UPR-1329-T
IFT-UAM/CSIC-24-85

Kerr Effective Black Hole Geometries
in Supergravity

Mirjam Cvetič<EMAIL_ADDRESS>, Nelson Hernández Rodríguez
<EMAIL_ADDRESS>,

Maria J. Rodriguez<EMAIL_ADDRESS>and Oscar Varela
<EMAIL_ADDRESS>
aDepartment of Physics and Astronomy and Department of Mathematics,

University of Pennsylvania, Philadelphia, PA 19104, USA

bCenter for Applied Mathematics and Theoretical Physics, University of
Maribor, Maribor, SI2000, Slovenia

cInstituto de Física Teórica UAM-CSIC, Madrid, 28049, Spain

dBlack Hole Initiative, Harvard University, Cambridge, MA 02138, USA

eDepartment of Physics, Utah State University, Logan, UT 84322, USA

###### Abstract

We derive the explicit embedding of the effective Kerr spacetimes, which are
pertinent to the vanishing of static Love numbers, soft hair descriptions of
Kerr black holes, and low-frequency scalar-Kerr scattering amplitudes, as
solutions within $N=2$ supergravity. These spacetimes exhibit a hidden
$SL(2,R)\times U(1)$ or $SO(4,2)$ symmetry resembling the so called subtracted
geometries with $SL(2,R)\times SL(2,R)$ symmetry, which accurately represent
the near-horizon geometry of Kerr black holes and, as we will argue most
accurately represents the internal structure of the Kerr black hole. To
quantify the differences among the effective Kerr spacetimes, we compare their
physical quantities, internal structures, and geodesic equations. Although
their thermodynamic properties, including entropy, match those of Kerr, our
study uncovers significant differences in the interiors of these effective
Kerr solutions. A careful examination of the internal structure of the
spacetimes highlights the distinctions between various effective Kerr
geometries and their quasinormal spectra.

###### Contents

  1. 1 Introduction
  2. 2 Kerr Effective Black Hole Geometries
    1. 2.1 Definitions
    2. 2.2 Effective $SL(2,R)\times SL(2,R)$ Black Hole Geometries
    3. 2.3 Effective $SL(2,R)\times U(1)$ Black Hole Geometries
    4. 2.4 Effective $SO(4,2)$ Black Hole Geometries
    5. 2.5 Thermodynamic properties
  3. 3 Matter Supporting the KEGs
    1. 3.1 The supergravity model
    2. 3.2  Static case
    3. 3.3 Rotating case
  4. 4 Monodromies and Internal Structure
  5. 5 Quasi-normal modes
  6. 6 Separability in KEG
  7. 7 Discussion
  8. A STU black holes and subtracted geometries
    1. A.1 Asymptotically flat black hole solution
    2. A.2 Scaling procedure for KEG with $SL(2,R)\times SL(2,R)$ symmetry
  9. B Harrison Transformation

##  1 Introduction

Exploring low-frequency physics within black hole backgrounds has proven
highly fruitful for the soft hair of Kerr black holes descriptions, low-
frequency scalar-Kerr scattering amplitudes and vanishing of the gravitational
static Love numbers for rotating black holes. The guiding principle in this
approach has been the observation that for certain low frequency regimes,
$\omega\ll 1/M$, with $M$ the black hole mass, one can explicitly realize a
hidden $SL(2,R)\times SL(2,R)$ symmetry in the massless Klein-Gordon (KG) wave
equation for a propagating field. This symmetry does not manifest itself as an
isometry of the background metric

One of the significant effective metrics associated with Kerr black holes is
the so-called subtracted geometry [1]. This is specifically connected to the
hidden symmetries of Kerr black hole spacetimes via Teukolsky’s equation in
the theory of gravitational perturbations as employed in [2]. The key feature
is that these geometries provide an approximation in the near-horizon region
of a Kerr black hole, maintaining its thermodynamic properties at the outer
event horizon and its internal structure.

An additional collection of fundamental geometries pertinent to black holes
are the effective near-zone metrics that play a crucial role in describing
gravitational tidal deformations, or Love symmetries [3, 4, 5]. These
different geometries arise from the ambiguity in how one defines the effective
Kerr near zone expansion associated with a freedom to move frequency dependent
terms in the Klein-Gordon (KG) equation. For each choice of near zone split,
the leading order massless wave equation enjoys a generalized local hidden
symmetry [6]. Two instances of Kerr Effective Geometry (KEG) approximations
have been recently proposed. On the one hand, there is a geometry preserving
an $SO(4,2)$ found in [7] and, on the other hand, an effective geometry
preserving an $SL(2,R)\times U(1)$ as shown in [8]. These are more mysterious
than the $SL(2,R)\times SL(2,R)$; even-though these alternative KEGs preserve
the same thermodynamic properties of Kerr, they also realize distinct hidden
symmetries of the KG wave equation which could explain the vanishing of the
static Love numbers for Kerr black holes. A surprising property of the KEG
with $SO(4,2)$ or $SL(2,R)\times U(1)$ hidden symmetries, is that these do not
capture the dynamical Love numbers for Kerr. As recently shown in [9], the KEG
with $SL(2,R)\times SL(2,R)$, seems to much closely capture the dynamical Love
numbers for Kerr in the low frequency regime.

The lack of uniqueness among these effective geometries poses a significant
issue, as each one is associated with distinct hidden symmetries that are
presumably unrelated to one another. The goal of the present paper is to
quantify the differences among these effective geometries by contrasting their
physical quantities and structures. In this context, we will refer to the
geometries that lead to local symmetries in the solution space of low
frequency massless field configurations as KEGs.

KEGs are no longer asymptotically flat. Moreover, these metrics with the
asymptotic geometry removed in this manner, no longer satisfy Einstein’s
equations in vacuum. Kerr black holes emerge as solutions within the vacuum
Einstein equations, whereas KEGs rely on additional matter content for
support. This has been shown in [10] for the subtracted $SL(2,R)\times
SL(2,R)$ KEG. In this paper, we will demonstrate explicitly the embedding of
more general KEGs as solutions to $N=2$ supergravity in four dimensions.

The structure of this paper is as follows. Section 2 reviews three distinct
effective Kerr metrics, namely, those whose associated leading order KG
equation enjoys a $SL(2,R)\times SL(2,R)$, $SO(4,2)$ or $SL(2,R)\times U(1)$
local hidden symmetry. We analyze the structure of these metrics and propose
different methods which enable us to calculate the thermodynamics of the KEGs.
The field configurations that define the KEGs as solutions of $N=2$
supergravity can be found in Section 3. In Section 4, additional evidence is
presented supporting the proposal that certain KEGs will replicate certain
aspects of the physics of Kerr black holes. This section also addresses issues
related to the monodromy and internal structure of the spacetimes. Remarkably
simple, explicit expressions for the quasinormal mode frequencies of the KEGs
are obtained in Section 5. The study of the separability of the wave equation,
existence of exact Killing-Stäckel tensors and geodesic equations is presented
in Section 6. We discuss our results and conclude by presenting some future
directions in Section 7. The paper also includes an Appendix that compiles the
necessary results for the calculations discussed earlier, presented in a
consistent notation. We use geometrized units where $G=c=k_{B}=\hbar=1$.

##  2 Kerr Effective Black Hole Geometries

In this section we focus on the study of physical properties of the KEGs. The
main observation about the KEGs is that these preserve the internal structure
of the Kerr black hole as well as the thermodynamic properties. While Kerr
black holes arise as solutions of the vacuum Einstein equations, KEGs are
supported by supplementary matter content. These fields supporting such
geometries vary as a function of radial distance at spatial infinity, instead
of becoming constant. KEGs exhibit distinct asymptotic structures when
contrasted with the asymptotically flat and asymptotically AdS scenarios.
Moreover, a comprehensive exploration of their conserved charges is lacking,
and new insights would provide a starting point for the study of gravity/field
theory duality for KEGs.

###  2.1 Definitions

The spacetime metrics can be treated physically as black holes confined in a
box. The KEG geometry is

$ds_{KEG}^{2}=-\Delta_{0}^{-1/2}G_{0}\,(dt+A_{0}\,d\phi)^{2}+\Delta_{0}^{1/2}\left(\frac{dr^{2}}{\Delta}+d\theta^{2}+\frac{\Delta}{G_{0}}\sin^{2}\theta\,d\phi^{2}\right)$
(2.1)

with $\Delta_{0}$ a function of $r$ and,

$\displaystyle\left.A_{0}\,\,\right|_{r=r_{+}}=-(r_{+}^{2}+a^{2})\,,\qquad\left.G_{0}\,\,\right|_{r=r_{+}}=-a^{2}\sin^{2}\theta\,,$
(2.2)

such that effectively the near black hole horizon Kerr region is preserved.
The corresponding asymptotic form of the (static) geometry reads

$ds_{AsympKEG}^{2}=-Y^{2p}dt^{2}+B^{2}\frac{dY^{2}}{Y^{2s}}+Y^{2q}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\,,$ (2.3)

where $(p,q,s,B)$ are constants; in particular KEGs are not asymptotically
flat implying $p,s\neq 0$, and $q,B\neq 1$.

For Kerr black holes in standard Boyer-Lindquist coordinates, the warp factor
$\Delta_{0}$, the function $\Delta$ and angular potential $A_{0}$ are given by

$\displaystyle G^{Kerr}_{0}=\Delta-a^{2}\sin^{2}\theta\,,\qquad
A^{Kerr}_{0}=\frac{2Ma\,r\sin^{2}\theta}{G^{Kerr}_{0}}\,,$ (2.4)
$\displaystyle\Delta^{Kerr}_{0}=(r^{2}+a^{2}\cos^{2}\theta)^{2}\,,\qquad\Delta^{Kerr}=r^{2}+a^{2}-2Mr\,.$
(2.5)

However, asymptotically the geometry is flat, taking the form (2.3) with
$s,p=0$ and $q,B=1$.

