# Near-Zone Symmetries of Kerr Black Holes

Lam Hui  Center for Theoretical Physics, Department of Physics,
Columbia University, New York, NY 10027, USA  Austin Joyce  Department of
Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA
Riccardo Penco  Department of Physics, Carnegie Mellon University, Pittsburgh,
PA 15213, USA  Luca Santoni  ICTP, International Centre for Theoretical
Physics, Strada Costiera 11, 34151, Trieste, Italy  Adam R. Solomon
Department of Physics and Astronomy, McMaster University, Hamilton, ON, L8S
4M1, Canada  Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L
2Y5, Canada

###### Abstract

We study the near-zone symmetries of a massless scalar field on four-
dimensional black hole backgrounds. We provide a geometric understanding that
unifies various recently discovered symmetries as part of an ${\rm SO}(4,2)$
group. Of these, a subset are exact symmetries of the static sector and give
rise to the ladder symmetries responsible for the vanishing of Love numbers.
In the Kerr case, we compare different near-zone approximations in the
literature, and focus on the implementation that retains the symmetries of the
static limit. We also describe the relation to spin-1 and 2 perturbations.

Introduction: Black hole perturbation theory has a long history dating back to
the work of Regge and Wheeler Regge and Wheeler (1957) and Zerilli Zerilli
(1970). Interestingly, recent investigations suggest the subject has depths
yet to be plumbed. A case in point is a number of symmetries discovered in the
past year Charalambous  _et al._ (2021a); Hui  _et al._ (2022), which shed
light on the well-known vanishing of black hole Love numbers (characterizing a
black hole’s static, non-dissipative tidal response) Fang and Lovelace (2005);
Damour and Nagar (2009); Binnington and Poisson (2009); Damour and Lecian
(2009); Kol and Smolkin (2012); Le Tiec and Casals (2021); Le Tiec  _et al._
(2021); Chia (2021); Hui  _et al._ (2021); Charalambous  _et al._ (2021b). In
this paper, we present a synthesis of these symmetries, and show how they fit
within a larger group containing further symmetries. Some of these are
familiar symmetries of the exact dynamics. The rest are approximate symmetries
in the low frequency regime. Of these, a subset are exact symmetries of the
static sector, and give rise to the ladder symmetries discussed in Hui  _et
al._ (2022). To keep the discussion simple, we focus largely on symmetries of
a massless scalar, first on a Schwarzschild, then Kerr, background. The
connection to spin-1 and 2 perturbations, via a spin ladder, is discussed in
the Supplemental Material.

Effective near-zone metric: We begin by considering the Schwarzschild case.
Our starting point is a free massless scalar field $\phi$ on a fixed 4D
Schwarzschild background:

${\rm d}s^{2}=-f(r){\rm d}t^{2}+\frac{{\rm d}r^{2}}{f(r)}+r^{2}{\rm
d}\Omega^{2}_{S^{2}},~{}\,f(r)=1-\frac{r_{s}}{r},$ (1)

with ${\rm d}\Omega_{S^{2}}^{2}\equiv{\rm d}\theta^{2}+\sin^{2}\theta\,{\rm
d}\varphi^{2}$ and where $r_{s}\equiv 2GM$ is the Schwarzschild radius. The
scalar’s action can be written explicitly as

$S=\frac{1}{2}\int{\rm d}t{\rm d}r{\rm
d}\Omega\left[\frac{r^{4}}{\Delta}(\partial_{t}\phi)^{2}-\Delta(\partial_{r}\phi)^{2}+\phi\nabla^{2}_{\Omega}\phi\right],$
(2)

with $\Delta(r)\equiv r(r-r_{s})$ and
$\nabla^{2}_{\Omega}\equiv(1/\sin\theta)\partial_{\theta}(\sin\theta\,\partial_{\theta})+(1/\sin^{2}\theta)\partial^{2}_{\varphi}$.
In frequency space ($\phi\propto e^{-i\omega t}$), we wish to focus on long-
wavelength perturbations satisfying $r_{s}\ll 1/\omega$. The behavior of
$\phi$ in the _near-zone_ region defined by $r_{s}\leq r\ll 1/\omega$ is
described by approximating the coefficient in front of the kinetic term as
follows: $r^{4}/\Delta(r)\simeq r_{s}^{4}/\Delta(r)$ Starobinskiǐ (1973);
Starobinskiǐ and Churilov (1974). Doing so means in the $\phi$ equation of
motion, the time derivative term $(r^{4}/\Delta)\partial_{t}^{2}\phi$ is
replaced by $(r_{s}^{4}/\Delta)\partial_{t}^{2}\phi$. This has the virtue of
preserving the correct singularity as $r\rightarrow r_{s}$, while still
accurately capturing the dynamics at larger $r$, as long as $\omega r\ll
1$.111These criteria do not uniquely fix the near-zone approximation for
finite $\omega$. Nevertheless, the near-zone approximation of Schwarzschild we
use is standard in the literature. The existence of the symmetries that we
will discuss below can be viewed as an _a posteriori_ motivation for this
particular implementation.

