_{[A}B_{B]}$
$\displaystyle=\underbrace{\left(\frac{1}{2r^{2}}\nabla^{b}\left(r^{4}F_{ab}\right)+\frac{(\ell+2)(\ell-1)}{2}h_{a}\right)}_{\delta
R^{B}_{a}}B_{A},$ (A.1)

where

$F_{ab}=\partial_{a}h_{b}-\partial_{b}h_{a}.$ (A.2)

In going to the second line we used the identity

$D^{B}D_{[A}B_{B]}=\frac{1}{2}\ell(\ell+1)B_{A}.$ (A.3)

To prove this, notice that $D_{[A}B_{B]}\propto\epsilon_{AB}$ by symmetry,
where the coefficient is

$\displaystyle D_{[A}B_{B]}$
$\displaystyle=\frac{\epsilon^{CD}D_{C}B_{D}}{2}\epsilon_{AB}$
$\displaystyle=\frac{\ell(\ell+1)}{2}\epsilon_{AB},$ (A.4)

where we have used the definition (2.10b) of $B_{A}$. Taking a derivative and
using the definition again, the result follows.

###  A.2 Even perturbations

To compute the Lagrangian (5.16a) we need the following components of the
perturbed Ricci tensor:

$\delta R_{ab},\quad\Omega^{AB}\delta R_{AB},\quad\delta R^{E}_{a}.$ (A.5)

Note that we do not need the piece of $\delta R^{E}_{a}$ involving $K$. The
relevant pieces of the relevant components are

$\displaystyle\delta R_{ab}$
$\displaystyle=\left(\frac{1}{2}r^{-2}\nabla_{c}\left(r^{2}\nabla_{d}\hat{h}^{cd}\right)-\frac{1}{4}\Box
h\right)g_{ab}+\frac{2}{r}r_{c}\hat{C}^{c}{}_{\langle{ab}\rangle}+\frac{1}{r}r_{\langle
a}\partial_{b\rangle}h$
$\displaystyle\hphantom{{}=}-r^{-2}\nabla_{(a}\left(r^{2}\nabla_{b)}K\right)-\frac{\ell(\ell+1)}{r^{2}}\nabla_{(a}\left(r^{2}r_{b)}\alpha\right)+\frac{1}{2}R\hat{h}_{ab}+\frac{\ell(\ell+1)}{2r^{2}}h_{ab},$
(A.6a) $\displaystyle\Omega^{AB}\delta R_{AB}$
$\displaystyle=2\nabla_{a}(rr_{b}\hat{h}^{ab})+\frac{\ell(\ell+1)+2}{2}h-r^{2}\nabla_{a}\left(r^{4}\nabla^{a}K\right)+(\ell+2)(\ell-1)K$
$\displaystyle\hphantom{{}=}-\ell(\ell+1)\left[r^{2}r^{a}\partial_{a}\alpha+(1+3f)r\alpha\right],$
(A.6b) $\displaystyle\delta R^{E}_{a}$
$\displaystyle=\frac{1}{2}\nabla^{b}\hat{h}_{ab}-\frac{1}{4}\nabla_{a}h+\frac{1}{2}\partial_{a}\ln
rh-\frac{1}{r^{2}}\nabla^{b}\left[r^{4}r_{[a}\nabla_{b]}\alpha\right]-r_{a}\alpha.$
(A.6c)

Angular brackets denote tracefree symmetrization,

$T_{\langle{ab}\rangle}=T_{(ab)}-\frac{1}{2}T^{c}{}_{c}g_{ab}.$ (A.7)

In the above we have defined

$\displaystyle\hat{C}^{c}{}_{ab}=\nabla_{(a}\hat{h}^{c}_{b)}-\frac{1}{2}\nabla^{c}\hat{h}_{ab}.$
(A.8)