The main observation leading to the KEG proposals is that the thermodynamic
properties of the Kerr black hole solution is completely independent of the
warping factor $\Delta_{0}$. A fact that suggests that modifying
$\Delta^{Kerr}_{0}$ in (2.3) preserves the internal near structure and retains
the same thermodynamic properties of Kerr black holes.

Concrete KEGs of the form (2.1) arise for different choices of the metric
functions. All three of them that we will consider in this paper lead to a
massless wave equation,

$\displaystyle\Delta^{-1/2}_{0}\left[\partial_{r}\Delta\partial_{r}+\frac{1}{\sin\theta}\partial_{\theta}\sin\theta\partial_{\theta}-\frac{\Delta_{0}}{G_{0}}\partial^{2}_{t}+\frac{G_{0}}{\Delta\sin^{2}\theta}(\partial_{\phi}-A_{0}\partial_{t})^{2}\right]\psi=0\
,$ (2.6)

with at least $SL(2,R)$ hidden symmetry [6]. The first such geometry allows
for modifications with respect to Kerr only in the warping factor
$\Delta_{0}$, leaving the angular potential intact, $A_{0}=A^{Kerr}_{0}$. This
geometry is the subtracted geometry, also called black hole a box, introduced
in [1, 10]. The subtracted geometry is asymptotically conical with a Lifshitz
type symmetry (a diffeomorphism under which the pull-back metric goes into a
constant multiple of itself), with time and radial distance scaling
differently. Two other KEG geometries arise that alter both the warping factor
$\Delta_{0}$ in (2.1) and the angular potential $A_{0}$ with respect to Kerr.
These lead to wave equations (2.6) that display hidden $SL(2,R)\times U(1)$
[8] and $SO(4,2)$ symmetries [7], respectively, and are relevant in the
context of tidal Love numbers. These have different asymptotics. It is worth
emphasizing that all these hidden symmetries do not correspond to isometries
of the background metric (2.1) but rather manifest themselves in the massless
scalar wave equation (2.6). The next subsections review these KEGs and compare
the corresponding thermodynamic properties in each case.

###  2.2 Effective $SL(2,R)\times SL(2,R)$ Black Hole Geometries

Let us review the effective geometry that generates the near region in [2]
that captures the $SL(2,R)\times SL(2,R)$ symmetry of the near-region scalar
field equation. In terms of the $(t,r,\theta,\phi)$ coordinates, the
corresponding effective metric is (2.1) with

$\displaystyle G_{0}=\Delta-a^{2}\sin^{2}\theta\,,\qquad
A_{0}=\frac{2ma\,r\sin^{2}\theta}{G_{0}}\,,$
$\displaystyle\Delta_{0}=4m^{2}(2mr-a^{2}\cos^{2}\theta)\,,\qquad\Delta=r^{2}+a^{2}-2mr\,.$
(2.7)

This metric coincides with the one presented in [11] in the chargeless limit.
At the specific points where $\Delta_{0}=0$, the Ricci scalar diverges,
suggesting a singularity in the curvature of the metric. This curvature
singularity lies behind the black hole horizons.

It is interesting to study the behavior of these asymptotically conical
metrics as we approach the limit where $r$ tends to infinity. At leading
order, the following expression emerges,

$\displaystyle ds^{2}(Asymp)_{SL(2,R)\times SL(2,R)}\sim$
$\displaystyle-\sqrt{\frac{r^{3}}{8m^{3}}}dt^{2}-2a\sin^{2}\theta\sqrt{\frac{r}{2m}}dtd\phi$
$\displaystyle+\sqrt{8m^{3}r}\,\left(\frac{dr^{2}}{r^{2}}+d\theta^{2}-\sin^{2}\theta
d\phi^{2}\right).$ (2.8)

A coordinate transformation $\phi\rightarrow\varphi+At$ where $A=a/(4m^{2})$
simplifies the metric corresponding to the asymptotic form (2.3). In this
case, $p=3,q=1,s=0,B=4$ and $Y=(8m^{3}\,r)^{1/4}$ as shown in [11].

###  2.3 Effective $SL(2,R)\times U(1)$ Black Hole Geometries

One of the metrics that has garnered increasing interest for its fundamental
role in describing Love symmetries is the one found in [8]. In this particular
case, the hidden “Love” symmetry possessed by this metric corresponds to
$SL(2,R)\times U(1)$. Once again, the line element is defined by (2.1), but in
this case the functions are

$\displaystyle G_{0}$ $\displaystyle=\Delta-a^{2}\sin^{2}\theta,\qquad
A_{0}=\frac{a\sin^{2}\theta}{G_{0}}(r_{+}^{2}+a^{2}+\beta(r-r_{+})),$
$\displaystyle\Delta_{0}$
$\displaystyle=(r_{+}^{2}+a^{2})^{2}(1+\beta^{2}\Omega^{2}\sin^{2}\theta),\qquad\Delta=r^{2}+a^{2}-2mr,$
(2.9)

where

$\displaystyle\beta=\frac{1}{2\pi T}\,,\qquad\text{and}\qquad
T\equiv\frac{\kappa_{+}}{2\pi}=\frac{r_{+}-r_{-}}{8\pi Mr_{+}}\,$ (2.10)

is the temperature of the black hole. The corresponding Ricci scalar exhibits
a divergence at $\Delta_{0}=0$, indicating a curvature singularity in the
metric.

The behavior at infinity of this metrics differs completely from the other
geometries. As we approach the limit of $r$, the metric of the black hole
behaves as

$ds^{2}(Asymp)_{SL(2,R)\times
U(1)}\sim-\frac{r^{2}}{\Gamma}\left(dt+\frac{a\beta\sin^{2}\theta
d\phi}{r}\right)^{2}+\Gamma\left(\frac{dr^{2}}{r^{2}}+d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$ (2.11)

where the function $\Gamma$ depends on $\theta$ as

$\Gamma=(a^{2}+r_{+}^{2})\sqrt{1+\beta^{2}\Omega^{2}\sin^{2}\theta}.$ (2.12)

Note that only in the static case, for $a=0$, the geometry is of the form
(2.3). In this static limit the asymptotic geometry has a scaling symmetry
$ds^{2}_{SL(2,R)\times U(1)}\rightarrow\epsilon^{0}\,ds^{2}_{SL(2,R)\times
U(1)}$ implemented by taking $r\rightarrow\epsilon r$ and
$t\rightarrow\epsilon^{-1}t$.

###  2.4 Effective $SO(4,2)$ Black Hole Geometries

Finally, the last metric we will consider was presented in [7], and has a
“Starobinsky” hidden symmetry $SO(4,2)$. Similar to the previous cases, the
line element will be (2.1); however, the functions will depend on how we
perform our effective near-zone approximation. Specifically, we consider

$\displaystyle G_{0}$ $\displaystyle=\Delta-a^{2}\sin^{2}\theta,\qquad
A_{0}=\frac{a\sin^{2}\theta}{G_{0}}(r_{+}^{2}+a^{2}),$
$\displaystyle\Delta_{0}$
$\displaystyle=(r_{+}^{2}+a^{2})^{2},\qquad\Delta=r^{2}+a^{2}-2mr.$ (2.13)

The metric is a conformally flat spacetime, has vanishing Ricci scalar (though
not vanishing Ricci tensor), and represents a solution supported by external
fields described in section 3.

The asymptotic behavior of the metric in the limit $r\rightarrow\infty$ is

$\displaystyle ds^{2}(Asymp)_{SO(4,2)}\sim$
$\displaystyle-\frac{r^{2}}{(r^{2}_{+}+a^{2})}dt^{2}-2a\sin^{2}\theta
dt\,d\phi$
$\displaystyle+(r^{2}_{+}+a^{2})\left(\frac{dr^{2}}{r^{2}}+d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$ (2.14)

A coordinate transformation $\phi\rightarrow\varphi+At$ with
$A=-\frac{a}{r_{+}^{2}+a^{2}}$ simplifies the metric corresponding to the
asymptotic form (2.3) with the choice of parameters $s=p=1$, $B=1$ and $q=0$,
which locally is equivalent to $\text{AdS}_{2}\times S^{2}$.

###  2.5 Thermodynamic properties

We have determined explicitly the thermodynamic quantities of the KEGs of the
previous subsections. The entropy, temperature and angular velocity of the
original Kerr black hole remain unchanged. Explicitly, the entropy of the KEGs
(2.1) is given by

$S=\frac{A}{4}=\left.\int_{0}^{2\pi}\int_{0}^{\pi}\sqrt{g_{\theta\theta}g_{\phi\phi}}\,\,\right|_{r=r_{+}}d\theta
d\phi=\pi\,(r_{+}^{2}+a^{2}).$ (2.15)

The temperature and angular velocity on the event horizon are respectively
defined as

$T\equiv\frac{\kappa}{2\pi}=\left.\frac{(N^{2})^{\prime}}{4\pi\sqrt{N^{2}g_{rr}}}\right|_{r=r_{+}}=\frac{\Delta^{{}^{\prime}}(r_{+})}{4\pi(r_{+}^{2}+a^{2})},$
(2.16)

where $\kappa$ is the surface gravity, $N$ is the lapse function, and
$\Delta^{{}^{\prime}}(r_{+})$ is the derivative of function $\Delta$ evaluated
at the outer horizon, and

$\Omega=-\left.\frac{g_{t\phi}}{g_{\phi\phi}}\,\,\right|_{r=r_{+}}=\frac{a}{r_{+}^{2}+a^{2}}.$
(2.17)

The black hole outer and inner horizons ($\Delta=0$) remain at

$r_{\pm}=m\pm\sqrt{m^{2}-a^{2}}\,.$ (2.18)

Likewise, the ergosphere defined by $g_{tt}=0$, is only determined by $G_{0}$
for this type of black holes. In this sense, for all KEGs considered here, the
black hole’s internal structure is preserved and is determined by the
following expression

$r^{erg}_{\pm}=m\pm\sqrt{m^{2}-a^{2}\cos^{2}\theta}\,.$ (2.19)

The definitions of mass and angular momentum are significantly influenced by
the asymptotic properties of the curved geometry. Since the KEGs do not
exhibit asymptotic flatness, calculating these conserved quantities becomes
quite challenging.