In this limit, the action (2) is the same as that of a massless scalar
minimally coupled to an _effective near-zone metric_ :

${\rm d}s^{2}_{\text{near-zone}}=-\frac{\Delta}{r_{s}^{2}}{\rm
d}t^{2}+\frac{r_{s}^{2}}{\Delta}{\rm d}r^{2}+r_{s}^{2}{\rm
d}\Omega^{2}_{S^{2}}.$ (3)

In the static limit ($\omega=0$) the scalar behaves identically on this metric
as on the original Schwarzschild background. Nevertheless, it is advantageous
to work with the near-zone geometry, both because it allows us to go beyond
the strictly static sector, and because it has a richer symmetry structure. In
fact, the metric (3) is that of ${\rm AdS}_{2}\times S^{2}$. (The $(t,r)$
coordinates are a somewhat nonstandard covering of a portion of ${\rm
AdS}_{2}$, which we describe in the Supplemental Material.) This immediately
implies that the near-zone metric (3) has 6 Killing vectors (KVs), in contrast
to Schwarzschild, which has only 4.

Another advantage of the near-zone metric is that it describes a conformally
flat spacetime, unlike Schwarzschild. This implies that the metric (3) has 9
additional conformal Killing vectors (CKVs). The near-zone metric also has a
vanishing Ricci scalar (though not Ricci tensor) because the curvature radii
of ${\rm AdS}_{2}$ and $S^{2}$ are identical. This means that the scalar
$\phi$ is effectively conformally coupled, guaranteeing the CKVs generate
symmetries of the action in the near zone. We now turn to the study of these
symmetries and their physical consequences.

Near-zone symmetries: The Killing vectors of ${\rm AdS}_{2}\times S^{2}$ in
$(t,r,\theta,\varphi)$ coordinates are

$\displaystyle T$ $\displaystyle=2r_{s}\,\partial_{t},$ (4a) $\displaystyle
L_{\pm}$ $\displaystyle=e^{\pm
t/2r_{s}}(2r_{s}\,\partial_{r}\sqrt{\Delta}\partial_{t}\mp\sqrt{\Delta}\partial_{r}),$
(4b) $\displaystyle J_{23}$ $\displaystyle=\partial_{\varphi},$ (4c)
$\displaystyle J_{12}$
$\displaystyle=\cos\varphi\,\partial_{\theta}-\cot\theta\sin\varphi\,\partial_{\varphi},$
(4d) $\displaystyle J_{13}$
$\displaystyle=\sin\varphi\,\partial_{\theta}+\cot\theta\cos\varphi\,\partial_{\varphi}.$
(4e)

The Killing vectors $L_{0}\equiv T$ and $L_{\pm}$ were first introduced in
Bertini  _et al._ (2012) and coincide with the zero-spin limit of the
symmetries discovered for Kerr in Charalambous  _et al._ (2021a). More
recently they were encountered in the context of rotating STU supergravity
black holes Cvetic  _et al._ (2021).

The near-zone metric (3) also possesses 9 conformal Killing vectors:

$\displaystyle J_{01}$
$\displaystyle=-\tfrac{2\Delta}{r_{s}}\cos\theta\,\partial_{r}-\tfrac{\partial_{r}\Delta}{r_{s}}\sin\theta\,\partial_{\theta},$
(5a) $\displaystyle J_{02}$
$\displaystyle=-\cos\varphi\left[\tfrac{2\Delta}{r_{s}}\sin\theta\,\partial_{r}+\tfrac{\partial_{r}\Delta}{r_{s}}\left(\tfrac{\tan\varphi}{\sin\theta}\partial_{\varphi}-\cos\theta\partial_{\theta}\right)\right],$
(5b) $\displaystyle J_{03}$
$\displaystyle=-\sin\varphi\left[\tfrac{2\Delta}{r_{s}}\sin\theta\,\partial_{r}-\tfrac{\partial_{r}\Delta}{r_{s}}\left(\tfrac{\cot\varphi}{\sin\theta}\partial_{\varphi}+\cos\theta\partial_{\theta}\right)\right],$
(5c) $\displaystyle K_{\pm}$ $\displaystyle=e^{\pm
t/2r_{s}}\tfrac{\sqrt{\Delta}}{r_{s}}\cos\theta\left(\tfrac{r_{s}^{3}}{\Delta}\partial_{t}\mp\partial_{r}\Delta\partial_{r}\mp
2\tan\theta\partial_{\theta}\right),$ (5d) $\displaystyle M_{\pm}$
$\displaystyle=e^{\pm
t/2r_{s}}\cos\varphi\left[\tfrac{r_{s}^{2}}{\sqrt{\Delta}}\sin\theta\partial_{t}\mp\tfrac{\sqrt{\Delta}\partial_{r}\Delta\sin\theta}{r_{s}}\partial_{r}\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.\pm\tfrac{2\sqrt{\Delta}}{r_{s}}\cos\theta\partial_{\theta}\mp\tfrac{2\sqrt{\Delta}}{r_{s}}\tfrac{\tan\varphi}{\sin\theta}\partial_{\varphi}\right],$
(5e) $\displaystyle N_{\pm}$ $\displaystyle=e^{\pm
t/2r_{s}}\sin\varphi\left[\tfrac{r_{s}^{2}}{\sqrt{\Delta}}\sin\theta\partial_{t}\mp\tfrac{\sqrt{\Delta}\partial_{r}\Delta\sin\theta}{r_{s}}\partial_{r}\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.\pm\tfrac{2\sqrt{\Delta}}{r_{s}}\cos\theta\partial_{\theta}\pm\tfrac{2\sqrt{\Delta}}{r_{s}}\tfrac{\cot\varphi}{\sin\theta}\partial_{\varphi}\right].$
(5f)

