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# The SOXS Data-Reduction Pipeline

David R. Young  Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University Belfast, Belfast BT7 1NN, UK  Marco Landoni  INAF
– Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807, Merate, Italy
Stephen J. Smartt  Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University Belfast, Belfast BT7 1NN, UK  Sergio Campana  INAF
– Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807, Merate, Italy
Riccardo Claudi  INAF – Osservatorio Astronomico di Padova, Vicolo
dell’Osservatorio 5, I-35122, Padua, Italy  Pietro Schipani  INAF –
Osservatorio Astronomico di Capodimonte, Sal. Moiariello 16, I-80131, Naples,
Italy  Matteo Aliverti  INAF – Osservatorio Astronomico di Brera, Via Bianchi
46, I-23807, Merate, Italy  Andrea Baruffolo  INAF – Osservatorio Astronomico
di Padova, Vicolo dell’Osservatorio 5, I-35122, Padua, Italy  Sagi Ben-Ami
Harvard-Smithsonian Center for Astrophysics, Cambridge, USA  Federico Biondi
Astrophysics Research Centre, School of Mathematics and Physics, Queen’s
University Belfast, Belfast BT7 1NN, UK  Max-Planck-Institut für
Extraterrestrische Physik, Giessenbachstr. 1, D-85748 Garching, Germany
Giulio Capasso  INAF – Osservatorio Astronomico di Capodimonte, Sal.
Moiariello 16, I-80131, Naples, Italy  Rosario Cosentino  FGG-INAF, TNG,
Rambla J.A. Fernández Pérez 7, E-38712 Breña Baja (TF), Spain  Francesco
D’Alessio  INAF – Osservatorio Astronomico di Roma, Via Frascati 33, I-00078
M. Porzio Catone, Italy  Paolo D’Avanzo  ESO, Karl Schwarzschild Strasse 2,
D-85748, Garching bei München, Germany  Ofir Hershko  Weizmann Institute of
Science, Herzl St 234, Rehovot, 7610001, Israel  Hanindyo Kuncarayakti
Finnish Centre for Astronomy with ESO (FINCA), FI-20014 University of Turku,
Finland  Tuorla Observatory, Dept. of Physics and Astronomy, FI-20014
University of Turku, Finland  Matteo Munari  INAF – Osservatorio Astrofisico
di Catania, Via S. Sofia 78 30, I-95123 Catania, Italy  Giuliano Pignata
Universidad Andres Bello, Avda. Republica 252, Santiago, Chile  Millennium
Institute of Astrophysics (MAS)  Adam Rubin  Weizmann Institute of Science,
Herzl St 234, Rehovot, 7610001, Israel  Salvatore Scuderi  INAF – Osservatorio
Astrofisico di Catania, Via S. Sofia 78 30, I-95123 Catania, Italy  Fabrizio
Vitali  INAF – Osservatorio Astronomico di Roma, Via Frascati 33, I-00078 M.
Porzio Catone, Italy  Jani Achrén  Incident Angle Oy, Capsiankatu 4 A 29,
FI-20320 Turku, Finland  José Antonio Araiza-Duran  Universidad Andres Bello,
Avda. Republica 252, Santiago, Chile  Millennium Institute of Astrophysics
(MAS)  Iair Arcavi  Tel Aviv University, Department of Astrophysics, 69978 Tel
Aviv, Israel  Anna Brucalassi  ESO, Karl Schwarzschild Strasse 2, D-85748,
Garching bei München, Germany  Rachel Bruch  Weizmann Institute of Science,
Herzl St 234, Rehovot, 7610001, Israel  Enrico Cappellaro  Astrophysics
Research Centre, School of Mathematics and Physics, Queen’s University
Belfast, Belfast BT7 1NN, UK  Mirko Colapietro  INAF – Osservatorio
Astronomico di Capodimonte, Sal. Moiariello 16, I-80131, Naples, Italy
Massimo Della Valle  INAF – Osservatorio Astronomico di Capodimonte, Sal.
Moiariello 16, I-80131, Naples, Italy  Marco De Pascale  INAF – Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122, Padua, Italy
Rosario Di Benedetto  INAF – Osservatorio Astrofisico di Catania, Via S. Sofia
78 30, I-95123 Catania, Italy  Sergio D’Orsi  INAF – Osservatorio Astronomico
di Capodimonte, Sal. Moiariello 16, I-80131, Naples, Italy  Avishay Gal-Yam
Weizmann Institute of Science, Herzl St 234, Rehovot, 7610001, Israel  Matteo
Genoni  INAF – Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807,
Merate, Italy  Marcos Hernandez  FGG-INAF, TNG, Rambla J.A. Fernández Pérez 7,
E-38712 Breña Baja (TF), Spain  Jari Kotilainen  Finnish Centre for Astronomy
with ESO (FINCA), FI-20014 University of Turku, Finland  Tuorla Observatory,
Dept. of Physics and Astronomy, FI-20014 University of Turku, Finland
Gianluca Li Causi  INAF - Istituto di Astrofisica e Planetologia Spaziali,
Rome, Italy  Seppo Mattila  Tuorla Observatory, Dept. of Physics and
Astronomy, FI-20014 University of Turku, Finland  Michael Rappaport  Weizmann
Institute of Science, Herzl St 234, Rehovot, 7610001, Israel  Kalyan
Radhakrishnan  INAF – Osservatorio Astronomico di Padova, Vicolo
dell’Osservatorio 5, I-35122, Padua, Italy  Davide Ricci  INAF – Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122, Padua, Italy
Marco Riva  INAF – Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807,
Merate, Italy  Bernardo Salasnich  INAF – Osservatorio Astronomico di Padova,
Vicolo dell’Osservatorio 5, I-35122, Padua, Italy  Ricardo Zanmar Sanchez
INAF – Osservatorio Astrofisico di Catania, Via S. Sofia 78 30, I-95123
Catania, Italy  Maximilian Stritzinger  Aarhus University, Ny Munkegade 120,
D-8000 Aarhus, Denmark  Hector Ventura  FGG-INAF, TNG, Rambla J.A. Fernández
Pérez 7, E-38712 Breña Baja (TF), Spain

###### Abstract

The Son-Of-X-shooter (SOXS) is a dual arm spectrograph (UV-VIS & NIR) and
Acquisition Camera (AC) due to mounted on the European Southern Observatory
(ESO) 3.6m New Technology Telescope (NTT) in La Silla. Designed to
simultaneously cover the optical and NIR wavelength range from 350-2050 nm,
the instrument will be dedicated to the study of transient and variable events
with many Target of Opportunity requests expected.

The goal of the SOXS Data Reduction pipeline is to use calibration data to
remove all instrument signatures from the SOXS scientific data frames for each
of the supported instrument modes, convert this data into physical units and
deliver them with their associated error bars to the ESO Science Archive
Facility (SAF) as Phase 3 compliant science data products, all within 30
minutes. The primary reduced product will be a detrended, wavelength and flux
calibrated, telluric corrected 1D spectrum with UV-VIS + NIR arms stitched
together. The pipeline will also generate Quality Control (QC) metrics to
monitor telescope, instrument and detector health.

The pipeline is written in Python 3 and has been built with an agile
development philosophy that includes adaptive planning and evolutionary
development. The pipeline is to be used by the SOXS consortium and the general
user community that may want to perform tailored processing of SOXS data. Test
driven development has been used throughout the build using ‘extreme’ mock
data. We aim for the pipeline to be easy to install and extensively and
clearly documented.