The corresponding intensive quantities are defined at the inner Cauchy horizon
for the KEG. References [12, 13] studied these relations for black holes, and
in particular for a Kerr black hole the angular velocity evaluated at the
Cauchy horizon was found in [14]

$\displaystyle\Omega^{Kerr}_{-}=-\left.\frac{g_{t\phi}}{g_{\phi\phi}}\,\,\right|_{r=r_{-}}=\frac{a}{r_{-}^{2}+a^{2}}$
(2.20)

Extending this definition for other geometries we can find the angular
velocity at the inner Cauchy horizon for the KEGs. We find

$\displaystyle\Omega_{-}^{SL(2,R)\times
SL(2,R)}=\frac{a}{r_{-}^{2}+a^{2}}=\Omega^{Kerr}_{-},$ (2.21)
$\displaystyle\Omega_{-}^{SO(4,2)}=-\Omega_{-}^{SL(2,R)\times
U(1)}=\frac{a}{r_{+}^{2}+a^{2}}=\Omega.$ (2.22)

As for the temperature of the Cauchy horizon, the Kerr black hole has

$T_{-}^{Kerr}\equiv-\frac{\kappa_{-}}{2\pi}=-\left.\frac{(N^{2})^{\prime}}{4\pi\sqrt{N^{2}g_{rr}}}\right|_{r=r_{-}}=-\frac{r_{+}-r_{-}}{8\pi
Mr_{-}}.$ (2.23)

For the KEGs we find

$\displaystyle T_{-}^{SL(2,R)\times SL(2,R)}$
$\displaystyle=-\frac{r_{+}-r_{-}}{8\pi Mr_{-}}=T_{-}^{Kerr}$ (2.24)
$\displaystyle T_{-}^{SL(2,R)\times U(1)}$
$\displaystyle=-\frac{r_{+}-r_{-}}{8\pi Mr_{+}}=-T_{-}^{SO(4,2)}$ (2.25)

##  3 Matter Supporting the KEGs

The KEGs under consideration do not satisfy Einstein’s equations of motion in
vacuum. In this section we explicitly identify the matter configurations that
support these geometries.

###  3.1 The supergravity model

Any geometry is a solution to Einstein’s equations if the energy momentum
tensor is chosen as the Einstein tensor of the geometry. A standard criterion
for the physical viability of that matter content involves specifying suitable
energy conditions for the stress tensor. For our KEGs, all energy conditions
are satisfied. In fact, all KEGs arise as solutions of $N=2$ supergravity
coupled to three vector multiplets, namely, the so-called STU model. This was
shown in [10] for the subtracted geometries with hidden $SL(2,R)\times
SL(2,R)$ symmetry. Here, we will extend that proof to the other KEGs under
consideration.

The bosonic Lagrangian of the STU model is [15]

$\displaystyle{\cal L}_{4}$ $\displaystyle=$ $\displaystyle R\,{*{\bf
1}}-\frac{1}{2}{*d\varphi_{i}}\wedge
d\varphi_{i}-\frac{1}{2}e^{2\varphi_{i}}\,{*d\chi_{i}}\wedge
d\chi_{i}-\frac{1}{2}e^{-\varphi_{1}}\,(e^{\varphi_{2}-\varphi_{3}}\,{*F_{(2)1}}\wedge
F_{(2)1}$ (3.1) $\displaystyle+$ $\displaystyle
e^{\varphi_{2}+\varphi_{3}}\,{*F_{(2)2}}\wedge
F_{(2)2}+e^{-\varphi_{2}+\varphi_{3}}\,{*{\cal F}^{1}_{(2)}}\wedge{\cal
F}^{1}_{(2)}+e^{-\varphi_{2}-\varphi_{3}}\,{*{\cal F}^{2}_{(2)}}\wedge{\cal
F}^{2}_{(2)})$ $\displaystyle-$ $\displaystyle\chi_{1}\,(F_{(2)1}\wedge{\cal
F}^{1}_{(2)}+F_{(2)2}\wedge{\cal F}^{2}_{(2)})\,.$

The index $i=1,2,3$ labels the vector multiplets. The dilaton and axion scalar
fields are respectively denoted as $\varphi_{i}$ and $\chi_{i}$. Finally, the
model also contains four $U(1)$ gauge fields, $A_{(1)1}$, $A_{(1)2}$, ${\cal
A}^{1}_{(1)}$ and ${\cal A}^{2}_{(1)}$, with field strengths defined as

$\displaystyle F_{(2)1}$ $\displaystyle=$ $\displaystyle
dA_{(1)1}-\chi_{2}\,d{\cal A}^{2}_{(1)},$ $\displaystyle F_{(2)2}$
$\displaystyle=$ $\displaystyle dA_{(1)2}+\chi_{2}\,d{\cal
A}^{1}_{(1)}-\chi_{3}\,dA_{(1)1}+\chi_{2}\,\chi_{3}\,d{\cal A}^{2}_{(1)},$
$\displaystyle{\cal F}^{1}_{(2)}$ $\displaystyle=$ $\displaystyle d{\cal
A}^{1}_{(1)}+\chi_{3}\,d{\cal A}^{2}_{(1)},$ $\displaystyle{\cal F}^{2}_{(2)}$
$\displaystyle=$ $\displaystyle d{\cal A}^{2}_{(1)}.$ (3.2)

The $SL(2,R)\times SL(2,R)$ subtracted geometry, (2.1) with (2.2), was shown
in [10] to arise as a solution of the STU model (3.1). This was done through a
rescaling limit that we briefly review in Appendix A.2. As we will show next,
the other KEGs are also solutions of (3.1). The starting point is the
asymptotically flat black hole solution [16, 15] (see also [17])

$\displaystyle d{s}^{2}_{4}$
$\displaystyle=-{\Delta}^{-1/2}_{0}{G}(d{\bar{t}}+{{\cal
A}})^{2}+{\Delta}^{1/2}_{0}\left(\frac{d\bar{r}^{2}}{X}+d\theta^{2}+\frac{X}{G}\sin^{2}\theta
d\bar{\phi}^{2}\right)\leavevmode\nobreak\ ,$ (3.3)

where

$\displaystyle{X}$ $\displaystyle=$
$\displaystyle{\bar{r}}^{2}-2{\bar{m}}{\bar{r}}+{\bar{a}}^{2}\leavevmode\nobreak\
,$ (3.4) $\displaystyle{G}$ $\displaystyle=$
$\displaystyle{\bar{r}}^{2}-2{\bar{m}}{\bar{r}}+{\bar{a}}^{2}\cos^{2}\theta\leavevmode\nobreak\
,$ (3.5) $\displaystyle{{\cal A}}$ $\displaystyle=$
$\displaystyle\frac{2\bar{m}\bar{a}\sin^{2}\theta}{G}\left[({\Pi_{c}}-{\Pi_{s}}){\bar{r}}+2{\bar{m}}{\Pi}_{s}\right]d\bar{\phi}\leavevmode\nobreak\
,$ (3.6)

and

$\displaystyle{{\Delta}}_{0}=$
$\displaystyle\prod_{I=1}^{4}({\bar{r}}+2{\bar{m}}\sinh^{2}{\bar{\delta}}_{I})+2{\bar{a}}^{2}\cos^{2}\theta[{\bar{r}}^{2}+{\bar{m}}{\bar{r}}\sum_{I=1}^{4}\sinh^{2}{\bar{\delta}_{I}}+\,4{\bar{m}}^{2}({\Pi}_{c}-{\Pi}_{s}){\Pi}_{s}$
(3.8)
$\displaystyle-2{\bar{m}}^{2}\sum_{I<J<K}\sinh^{2}{\bar{\delta}}_{I}\sinh^{2}{\bar{\delta}}_{J}\sinh^{2}{\bar{\delta}}_{K}]+{\bar{a}}^{4}\cos^{4}\theta\leavevmode\nobreak\
.$

We are employing the following abbreviations:

${\Pi}_{c}\equiv\prod_{I=1}^{4}\cosh{\bar{\delta}}_{I}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{\Pi}_{s}\equiv\prod_{I=1}^{4}\sinh{\bar{\delta}}_{I}\leavevmode\nobreak\ .$
(3.9)

Please refer to appendix A for the explicit expressions of the scalars and
gauge fields that support the black hole geometry (3.3).

Let us first implement the scaling limit in the static $SL(2,R)\times U(1)$
and $SO(4,2)$ cases, before moving to the rotating cases.