Expressing each of the Killing and conformal Killing generators as
$\xi^{\mu}\partial_{\mu}$, the symmetries act on the scalar as

$\delta\phi=\xi^{\mu}\partial_{\mu}\phi+\frac{1}{4}\nabla_{\mu}\xi^{\mu}\phi\,.$
(6)

The time translation $T$ and spatial rotations $J_{ij}$ ($i,j=1,2,3$) are the
familiar symmetries of the exact dynamics. In addition, the symmetry
generators $J_{0i}$ ($i=1,2,3$) have a somewhat privileged status: they
generate symmetries of the exact system in the static limit, $\omega=0$ Hui
_et al._ (2022); see also Achour  _et al._ (2022) for a related discussion.
The other (C)KVs do not give rise to exact symmetries in this limit. Each
contains a factor of $e^{\pm t/2r_{s}}$, and thus when applied to a static
scalar generates a solution with $\omega=\pm i/2r_{s}$ (which also means the
resulting scalar has an $|\omega|$ outside the regime of validity of the near-
zone approximation). A corollary is that these other (C)KVs are not well-
defined in the flat space ($r_{s}\to 0$) limit. Nonetheless, these generators
can still be used to infer properties of exact static solutions Charalambous
_et al._ (2021a). On the other hand, the generators $J_{0i}$ have an overall
factor of $1/r_{s}$ which can be removed without trouble, and thus do have a
well-defined flat space limit.

All together, the algebra of the Killing (4) and conformal Killing (5)
symmetries is ${\rm so}(4,2)$, as expected because the metric (3) is
conformally flat. There are a number of subalgebras of interest. Firstly, the
generators $J_{0i},J_{ij}$ ($i,j=1,2,3$) form an ${\rm so}(3,1)$ subalgebra
Hui  _et al._ (2022). In addition, each pair of vectors labeled with the
subscripts $\pm$ in eqs. (4) and (5) forms a subgroup together with the
generator $T$. More precisely, denoting $X=\\{L,K,M,N\\}$, we have

$[T,X_{\pm}]=\pm X_{\pm},\qquad\quad[X_{+},X_{-}]=2\sigma_{X}T,$ (7)

with $\sigma_{L}=-1$ and $\sigma_{K}=\sigma_{M}=\sigma_{N}=+1$, giving
different ${\rm sl}(2,\mathbb{R})$ subalgebras. To the best of our knowledge,
the consequences of the symmetries $K_{\pm},M_{\pm}$, and $N_{\pm}$ for
perturbations around Schwarzschild have not been explored in the literature.

Effective Kerr near-zone metric: The Kerr line element in Boyer–Lindquist
coordinates is:

$\displaystyle{\rm d}s^{2}=$ $\displaystyle-{\rho^{2}-r_{s}r\over\rho^{2}}{\rm
d}t^{2}-{2ar_{s}r\sin^{2}\theta\over\rho^{2}}{\rm d}t{\rm
d}\varphi+{\rho^{2}\over\Delta}{\rm d}r^{2}$ (8) $\displaystyle+\rho^{2}{\rm
d}\theta^{2}+{(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta\over\rho^{2}}\sin^{2}\theta{\rm
d}\varphi^{2},$

where we have defined the quantities

$\rho^{2}\equiv r^{2}+a^{2}\cos^{2}\theta\,,\qquad\Delta\equiv
r(r-r_{s})+a^{2}\,.$ (9)

The Schwarzschild radius $r_{s}$ and the spin parameter $a$ are related to the
outer and inner horizons $r_{\pm}$, i.e., the radii where $\Delta=0$, via
$r_{\pm}\equiv r_{s}/2\pm\sqrt{(r_{s}/2)^{2}-a^{2}}$.

The Klein–Gordon equation on the Kerr background is

$\displaystyle\partial_{r}(\Delta\partial_{r}\phi)$
$\displaystyle+\nabla^{2}_{\Omega}\phi-{a^{2}\over\Delta}\partial_{\varphi}^{2}\phi$
(10) $\displaystyle-{1\over\Delta}\Big{[}(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\Big{]}\partial_{t}^{2}\phi$
$\displaystyle-{2a\over\Delta}\Big{[}(r^{2}+a^{2})-\Delta\Big{]}\partial_{t}\partial_{\varphi}\phi=0\,.$

We define the near-zone region using the same approximation as in the
Schwarzschild case. We choose to implement this approximation in
Boyer–Lindquist coordinates because they are inertial at infinity. Wherever
there are time derivatives, we keep terms that go as $1/\Delta$ to preserve
the singularity as $r$ approaches the horizon, and set $r\rightarrow r_{+}$ in
the corresponding numerators. This ensures any corrections are subdominant at
the horizon and are small away from it in the low frequency regime: $\omega
a\leq\omega r_{+}\leq\omega r\ll 1$. Thus the near-zone scalar equation is

$\partial_{r}(\Delta\partial_{r}\phi)+\nabla^{2}_{\Omega}\phi-{1\over\Delta}\big{[}(r_{+}^{2}+a^{2})\partial_{t}+a\partial_{\varphi}\big{]}^{2}\phi=0\,.$
(11)

Note that, unlike some near-zone approximations of Kerr put forward in the
literature, this approximation remains well defined even in the extremal limit
$a\to r_{s}/2$. Expanding in frequency space and spherical harmonics222For
non-static perturbations one should in general use spheroidal harmonics.
However, at the order we are working with in the near-zone approximation, it
is consistent to decompose the field in terms of spherical harmonics Teukolsky
(1973). $\phi=e^{-i\omega t}Y_{\ell m}(\theta,\varphi)R(r)$ (where the $\ell$
and $m$ dependence of $R$ is suppressed), and using
$r_{+}^{2}+a^{2}=r_{s}r_{+}$, the near-zone equation reads Starobinskiǐ
(1973); Starobinskiǐ and Churilov (1974); Page (1976); Maldacena and
Strominger (1997)333Note that there is a typo in eqs. (2.11) and (2.12) of
Maldacena and Strominger (1997), where the factor $r_{+}^{4}$ should be
replaced by $r_{s}^{2}r_{+}^{2}$.