###### keywords:

SOXS, Pipeline, Data Reduction, Spectroscopy, Imaging

##  1 INTRODUCTION

The SOXS (Son Of X-Shooter) instrument is a new medium resolution spectrograph
($R\simeq 4500$) capable of simultaneously observing 350-2000nm (U- to H-band)
to a limiting magnitude of R $\sim 20$ (3600sec, S/N $\sim$ 10). It shall be
hosted at the Nasmyth focus of the NTT at La Silla Observatory, Chile (see [1]
for an overview). This paper describes the design of the SOXS data-reduction
pipeline and data-products it shall generate and is part of a series of
contributions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] describing the
current development status of the SOXS subsystems.

Section 2 gives details about the individual instruments that the SOXS
pipeline shall receive data from. Section 3 details the various observation
modes SOXS will be able to operate in and the products the pipeline is
expected to generate for each of those modes. In Section 4 we propose how we
will automate much of the data-reductions for SOXS, followed by how users in
will be able to interact with the pipeline on their own machines in Section 5,
the development environment we are using in Section 6 and finally giving an
overview of the architecture of the pipeline code-base in Section 7.

##  2 Three SOXS Detectors

SOXS will host three instruments; the UV-VIS and NIR spectrographs and the AC.
Figure 1 shows how these instruments are to be mounted on the NTT’s Nasmyth
focus rotator-flange. The SOXS pipeline will be capable of reducing the pixel-
data collected by each of these instruments as well as outputting Quality
Control metrics used to monitor the health of each unit.

---
Figure 1:  A face-on view of SOXS on the NTT nasmyth-focus rotator flange.

###  2.1 UV-VIS Spectrograph

---
Figure 2:  The four quasi-orders of the SOXS UV-VIS spectral format realised
on the UV-VIS detector plain.

The UV-VIS spectrograph [3] employs 4 ion-etched transmission gratings in the
first order ($m=1$) to record spectra in the 350-850nm wavelength range
(providing a 50nm overlap with the NIR arm for cross-calibration). The
spectral band is split into four poly-chromatic channels and sent to their
specific grating. The spectral format of these four quasi-orders will be
linear as shown in Figure 2. The main characteristics of the UV-VIS
spectrograph and the e2V CCD44-82 detector [10] can be found in Table 1.

Table 1: Characteristics of the SOXS UV-VIS Spectrograph & CCD. Detector | e2V CCD44-82
---|---
Pixel-Size | $15\text{\,}\mathrm{\SIUnitSymbolMicro m}$
Array-Size | 2048$\times$4096px; 30.7$\times$61.4mm
Array-Scale | 0.28 arcsec/px
Peak Signal | 200,000 e-/px
Gain |  Slow: 0.6 ± 0.1 e-/ADU Fast: 2 ± 0.2 e-/ADU
Read noise (rms) |  Slow: $<$ 3e- Fast: $<$8 e-
Dark current @ 153K |  $<$0.00001 e-/s/px
Resolution (R) | 3500-7000 ($\simeq$ 4500 mean)
Wavelength Range | 350-850nm
Slit Widths | 0.5, 1.0, 1.5, 5.0 arcsec
Slit Length | 12 arcsec
Grating Blaze Angle | 41°
Orders (quasi) | 4

###  2.2 NIR Spectrograph

The SOXS NIR spectrograph [14] is a cross-dispersed echelle that employs the
‘4C’ (Collimator Correction of Camera Chromatism) to obtain spectra in
800-2000nm wavelength range in 15 orders (providing a 50nm overlap with the
UV-VIS arm for cross-calibration). Unlike the UV-VIS spectrograph the NIR
orders display a curvature as evidenced by the spectral format of the NIR
orders shown in Figure 3.

---
Figure 3:  The 15 orders of the SOXS NIR spectral format realised on the NIR array plain Table 2: Characteristics of the SOXS NIR Spectrograph & Detector Array Detector | Teledyne H2RG
---|---
Pixel-Size | $18\text{\,}\mathrm{\SIUnitSymbolMicro m}$
Array-Size | 2048$\times$2048px
Array-Scale | 0.25 arcsec/px
Read noise (rms) |  Double correlated: $<$20 e- 16 Fowler pairs $<$7 e-
Dark current @ 40K |  $<$0.005 e-/s/px
Resolution (R) |  $\simeq$ 5000 (1 arcsec slit)
Wavelength Range | 800-2000 nm
Slit Widths | 0.5, 1.0, 1.5, 5.0 arcsec
Slit Length | 12 arcsec
Grating Blaze Angle | 44°
Detector Operating Temp | 40K
Spectrograph Operating Temp | 150K
Orders | 15

###  2.3 Acquisition Camera

The Acquisition Camera [8] with a $3.5^{\prime}\times 3.5^{\prime}$ FOV will
also allow for science-grade multi-band photometry.

The primary use of the acquisition camera is to acquire spectral targets and
centre them on the slits. In addition, the camera’s $3.5^{\prime}\times
3.5^{\prime}$ FOV and 0.205 arcsec/px scale will also allow for science-grade
multi-band photometry. Observers will be able to select from 7 filters; the
LSST $u,g,r,i,z,y$ set and Johnson $V$.

Table 3: Characteristics of the SOXS AC. Camera | Andor iKon M934
---|---
Detector | BEX2-DD
Pixel-Size | $13\text{\,}\mathrm{\SIUnitSymbolMicro m}$
Array-Size | 1024$\times$1024; 13.3$\times 13.3$mm
Peak Signal | 130000 e-/px
Dark Current @ 173 K | 0.00012 e-/s/px
Read noise (rms) | 2.9 e-
Filters | _u, g, r, i, z, y, V_

##  3 Observation Modes and Pipeline Data-Products

The challenge faced by any pixel-based data reduction pipeline is to identify
and remove (or minimise) all sources of noise from the raw data so as to be
able to best extract the scientific information contained on the frames.
Alongside the standard detrending stages of calibration needed to remove
instrument signatures (bias, dark-removal and flat-field correcting), the
pipeline will calculate and apply accurate wavelength- and flux-calibration
solutions to the spectra. For observations taken in the standard stare-mode it
must also solve the notoriously difficult problem of removing the unwanted
signal of diffuse light scattered in the earth’s atmosphere (the _sky-
background_). Finally, the pipeline must provided the means to extract the
object spectra from the 2D image frames in a manner that maximises the signal-
to-noise of the data; otherwise known as _optimal-extraction_.

SOXS will be able to operate in 6 different spectroscopic observation modes:

  1. 1.

Stare-mode. Standard ‘point-and-shoot’ observation.

  2. 2.

Stare-mode, synchronised. Standard ‘point-and-shoot’ observations where the
mid-point of the UV-VIS and NIR exposures are matched.

  3. 3.

Nodding-mode. The telescope ‘nods’ between two-positions along the slit
throughout exposure, allowing for on-the-fly sky background removal.

  4. 4.

Fixed sky-offset mode. This mode is for extended objects where not enough
uncontaminated sky-background is seen within the 12′′ slit to allow for
measurement and removal.

  5. 5.

Generic sky-offset mode. User defined pattern of telescope offsets.

  6. 6.

Mapping-mode. Used to ‘map’ an object or location.

Together with these spectroscopic modes, an imaging observation mode will also
be available via the AC. The pipeline will be able to reduced data from this
imaging-mode alongside the first 3 spectroscopic modes in a completely
automated fashion (see Section 4).

Table 4 details each of the final data-products the pipeline shall produce.
These products shall meet ESO’s Phase III data-products standards and are
uploaded and archived on the ESO SAF [15] for dissemination to data owners.