###  3.2  Static case

Consider first the static, $a=0$, limit, of the black hole (3.3). This metric
is a solution to the field equations that follow from (3.1), together with the
scalars and vectors specified in appendix A with $a=0$. For example, the
scalar fields take the form

$\displaystyle\chi_{i}=0\quad(i=1,2,3)$ $\displaystyle
e^{\varphi_{1}}=\left[\frac{(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{1})(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{3})}{(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{2})(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{4})}\right]^{\frac{1}{2}}\,e^{\varphi_{2}}=\left[\frac{(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{2})(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{3})}{(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{1})(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{4})}\right]^{\frac{1}{2}}$
$\displaystyle
e^{\varphi_{3}}=\left[\frac{(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{1})(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{2})}{(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{3})(\bar{r}+2\bar{m}\sinh^{2}\bar{\delta}_{4})}\right]^{\frac{1}{2}}.$
(3.10)

We have put bars over all variables in the original $N=2$ supergravity
solution. Next, we choose the charge parameters to be all equal, as
$\bar{\delta}_{1}=\bar{\delta}_{2}=\bar{\delta}_{3}=\bar{\delta}_{4}\equiv\bar{\delta}$.
This reduces the solution to the the Reissner-Nordström black hole, as all
scalars become zero and the gauge fields become related to a single Einstein-
Maxwell gauge field or its Hodge dual. Finally, we rescale the original
variables as

$\displaystyle\bar{r}=r\epsilon\,,\qquad\bar{t}=t\epsilon^{-1}\,,\qquad\bar{\phi}=\phi\,,\qquad\bar{m}=m\,\epsilon\,,$
$\displaystyle 2\,\bar{m}\sinh^{2}\bar{\delta}\equiv
Q\,,\qquad\sinh^{2}\bar{\delta}\equiv\epsilon^{-1}\sinh^{2}{\delta}\,,$ (3.11)

and take the $\epsilon\rightarrow 0$ limit. This limit is finite in our new
cases, in contrast to the formally infinite limiting case considered in [10].

In the limit, the metric becomes the (static) KEG geometry (2.1), with the
functions defined in either Section 2.3 or Section 2.4. The scalars (3.2)
vanish (even before taking the limit, as noted above)

$\displaystyle\chi_{1}=\chi_{2}=\chi_{3}=0,\qquad{\varphi_{1}}={\varphi_{2}}={\varphi_{3}}=0\,,$
(3.12)

the electric gauge fields become

$\displaystyle
A_{(1)2}^{Star}=(\mathcal{A}_{(1)}^{2})^{Star}=\frac{(r-m)}{2\,m\sinh^{2}\delta}\,dt$
(3.13)

and the magnetic ones

$\displaystyle
A_{(1)1}^{Star}=\mathcal{A}_{(1)1}^{Star}=-2\,m\cos\theta\sinh^{2}\delta\,d\phi$
(3.14)

The vectors (3.14) are dual to (3.13) and have constant electric field
strength. The limiting configuration turns out to be the Bertotti-Robinson
solution AdS${}_{2}\times S^{2}$ of $N=2$ pure supergravity, namely, Einstein-
Maxwell theory.

The realization of these geometric configurations as solutions within the same
theory as the original black holes implies a more immediate connection between
these theories. We show that the KEG can be obtained by a solution generating
technique (within the STU-model) emerging as a distinct Harrison
transformation within the framework of Dilaton-Maxwell-Einstein gravity theory
in Appendix B. Recognizing the transformation in the non-rotational scenario,
as we have, might also offer a practical approach for extending these ideas to
cases involving rotations following [18, 19, 20].

###  3.3 Rotating case

We now derive the rotating KEGs as solutions of the $N=2$ supergravity model
of section 3.1. The metric is (2.1) above, but now we write all the quantities
in the original geometry with tildes, for convenience. We also set

$\tilde{\delta}_{1}=\tilde{\delta}_{2}=\tilde{\delta}_{3}\equiv\tilde{\delta}\;,\quad\mathrm{and}\quad\tilde{\delta}_{4}\equiv\tilde{\delta}_{0}\;.$
(3.15)

It is useful to note that, after these changes, the scalar fields take on the
form

$\displaystyle\chi_{1}=\chi_{2}=\chi_{3}=\frac{2{\bar{m}}\,{\bar{a}}\cos\theta\,\cosh{\bar{\delta}}\sinh{\bar{\delta}}(\cosh{\bar{\delta}}\sinh{\bar{\delta}}_{0}-\sinh{\bar{\delta}}\cosh{\bar{\delta}}_{0})}{({\bar{r}}+2{\bar{m}}\sinh^{2}{\bar{\delta}})^{2}+{\bar{a}}^{2}\cos^{2}\theta},$
(3.16) $\displaystyle
e^{\varphi_{1}}=e^{\varphi_{2}}=e^{\varphi_{3}}=\frac{({\bar{r}}+2{\bar{m}}\sinh^{2}{\bar{\delta}})^{2}+{\bar{a}}^{2}\cos^{2}\theta}{{\bar{\Delta}_{0}}^{1/2}}\,$
(3.17)

The corresponding gauge potentials follow from the expressions given in
Appendix A.

### Starobinski Scaling

Let us first the rescaling corresponding to the $SO(4,2)$ Starobinski
scenario. With the choice of charges (3.15), we next adopt the rescaling

$\displaystyle\bar{r}=r\epsilon\,,\qquad\bar{t}=t\epsilon^{-1}\,,\qquad\bar{\phi}=\epsilon\phi\,,\qquad\bar{m}=m\,\epsilon\,,$
$\displaystyle\bar{a}=a\,\epsilon\,,\qquad\sinh\bar{\delta}=\sinh\bar{\delta_{0}}\equiv\epsilon^{-1/2}\sinh{\delta}$
(3.18)

and then send $\epsilon\rightarrow 0$. In the limit, the metric (2.1) with
(2.4) is recovered, provided one identifies $\sinh^{4}{\delta}\equiv
r_{+}/(r_{+}+r_{-})$. This configuration has again vanishing scalar fields,

$\displaystyle\chi_{1}=\chi_{2}=\chi_{3}=0,\qquad{\varphi_{1}}={\varphi_{2}}={\varphi_{3}}=0\,,$
(3.19)

but the gauge fields become

$\displaystyle
A_{(1)2}^{Star}=(\mathcal{A}_{(1)}^{2})^{Star}=\frac{(r-m)}{2m\sinh^{2}\delta}\,dt$
(3.20)

and

$\displaystyle
A_{(1)1}^{Star}=\mathcal{A}_{(1)1}^{Star}=\frac{a\,\cos\theta}{2m\sinh^{2}\delta}\,dt-2m\cos\theta\sinh^{2}\delta\,d\phi\;.$
(3.21)

We have explicitly verified that this configuration solves the field equations
that derive from the Lagrangian (3.1).

Both the original $N=2$ Supergravity solution and the $SO(4,2)$ KEG are
parameterized respectively by charge variables $\bar{\delta}$ and $\delta$. We
find that there is an interesting enhancement of symmetry, that resembles the
Kerr-Newman black hole solution, as all four charges are identical.

### Love Scaling

In order to recover the $SL(2,R)\times U(1)$ geometry, we again set the charge
parameters as in 3.15, but now rescale the original quantities as

$\displaystyle\bar{r}=r\epsilon\,,\qquad\bar{t}=t\epsilon^{-1}\,,\qquad\bar{m}=m\,\epsilon\,,$
(3.22)
$\displaystyle\bar{a}=a\,\epsilon\,,\qquad\sinh\bar{\delta}=-\sinh\bar{\delta_{0}}\equiv\epsilon^{-1/2}\sinh{\delta}.$

We recover the Love geometry, (2.1) with (2.3), in the limit
$\epsilon\rightarrow 0$ upon setting $\sinh^{4}{\delta}=r_{+}/(r_{+}-r_{-})$.
The scalar fields become

$\displaystyle\chi_{1}=-\chi_{2}=\chi_{3}=-\frac{a}{m}\cos\theta\,,$ (3.23)
$\displaystyle
e^{\varphi_{1}}=e^{\varphi_{2}}=e^{\varphi_{3}}=\frac{m}{\sqrt{m^{2}-a^{2}\cos^{2}\theta}}\,.$
(3.24)

Finally, the gauge potentials are given by

$\displaystyle
A_{(1)2}^{Love}=-(\mathcal{A}_{(1)}^{2})^{Love}=\frac{m(m-r)\hskip
2.84544pt}{2\sinh^{2}\delta\,(m^{2}-a^{2}\cos^{2}\theta)}\,dt-\frac{2am^{2}\,\sinh^{2}\delta\sin^{2}\theta}{(m^{2}-a^{2}\cos^{2}\theta)}\,d\phi$
(3.25)

and

$\displaystyle
A_{(1)1}^{Love}=(\mathcal{A}_{(1)}^{1})^{Love}=-\frac{a(m-r)\cos\theta\hskip
2.84544pt}{2\sinh^{2}\delta\,(m^{2}-a^{2}\cos^{2}\theta)}\,dt-\frac{2m(m^{2}-a^{2})\,\sinh^{2}\delta\cos\theta}{(m^{2}-a^{2}\cos^{2}\theta)}\,d\phi.$
(3.26)

We have again verified that this configuration solves the equations of motion
that follow from the Lagrangian (3.1).

We can now write the electric charges

$\displaystyle Q=-\mathcal{Q}=\frac{1}{2}m\,\sinh^{2}\delta\,.$ (3.27)

An important point to notice here is that the dilatons and axions have a
$\theta$-dependence. Interestingly, as for the subtracted KEG in [10], the
gauge fields supporting such KEG geometries vary as a function of radial
distance at infinity. This fact will become relevant in the definitions of the
asymptotic charges.

Remarkably, we have successfully recovered the previously established scaling
limit (3.2) when the spin parameter vanishes – it is static, and the spin is
set to zero $a=0$. This not only underscores the robustness of our rescaling
but also offers valuable insights into the behavior of the functions under
specific scaling conditions.