$\partial_{r}(\Delta\partial_{r}R)+\left[\frac{r_{s}^{2}r_{+}^{2}}{\Delta}(\omega-m\Omega_{+})^{2}-\ell(\ell+1)\right]R=0\,,$
(12)

where for convenience we have introduced $\Omega_{+}\equiv a/(r_{s}r_{+})$.

It is straightforward to show that eq. (11) is the equation of motion for a
massless scalar propagating in the following effective near-zone metric:444We
stress that the near-zone metric (13) should not be confused with the near-
horizon limit encountered in the context of the extremal Kerr metric (see,
e.g., Bardeen and Horowitz (1999)). This near-horizon limit is defined by a
rescaling of the radial and time coordinates which keeps the coordinate
$\varphi^{\prime}$ fixed. As a byproduct, it is suited to study modes with
$\omega\sim m\Omega_{+}$, rather than the static regime.

$\displaystyle{\rm d}s^{2}_{\text{near-zone}}=$
$\displaystyle-\frac{\Delta-a^{2}\sin^{2}\theta}{r_{s}r_{+}}{\rm
d}t^{2}-2a\sin^{2}\theta{\rm d}t{\rm d}\varphi$ (13)
$\displaystyle+\frac{r_{s}r_{+}}{\Delta}{\rm d}r^{2}+r_{s}r_{+}{\rm
d}\Omega^{2}_{S^{2}}\,.$

This metric reduces to (3) in the limit $a\to 0$, and moreover is conformally
flat, and therefore has the same number of CKVs. A coordinate transformation
$\varphi^{\prime}=\varphi-(a/r_{s}r_{+})t$ simplifies the metric to

${\rm d}s^{2}_{\text{near-zone}}=-\frac{\Delta}{r_{s}r_{+}}{\rm
d}t^{2}+\frac{r_{s}r_{+}}{\Delta}{\rm d}r^{2}+r_{s}r_{+}{\rm
d}\Omega^{\prime}_{S^{2}}{}^{2}.$ (14)

In the extremal limit where $\Delta=(r-r_{+})^{2}=(r-r_{-})^{2}$, one can see
the $(t,r)$ subspace is ${\rm AdS}_{2}$ in Poincaré coordinates upon
redefining $r-r_{-}$ as the new radial coordinate. Interestingly, the extremal
near-zone metric coincides with the _near-horizon_ limit of the extremal
Reissner–Nördstrom solution with $r_{s}^{2}\to r_{s}r_{+}$.

Away from the extremal limit, the simplest way to deduce the symmetries is to
recognize that the effective near-zone metric for Kerr is in fact equivalent
to the near-zone metric for Schwarzschild. To see this, redefine
$t^{\prime}=(r_{*}/\sqrt{r_{s}r_{+}})t$ and
$r^{\prime}=\sqrt{r_{s}r_{+}}(r-r_{-})/r_{*}$, with $r_{*}\equiv r_{+}-r_{-}$.
The Kerr near-zone metric is then rewritten as:

${\rm d}s^{2}_{\text{near-zone}}=-\frac{\tilde{\Delta}}{r_{s}r_{+}}{\rm
d}t^{\prime 2}+\frac{r_{s}r_{+}}{\tilde{\Delta}}{\rm d}r^{\prime
2}+r_{s}r_{+}{\rm d}\Omega^{\prime 2}\,,$ (15)

with $\tilde{\Delta}\equiv
r^{\prime}(r^{\prime}-\sqrt{r_{s}r_{+}})=(r_{s}r_{+}/r_{*}^{2})\Delta$. This
has the same form as the Schwarzschild near-zone metric in (3) with
$r_{s}\to\sqrt{r_{s}r_{+}}$.

The 15 (conformal) Killing vectors for a spinning black hole in the near-zone
can thus be obtained from their Schwarzschild counterparts (4)–(5) using the
coordinate transformations discussed above and replacing
$r_{s}\to\sqrt{r_{s}r_{+}}$. For completeness, we report their explicit
expressions in the Supplemental Material. There are several ${\rm
sl}(2,\mathbb{R})$ subalgebras, just as in Schwarzschild, taking the same form
as in eq. (7). Spatial rotations $J_{ij}$ and boosts $J_{0i}$ ($i,j=1,2,3$)
form an ${\rm so}(3,1)$ subalgebra just as before. However, only $J_{23}$
among the spatial rotations is an exact symmetry of the static sector, while
$J_{12}$ and $J_{13}$ do not preserve the static nature of field
configurations.555By a static configuration, we mean $\partial_{t}\phi=0$
keeping $\varphi$ (as well as $r$ and $\theta$) fixed, as opposed to keeping
$\varphi^{\prime}$ fixed. This choice is dictated by the fact that $\varphi$
is an inertial coordinate at infinity. Physically, this is because the Kerr
metric has a preferred direction. Similarly, of the three boosts, only
$J_{01}$ is a symmetry of the exact system in the static limit.666This is
reflected by the fact that only $J_{01}$ among them has no time dependence
when expressed in Boyer–Lindquist coordinates.