Table 4: Final data-products generated by the SOXS pipeline. Product | Description
---|---
1D Source Spectra |  1D spectra in FITS binary table format, one for each arm. Each FITS spectrum file will contain 4 extensions: 1\. Wavelength- and flux-calibrated spectra with absolute flux correction via scaling to acquisition image source photometry, 2\. an additional spectrum with correction for telluric absorption via MOLECFIT, 3\. the variance array and 4\. the sky-background spectra.
1D Merged Source Spectrum |  1D UV-VIS & NIR merged spectrum in FITS binary table format with PDF visualisation. This spectrum will be rebinned to a common pixel scale for each arm. This spectrum file will also have the same 4 extensions described above.
2D Source Spectra |  A 2D FITS image for each spectral arm containing wavelength and flux calibrated spectra (no other corrections applied) allowing users to perform source extraction with their tool of choice. Note that rectification of the curved orders in the NIR introduces a source of correlated noise not present in extractions performed on the unstraightened orders as done by the pipeline.
Acquisition Camera Images |  _ugrizy_ astrometrically and photometrically (_griz_ only) calibrated to Refcat2 [16]

##  4 Automated Data-Reduction and Data flow

The pipeline is being designed to reduce all SOXS data to ESO Phase III
standards ‘out-of-the-box’ with the aim of reducing most of the SOXS data and
delivering the final-products to the ESO SAF without the need for human-
interaction. The pipeline will run automatically on all point-source targets
above an AB magnitude of $r=19$ (with the stretch goal of $r=20$). Below this
magnitude, the pipeline will _attempt_ to run automatically but it may require
some user interaction to optimise object extraction. The pipeline may also
struggle to provide an optimal absolute flux calibration and/or source
extraction for sources with in crowded fields or with complex backgrounds.

The current design for the flow of data from the telescope to the archive is
as follows:

  1. 1.

Data is acquired by NTT & SOXS on the summit at La Silla, Chile.

  2. 2.

Raw data is transferred within 10 minutes to the ESO SAF (Garching, Germany).

  3. 3.

Raw data is downloaded from the SAF to the SOXS Data Reduction And WEllness
Reporter (DRAWER) cluster in Belfast, UK.

  4. 4.

Data is reduced on the cluster by the SOXS pipeline and streamed back to the
SAF where it can be accessed by data rights owners.

Our goal is to populate the ESO SAF with the fully reduced data products
within 30 min of the raw exposures appearing in the ESO SAF (15 min stretch
goal). This is possible thanks to the fixed format of SOXS (apart from the
exchangeable slit) allowing calibration frames to be prepared ahead-of-time
before science data reductions.

Access to the ‘open stream’ method of shipping reduced data immediately to the
ESO SAF will initially require the ESO Quality Control Group to review and
verify a moderately sized collection of SOXS-pipeline reduced data. Once the
quality and content of the data produced by the pipeline has met with ESO
Phase III standards, we will then be allowed to upload data-products to the
archive without further need of passing through a gatekeeper.

##  5 Pipeline Usage

As stated in Section 4, the SOXS pipeline will be designed to reduce most SOXS
data in a non-interactive, automated fashion. However, users will be able to
download and install the pipeline on their own machines and drive the pipeline
to reduce the data to their own preferences. We foresee that the pipeline
functionality will be accessible via at least two user-interfaces; a Command-
Line Interface (CLI) and via ESO’s Reflex GUI.

The CLI being developed has a syntax similar to ESO’s Common Pipeline Library
(CPL) interface _Esorex_ [17]. Each recipe can be called with an individual
command and takes as input a set-of-files (SOF) list containing the required
file input (raw data, static calibration products, etc.) and the path to a
settings file containing the parameter values for the recipes such as sigma-
clipping thresholds, polynomial orders for fitting, etc.

⬇

Usage:

soxspipe init

soxspipe mbias <inputFrames> [-s <pathToSettingsFile>]

soxspipe mdark <inputFrames> [-s <pathToSettingsFile>]

soxspipe mflat <inputFrames> [-s <pathToSettingsFile>]

soxspipe disp_sol <inputFrames> [-s <pathToSettingsFile>]

soxspipe order_centres <inputFrames> [-s <pathToSettingsFile>]

Options:

init setup the soxspipe settings file for the first time

mbias the master bias recipe

mdark the master dark recipe

mflat the master flat recipe

disp_sol the disp solution recipe

order_centres the order centres recipe

inputFrames path to a directory of frames or a set-of-files file

-h, \--help show this help message

The most recent version of ESO’s _ESOReflex_ [18] GUI now includes a Python
based plugin API that shall allow us to integrate the SOXS pipeline into this
data-reduction control and visualisation tool (see Figure 4). Those
experienced with the X-Shooter data reductions will be familiar with this tool
and should easily transition to reducing SOXS data with the same tool.
_ESOReflex_ integration will be included in the final version of the pipeline,
released after the SOXS’s Preliminary Acceptance Chile (PAC).

Figure 4: The ESO Reflex environment for X-shooter [18].

##  6 Development Environment

At this current point in time the astronomical community have overwhelmingly
adopted Python as their scripting language of choice and there a plethora of
well maintained, mature python packages to help with basic data-reduction
routines not to mention visualisation, user-interaction and data manipulation.
To this end we have opted to develop the SOXS pipeline in Python 3.

The pipeline is being built with in agile development philosophy that includes
adaptive planning and evolutionary development. As with any project, one of
the greatest risks is knowledge loss due to a team member leaving the before
project completion. To mitigate this risk we employ pair-programming
techniques throughout to share knowledge, both explicit and tacit, between two
developers. In times of travel bans and remote working a JupterHub server with
Python based notebooks and shared screens within video conferencing tools have
been invaluable to execute these techniques.

We aim for the pipeline to be easy to install and extensively and clearly
documented. Installing the pipeline is as simple as running the command:

⬇

pip install soxspipe

Pipeline documentation is being written in parallel with the code in
docstrings and stand-alone markup files. A push of new code to github triggers
a new build of the documentation on readthedocs111readthedocs
https://soxspipe.readthedocs.io/ using the back-end sphinx documentation
engine.

###  6.1 Test Driven Development

To verify the pipeline is not only able to reduce a typical data-set but also
data that is far from ideal we are using test driven development throughout
the build using ‘extreme’ mock data. This data helps push the pipeline to the
limits of it capabilities and allows us to develop for possible edge-case
scenarios that the pipeline will experience in real-time production.

This mock data is generated via the SOXS End-to-End (E2E) simulator [5] which
is able to synthesise 2D images that take into account the main optical
behaviour of the system (grating dispersion, sampling, PSF, noises and
position of various resolution elements coming from full ray-tracing). An
example of the mock data is reported in Figures 5 and 6.

Figure 5: Simulated UV-VIS spectral image of a GOV star ($V=18$) object and
different tilts on the slit for each pseudo-order. Seeing 0.87′′, slit 1′′,
exposure 1800s. Figure 6: Full NIR arm displaying a synthetic image of black
body at 5800 K at $V=16$ mag point-source. Seeing 1′′, slit 1′′, 300 sec
exposure.

In particular, the plan is to generate mock data by varying different
observing conditions in terms of source magnitude (faint or bright continuum),
sky-brightness and calibration frames so as to provide a full suite of stress
tests for point-like sources down to the limit magnitude considered for SOXS.

The E2E model will be also able to provide synthetic simulated frames produced
by the acquisition camera with the same aim to test the data flow of the
reduction chain for the photometry performed with the AC.