##  4 Monodromies and Internal Structure

If one is interested in Kerr black holes far from extremality, an interesting
option to consider are KEGs. All three KEGs in this paper are contained in the
one-parameter family generating local symmetries in the solution space of low
frequency massless field perturbations in the general Kerr geometry [6]. These
possess an $SL(2,R)\times SL(2,R)$ symmetry, except at two special points
corresponding to a reduction of the algebra to an $SL(2,R)$ symmetry of the
Schwarzschild background (with $a=0$). One corresponding to the KEG with
$SL(2,R)\times U(1)$ hidden symmetry in Section 2.3 and $SO(4,2)$ in Section
2.4. A detailed analysis of the wave equation generated by the KEG in Section
2.2 can be found in [13]. The radial second order differential equation
reduces to

$\displaystyle\Bigg{[}\partial_{r}\left(\Delta_{r}\partial_{r}\right)+\frac{\left(2M\omega
r_{+}-am\right)^{2}}{(r-r_{+})(r_{+}-r_{-})}-\frac{\left(2M\omega
r_{-}-am\right)^{2}}{(r-r_{-})(r_{+}-r_{-})}-\hat{K}_{\ell,s}\Bigg{]}\hat{R}_{s}=0$
(4.1)

where the $\hat{K}_{\ell,s}$ is the separation constant. Unlike for Kerr with
an irregular singular point at $r=\infty$, the ODE has three regular singular
points located at $r=(r_{+},r_{-},\infty)$, and can be solved by
hypergeometric functions. The corresponding monodromy eigenvalues at each
horizon are defined as

$\displaystyle\alpha_{\pm}=\frac{\omega-\Omega_{\pm}m}{2(\pm\kappa_{\pm})}$
(4.2)

where $\kappa_{\pm}=\frac{r_{+}-r_{-}}{4Mr_{\pm}}$ is the surface gravity and
$\Omega_{\pm}=\frac{a}{2Mr_{\pm}}$ the angular velocity at the inner ($-$) and
outer ($+$) black hole horizons. In contrast, one can show that the wave
equations for the other KEGs with $SL(2,R)\times U(1)$ hidden symmetry and
$SO(4,2)$ reduce to

$\displaystyle\Bigg{[}\partial_{r}\left(\Delta_{r}\partial_{r}\right)+\frac{\left(2M\omega
r_{+}-am\right)^{2}}{(r-r_{+})(r_{+}-r_{-})}-\frac{\left(2Mk\omega
r_{+}-am\right)^{2}}{(r-r_{-})(r_{+}-r_{-})}-\hat{K}_{\ell,s}\Bigg{]}\hat{R}_{s}=0$
(4.3)

with $s=0$, separation constant $\hat{K}_{\ell,s}=\ell(1+\ell)$ and a
deformation parameter $k=\pm 1$ 111Alternatively, as shown in [6], one can set
$k=r_{-}/r_{+}$ such that the wave equation reduces to (4.1). The $k$
deformation is such that the linearized equation of motion simplifies to the
Schwarzschild equation as $\omega\rightarrow 0$, as demonstrated in [6].
Specifically, the case $k=+1$ corresponds to the KEG with $SO(4,2)$ symmetry,
while $k=-1$ corresponds to the KEG with hidden symmetry $SL(2,R)\times U(1)$.
While the location of the regular singular points remain invariant in
comparison to (4.1) the monodromies become

$\displaystyle\alpha^{k=\pm
1}_{+}=\frac{\omega-\Omega_{+}m}{2(+\kappa_{+})}\,,\qquad\alpha^{k=\pm
1}_{-}=\frac{k\omega-\Omega_{+}m}{2(-\kappa_{+})}$ (4.4)

Shifting the position of the monodromies in the black hole interior may a
priori not seem problematic. However, as shown in [9] the dynamical tidal
responses for Kerr black holes will capture the low frequency responses when
considering (4.1). The internal structure of the KEGs is in fact relevant to
invariant quantities characterizing the black holes.

Let us conclude this section with some additional observations. Capping the
inner black hole horizon region leading to (4.3) does not alter the
thermodynamics of the outer horizon but will impact the interpretation of the
inner black hole mechanics as explored in [13] as well as the product of areas
property $A_{+}A_{-}\in\mathbb{Z}$ depend only on the quantized charges (See
[12] and references therein).

In this context as well, the internal alterations in KEGs with $SL(2,R)\times
U(1)$ and $SO(4,2)$ hidden symmetry respect to Kerr will impact the “first
law” for the inner Cauchy horizons of black holes [13]. The areas of the inner
Cauchy black hole horizon at $r=r_{-}$ for KEGs are

$\displaystyle A_{-}^{SL(2,R)\times SL(2,R)}=\pi(r_{-}^{2}+a^{2})\equiv
A_{-}^{Kerr}\,,$ (4.5) $\displaystyle A_{-}^{SO(4,2)}=-A_{-}^{SL(2,R)\times
U(1)}=\pi(r_{+}^{2}+a^{2})\equiv A_{+}^{Kerr}\,.$ (4.6)

While the inner horizon area remains invariant in the first case respect to
Kerr, for the latter two cases there is a shift. Finally, we note that the
monodromy matrices $M_{\gamma}$ characterized by the monodromy eigenvalues
around the $r=\gamma$ singular point must obey the global trivial identity

$\displaystyle M_{r_{+}}M_{r_{-}}M_{\infty}=1\,.$ (4.7)

Hence, this relation may explain why the inner horizon data for each KEGs
appears in black hole scattering computations. Modifying the inner information
of the black hole, as in the Starobinsky and Love KEG, will result in a
corresponding modification in the scattering data.

##  5 Quasi-normal modes

In this section we will explicitly compute the quasinormal mode (QNM)
frequencies for the KEG. Following [21], we will argue that only the
$SL(2,R)\times SL(2,R)$ KEG in [1, 22] is continuously connected to Kerr black
holes.

Each KEG has a distinctive spectrum, characterized by the QNM frequencies that
represent the scattering resonances of the black-hole spacetime. The QNMs
correspond to specific boundary conditions where the waves are purely outgoing
at infinity and purely ingoing at the horizon. They thus correspond to poles
of the transmission and reflection amplitudes.

Our starting point is the radial equation (4.3). By making the following
change of coordinates, and redefinition of the functions

$\displaystyle
z=\frac{r-r_{+}}{r-r_{-}}\,,\qquad\hat{R}_{s}(r)=(r-r_{-})^{p}\,(r-r_{+})^{q}\,w\,,$
(5.1)

we can bring the equation to the form

$\displaystyle
z\,(1-z)\,\,\frac{d^{2}w}{dz^{2}}+[\mathfrak{c}-(\mathfrak{a}+\mathfrak{b}+1)\,z]\,\,\frac{dw}{dz}-\mathfrak{a}\mathfrak{b}\,\,w=0\,$
(5.2)

where we choose

$\displaystyle p=i\,\alpha_{+}-(1+\ell)\,,\qquad q=-i\alpha_{+}\,,$
$\displaystyle\mathfrak{a}=1+\ell+i(\alpha_{+}-\alpha_{-}),\qquad\mathfrak{b}=1+\ell-i(\alpha_{+}+\alpha_{-}),\qquad\mathfrak{c}=1+2i\alpha_{+}\,.$
(5.3)

A solution which is ingoing on the future horizon must be regular at the
horizon $z=0$, and this implies

$\displaystyle\hat{R}_{s}(z)=c_{1}(1-z)^{-(p+q)}\,z^{q}\,F[\mathfrak{a},\mathfrak{b},\mathfrak{c};z]\,.$
(5.4)

where $c_{1}$ is a normalization constant. We can then analyze the behavior at
large distances, for $z\rightarrow 1$

$\displaystyle F[\mathfrak{a},\mathfrak{b},\mathfrak{c};z]$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\mathfrak{c})\Gamma(\mathfrak{c}-\mathfrak{a}-\mathfrak{b})}{\Gamma(\mathfrak{c}-\mathfrak{a})\Gamma(\mathfrak{c}-\mathfrak{b})}F[\mathfrak{a},\mathfrak{b},\mathfrak{a}-\mathfrak{b}-\mathfrak{c}+1;1-z]$
$\displaystyle+(1-z)^{\mathfrak{c}-\mathfrak{a}-\mathfrak{b}}\,\frac{\Gamma(\mathfrak{c})\Gamma(\mathfrak{a}+\mathfrak{b}-\mathfrak{c})}{\Gamma(\mathfrak{a})\Gamma(\mathfrak{b})}F[\mathfrak{c}-\mathfrak{a},\mathfrak{c}-\mathfrak{b},\mathfrak{c}-\mathfrak{a}-\mathfrak{b}+1;1-z]\,.$

This vanishing of the the solution at the boundary, $z=1$ will only be
possible for a discrete set of complex frequencies $\omega$ called QNM
frequencies. We therefore must set

$\displaystyle\mathfrak{a}=-N_{L}\,,\qquad\mathfrak{b}=-N_{R}$ (5.5)

where $N_{L,R}=0,1,...$. This gives the following remarkably simple formulae
for the frequencies of the QNMs

$\displaystyle\omega\,(1-k)$ $\displaystyle=$
$\displaystyle-i\,4\pi\,T_{+}(1+\ell+N_{L})\,,$ $\displaystyle\omega\,(1+k)$
$\displaystyle=$ $\displaystyle-i\,4\pi\,T_{+}(1+\ell+N_{R})+2m\Omega_{+}$
(5.6)

As for the Kerr black hole, there exists a discrete spectrum of QNMs. These
modes consist of two branches: one is overdamped and the other is underdamped,
both exhibiting rotational splitting (for $k\neq\pm 1$). In the KEGs of
interest to this paper, we argue that both damped frequencies only exist in
the $SL(2,R)\times SL(2,R)$ KEG case for $k=r_{-}/r_{+}$. This is exactly what
was found ten years ago by inspecting the QNMs in a curved background
characterized by an $SL(2,R)\times SL(2,R)$ symmetry in [22]. It was argued
that the near-horizon geometry of Kerr black holes is accurately represented
by the KEG solution, characterized by the subtracted geometry. In fact, the
family of modes given by the second family of modes in (5) have identical
frequencies to those of [23] where the resonances take the form:

$\displaystyle\omega=-2i\pi T_{+}(l+1+n)+m\,\Omega_{+}\,,$ (5.7)

in the $Im(\omega)\ll Re(\omega)\ll 1/M$ regime. This spectrum requires a Kerr
black hole in the slow rotation, near-extremal limit $\sqrt{M^{2}-a^{2}}\ll
a\ll M$ and $m>0$.