Comparison of different near zones: The effective metric (13) captures the
near-zone dynamics (11) of massless scalar perturbations around a Kerr black
hole. For different reasons, various deformations of the near-zone
approximation (11) have been proposed in the literature. There are in fact
many ways of deforming (11) at subleading order in $(r-r_{+})/r_{+}$ Teukolsky
(1973); Lowe and Skanata (2012). It is instructive to briefly review some of
these possibilities and highlight the main differences with (11).

One notable example is given in Castro  _et al._ (2010). Supported by the
observation that the Cardy formula for a $\text{CFT}_{2}$ gives exactly the
Bekenstein–Hawking entropy of the Kerr solution, Castro  _et al._ (2010)
conjectured (see also Guica  _et al._ (2009); Bredberg  _et al._ (2010);
Bardeen and Horowitz (1999)) that a non-extremal Kerr black hole is dual to a
two-dimensional CFT and proposed a near-zone approximation with an
$\text{${\rm sl}(2,\mathbb{R})$}_{L}\times\text{${\rm sl}(2,\mathbb{R})$}_{R}$
symmetry. This can be obtained by adding to (12) the term $(\omega
r_{s}(\omega r_{s}^{2}-2ma)/(r-r_{-}))R$. The effect of this term—which is
small in the low frequency regime and subleading near the horizon compared to
the $1/\Delta$ term in (12)—is to break some of the symmetries while
introducing a new $\text{${\rm sl}(2,\mathbb{R})$}_{L}\times\text{${\rm
sl}(2,\mathbb{R})$}_{R}$ symmetry (which is not a subalgebra of our ${\rm
so}(4,2)$). These $\text{${\rm sl}(2,\mathbb{R})$}_{L}\times\text{${\rm
sl}(2,\mathbb{R})$}_{R}$ generators are singular in the Schwarzschild limit
(the near zone of Castro  _et al._ (2010) is not smoothly connected to the
Schwarzschild near zone above in the limit $a\to 0$) and they are not globally
defined, as they do not respect the $\varphi\to\varphi+2\pi$ periodicity.

A different near-zone approximation that overcomes these issues has recently
been proposed in Charalambous  _et al._ (2021a). The near zone of Charalambous
_et al._ (2021a) differs from the one in eq. (11) by the term
$(4\omega\Omega_{+}m(r-r_{+})/(r_{+}-r_{-}))R$. This too breaks some of the
symmetries, but keeps an ${\rm sl}(2,\mathbb{R})$ group with generators that
are both globally well defined and have a smooth Schwarzschild limit.

All three of these approximations are contained in the one-parameter family of
Lowe and Skanata (2012), which possesses an $\text{${\rm
sl}(2,\mathbb{R})$}_{L}\times\text{${\rm sl}(2,\mathbb{R})$}_{R}$ symmetry
except at two special points, one corresponding to (11) (where they identified
the generators $T$ (25a) and $L_{\pm}$ (25h)) and the other to Charalambous
_et al._ (2021a). See also Keeler  _et al._ (2021) for a recent discussion of
near-zone approximations and their relations to Killing tensor symmetries.

Each of these approximation schemes has benefits and drawbacks. For our
purposes, the main appeal of the approximation (11), besides having a smooth
Schwarzschild limit and globally well-defined generators, is that its
effective metric (13) contains in particular the symmetry generator $J_{01}$
in (25b). This is a symmetry of the exact dynamics for static field
configurations (and is in fact a CKV of an effective 3D metric Hui  _et al._
(2022)). Keeping $J_{01}$ as a symmetry is thus useful for a near-zone
approximation intended for low frequency phenomena. The effective metric (13)
has other special properties: since it is conformally flat, it possesses the
maximal number of CKVs (15 in $d=4$), and since its Ricci scalar vanishes, a
massless scalar is automatically conformally coupled, so that each of the
(C)KVs $\xi^{\mu}$ generates a symmetry acting on the scalar as in (6). Each
gives rise to a conserved current in the standard way:
$j^{\mu}=T^{\mu\nu}\xi_{\nu}$, where $T^{\mu\nu}$ is the (traceless) energy
momentum tensor of the scalar.

Ladder symmetries and tidal response: Up to an irrelevant constant factor, the
CKV associated with $J_{01}$ in (25b) can be written as
$\xi^{\mu}_{J_{01}}=(0,\Delta\cos\theta,\frac{1}{2}\Delta^{\prime}\sin\theta,0)$,
with $\Delta^{\prime}\equiv\partial_{r}\Delta$. As mentioned above, this CKV
is time independent and survives in the zero-frequency limit, recovering the
CKV that we showed in Hui  _et al._ (2022) to be associated with a ladder
structure for static perturbations around Kerr black holes, leading to
vanishing static response.777Note that the ${\rm sl}(2,\mathbb{R})$ involving
$L_{\pm}$ in the near zone (11) includes (25a) as a generator instead of
$\partial_{t}$, which precludes applying the arguments of Charalambous  _et
al._ (2021a) directly. Let us now recall how this works and extend the results
of Hui  _et al._ (2022) to non-zero frequencies.