Code is version controlled with git, hosted on github222github
https://github.com/thespacedoctor/soxspipe and linked to a Jenkins Continuous
Integration/Continuous Deployment server via webhooks. A push of new code to
any branch in the github pipeline repository triggers a new ‘build’ of the
code where all unit-tests are ran. If all tests pass then the branch is merged
into its parent branch (which in turn triggers a new testing of the parent
branch). If the master branch being tested and all tests pass then a new dot
release version of the code is shipped to PyPI333pypi
https://pypi.org/project/soxspipe/ for deployment.

##  7 Pipeline Architecture

The SOXS pipeline borrows the informative concept of ‘recipes’ employed by
ESO’s CPL to define the modular components of the pipeline. These recipes can
be strung together to create an end-to-end workflow that takes as input the
raw and calibration frames from the instrument and telescope and processes
them all the way through to fully reduced, calibrated, ESO Phase III compliant
science products.

We also employ the term ‘utilities’ to define reusable functions designed to
be called from multiple recipes. Recipes are named with the prefix ‘soxs’
(e.g. soxs_mbais) followed by a succinct description of the recipe.

Table 5: Index of SOXS Pipeline Recipes. Recipe | Reduction Stage
---|---
soxs_data_organiser | pre-processing
soxs_lingain | calibration
soxs_img_mflat | calibration
soxs_mbias | calibration
soxs_mdark | calibration
soxs_disp_solution | rectification
soxs_order_centres | rectification
soxs_spatial_solution | rectification
soxs_spec_mflat | rectification
soxs_straighten | rectification
soxs_line_check | rectification
soxs_nod | sky-subtraction
soxs_stare | sky-subtraction
soxs_offset | sky-subtraction
soxs_extract | extraction+
soxs_response | extraction+
soxs_merge | extraction+
soxs_astro_phot | extraction+

The reduction cascade of SOXS pipeline for spectroscopic data is reported in
Figure 7 while the cascade associated to acquisition camera imaging reduction
is depicted in Figure 8.

Figure 7: The SOXS Spectroscopic Data Reduction Cascade Figure 8: SOXS Imaging
Data Reduction Cascade

In particular, the association maps in Figures 7 and 8 show how spectral and
imaging data cascade flow through the SOXS pipeline. The input data,
calibration products required and the output frames are shown for each
pipeline recipe implemented in the pipeline. Each of the vertical lines in the
map depict a raw data frame, the specific recipe to applied to that frame and
the data product(s) output by that recipe. Horizontal lines show how those
output data products are used by subsequent pipeline recipes. Time loosely
proceeds from left-to-right (recipe order) and from top-to-bottom (recipe
processing steps) on the map. To the right of the grey dashed line in 7 there
are input calibration products generated from a separate pipeline processing
cascade.

The recipes required to reduce spectroscopic data from raw frames to 1D flux
calibrated spectra are:

  * •

soxs_lingain: Identify pixels that respond to varying levels of flux in a
different way compared to the typical pixel.

  * •

soxs_mbias: Used to create Master Bias frame and the first guess of bad pixels
on the detectors.

  * •

soxs_mdark: Remove the mean dark current from images and identify all hot or
cold pixels.

  * •

soxs_disp_solution: Generate first guess of the instruments spectral format
and dispersion solutions and central trace positions for each order. Results
from this recipe are shown in Figure 9.

  * •

soxs_order_centres: This recipe, starting from the traces first approximated
in the soxs_disp_solution accurately measures the central trace of each order.
An example of the output of the recipe and its ac hived performance is
reported in Figure 10.

  * •

soxs_spec_mflat: The recipe generates master flat-field frames, flag hot and
dead pixels. The spatial extension of the orders is also measured at this
stage. Figure 11 depicts results obtained against X-shooter NIR arm data.

  * •

soxs_spatial_solution: This recipe estimates the spatial- and dispersion-
solutions to generate a 2D map used later to translate curved order images
into a linear ($\lambda$, s) space. The recipe is similar in logic to
soxs_disp_solution but it also samples along the slit in the cross-dispersion
direction (i.e. Y-axis) by using the multi-pinhole slit frames illuminated by
the arc lamp.

  * •

soxs_straighten: This recipe uses the 2D map (modeled via polynomials)
computed in
soxs_spatial_solution to straighten (rectify) the curve order images and merge
them together. Users will be able (if they wish) to perform source extraction
on these rectified frames via the pipeline or using their own extraction
routines.

  * •

soxs_nod, soxs_stare, soxs_offset: These recipes are in charge of extracting
the spectra in counts (order by order) and, in the case of stare mode, produce
a model of the sky spectrum to be subtracted before extraction of the science
pixels.

  * •

soxs_response: Starting from observation of standard stars, this recipe
computes the efficiency curve and response curve of SOXS to flux calibrate
spectra for each order.

  * •

soxs_merge: The orders extracted and flux calibrated in the previous step are
merged together in order to obtain a unique, rectified and calibrated 1D
spectra of the source (with sky subtraction).

Figure 9: soxs_disp_solution recipe output. Top panel - Identified lines
against a raw frame single pinhole illuminated by calibration lamp. Central
panel - Survived spectral lines after sigma clipping and iteration on the
global fit. Bottom panel: residuals (in px)

.

Figure 10: soxs_order_centres recipe output. Top panel - Fitted position of
orders on the raw single-pinhole image. Central panel - Global fit obtained by
interpolating red points of top panel. Bottom panel: residuals (in px)

.

Figure 11: soxs_mflat recipe output. Top, Central and bottom panel are similar
to Figure 10

.

##  8 Conclusions

SOXS is expected to be delivered to Chile (La Silla) in mid 2022 and offered
to the community for the GTO and regular proposals from ESO by the end 2022.
To meet this goal, we designed an agile framework of development based on
Python 3 to develop a robust pipeline that can be used both automatically and
interactively by supporting standard Esoreflex workflow.
In the paper, we presented the current status of the pipeline by especially
focusing on the design architecture and development environment that foresee
the use of current state-of-the-art technologies in the context of software
development frameworks. At the moment of writing of this manuscript, the team
is almost on schedule for delivering the pipeline by end 2021 where first
tests of SOXS will take place in Europe. The adoption of agile framework,
presented in Section 6 and the use of Python 3 greatly helped on meeting the
time constraint while assuring high quality of the written software in terms
of maintainability, readability and robustness.