The first (overdamped) branch of (5) corresponds to negative imaginary
frequencies with absolute values significantly larger than those of the second
set that was not observed in [23]. Let us conclude this section with some
additional comments for the deformation $k=\pm 1$ associated with the Love and
Starobinsky KEG models. One could contemplate these scenarios and notice that
unlike the KEG with hidden $SL(2,R)\times SL(2,R)$ symmetry, these cases
feature only one branch of QNMs in each case. According to Keshet and Neitzke
[24], a CFT description is expected to emerge in the regime of large damping,
where the transmission and reflection amplitudes adopt a familiar CFT-like
form. It is interesting to note that for $k=+1$ in the KEG with $SO(4,2)$
hidden symmetry, there is only one damped frequency branch, which precisely
matches the expression for Kerr given by (5.7). Finally, we observe that when
$k=-1$, the absence of overdamped resonance may suggest that the KEG with
$SL(2,R)\times U(1)$ symmetry is not linked to a Kerr quasinormal spectrum and
might not be relevant in the context of a dual CFT description.

##  6 Separability in KEG

The rationale for the separability of the Teukolsky equation (2.6) in the Kerr
background has been understood for a considerable time now, and it is
attributed to the presence of a nontrivial Exact Killing-Stäckel Tensor (EKST)
$K_{\nu\rho}$ satisfying

$\displaystyle\nabla_{(\mu}K_{\nu\rho)}=0\;.$ (6.1)

The properties of Kerr do not readily extend to the broader class of KEGs.
Although the Kerr geometry has a nontrivial EKST satisfying (6.1) [25], other
geometries within the class we are examining will only exhibit a Conformal
Killing-Stäckel Tensor (CKST), which satisfies

$\displaystyle\nabla_{\mu}M_{\nu\rho}=g_{(\mu\nu}N_{\rho)}$ (6.2)

with nonzero $N_{\rho}$. See e.g. [1] for a review on the classification of
Killing tensors. In these geometries, $T^{\mu}T^{\nu}M_{\mu\nu}$ is only
conserved along affinely parameterized null geodesics. The existence of a
nontrivial CKST ensures separability of the massless scalar wave equation or
the Hamilton-Jacobi equation. In contrast, the separability of the massive
equation is only guaranteed by an exact Killing-Stäckel tensor solving (6.1).
We will see that in certain cases such a tensor can be found for KEGs.

The CKST for the KEG metric (2.1) satisfy the condition (6.2) with the tensors

$\displaystyle P^{\mu\nu}(\theta)\partial_{\mu}\partial_{\nu}$
$\displaystyle=\partial_{\theta}^{2}+\frac{1}{\sin^{2}\theta}\partial_{\phi}^{2}-\varrho_{\theta}\partial_{t}^{2}+2a\partial_{t}\partial_{\phi}$
(6.3) $\displaystyle S^{\mu\nu}(r)\partial_{\mu}\partial_{\nu}$
$\displaystyle=\Delta\partial^{2}_{r}-\varrho_{R}\partial_{t}^{2}-\frac{1}{\Delta}(\mathcal{A}_{red}\partial_{t}+a\partial_{\phi})^{2}-2a\partial_{t}\partial_{\phi}.$
(6.4)

where $\mathcal{A}_{red}=A_{0}\,G/(a\sin^{2}\theta)$ and

$\displaystyle\varrho\equiv\frac{\Delta_{0}-\mathcal{A}_{red}^{2}}{G_{0}}=\varrho_{\theta}(\theta)+\varrho_{R}(r)$
(6.5)

Thus $P^{\mu\nu}$ and $S^{\mu\nu}$ are CKSTs with inhomogeneous terms
respectively given by

$V^{\mu}=\frac{\partial_{\theta}(\Delta_{0}^{1/2})}{\Delta_{0}^{1/2}}\delta^{\mu}_{\theta},\qquad
U^{\mu}=\frac{\partial_{r}(\Delta_{0}^{1/2})\Delta}{\Delta_{0}^{1/2}}\delta^{\mu}_{r}.$
(6.6)

The CKSTs constructed can in certain cases be promoted to EKST satisfying
(6.1).

$\displaystyle
K_{\mu\nu}(r,\theta)=\frac{1}{\Delta_{0}^{1/2}}[h(r)P_{\mu\nu}(\theta)-f(\theta)S_{\mu\nu}(r)]$
(6.7)

where the functions $h,f$ are defined by

$\displaystyle\Delta_{0}(r,\theta)^{1/2}=f(\theta)+h(r)$ (6.8)

One of the KEGs of interest was constructed in [1], by demanding that they
promote approximate $SL(2,R)\times SL(2,R)$ near horizon symmetries to exact
symmetries. Despite the enhanced symmetry, as shown in [25], the CKSTs of
these geometries cannot be promoted to EKSTs.

Interestingly, we find that for other KEGs, such as the one exhibiting an
$SO(4,2)$ or $SL(2,R)\times U(1)$ hidden symmetry, one can promote the CKST to
EKST. Collecting formulae, we finally find the EKST for these cases:

$\displaystyle K_{SL(2,R)\times U(1)}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{[h(r)P_{\mu\nu}(\theta)-f(\theta)S_{\mu\nu}(r)]}{(r_{+}^{2}+a^{2})(1+\beta^{2}\Omega^{2}\sin^{2}\theta)^{1/2}}\,,$
(6.9)

where

$\displaystyle
f(\theta)+h(r)=(r_{+}^{2}+a^{2})(1+\beta^{2}\Omega^{2}\sin^{2}\theta)^{1/2}\,\qquad\varrho_{\theta}+\varrho_{R}=(r_{+}^{2}+a^{2})^{2}\beta^{2}\Omega^{2}/a^{2}$
(6.10)

and

$\displaystyle K_{SO(4,2)}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{[h(r)P_{\mu\nu}(\theta)-f(\theta)S_{\mu\nu}(r)]}{(r_{+}^{2}+a^{2})}\,.$
(6.11)

where

$\displaystyle
f(\theta)+h(r)=(r_{+}^{2}+a^{2})\,\qquad\varrho_{\theta}+\varrho_{R}=0$ (6.12)

These expressions can be recast in many equivalent ways by freely choose the
separation constants between the $f,h$ and $\varrho_{\theta},\varrho_{R}$.

The existence of EKST suggests that both of these geometries satisfy the
strongest form of separability of the Teukolsky equation, known as quantum
separability. This refers to the situation where

$\displaystyle[\nabla_{\mu}K_{\mu\nu}\nabla_{\nu},g_{\lambda\sigma}\nabla_{\lambda}\nabla_{\sigma}]=0$
(6.13)

This condition does not differentiate between massive and massless Klein-
Gordon equation, and it evidently implies the separability of the geodesic
equations.

##  7 Discussion

We have identified a matter configuration that supports these geometries
through a rescaling limit. Consequently, we argue that the effective field
theory description of Kerr black holes is encoded within the black holes of
the STU model rather than vacuum General Relativity. The scaling solution
generating technique that we detailed could be used for a wider class of
stationary solutions, beyond the KEG solutions that we used. While the
application of this method was known for the subtracted geometries [1], two
new examples, the Love and Starobinsky KEGs, were obtained via this method.
Under the assumption of static configurations, we have utilized the Harrison
transformations to embed the static KEG within the supergravity theories.

Our quantitative analysis compares physical quantities, internal structures,
and geodesic equations across different effective Kerr spacetimes. While their
thermodynamic properties align with those of Kerr, including entropy
calculations, a closer examination of their internal structures revealed
substantial differences. These differences point to distinct microscopic
configurations and dynamics within these spacetimes, despite their macroscopic
similarities.

We have given the quasinormal spectra of the KEGs, which provides further
insight into the behavior of these spacetimes under linear scalar
perturbations. We showed that the fundamental resonances can be expressed in
terms of the black hole temperature and angular velocity. We noted the
property that the generic KEGs are not isospectral to Kerr. This analysis
helps to further contrast the scope of the various KEGs as a Kerr black hole.
In this context, one interesting aspect that we will leave for future
exploration is study of the stability of the KEGs, presenting an intriguing
avenue of research.

We discussed the separability of the massless Klein–Gordon equations for KEGs
of interest, those with $SO(4,2)$ or $SL(2,R)\times U(1)$ hidden symmetry, but
a comprehensive examination of all KEGs remains to be done. We showed that
these KEGs admit an Exact Killing-Stäckel Tensor. The existence of this tensor
indicates that both of these geometries satisfy the strongest form of
separability of the Teukolsky equation, known as quantum separability. We can
further deduce that the Klein-Gordon equation, for both massive and massless
cases, is separable, and implies the separability of the geodesic equations.