The CKV $\xi^{\mu}_{J_{01}}$ corresponds to a symmetry of the scalar action,
which acts on the scalar field $\phi$ (in real space) as in (6). After
decomposing $\phi$ in spherical harmonics as $\phi=e^{-i\omega t}{Y_{\ell
m}}(\theta,\varphi)R_{\ell}(r)$, and extracting a convenient phase factor from
$R_{\ell}$,

$R_{\ell}\equiv
e^{-\frac{ir_{+}r_{s}}{r_{+}-r_{-}}\left(\omega-m\Omega_{+}\right)\log\left(\frac{r-r_{+}}{r-r_{-}}\right)}\psi_{\ell},$
(16)

the scalar equation (12) takes the form

$\left[\partial_{r}\Big{(}\Delta\partial_{r}-2ir_{s}r_{+}\left(\omega-m\Omega_{+}\right)\Big{)}-\ell(\ell+1)\right]\psi_{\ell}=0\,,$
(17)

where derivatives act on everything to their right, and the field
transformation takes the form

$\delta\psi_{\ell}=\mathcal{Q}_{\ell}D_{\ell-1}^{+}\psi_{\ell-1}-\mathcal{Q}_{\ell+1}D_{\ell+1}^{-}\psi_{\ell+1}\,.$
(18)

We have defined
$\mathcal{Q}_{\ell}\equiv\sqrt{(\ell^{2}-m^{2})/(4\ell^{2}-1)}$ and introduced
the operators

$\displaystyle D^{+}_{\ell}$
$\displaystyle\equiv-\Delta\partial_{r}-\frac{\ell+1}{2}\Delta^{\prime}+ir_{s}r_{+}(\omega-m\Omega_{+}),$
(19a) $\displaystyle D^{-}_{\ell}$
$\displaystyle\equiv\Delta\partial_{r}-\frac{\ell}{2}\Delta^{\prime}-ir_{s}r_{+}(\omega-m\Omega_{+}).$
(19b)

The $D^{\pm}_{\ell}$ are ladder operators in the sense that
$\delta\psi_{\ell\pm 1}=D_{\ell}^{\pm}\psi_{\ell}$ are solutions to the near-
zone equation (17) at level $\ell\pm 1$ if $\psi_{\ell}$ solves (17) at level
$\ell$. These operators are useful because they allow us to recursively define
an on-shell conserved charge at each $\ell$ Hui  _et al._ (2022):

$P_{\ell}=\alpha_{\ell}\big{[}\Delta\partial_{r}-2ir_{s}r_{+}\left(\omega-m\Omega_{+}\right)\big{]}D_{1}^{-}\cdots
D_{\ell}^{-}\psi_{\ell},$ (20)

where $\alpha_{\ell}\equiv-2^{2\ell-1}(\ell!)^{2}/[(2\ell)!(2\ell+1)!]$. These
conserved charges allow us to connect the behavior of solutions near the
horizon to the behavior at infinity without solving eq. (17) explicitly,
making it possible to infer the vanishing of static responses from symmetry
Hui  _et al._ (2022). Corresponding to each of these charges is an off-shell
symmetry of the action, which can be inferred as described in Hui  _et al._
(2022) (see also Compton and Morrison (2020) for a similar construction on de
Sitter backgrounds).

Evaluating $P_{\ell}$ on $\psi_{\ell}=D_{\ell-1}^{+}\cdots D_{0}^{+}\psi_{0}$,
where $\psi_{0}={\rm constant}$—which is the solution with the correct
infalling behavior at $r=r_{+}$—yields

$P_{\ell}=-iq(r_{+}-r_{-})^{2\ell+1}\frac{(\ell!)^{2}}{(2\ell)!(2\ell+1)!}\prod_{k=1}^{\ell}(k^{2}+4q^{2}),$
(21)

with $q\equiv r_{s}r_{+}(m\Omega_{+}-\omega)/(r_{+}-r_{-})$. The conserved
charges $P_{\ell}$ in (21) reproduce the induced multipole moments of a scalar
field on a fixed Kerr geometry Wong  _et al._ (2019) (see also Starobinskiǐ
(1973); Starobinskiǐ and Churilov (1974); Creci  _et al._ (2021)). In
particular, when $\omega=0$, one recovers the scalar’s static dissipative
response Le Tiec and Casals (2021); Le Tiec  _et al._ (2021); Chia (2021);
Charalambous  _et al._ (2021b); Hui  _et al._ (2022).

Acknowledgments: We would like to thank Albert Law and John Stout for helpful
discussions. LH is supported by the DOE DE-SC0011941 and a Simons Fellowship
in Theoretical Physics. The work of RP is supported in part by the National
Science Foundation under Grant No. PHY-1915611. ARS’s research was 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.