The expression for $\delta R_{ab}$ contains the term
$\nabla_{c}\hat{C}^{c}{}_{ab}$ (cf. Ref. [11]), which we have simplified using
the identity

$\nabla^{c}\nabla_{(a}p_{b)c}-\frac{1}{2}\Box
p_{ab}-\frac{1}{2}g_{ab}\nabla^{c}\nabla^{d}p_{cd}=\frac{R}{2}p_{ab}$ (A.9)

for symmetric traceless tensors $p_{ab}$ in $D=2$,

$\nabla_{c}\hat{C}^{c}{}_{ab}=\frac{1}{2}R\hat{h}_{ab}+\frac{1}{2}\nabla_{c}\nabla_{d}\hat{h}^{cd}g_{ab}.$
(A.10)

## References

  * Chandrasekhar [1983] Subrahmanyan Chandrasekhar,  _The Mathematical Theory of Black Holes_ (Oxford University Press, 1983).
  * Berti  _et al._ [2009] Emanuele Berti, Vitor Cardoso, and Andrei O. Starinets, “Quasinormal modes of black holes and black branes,” Class. Quant. Grav. 26, 163001 (2009), arXiv:0905.2975 [gr-qc] .
  * Agullo  _et al._ [2017] I. Agullo, A. del Rio, and J. Navarro-Salas, “Electromagnetic duality anomaly in curved spacetimes,” Phys. Rev. Lett. 118, 111301 (2017), arXiv:1607.08879 [gr-qc] .
  * Agullo  _et al._ [2018] Ivan Agullo, Adrian del Rio, and Jose Navarro-Salas, “Classical and quantum aspects of electric-magnetic duality rotations in curved spacetimes,” Phys. Rev. D 98, 125001 (2018), arXiv:1810.08085 [gr-qc] .
  * Kallosh and Rahman [2021] Renata Kallosh and Adel A. Rahman, “Quantization of gravity in the black hole background,” Phys. Rev. D 104, 086008 (2021), arXiv:2106.01966 [hep-th] .
  * Kallosh [2021] Renata Kallosh, “Quantization of gravity in spherical harmonic basis,” Phys. Rev. D 104, 086023 (2021), arXiv:2107.02099 [hep-th] .
  * Calkin [1965] MG Calkin, “An invariance property of the free electromagnetic field,” American Journal of Physics 33, 958–960 (1965).
  * Deser and Teitelboim [1976] Stanley Deser and C. Teitelboim, “Duality Transformations of Abelian and Nonabelian Gauge Fields,” Phys. Rev. D 13, 1592–1597 (1976).
  * Deser [1982] Stanley Deser, “Off-Shell Electromagnetic Duality Invariance,” J. Phys. A 15, 1053 (1982).
  * Sarbach and Tiglio [2001] Olivier Sarbach and Manuel Tiglio, “Gauge invariant perturbations of Schwarzschild black holes in horizon penetrating coordinates,” Phys. Rev. D 64, 084016 (2001), arXiv:gr-qc/0104061 .
  * Martel and Poisson [2005] Karl Martel and Eric Poisson, “Gravitational perturbations of the Schwarzschild spacetime: A Practical covariant and gauge-invariant formalism,” Phys. Rev. D71, 104003 (2005), arXiv:gr-qc/0502028 [gr-qc] .
  * Chaverra  _et al._ [2013] Eliana Chaverra, Néstor Ortiz, and Olivier Sarbach, “Linear perturbations of self-gravitating spherically symmetric configurations,” Phys. Rev. D 87, 044015 (2013), arXiv:1209.3731 [gr-qc] .
  * Frolov  _et al._ [2017] Valeri Frolov, Pavel Krtous, and David Kubiznak, “Black holes, hidden symmetries, and complete integrability,” Living Rev. Rel. 20, 6 (2017), arXiv:1705.05482 [gr-qc] .
  * Newman and Penrose [2009] E. (Ted) Newman and R. Penrose, “Spin-coefficient formalism,” Scholarpedia 4, 7445 (2009).