We studied the thermodynamical properties of a class of KEGs on the black hole
horizons. These geometries have very different asymptotic structure compared
to the asymptotically flat and asymptotically AdS case. The energy density of
such metrics typically falls off as inverse of the radial distance and thus
the geometry cannot have a finite total energy. Within the KEG family, only
for the $SL(2,R)\times SL(2,R)$ metric explicit expressions for the mass and
angular momentum are known [11]. For generic KEGs the corresponding asymptotic
charges, the mass and angular momentum have not been explored in detail. New
insights there would clarify the first law of thermodynamics both at the outer
and inner horizon and that the Smarr formula holds.

Acknowledgements

We would like to thank Austin Joyce, Malcolm Perry, Chris Pope, Luca Santoni,
Adam Solomon and Chiara Toldo for discussions. We gratefully thank the
Mitchell Family Foundation at Cook’s Branch workshop and the Centro de
Ciencias de Benasque Pedro Pascual for their warm hospitality. MC is supported
by the by DOE Award (HEP) DE-SC0013528, the Simons Foundation Collaboration
grant $\\#$724069, Slovenian Research Agency (ARRS No. P1-0306) and Fay R. and
Eugene L. Langberg Endowed Chair funds. NHR was supported by RYC-2016-21159
and CNS2022-135880. MJR was partially supported through the NSF grant
PHY-2309270, CNS2022-135880, CEX2020-001007-S and PID2021-123017NB-I00, funded
by MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe. OV is
supported by the NSF grant 2310223 and, partially, by CEX2020-001007-S and
PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by ERDF A
way of making Europe.

##  Appendix A STU black holes and subtracted geometries

###  A.1 Asymptotically flat black hole solution

Below is a summarized description of the general scaling procedure for this
type of effective Kerr solutions, as described in [15, 16]. For a detailed
explanation, which includes the reduction of the Kerr metric to 3 dimensions
and its subsequent conversion back to 4 dimensions using the Kaluza-Klein
reduction rules, please refer to the mentioned article. After this process,
the resulting metric for the 4-dimensional rotating black hole with 4 charges
is

$ds_{4}^{2}=-\frac{\rho^{2}-2mr}{W}(dt+\mathcal{B}_{(1)})^{2}+W\left(\frac{dr^{2}}{\Delta}+d\theta^{2}+\frac{\Delta\sin^{2}\theta
d\phi^{2}}{\rho^{2}-2mr}\right),$ (A.1)

where $B_{(1)}$ is the Kaluza-Klein vector from the reduction and $\rho$ and
$\Delta$ are the original functions from the Kerr metric,

$\displaystyle B_{(1)}=\hskip 2.84544pt$
$\displaystyle\frac{2m(a^{2}-u^{2})(rc_{123}-(r-2m)s_{1234})}{a(\rho^{2}-2mr)}\text{d}\phi$
$\displaystyle\rho^{2}\hskip 2.84544pt=$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta\hskip 56.9055pt\Delta\hskip
2.84544pt=r^{2}-2mr+a^{2},$ (A.2)

and the remaining elements are defined as

$\displaystyle W^{2}=\hskip 2.84544pt$ $\displaystyle r_{1}\hskip
2.84544ptr_{2}\hskip 2.84544ptr_{3}\hskip
2.84544ptr_{4}+u^{4}+u^{2}\\{2r^{2}+2mr\hskip
2.84544pt(s_{1}^{2}+s_{2}^{2}+s_{3}^{2}+s_{4}^{2})$
$\displaystyle+8m^{2}\hskip 2.84544ptc_{1234}\hskip
2.84544pts_{1234}-4m^{2}(s_{123}^{2}+s_{124}^{2}+s_{134}^{2}+s_{234}^{2}+2s_{1234}^{2}\\}$
$\displaystyle s_{i_{1}\cdot i_{n}}=\hskip 2.84544pt$
$\displaystyle\sinh{\delta_{i_{1}}}\cdot\sinh{\delta_{i_{n}}}\hskip
28.45274ptc_{i_{1}\cdot i_{n}}=\hskip
2.84544pt\cosh{\delta_{i_{1}}}\cdot\cosh{\delta_{i_{n}}}\hskip
28.45274ptu=a\hskip 2.84544pt\cos\theta,$ (A.3)

with $\delta_{i}$ being the charge parameter.

In this context, the 4-dimensional solution for 4 charges consists of 4 gauge
potentials, with 2 of them associated with the electric charges,

$\displaystyle A_{12}=\frac{2m}{aW^{2}}\\{($ $\displaystyle r_{1}\hskip
2.84544ptr_{3}\hskip 2.84544ptr_{4}+ru^{2})\left[c_{2}\hskip
2.84544pts_{2}\hskip 2.84544pta{dt}-(a^{2}-u^{2})(c_{134}\hskip
2.84544pts_{2}-s_{134}\hskip 2.84544ptc_{2})\hskip 2.84544ptd\phi\right]$
$\displaystyle+2mu^{2}\left[e_{2}\hskip 2.84544ptadt-(a^{2}-u^{2})\hskip
2.84544pts_{134}\hskip 2.84544ptc_{2}\hskip 2.84544ptd\phi\right]\\}$
$\displaystyle\mathcal{A}_{1}^{2}=\frac{2m}{aW^{2}}\\{($ $\displaystyle
r_{1}\hskip 2.84544ptr_{2}\hskip 2.84544ptr_{3}+ru^{2})\left[c_{4}\hskip
2.84544pts_{4}\hskip 2.84544ptadt-(a^{2}-u^{2})(c_{123}\hskip
2.84544pts_{4}-s_{123}\hskip 2.84544ptc_{4})\hskip 2.84544ptd\phi\right]$
$\displaystyle+2mu^{2}\left[e_{4}\hskip
2.84544ptadt-(a^{2}-u^{2})s_{123}\hskip 2.84544ptc_{4}\hskip
2.84544ptd\phi\right]\\},$ (A.4)

and the other 2, analytically more complex, associated with the magnetic
charges,

$\displaystyle A_{(1)1}=\frac{2mu}{aW^{2}}\\{($ $\displaystyle r\hskip
2.84544ptr_{1}+u^{2})\left[(c_{234}\hskip 2.84544pts_{1}-s_{234}\hskip
2.84544ptc_{1})\hskip 2.84544pta\text{dt}-c_{1}\hskip 2.84544pts_{1}\hskip
2.84544pta^{2}\text{d}\phi\right]+2m\hskip 2.84544ptr_{1}\hskip
2.84544pts_{234}\hskip 2.84544ptc_{1}\hskip 2.84544pta\text{dt}$
$\displaystyle-(c_{1}\hskip 2.84544pts_{1}\hskip 2.84544pt(r_{1}\hskip
2.84544ptr_{2}\hskip 2.84544ptr_{3}\hskip
2.84544ptr_{4}+u^{2}\left[r^{2}+2mr\hskip
2.84544pt(s_{2}^{2}+s_{3}^{2}+s_{4}^{2})-4m^{2}\hskip
2.84544pts_{234}^{2}\right])$ $\displaystyle+4m^{2}u^{2}\hskip
2.84544ptc_{234}\hskip 2.84544pts_{234}\hskip 2.84544pts_{1}^{2}+2m\hskip
2.84544pte_{1}\hskip 2.84544pt(a^{2}\hskip 2.84544ptr_{1}-r\hskip
2.84544ptu^{2}))\text{d}\phi\\},$
$\displaystyle\mathcal{A}_{(1)1}=\frac{2mu}{aW^{2}}\\{($ $\displaystyle
r\hskip 2.84544ptr_{3}+u^{2})\left[(c_{124}\hskip 2.84544pts_{3}-s_{124}\hskip
2.84544ptc_{3})\hskip 2.84544pta\text{dt}-c_{3}\hskip 2.84544pts_{3}\hskip
2.84544pta^{2}\text{d}\phi\right]+2m\hskip 2.84544ptr_{3}\hskip
2.84544pts_{124}\hskip 2.84544ptc_{3}\hskip 2.84544pta\text{dt}$
$\displaystyle-(c_{3}\hskip 2.84544pts_{3}\hskip 2.84544pt(r_{1}\hskip
2.84544ptr_{2}\hskip 2.84544ptr_{3}\hskip
2.84544ptr_{4}+u^{2}\left[r^{2}+2mr\hskip
2.84544pt(s_{1}^{2}+s_{2}^{2}+s_{4}^{2})-4m^{2}\hskip
2.84544pts_{124}^{2}\right])$ $\displaystyle+4m^{2}u^{2}\hskip
2.84544ptc_{124}\hskip 2.84544pts_{124}\hskip 2.84544pts_{3}^{2}+2m\hskip
2.84544pte_{3}\hskip 2.84544pt(a^{2}\hskip 2.84544ptr_{3}-r\hskip
2.84544ptu^{2}))\text{d}\phi\\},$ (A.5)

where $r_{i}=r+2ms_{i}^{2}$ in the solutions shown above. Likewise, the
expressions for the $e_{i}$ are defined as follows,

$\displaystyle e_{1}=$ $\displaystyle c_{234}\hskip
2.84544pts_{234}(c_{1}^{2}+s_{1}^{2})-c_{1}\hskip
2.84544pts_{1}(s_{23}^{2}+s_{24}^{2}+s_{34}^{2}+2s_{234}^{2})$ $\displaystyle
e_{2}=$ $\displaystyle c_{134}\hskip
2.84544pts_{134}(c_{2}^{2}+s_{2}^{2})-c_{2}\hskip
2.84544pts_{2}(s_{13}^{2}+s_{14}^{2}+s_{34}^{2}+2s_{134}^{2})$ $\displaystyle
e_{3}=$ $\displaystyle c_{124}\hskip
2.84544pts_{124}(c_{3}^{2}+s_{3}^{2})-c_{3}\hskip
2.84544pts_{3}(s_{12}^{2}+s_{24}^{2}+s_{14}^{2}+2s_{124}^{2})$ $\displaystyle
e_{4}=$ $\displaystyle c_{123}\hskip
2.84544pts_{123}(c_{4}^{2}+s_{4}^{2})-c_{4}\hskip
2.84544pts_{4}(s_{23}^{2}+s_{12}^{2}+s_{13}^{2}+2s_{123}^{2}).$ (A.6)