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## Supplemental material

###  .1 Near-zone AdS2 geometry

Here we elaborate on the geometry of the $(t,r)$ subspace of the near-zone
metric (3). This metric describes ${\rm AdS}_{2}$ in de Sitter-slice
coordinates. In order to see this, we make the coordinate transformation

$\tau=\frac{t}{2r_{s}}\,,\qquad\quad\xi=\cosh^{-1}\left(\frac{2r}{r_{s}}-1\right)\,,$
(22)

so that the two-dimensional metric ${\rm d}s^{2}=-\frac{\Delta}{r_{s}^{2}}{\rm
d}t^{2}+\frac{r_{s}^{2}}{\Delta}{\rm d}r^{2}$ becomes

$\displaystyle{\rm d}s^{2}=r_{s}^{2}\left({\rm d}\xi^{2}-\sinh^{2}\\!\xi\,{\rm
d}\tau^{2}\right),$ (23)

with $\tau\\!\in\\!(-\infty,+\infty),\,\xi\\!\in\\![0,+\infty)$. Notice in
particular that $\xi=0$ corresponds to $r=r_{s}$. This is an AdS2 metric. To
see this explicitly, we note that these coordinates correspond to the
embedding

$\displaystyle X_{0}=r_{s}\cosh\xi,\qquad\qquad
X_{1}=r_{s}\sinh\xi\sinh\tau,\qquad\qquad X_{2}=r_{s}\sinh\xi\cosh\tau,$ (24)

which satisfy $-X_{0}^{2}-X_{1}^{2}+X^{2}_{2}=-r_{s}^{2}$, and cover the
region of this hyperboloid that satisfies $X_{0}\geq r_{s},X_{2}\geq 0$. This
portion of AdS2 is depicted in Figure 1.

Figure 1: Portion of ${\rm AdS}_{2}$ covered by the de Sitter slice
coordinates $(\tau,\xi)$, along with lines of constant $\tau$ and $\xi$.

###  .2 (Conformal) Killing vectors of near-zone Kerr geometry

The (conformal) Killing vectors of the near-zone Kerr metric in eq. (13) (or,
equivalently, eq. (14)) are:

$\displaystyle T$
$\displaystyle=\mathcal{R}\partial_{t}+\tfrac{2a}{r_{\star}}\partial_{\varphi},$
(25a) $\displaystyle J_{01}$
$\displaystyle=-\tfrac{2\Delta}{r_{\star}}\cos\theta\,\partial_{r}-\tfrac{\partial_{r}\Delta}{r_{\star}}\sin\theta\,\partial_{\theta},$
(25b) $\displaystyle J_{02}$
$\displaystyle=-\cos\varphi^{\prime}\left[\tfrac{2\Delta}{r_{\star}}\sin\theta\,\partial_{r}+\tfrac{\partial_{r}\Delta}{r_{\star}}\left(\tfrac{\tan\varphi^{\prime}}{\sin\theta}\partial_{\varphi}-\cos\theta\partial_{\theta}\right)\right],$
(25c) $\displaystyle J_{03}$
$\displaystyle=-\sin\varphi^{\prime}\left[\tfrac{2\Delta}{r_{\star}}\sin\theta\,\partial_{r}-\tfrac{\partial_{r}\Delta}{r_{\star}}\left(\tfrac{\cot\varphi^{\prime}}{\sin\theta}\partial_{\varphi}+\cos\theta\partial_{\theta}\right)\right],$
(25d) $\displaystyle J_{12}$
$\displaystyle=\cos\varphi^{\prime}\partial_{\theta}-\cot\theta\sin\varphi^{\prime}\,\partial_{\varphi},$
(25e) $\displaystyle J_{13}$
$\displaystyle=\sin\varphi^{\prime}\partial_{\theta}+\cot\theta\cos\varphi^{\prime}\,\partial_{\varphi},$
(25f) $\displaystyle J_{23}$ $\displaystyle=\partial_{\varphi},$ (25g)
$\displaystyle L_{\pm}$ $\displaystyle=e^{\pm
t/\mathcal{R}}\left[\mathcal{R}(\partial_{r}\sqrt{\Delta})\partial_{t}\mp\sqrt{\Delta}\partial_{r}+\tfrac{2a}{r_{\star}}(\partial_{r}\sqrt{\Delta})\partial_{\varphi}\right]$
(25h) $\displaystyle K_{\pm}$ $\displaystyle=e^{\pm t/{\cal
R}}\tfrac{\sqrt{\Delta}}{r_{\star}}\cos\theta\left(\tfrac{r_{s}r_{+}r_{\star}}{\Delta}\partial_{t}\mp\partial_{r}\Delta\partial_{r}\mp
2\tan\theta\partial_{\theta}+\tfrac{ar_{\star}}{\Delta}\partial_{\varphi}\right),$
(25i) $\displaystyle M_{\pm}$ $\displaystyle=e^{\pm t/{\cal
R}}\cos\varphi^{\prime}\left[\tfrac{r_{s}r_{+}}{\sqrt{\Delta}}\sin\theta\partial_{t}\mp\tfrac{\sqrt{\Delta}\partial_{r}\Delta\sin\theta}{r_{\star}}\partial_{r}\pm\tfrac{2\sqrt{\Delta}}{r_{\star}}\cos\theta\partial_{\theta}+\left(\tfrac{a\sin\theta}{\sqrt{\Delta}}\mp\tfrac{2\sqrt{\Delta}}{r_{\star}}\tfrac{\tan\varphi^{\prime}}{\sin\theta}\right)\partial_{\varphi}\right],$
(25j) $\displaystyle N_{\pm}$ $\displaystyle=e^{\pm t/{\cal
R}}\sin\varphi^{\prime}\left[\tfrac{r_{s}r_{+}}{\sqrt{\Delta}}\sin\theta\partial_{t}\mp\tfrac{\sqrt{\Delta}\partial_{r}\Delta\sin\theta}{r_{\star}}\partial_{r}\pm\tfrac{2\sqrt{\Delta}}{r_{\star}}\cos\theta\partial_{\theta}+\left(\tfrac{a\sin\theta}{\sqrt{\Delta}}\pm\tfrac{2\sqrt{\Delta}}{r_{\star}}\tfrac{\cot\varphi^{\prime}}{\sin\theta}\right)\partial_{\varphi}\right],$
(25k)

where we have defined $\mathcal{R}=\frac{2r_{s}r_{+}}{r_{\star}}$,
$\varphi^{\prime}=\varphi-\frac{a}{r_{s}r_{+}}t$ and $r_{\star}\equiv
r_{+}-r_{-}$. Note that these reduce to (4) and (5) in the limit $a\to 0$. The
generators (25) satisfy the ${\rm so}(4,2)$ algebra. We can make this explicit
by defining $J_{54}=T$ along with