###### Acknowledgements.

## References

  * [1] Schipani, P. et al., “Development status of the SOXS spectrograph for the ESO-NTT telescope Conference,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [2] Landoni, M. et al., “The SOXS Scheduler for remote operation at LaSilla: concept and design,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [3] Rubin, A. et al., “Progress on the UV-VIS arm of SOXS,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [4] Biondi, F. et al., “The AIV strategy of the Common Path of Son of X-Shooter,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [5] Genoni, M. et al., “SOXS End-to-End simulator: development and applications for pipeline design,” in [Modeling, Systems Engineering, and Project Management for Astronomy IX  ], Proc. SPIE 11450 (2020).
  * [6] Kuncarayakti, H. et al., “Design and development of the SOXS calibration unit,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [7] Young, D. et al., “The SOXS Data Reduction Pipeline,” in [ Software and Cyberinfrastructure for Astronomy VI ], Proc. SPIE 11452 (2020).
  * [8] Brucalassi, A. et al., “Final Design and development status of the Acquisition and Guiding System for SOXS,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [9] Colapietro, M. et al., “Progress and tests on the Instrument Control Electronics for SOXS,” in [Software and Cyberinfrastructure for Astronomy VI  ], Proc. SPIE 11452 (2020).
  * [10] Cosentino, R. et al., “Development status of the UV-VIS detector system of SOXS for the ESO-NTT telescope,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [11] Claudi, R. et al., “Operational modes and efficiency of SOXS,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [12] Aliverti, M. et al., “Manufacturing, integration, and mechanical verification of SOXS,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [13] Sánchez, R. Z. et al., “Effects on optical performances due to gravity flexures, temperature variations and subsystems alignment,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [14] Vitali, F. et al., “The development status of the NIR spectrograph for the new SOXS instrument at the NTT,” in [Ground-based and Airborne Instrumentation for Astronomy VIII ], Proc. SPIE 11447 (2020).
  * [15] Romaniello, M., “ESO’s Science Archive Facility,” in [American Astronomical Society Meeting Abstracts #218 ], American Astronomical Society Meeting Abstracts 218, 305.03 (May 2011).
  * [16] Tonry, J. L., Denneau, L., Flewelling, H., Heinze, A. N., Onken, C. A., Smartt, S. J., Stalder, B., Weiland, H. J., and Wolf, C., “The ATLAS All-Sky Stellar Reference Catalog,” ApJ 867, 105 (Nov. 2018).
  * [17] Modigliani, A., Goldoni, P., Royer, F., Haigron, R., Guglielmi, L., François, P., Horrobin, M., Bristow, P., Vernet, J., Moehler, S., Kerber, F., Ballester, P., Mason, E., and Christensen, L., “The X-shooter pipeline,” in [Observatory Operations: Strategies, Processes, and Systems III ], Silva, D. R., Peck, A. B., and Soifer, B. T., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 7737, 773728 (July 2010).
  * [18] Freudling, W., Romaniello, M., Bramich, D. M., Ballester, P., Forchi, V., García-Dabló, C. E., Moehler, S., and Neeser, M. J., “Automated data reduction workflows for astronomy - the eso reflex environment,” A&A 559, A96 (2013).

  *[SOXS]: Son-Of-X-shooter
  *[AC]: Acquisition Camera
  *[ESO]: European Southern Observatory
  *[NTT]: New Technology Telescope
  *[SAF]: Science Archive Facility
  *[QC]: Quality Control
  *[DRAWER]: Data Reduction And WEllness Reporter
  *[CLI]: Command-Line Interface
  *[CPL]: Common Pipeline Library


UPR-1320-T CERN-TH-2022-102 MI-HET-779

Mode Stability For Massless Scalars In Five-Dimensional Black Hole Backgrounds

M. Cvetič1,2,3,4, C.N. Pope5,6, B.F. Whiting7 and Haoyu Zhang5

$\,{}^{1}$Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA

$\,{}^{2}$Department of Mathematics, University of Pennsylvania, Philadelphia,
PA 19104, USA

$\,{}^{3}$Center for Applied Mathematics and Theoretical Physics,
University of Maribor, SI2000 Maribor, Slovenia

$\,{}^{4}$CERN Theory Department, CH-1211 Geneva, Switzerland

$\,{}^{5}$George P. & Cynthia Woods Mitchell Institute for Fundamental Physics
and Astronomy,
Texas A&M University, College Station, TX 77843, USA

$\,{}^{6}$DAMTP, Centre for Mathematical Sciences, Cambridge University,
Wilberforce Road, Cambridge CB3 OWA, UK

$\,{}^{7}$Department of Physics, P.O. Box 118440, University of Florida,
Gainsville, FL 32611-8440, USA

###### Abstract

The mode stability of the Kerr black hole in four dimensions was demonstrated
by Whiting in 1989, by separating the Teukolsky equation that describes
gravitational perturbations and then transforming the radial and angular
equations in such a way that the problem can be reformulated as a wave
equation in an auxiliary spacetime in which the proof of stability is greatly
simplified, owing to the absence of an ergoregion. As a preliminary step
towards extending these ideas to higher-dimensional black holes, we study the
mode stability of the massless scalar wave equation in the five-dimensional
black hole solutions of Einstein gravity and supergravity. We show how the
wave equation can again be mapped into one in an auxiliary spacetime in which
there is no ergoregion, allowing us to give a proof of the mode stability of
the solutions of the scalar wave equation.

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Massless Scalar Wave Equation in Five-Dimensional Black Hole Background
    1. 2.1 Five-dimensional Myers-Perry black hole
    2. 2.2 Massless scalar wave equation
    3. 2.3 Transforming the radial equation
    4. 2.4 Behaviour of unstable modes
  3. 3 Modes in the Transformed Spacetime
    1. 3.1 Combining angular and transformed radial equations
  4. 4 3-Charge Five-Dimensional STU Supergravity Black Holes
  5. 5 Discussion
  6. 6 Conclusion
  7. A Global Structure of the Myers-Perry Black Hole

##  1 Introduction

Establishing results on the stability of black hole solutions has been a major
activity in the general relativity community for many years. This is
especially subtle, and also important, in the case of stationary rotating
black holes. Many different approaches have been followed, but one of the most
fruitful has involved finding an integral transformation that maps the
difficult analysis in the original variables into a considerably simpler
analysis in the transformed variables. The idea was first developed in [1],
where it was employed to establish the mode stability of the Kerr black hole
[1]. The technique has been developed further in recent years, and found
application in studies such as the global stability of black holes (see, for
example, [2, 3]), the stability of extremal black holes [4], and mode
stability on the real frequency axis [5, 6].

Establishing the mode stability of the Kerr black hole involved studying the
properties of the mode functions in the separation of variables for the
Teukolsky equation, which provides a gauge-invariant description of the
perturbations around the Kerr background. In [1] the generalised Teukolsky
equation with a spin parameter $s$ was studied, with $s=\pm 2$ corresponding
to the actual case of interest in which the equation describes the
gravitational perturbations themselves. The case $s=0$ corresponds to the
massless scalar wave equation (the massless Klein-Gordon equation), while the
$s=\pm 1$ case governs gauge-invariant components of the Maxwell field.

The techniques for analysing the Teukolsky equation for spin $s$ that were
developed in [1] were broadly similar for all $s$, and in fact the essential
features associated with the stability of the solutions could already be seen
in the $s=0$ case. This is a useful observation because if one looks at more
complicated situations than black holes in pure Einstein gravity, such as
black holes in Einstein-Maxwell theory or in supergravity, the analogue of the
Teukolsky analysis has not been implemented. In the absence of a gauge-
invariant treatment of the perturbations of the black holes in these more
complicated situations, one can at least study the analogue of $s=0$ Teukolsky
equation, that is, the massless scalar wave equation in the black hole
background. One may hope that this can provide a “proxy” for the full
analysis, and that establishing stability results for the solutions of the
massless wave equation may be indicative of what one might find in a more
elaborate and complete perturbative analysis. This approach was adopted in
[7], where the techniques of [1] were applied to the study of the mode
stability for solutions of the massless scalar wave equation in the background
of a class of four-dimensional supergravity black holes carrying four
independent electric charges [8, 9].

In this paper, we extend some of the four-dimensional techniques for mode-
stability analysis that were developed in [1] to the case of five dimensions.
Even in the pure Einstein case, the analogue of the four-dimensional Teukolsky
analysis is unknown. The stumbling block is that the four-dimensional analysis
depended heavily upon the use of the Newman-Penrose formalism, and no
particularly useful extension of this to five or higher dimensions has been
constructed.111However, see [10]. Thus for now, our approach will mirror the
one that can be followed for more complicated theories in four dimensions,
namely, we shall focus attention on establishing stability results for
solutions of the five-dimensional massless scalar wave equation. This already
allows us to develop a rather non-trivial generalisation of the techniques
that were employed in four dimensions, and it reveals ways in which the
integral transformation that allows us to establish mode-stability results is
substantially different from the one in four dimensions.