  * Carter and Hartle [1987] B. Carter and J. B. Hartle, eds.,  _Gravitation in Astrophysics: Proceedings, NATO Advanced Study Institute, Cargese, France, July 15-31, 1986_, Vol. 156 (1987).
  * Kinnersley [1969] William Kinnersley, “Type D Vacuum Metrics,” J. Math. Phys. 10, 1195–1203 (1969).
  * Pound and Wardell [2020] Adam Pound and Barry Wardell, “Black hole perturbation theory and gravitational self-force,” in  _Handbook of Gravitational Wave Astronomy_, edited by Cosimo Bambi, Stavros Katsanevas, and Konstantinos D. Kokkotas (Springer Singapore, Singapore, 2020) pp. 1–119, arXiv:2101.04592 [gr-qc] .
  * Hui  _et al._ [2021] Lam Hui, Austin Joyce, Riccardo Penco, Luca Santoni, and Adam R. Solomon, “Static response and Love numbers of Schwarzschild black holes,” JCAP 04, 052 (2021), arXiv:2010.00593 [hep-th] .
  * Lagos  _et al._ [2014] Macarena Lagos, Máximo Bañados, Pedro G. Ferreira, and Sebastián García-Sáenz, “Noether Identities and Gauge-Fixing the Action for Cosmological Perturbations,” Phys. Rev. D89, 024034 (2014), arXiv:1311.3828 [gr-qc] .
  * Motohashi  _et al._ [2016] Hayato Motohashi, Teruaki Suyama, and Kazufumi Takahashi, “Fundamental theorem on gauge fixing at the action level,” Phys. Rev. D94, 124021 (2016), arXiv:1608.00071 [gr-qc] .
  * Wardell and Kavanagh [2021] Barry Wardell and Chris Kavanagh, “Separable electromagnetic perturbations of rotating black holes,” Phys. Rev. D 103, 104049 (2021), arXiv:2011.03548 [gr-qc] .
  * Wald [1978] Robert M. Wald, “Construction of Solutions of Gravitational, Electromagnetic, Or Other Perturbation Equations from Solutions of Decoupled Equations,” Phys. Rev. Lett. 41, 203–206 (1978).
  * Fackerell and Ipser [1972] E. D. Fackerell and J. R. Ipser, “Weak electromagnetic fields around a rotating black hole,” Phys. Rev. D 5, 2455–2458 (1972).
  * Bini  _et al._ [2002] Donato Bini, Christian Cherubini, Robert T Jantzen, and Remo J. Ruffini, “Teukolsky master equation: De Rham wave equation for the gravitational and electromagnetic fields in vacuum,” Prog. Theor. Phys. 107, 967–992 (2002), arXiv:gr-qc/0203069 .
  * Krasnov and Shaw [2023] Kirill Krasnov and Adam Shaw, “Weyl curvature evolution system for GR,” Class. Quant. Grav. 40, 075013 (2023), arXiv:2212.12273 [gr-qc] .
  * Teukolsky [1972] S. A. Teukolsky, “Rotating black holes - separable wave equations for gravitational and electromagnetic perturbations,” Phys. Rev. Lett. 29, 1114–1118 (1972).
  * Tóth [2018] Gábor Zsolt Tóth, “Noether currents for the Teukolsky master equation,” Class. Quant. Grav. 35, 185009 (2018), arXiv:1801.04710 [gr-qc] .
  * Cohen and Kegeles [1974] J. M. Cohen and L. S. Kegeles, “Electromagnetic fields in curved spaces - a constructive procedure,” Phys. Rev. D 10, 1070–1084 (1974).
  * Stewart [1979] John M. Stewart, “Hertz-Bromwich-Debye-Whittaker-Penrose Potentials in General Relativity,” Proc. Roy. Soc. Lond. A 367, 527–538 (1979).