$\displaystyle J_{04}$ $\displaystyle=\frac{L_{+}-L_{-}}{2},$ $\displaystyle
J_{05}$ $\displaystyle=\frac{L_{+}+L_{-}}{2},$ (26a) $\displaystyle J_{14}$
$\displaystyle=\frac{K_{+}-K_{-}}{2},$ $\displaystyle J_{15}$
$\displaystyle=\frac{K_{+}+K_{-}}{2},$ (26b) $\displaystyle J_{24}$
$\displaystyle=\frac{M_{+}-M_{-}}{2},$ $\displaystyle J_{25}$
$\displaystyle=\frac{M_{+}+M_{-}}{2},$ (26c) $\displaystyle J_{34}$
$\displaystyle=\frac{N_{+}-N_{-}}{2},$ $\displaystyle J_{35}$
$\displaystyle=\frac{N_{+}+N_{-}}{2},$ (26d)

which then have the ${\rm so}(4,2)$ commutation relations

$[J_{AB},J_{CD}]=\eta_{AD}J_{BC}+\eta_{BC}J_{AD}-\eta_{AC}J_{BD}-\eta_{BD}J_{AC},$
(27)

where $\eta_{AB}=\text{diag}(-1,1,1,1,1,-1)$. A few comments are in order.
First, note that only $J_{01}$ and $J_{23}$ are time-independent (when
expressing all quantities in Boyer–Lindquist coordinates) and are exact
symmetries of the static sector Hui  _et al._ (2022). The other generators
depend explicitly on time and are not (C)KVs of the effective 3D Kerr metric
of Hui  _et al._ (2022). Second, the generators $L_{\pm}$ differ from the ones
introduced in Charalambous  _et al._ (2021a) for non-zero values of the spin
parameter $a$. Interestingly, though, they coincide (up to a rescaling of the
time coordinate) in the region close to the horizon defined by
$(r-r_{+})/r_{+}\ll 1$. This is a manifestation of the fact that all the near-
zone approximations of Kerr put forward in the literature actually coincide in
this limit. Note also that some of the generators in (25) are manifestly well
defined in the extremal limit ($a\to r_{s}/2$, $r_{\star}\rightarrow 0$),
while others look singular in this limit. This is not a problem because one
can consistently recover all the (C)KVs of the metric (13) at extremality by
multiplying with suitable powers of $r_{\star}$ and taking linear combinations
of the generators (25). For instance, in addition to $J_{ij}$ and $T$ (after
extracting a $1/r_{\star}$ factor), the other two KVs of (13) in the extremal
limit are obtained by expanding $J_{04}$ and the combination
$(T-J_{05})/r_{\star}$ at leading order in $r_{\star}$.

###  .3 Ladder in spin and finite frequency

A convenient near-zone approximation that describes the dynamics of particles
of generic spin, $s$, in the limit $r_{+}\leq r\ll 1/\omega$ is Page (1976);
Starobinskiǐ and Churilov (1974)

$x(x+1)\partial_{x}^{2}R+(s+1)(2x+1)\partial_{x}R+\left[-(\ell-s)(\ell+s+1)+\frac{q^{2}+isq(2x+1)}{x(x+1)}\right]R=0,$
(28)

where $q\equiv r_{s}r_{+}(m\Omega_{+}-\omega)/(r_{+}-r_{-})$ and

$x\equiv\frac{r-r_{+}}{r_{+}-r_{-}}.$ (29)

It is straightforward to show that eq. (28) admits the following set of spin
raising and lowering operators:

$E^{+}_{s}=\left(\partial_{r}-\frac{ir_{s}r_{+}(\omega-m\Omega_{+})}{\Delta}\right)R\,,\qquad\qquad
E^{-}_{s}=\Delta^{-s+1}\left(\partial_{r}+\frac{ir_{s}r_{+}(\omega-m\Omega_{+})}{\Delta}\right)\Delta^{s}R\,,$
(30)

which generate solutions with spin $s+1$ and $s-1$ respectively, i.e.,
$R^{(s+1)}=E^{+}_{s}R^{(s)}$ and $R^{(s-1)}=E_{s}^{-}R^{(s)}$, where $R^{(s)}$
solves (28) with spin $s$. The operators (30) generalize the
Teukolsky–Starobinsky identities Press and Teukolsky (1973); Starobinskiǐ and
Churilov (1974); Chandrasekhar (1985) in the near-zone regime by connecting
solutions with consecutive spin $s$ and $s\pm 1$. These spin raising and
lowering operators provide a simple way of extending the results discussed
above for spin-0 fields to spin-1 and spin-2 particles described by the
Teukolsky equation Hui  _et al._ (2022).