We shall describe in the subsequent sections how one can establish mode
stability results for solutions of the massless wave equation in the
background of the five-dimensional Myers-Perry rotating black hole [11], and
also in the background of general 3-charge rotating black holes in the five-
dimensional STU supergravity theory [12].

##  2 Massless Scalar Wave Equation in Five-Dimensional Black Hole Background

###  2.1 Five-dimensional Myers-Perry black hole

The natural generalisation of the four-dimensional rotating Kerr [13] black
hole to higher spacetime dimensions is provided by the Myers-Perry black hole
solutions [11]. These vacuum solutions of the $D$-dimensional Einstein
equations are characterised by their mass $M$ and by $[(D-1)/2]$ independent
angular momenta, reflecting the fact that independent rotations can occur in
each orthogonal spatial 2-plane. In this paper we shall be concerned
specifically with the example of the five-dimensional rotating black hole. Its
metric is given by [11]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-{\frac{\Delta}{\rho^{2}}}\,(dt-a\sin^{2}\theta\,d\phi-b\cos^{2}\theta\,d\psi)^{2}+{\frac{\rho^{2}}{\Delta}}\,dr^{2}+\rho^{2}\,d\theta^{2}$
(2.1)
$\displaystyle+{\frac{\sin^{2}\theta}{\rho^{2}}}\,[adt-(r^{2}+a^{2})\,d\phi]^{2}+{\frac{\cos^{2}\theta}{\rho^{2}}}\,[bdt-(r^{2}+b^{2})\,d\psi]^{2}$
$\displaystyle+{\frac{1}{r^{2}\,\rho^{2}}}\,[abdt-b(r^{2}+a^{2})\sin^{2}\theta
d\phi-a(r^{2}+b^{2})\cos^{2}\theta\,d\psi]^{2}$

where

$\displaystyle\Delta={\frac{(r^{2}+a^{2})(r^{2}+b^{2})}{r^{2}}}-2M\,,\qquad\rho^{2}=r^{2}+a^{2}\,\cos^{2}\theta+b^{2}\,\sin^{2}\theta\,.$
(2.2)

Here $a$ and $b$ are the two independent rotation parameters, with $\phi$ and
$\psi$ being the two associated azimuthal angles (each with period $2\pi$).
The latitude coordinate $\theta$ ranges over
$0\leq\theta\leq{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}\pi$.

###  2.2 Massless scalar wave equation

Our focus will be the investigation of solutions of the massless scalar wave
equation $\square\Psi=0$ in the Myers-Perry background, with the goal of
establishing that modes with time dependence $e^{-{{\rm i\,}}\omega t}$ that
are ingoing on the future horizon and outgoing at future null infinity cannot
have a frequency $\omega$ with a positive imaginary part. In other words, we
seek to show that there cannot exist spatially regular modes that would give
rise to instabilities growing exponentially in time.

One can define new radial and latitude coordinates ${{\tilde{x}}}$ and $y$ by
writing

${{\tilde{x}}}=r^{2}\,,\qquad y=\cos^{2}\theta\,.$ (2.3)

Using these, we have

$\sqrt{-g}={\frac{{{\tilde{x}}}+a^{2}\,y+b^{2}\,(1-y)}{4}}\,,$ (2.4)

and defining the quantity $G^{\mu\nu}\equiv\sqrt{-g}\,g^{\mu\nu}$, we have

$\displaystyle G^{00}$ $\displaystyle=$
$\displaystyle-{\frac{({{\tilde{x}}}+2M+b^{2})}{4}}-{\frac{M^{2}\,{{\tilde{x}}}}{D}}+{\frac{(b^{2}-a^{2})y}{4}}\,,\quad
G^{01}=G^{02}=0\,,\quad G^{03}=-{\frac{aM({{\tilde{x}}}+b^{2})}{2D}}\,,$
$\displaystyle G^{04}$ $\displaystyle=$
$\displaystyle-{\frac{bM({{\tilde{x}}}+a^{2})}{2D}}\,,\quad
G^{11}={{\tilde{x}}}^{2}+(a^{2}+b^{2}-2M){{\tilde{x}}}+a^{2}b^{2}\,,\quad
G^{12}=G^{13}=G^{14}=0\,,$ $\displaystyle G^{22}$ $\displaystyle=$
$\displaystyle y(1-y)\,,\quad G^{23}=G^{24}=0\,,$ $\displaystyle G^{33}$
$\displaystyle=$
$\displaystyle{\frac{(b^{2}-a^{2}){{\tilde{x}}}+b^{2}\,(b^{2}-a^{2}-2M)}{4D}}+{\frac{1}{4(1-y)}}\,,\quad
G^{34}=-{\frac{abM}{2D}}\,,$ $\displaystyle G^{44}$ $\displaystyle=$
$\displaystyle{\frac{(a^{2}-b^{2}){{\tilde{x}}}+a^{2}\,(a^{2}-b^{2}-2M)}{4D}}+{\frac{1}{4y}}\,,$
(2.5)

where

$D=({{\tilde{x}}}+a^{2})({{\tilde{x}}}+b^{2})-2M{{\tilde{x}}}\,.$ (2.6)

It can be shown [14] that the massless scalar wave equation $\square\Psi=0$,
which may be written as
${\partial}_{\mu}(G^{\mu\nu}\,{\partial}_{\nu}\Psi)=0$, is separable.

We can obtain the separated ${{\tilde{x}}}$ and $y$ equations in Schrödinger
form by writing 222For the separated form of the equations before casting them
in the Schrödinger form, see [14]. There the separation was already performed
for the 5d STU black holes [12].

$\Psi=e^{-{{\rm i\,}}\omega t+{{\rm i\,}}m\phi+{{\rm
i\,}}n\psi}\,R({{\tilde{x}}})\,S(y)$ (2.7)

and then defining

$R({{\tilde{x}}})={\frac{X({{\tilde{x}}})}{\sqrt{D}}}\,,\qquad
S(y)={\frac{Y(y)}{\sqrt{y(1-y)}}}\,.$ (2.8)

This gives the separated equations

${\frac{X^{\prime\prime}}{X}}+U_{{\tilde{x}}}-{\frac{\sigma}{D}}=0\,,\qquad{\frac{Y^{\prime\prime}}{Y}}+U_{y}+{\frac{\sigma}{y(1-y)}}=0\,.$
(2.9)

It is convenient now to make a further change of the radial variable. Defining
${\epsilon}_{+}$ and ${\epsilon}_{-}$ by

$\displaystyle 2M-(a+b)^{2}=2M{\epsilon}_{+}^{2}\,,\qquad
2M-(a-b)^{2}=2M{\epsilon}_{-}^{2}\,,$ (2.10)

a further coordinate transformation from ${{\tilde{x}}}$ to $x$, given by

$\displaystyle{{\tilde{x}}}=2M{\epsilon}_{+}\,{\epsilon}_{-}\,x+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}M({\epsilon}_{+}-{\epsilon}_{-})^{2}\,,$ (2.11)

implies that $D$, defined in eqn (2.6), becomes

$\displaystyle D=4M^{2}\,{\epsilon}_{+}^{2}\,{\epsilon}_{-}^{2}\,x(x-1)\,.$
(2.12)

The outer horizon $r=r_{+}$ is located at $x=1$, and the inner horizon
$r=r_{-}$ at $x=0$. The region where $r$ goes to infinity corresponds to $x$
goes to infinity.