  * Dolan [2019] Sam R. Dolan, “Electromagnetic fields on Kerr spacetime, Hertz potentials and Lorenz gauge,” Phys. Rev. D 100, 044044 (2019), arXiv:1906.04808 [gr-qc] .
  * Ripley and Yagi [2018] Justin L. Ripley and Kent Yagi, “Black hole perturbation under a 2 + 2 decomposition in the action,” Phys. Rev. D 97, 024009 (2018), arXiv:1705.03068 [gr-qc] .
  * De Felice  _et al._ [2011] Antonio De Felice, Teruaki Suyama, and Takahiro Tanaka, “Stability of Schwarzschild-like solutions in f(R,G) gravity models,” Phys. Rev. D83, 104035 (2011), arXiv:1102.1521 [gr-qc] .
  * Kobayashi  _et al._ [2014] Tsutomu Kobayashi, Hayato Motohashi, and Teruaki Suyama, “Black hole perturbation in the most general scalar-tensor theory with second-order field equations II: the even-parity sector,” Phys. Rev. D 89, 084042 (2014), arXiv:1402.6740 [gr-qc] .
  * Regge and Wheeler [1957] Tullio Regge and John A. Wheeler, “Stability of a Schwarzschild singularity,” Phys. Rev. 108, 1063–1069 (1957).
  * Cunningham  _et al._ [1978] C. T. Cunningham, R. H. Price, and V. Moncrief, “Radiation from collapsing relativistic stars. I - Linearized odd-parity radiation,” Astrophys. J. 224, 643 (1978).
  * Woodard [2015] Richard P. Woodard, “Ostrogradsky’s theorem on Hamiltonian instability,” Scholarpedia 10, 32243 (2015), arXiv:1506.02210 [hep-th] .
  * Zerilli [1970] Frank J. Zerilli, “Effective potential for even parity Regge-Wheeler gravitational perturbation equations,” Phys. Rev. Lett. 24, 737–738 (1970).
  * Chandrasekhar and Detweiler [1975] S. Chandrasekhar and Steven L. Detweiler, “The quasi-normal modes of the Schwarzschild black hole,” Proc. Roy. Soc. Lond. A344, 441–452 (1975).
  * Chandrasekhar [1975] S. Chandrasekhar, “On the Equations Governing the Perturbations of the Schwarzschild Black Hole,” Proceedings of the Royal Society of London Series A 343, 289–298 (1975).
  * Chandrasekhar [1984] S. Chandrasekhar, “On Algebraically Special Perturbations of Black Holes,” Proceedings of the Royal Society of London Series A 392, 1–13 (1984).
  * Wald [1973] Robert M. Wald, “On perturbations of a Kerr black hole,” J. Math. Phys. 14, 1453–1461 (1973).
  * Aksteiner [2014] Steffen Aksteiner,  _Geometry and analysis on black hole spacetimes_ , Ph.D. thesis, Leibniz U., Hannover (2014).
  * Pereñiguez [2023] David Pereñiguez, “Black hole perturbations and electric-magnetic duality,” Phys. Rev. D 108, 084046 (2023), arXiv:2302.10942 [gr-qc] .
  * Nichols  _et al._ [2012] David A. Nichols, Aaron Zimmerman, Yanbei Chen, Geoffrey Lovelace, Keith D. Matthews, Robert Owen, Fan Zhang, and Kip S. Thorne, “Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes III. Quasinormal Pulsations of Schwarzschild and Kerr Black Holes,” Phys. Rev. D 86, 104028 (2012), arXiv:1208.3038 [gr-qc] .
  * Cooper  _et al._ [1995] Fred Cooper, Avinash Khare, and Uday Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rept. 251, 267–385 (1995), arXiv:hep-th/9405029 .