On the other hand, The remaining 4-dimensional fields, namely, the three
dilatons $(\varphi_{1},\varphi_{2},\varphi_{3})$ and the three axions
$(\chi_{1},\chi_{2},\chi_{3})$, are determined by the following expressions,

$\displaystyle\chi_{1}=$
$\displaystyle\frac{2mu(c_{13}s_{24}-c_{24}s_{13})}{r_{1}r_{3}+u^{2}},\hskip
28.45274pt\chi_{2}=\frac{2mu(c_{14}s_{23}-c_{23}s_{14})}{r_{2}r_{3}+u^{2}}$
(A.7) $\displaystyle\chi_{3}=$
$\displaystyle\frac{2mu(c_{12}s_{34}-c_{34}s_{12})}{r_{1}r_{2}+u^{2}},\hskip
25.6073pte^{\varphi_{1}}=\frac{r_{1}r_{3}+u^{2}}{W}$ (A.8) $\displaystyle
e^{\varphi_{2}}=$ $\displaystyle\frac{r_{2}r_{3}+u^{2}}{W},\hskip
88.2037pte^{\varphi_{3}}=\frac{r_{1}r_{2}+u^{2}}{W}.$ (A.9)

###  A.2 Scaling procedure for KEG with $SL(2,R)\times SL(2,R)$ symmetry

The results presented above in the limit
$\delta_{1}=\delta_{2}=\delta_{3}=\delta_{4}$ lead to the dyonic Kerr-Newman
solution; however, in this subsection, another arrangement of the charges is
discussed as proposed in [22].

####  A.2.1 Static case

If a symmetry of type $SL(2,R)\times SL(2,R)$ exists, we will experience
variations in the charges, potentials, and parameters used in Section 3. In
this case, we will employ the fields defined in equations (A.1) and (A.1) and
the choice of charge parameters
$\bar{\delta}_{1}=\bar{\delta}_{2}=\bar{\delta}_{3}=\bar{\delta}$ and
$\bar{\delta}_{4}=\bar{\delta}_{0}$. The limit is implemented by means of the
following scalings:

$\displaystyle\bar{r}=r\epsilon,\quad\bar{t}=t{\epsilon^{-1}},\quad\bar{m}=m\epsilon\,,\
$ (A.10) $\displaystyle 2\bar{m}\sinh^{2}\bar{\delta}\equiv
Q={2m}{\epsilon^{-1/3}}(\Pi_{c}^{2}-\Pi_{s}^{2})^{1/3},\quad\sinh^{2}\bar{\delta}_{0}=\frac{\Pi_{s}^{2}}{\Pi_{c}^{2}-\Pi_{s}^{2}}\,,\
$ (A.11)

Considering $\epsilon\rightarrow 0$, the gauge potentials and gauge fields
related to the metric take the following form,

$\displaystyle\chi_{1}=\chi_{2}=\chi_{3}=0,\ \
e^{\varphi_{1}}=e^{\varphi_{2}}=e^{\varphi_{3}}=\frac{Q^{2}}{\Delta_{s}^{\frac{1}{2}}},$
(A.12) $\displaystyle A=-\frac{r}{Q}\,dt,\ \ {\cal
A}=\frac{Q^{3}(2m)\Pi_{c}\Pi_{s}}{(\Pi_{c}^{2}-\Pi_{s}^{2})\Delta_{s}}\,dt,$
(A.13)

where the corresponding field strengths are

$\displaystyle F_{t\,r}=\frac{1}{Q},\quad{\cal
F}_{t\,r}=\frac{Q^{3}(2m)^{4}\Pi_{c}\Pi_{s}}{\Delta_{s}^{2}}\,.$ (A.14)

where

$\Delta_{0s}\to\Delta_{s}=(2m)^{3}r(\Pi_{c}^{2}-\Pi_{s}^{2})+(2m)^{4}\Pi^{2}_{s}\
.$ (A.15)

It is important to note that the static case has no axions.

####  A.2.2 Rotating case

Now let us perform the same calculation taking into account rotation, which
means that the spin parameter is not zero, i.e., $a\neq 0$. Under these
assumptions, the result obtained for the scalar fields associated with the
metric is,

$\displaystyle\chi_{1}=\chi_{2}=\chi_{3}=-\frac{2ma(\Pi_{c}-\Pi_{s})\cos{\theta}}{Q^{2}},$
$\displaystyle
e^{\varphi_{1}}=e^{\varphi_{2}}=e^{\varphi_{3}}=\frac{Q^{2}}{\Delta^{1/2}},$

where the appearance of axions, which are nonexistent in the static solution,
is highlighted. On the other hand, the gauge fields of this solution
correspond to

$\displaystyle
A=-\frac{r}{Q}dt+\frac{(2m)^{2}a^{2}[2m\Pi_{s}^{2}-r(\Pi_{c}-\Pi_{s})^{2}]\cos^{2}\theta}{Q\Delta}dt$
$\displaystyle-\frac{2m\,a(\Pi_{c}-\Pi_{s})\sin^{2}\theta}{Q}\left(1+\frac{(2m)^{2}a^{2}(\Pi_{c}-\Pi_{s})^{2}\cos^{2}\theta}{\Delta}\right)d\phi\,,$
(A.16) $\displaystyle{\cal
A}=\frac{Q^{3}[(2m)^{2}\Pi_{c}\Pi_{s}+a^{2}(\Pi_{c}-\Pi_{s})^{2}\cos^{2}\theta]}{2m(\Pi_{c}^{2}-\Pi_{s}^{2})\Delta}\,dt\,+\frac{Q^{3}2m\,a(\Pi_{c}-\Pi_{s})\sin^{2}\theta}{\Delta}\,d\phi.$

##  Appendix B Harrison Transformation

Given a solution of four-dimensional Einstein-Maxwell dilation theory of the
form

$ds^{2}=-e^{2U}dt^{2}+e^{-2U}\gamma_{ij}dx^{i}dx^{j},\hskip
56.9055ptF_{i0}=\partial_{i}\Psi,\hskip 28.45274pte^{-2\varphi}$ (B.1)

We can derive an effective three-dimensional action by performing a
dimensional reduction along a timelike Killing vector instead of a spacelike
one. This approach allows us to define

$P=e^{(\alpha-1)U}e^{-(\alpha+1)\varphi}\left(\begin{array}[]{cc}e^{2(U+\varphi\alpha)}-(1+\alpha^{2})\Psi^{2}&-\sqrt{1+\alpha^{2}}\Psi\\\
\sqrt{1+\alpha^{2}}\Psi&-1\end{array}\right)$ (B.2)

and we can express the Lagrangian density, which is invariant under
$GL(2;\mathbb{R})$, in the following form

$\mathcal{L}_{3}=\mathcal{R}+\frac{1}{1+\alpha^{2}}\gamma_{ij}\textrm{Tr}(\partial_{i}P\partial_{j}P^{-1}).$
(B.3)

By applying a specific element of SO(1; 1), corresponding to a Harrison
transformation

$H=\left(\begin{array}[]{cc}1&0\\\ \beta&1\end{array}\right)$ (B.4)

on $P$, we can find $P^{{}^{\prime}}=HPH^{-1}$ with new potentials, and hence
a new four-dimensional solution

$ds^{2}_{new}=-e^{2U^{\prime}}dt^{2}+e^{-2U^{\prime}}\gamma_{ij}dx^{i}dx^{j},\hskip
56.9055ptF_{i0}^{{}^{\prime}}=\partial_{i}\Psi^{{}^{\prime}},\hskip
28.45274pte^{-2\varphi^{\prime}},$ (B.5)

where

$\displaystyle e^{2U^{\prime}}$
$\displaystyle=\Lambda^{\frac{-2}{(1+\alpha^{2}}}e^{2U},\hskip
34.14322pte^{-2\varphi^{\prime}}=\Lambda^{\frac{-2\alpha}{(1+\alpha^{2}}}e^{-2\varphi}$
(B.6) $\displaystyle\Psi^{{}^{\prime}}$
$\displaystyle=\Lambda^{-1}\left(\Psi+\frac{\beta(e^{2(U+\alpha\varphi)}-(1+\alpha^{2})\Psi^{2})}{\sqrt{1+\alpha^{2}}}\right)$
(B.7) $\displaystyle\Lambda$
$\displaystyle=(\beta\psi+1)^{2}-\beta^{2}e^{2}(U+\alpha\varphi).$ (B.8)

If we set $\alpha=0$, we obtain the expected result for the Love and
Starobinsky metrics in the spinless limit, $a\rightarrow 0$. The initial
solution is

$\textrm{Schwar}=\left\\{\begin{array}[]{ll}e^{2U}&=1-\frac{2M}{r},\hskip
28.45274pt\alpha=0\\\ \varphi&=0,\hskip 56.9055pt\Psi=0\end{array}\right.$
(B.9)

and after the transformation the new solution is characterized by

$\textrm{Schwar
New}=\left\\{\begin{array}[]{ll}e^{2U^{\prime}}&=-\frac{r(2M-r)}{4M^{2}},\hskip
14.22636pt\alpha=0,\hskip 14.22636pt\Lambda=\frac{2M}{r}\\\
\varphi^{{}^{\prime}}&=0,\hskip
50.3615pt\Psi^{{}^{\prime}}=-1+\frac{r}{2M}\end{array}\right.$ (B.10)

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