In terms of the variable $x$, we can now write the radial equation in the form

$\displaystyle{\frac{X^{\prime\prime}}{X}}+{\frac{\kappa+\Lambda}{x}}+{\frac{\kappa-\Lambda}{x-1}}+{\frac{{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}-\beta^{2}}{x^{2}}}+{\frac{{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}-\gamma^{2}}{(x-1)^{2}}}=0\,,$ (2.13)

finding

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}M{\epsilon}_{+}\,{\epsilon}_{-}\,\omega^{2}\,,$ $\displaystyle\beta$
$\displaystyle=$ $\displaystyle{\frac{{{\rm
i\,}}}{4\sqrt{2M}}}\,\Big{[}{\frac{2M\omega-(a+b)(m+n)}{{\epsilon}_{+}}}-{\frac{2M\omega-(a-b)(m-n)}{{\epsilon}_{-}}}\Big{]}\,,$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle{\frac{{{\rm
i\,}}}{4\sqrt{2M}}}\,\Big{[}{\frac{2M\omega-(a+b)(m+n)}{{\epsilon}_{+}}}+{\frac{2M\omega-(a-b)(m-n)}{{\epsilon}_{-}}}\Big{]}\,,$
(2.14) $\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle\sigma+{\frac{1}{2}}-{\frac{3}{4}}M\omega^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
8}}}(a^{2}-b^{2})\omega^{2}+{\frac{[2M\omega-(a+b)(m+n)]^{2}}{16M{\epsilon}_{+}^{2}}}+{\frac{[2M\omega-(a-b)(m-n)]^{2}}{16M{\epsilon}_{-}^{2}}}\,.$

In the angular direction, the equation for $Y$ in (2.9) is of the form

$\displaystyle{\frac{Y^{\prime\prime}}{Y}}+{\frac{\hat{\kappa}+\hat{\Lambda}}{y}}+{\frac{\hat{\kappa}-\hat{\Lambda}}{y-1}}+{\frac{{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
4}}}-\hat{\beta}^{2}}{y^{2}}}+{\frac{{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}-\hat{\gamma}^{2}}{(y-1)^{2}}}=0\,,$ (2.15)

with

$\displaystyle\hat{\kappa}$ $\displaystyle=$
$\displaystyle-{\frac{1}{8}}(a^{2}-b^{2})\omega^{2}\,,\qquad\hat{\beta}={\frac{n}{2}}\,,\qquad\hat{\gamma}={\frac{m}{2}}$
$\displaystyle\hat{\Lambda}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}+\sigma-{\frac{(m^{2}+n^{2})}{4}}+{\frac{(a^{2}-b^{2})\omega^{2}}{8}}\,.$
(2.16)

###  2.3 Transforming the radial equation

We start with the radial equation given by eqn (2.13), with the constants
given in (2.14). Next, introduce a new function $f(x)$, related to $X(x)$ by

$\displaystyle
X(x)=x^{{\frac{1}{2}}-{\epsilon}_{1}\,\beta}\,(x-1)^{{\frac{1}{2}}-{\epsilon}_{2}\,\gamma}\,f(x)\,,$
(2.17)

where ${\epsilon}_{1}^{2}={\epsilon}_{2}^{2}=1$. The function $f$ therefore
satisfies

$\displaystyle[x(x-1)\,{\partial}^{2}_{x}+(bx+c)\,{\partial}_{x}+dx+e]f(x)=0\,,$
(2.18)

with

$\displaystyle b$ $\displaystyle=$ $\displaystyle
2(1-{\epsilon}_{1}\,\beta-{\epsilon}_{2}\,\gamma)\,,\qquad
c=-1+2{\epsilon}_{1}\,\beta\,,$ $\displaystyle d$ $\displaystyle=$
$\displaystyle 2\kappa\,,\qquad e={\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}-\kappa-{\epsilon}_{1}\,\beta-{\epsilon}_{2}\,\gamma+2{\epsilon}_{1}\,{\epsilon}_{2}\,\beta\,\gamma-\Lambda\,.$
(2.19)

Next, we make an integral transform to a new radial variable $h(z)$, by
defining

$\displaystyle
h(z)=e^{-{{\tilde{\alpha}}}\,z}\,e^{{{\tilde{\nu}}}/z}\,z^{1+{{\tilde{\gamma}}}}\,\int_{1}^{\infty}e^{2{{\tilde{\alpha}}}\,x\,z}\,f(x)\,dx$
(2.20)

Now, multiplying eqn (2.18) by $e^{2{{\tilde{\alpha}}}\,x\,z}$ and integrating
gives, after integration by parts,

$\displaystyle
0=B+\int_{1}^{\infty}[4{{\tilde{\alpha}}}^{2}\,x(x-1)\,z^{2}+2{{\tilde{\alpha}}}\,(4x-bx-2-c)z+dx+2-b+e]\,f(x)\,dx\,,$
(2.21)

where $B$ is the boundary term

$\displaystyle
B=\Big{[}e^{2{{\tilde{\alpha}}}\,x\,z}x(x-1)\,{\partial}_{x}f-{\partial}_{x}(e^{2{{\tilde{\alpha}}}\,x\,z}\,x(x-1))\,f+e^{2{{\tilde{\alpha}}}\,x\,z}(bx+c)\,f\Big{]}_{1}^{\infty}\,.$
(2.22)

Note that eqn (2.21) can be written as

$\displaystyle 0=B+\int_{1}^{\infty}f(x){\cal
O}_{z}(e^{2{{\tilde{\alpha}}}\,x\,z})\,dx\,,$ (2.23)

where

$\displaystyle{\cal
O}_{z}=z^{2}\,{\partial}^{2}_{z}+\left(4z-2{{\tilde{\alpha}}}\,z^{2}-bz+{\frac{d}{2{{\tilde{\alpha}}}}}\right)\,{\partial}_{z}+2-2{{\tilde{\alpha}}}\,c\,z-4{{\tilde{\alpha}}}\,z-b+e\,.$
(2.24)

It can then be seen that provided the boundary term $B$ vanishes, a question
to which we shall return later, then $h(z)$ defined in eqn (2.20) satisfies
${\cal
O}_{z}\Big{(}e^{{{\tilde{\alpha}}}\,z}\,e^{-{{\tilde{\nu}}}/z}\,z^{-1-{{\tilde{\gamma}}}}\,h(z)\Big{)}=0$,
and thus

$\displaystyle\Big{(}{\partial}^{2}_{z}-{{\tilde{\alpha}}}^{2}+{\frac{2{{\tilde{\alpha}}}\,{{\tilde{\kappa}}}}{z}}+{\frac{{{\tilde{\Lambda}}}}{z^{2}}}+{\frac{2{{\tilde{\gamma}}}\,{{\tilde{\nu}}}}{z^{3}}}-{\frac{{{\tilde{\nu}}}^{2}}{z^{4}}}\Big{)}h(z)=0\,,$
(2.25)

where the constants are chosen so that

$\displaystyle{{\tilde{\kappa}}}$ $\displaystyle=$
$\displaystyle-{\epsilon}_{1}\,\beta+{\epsilon}_{2}\,\gamma\,,\qquad{{\tilde{\Lambda}}}={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}-\beta^{2}-\gamma^{2}-\Lambda\,,$
$\displaystyle{{\tilde{\gamma}}}$ $\displaystyle=$
$\displaystyle{\epsilon}_{1}\,\beta+{\epsilon}_{2}\,\gamma\,,\qquad{{\tilde{\nu}}}=-{\frac{\kappa}{2{{\tilde{\alpha}}}}}\,.$
(2.26)