  * Aksteiner and Andersson [2011] Steffen Aksteiner and Lars Andersson, “Linearized gravity and gauge conditions,” Class. Quant. Grav. 28, 065001 (2011), arXiv:1009.5647 [gr-qc] .
  * Henneaux and Teitelboim [2005] Marc Henneaux and Claudio Teitelboim, “Duality in linearized gravity,” Phys. Rev. D71, 024018 (2005), arXiv:gr-qc/0408101 [gr-qc] .
  * Julia  _et al._ [2005] Bernard Julia, Jerome Levie, and Sebastien Ray, “Gravitational duality near de Sitter space,” JHEP 11, 025 (2005), arXiv:hep-th/0507262 .
  * Hörtner [2019] Sergio Hörtner, “Manifest gravitational duality near anti de Sitter space-time,” Front. in Phys. 7, 188 (2019), arXiv:1903.07763 [hep-th] .
  * Damour and Nagar [2009] Thibault Damour and Alessandro Nagar, “Relativistic tidal properties of neutron stars,” Phys. Rev. D80, 084035 (2009), arXiv:0906.0096 [gr-qc] .
  * Binnington and Poisson [2009] Taylor Binnington and Eric Poisson, “Relativistic theory of tidal Love numbers,” Phys. Rev. D 80, 084018 (2009), arXiv:0906.1366 [gr-qc] .
  * Porto [2016] Rafael A. Porto, “The Tune of Love and the Nature(ness) of Spacetime,” Fortsch. Phys. 64, 723–729 (2016), arXiv:1606.08895 [gr-qc] .
  * Chia [2021] Horng Sheng Chia, “Tidal deformation and dissipation of rotating black holes,” Phys. Rev. D 104, 024013 (2021), arXiv:2010.07300 [gr-qc] .
  * Charalambous  _et al._ [2021a] Panagiotis Charalambous, Sergei Dubovsky, and Mikhail M. Ivanov, “On the Vanishing of Love Numbers for Kerr Black Holes,” JHEP 05, 038 (2021a), arXiv:2102.08917 [hep-th] .
  * Hui  _et al._ [2022a] Lam Hui, Austin Joyce, Riccardo Penco, Luca Santoni, and Adam R. Solomon, “Ladder symmetries of black holes. Implications for love numbers and no-hair theorems,” JCAP 01, 032 (2022a), arXiv:2105.01069 [hep-th] .
  * Hui  _et al._ [2022b] Lam Hui, Austin Joyce, Riccardo Penco, Luca Santoni, and Adam R. Solomon, “Near-zone symmetries of Kerr black holes,” JHEP 09, 049 (2022b), arXiv:2203.08832 [hep-th] .
  * Charalambous  _et al._ [2021b] Panagiotis Charalambous, Sergei Dubovsky, and Mikhail M. Ivanov, “Hidden Symmetry of Vanishing Love Numbers,” Phys. Rev. Lett. 127, 101101 (2021b), arXiv:2103.01234 [hep-th] .
  * Charalambous  _et al._ [2022] Panagiotis Charalambous, Sergei Dubovsky, and Mikhail M. Ivanov, “Love symmetry,” JHEP 10, 175 (2022), arXiv:2209.02091 [hep-th] .
  * Yagi and Yunes [2013] Kent Yagi and Nicolas Yunes, “I-Love-Q Relations in Neutron Stars and their Applications to Astrophysics, Gravitational Waves and Fundamental Physics,” Phys. Rev. D 88, 023009 (2013), arXiv:1303.1528 [gr-qc] .
  * Yagi and Yunes [2017] Kent Yagi and Nicolás Yunes, “Approximate Universal Relations for Neutron Stars and Quark Stars,” Phys. Rept. 681, 1–72 (2017), arXiv:1608.02582 [gr-qc] .
  * Kol and Smolkin [2012] Barak Kol and Michael Smolkin, “Black hole stereotyping: Induced gravito-static polarization,” JHEP 02, 010 (2012), arXiv:1110.3764 [hep-th] .