The constant ${{\tilde{\alpha}}}$ is arbitrary at this stage (it really just
sets the scale of the new radial variable $z$ that is introduced in eqn
(2.20)). We shall find it convenient to define it to be

$\displaystyle{{\tilde{\alpha}}}={\frac{{{\rm
i\,}}\sqrt{M}\,{\epsilon}_{-}\,\omega}{2\sqrt{2}}}\,.$ (2.27)

###  2.4 Behaviour of unstable modes

If they existed, unstable modes would be solutions than were purely outgoing
at $\mathscr{I}$ and purely ingoing at the horizon ${\cal H}$, that is,
asymptotically, they would have support only at $\mathscr{I}^{+}$ and ${\cal
H}^{+}$. We are interested in establishing the non-existence of unstable modes
arising from frequencies in the upper half of the complex $\omega$-plane, that
is for $\omega=\omega_{0}+i\omega_{1}$, with $\omega_{1}>0$, since these would
grow exponentially in the future. Thus, near $\mathscr{I}^{+}$, we must have:

$\Psi\sim\exp{-i\omega(t-r_{*})}\,,\qquad\hbox{as}\quad\\{t,r_{*}\\}\rightarrow\infty\,,$
(2.28)

and, near ${\cal H}^{+}$, we must have:

$\Psi\sim\exp{-i\omega(t+r_{*})}\,,\qquad\hbox{as}\quad\\{-t,r_{*}\\}\rightarrow-\infty\,.$
(2.29)

From the definition of the $r_{*}$ coordinate in eqn (A.1), it can be seen
that in terms of the radial coordinate $x$ introduced in eqn (2.11), we shall
have asymptotically

$\displaystyle\hbox{Near}\quad\mathscr{I}:$ $\displaystyle\qquad
r_{*}=\sqrt{2M{\epsilon}_{+}\,{\epsilon}_{-}}\,x^{{\frac{1}{2}}}+{\cal
O}\big{(}x^{-1/2}\big{)}\,,$ (2.30) $\displaystyle\hbox{Near}\quad{\cal H}:$
$\displaystyle\qquad
r_{*}={\frac{\sqrt{M}\,({\epsilon}_{+}+{\epsilon}_{-})}{2\sqrt{2}\,{\epsilon}_{+}\,{\epsilon}_{-}}}\,\log(x-1)+{\cal
O}\big{(}x-1\big{)}\,.$ (2.31)

From the radial equation (2.13), it can be seen that near the horizon $x=1$,
the solutions will have the form

$\displaystyle X(x)=(x-1)^{{\frac{1}{2}}\pm\gamma}\,F^{(1)}_{\pm}(x-1)\,,$
(2.32)

where the functions $F^{(1)}_{\pm}(x-1)$ are analytic around $x=1$, with
$F^{(1)}_{\pm}(0)$ a non-zero constant. As noted above, an unstable mode would
have $\omega$ dependence of the form $e^{-{{\rm i\,}}\omega r_{*}}$ near H,
and hence from (2.31) and (2.14) it would correspond to the minus-sign choice
in eqn (2.32). Thus for an unstable mode $X_{um}$, we must have the near-
horizon behaviour

$\displaystyle\hbox{Near}\quad{\cal H}:\qquad
X_{um}(x)=(x-1)^{{\frac{1}{2}}-\gamma}\,F^{(1)}_{-}(x-1)\,.$ (2.33)

In the asymptotic region near $x=\infty$, it can be seen from the radial
equation (2.13) that the asymptotic form of the solutions will be

$\displaystyle X(x)=x^{{\frac{1}{4}}}\,e^{\pm 2{{\rm
i\,}}\sqrt{2\kappa}\,\sqrt{x}}\,F^{(\infty)}_{\pm}\Big{(}{\frac{1}{\sqrt{x}}}\Big{)}\,,$
(2.34)

with the functions $F^{(\infty)}_{\pm}\Big{(}{\frac{1}{\sqrt{x}}}\Big{)}$
being asymptotic series with $F^{(\infty)}_{\pm}(0)$ a non-zero constant. Now,
as we saw, an unstable mode would correspond to a having an $\omega$
dependence of the form $e^{{{\rm i\,}}\omega r_{*}}$ near infinity, and so
from (2.30) and (2.14) it would correspond to the plus-sign choice in eqn
(2.34). Thus, for an unstable mode $X_{um}(x)$, we must have the asymptotic
behaviour

$\displaystyle\hbox{Near}\quad\mathscr{I}:\qquad
X_{um}(x)=x^{{\frac{1}{4}}}\,e^{2{{\rm
i\,}}\sqrt{2\kappa}\,\sqrt{x}}\,F^{(\infty)}_{+}\Big{(}{\frac{1}{\sqrt{x}}}\Big{)}\,.$
(2.35)

Before going on, we point out that infinitely many modes with these analytic
properties do exist with $\omega_{1}$, the imaginary part of $\omega$, being
$<0$. These modes decay exponentially in time, have been extensively studied
[15], and are known as the quasi-normal modes which, taken together, uniquely
characterise a black hole spacetime to which they correspond.

We turn now to the behaviour of the transformed radial function $h(z)$,
defined by eqn (2.20). In particular, we shall be concerned with the behaviour
in the coordinate range $0\leq z\leq\infty$.

Near $z=0$, it can be seen from eqn (2.25) that the leading-order behaviour of
the solutions will be

$\displaystyle
h(z)=e^{\pm{{\tilde{\nu}}}/z}\,z^{1\pm{{\tilde{\gamma}}}}\,G^{(0)}_{\pm}(z)\,,$
(2.36)

where $G^{(0)}_{\pm}(z)$ are analytic functions with $G^{(0)}_{\pm}(0)$ being
non-zero constants. It is evident from (2.20) that at $z=0$ the integrand for
an unstable mode is just an analytic function of $x$, and so the leading-order
behaviour of $h(z)$ near $z=0$ will be given by the pre-factor functions
$e^{{{\tilde{\nu}}}/z}\,z^{1+{{\tilde{\gamma}}}}$. In other words, the
unstable mode corresponds to the plus-sign choice in eqn (2.36):

$\displaystyle\hbox{Near}\quad z=0:\qquad
h_{um}(z)=e^{{{\tilde{\nu}}}/z}\,z^{1+{{\tilde{\gamma}}}}\,G^{(0)}_{+}(z)\,,$
(2.37)

As a check, we see from eqns (2.14), (2.26) and (2.27) that

$\displaystyle{{\tilde{\nu}}}={\frac{{{\rm
i\,}}\sqrt{M}\,{\epsilon}_{+}\,\,\omega}{2\sqrt{2}}}\,,$ (2.38)

whose real part is negative when $\omega$ has a positive imaginary part, thus
implying that $h_{um}(z)$ in eqn (2.37) is finite, and goes to zero, as $z$
goes to zero. (This justifies the sign choice in the definition of
${{\tilde{\alpha}}}$ in eqn (2.27).)

It can be seen from eqn (2.25) that near $z=\infty$, the function $h(z)$ has
the behaviour

$\displaystyle
h(z)=e^{\pm{{\tilde{\alpha}}}z}\,z^{\mp{{\tilde{\kappa}}}}\,G^{(\infty)}_{\pm}\Big{(}{\frac{1}{z}}\Big{)}\,,$
(2.39)

where the functions $G^{(\infty)}_{\pm}\Big{(}{\frac{1}{z}}\Big{)}$ are
asymptotic in $z^{-1}$ with $G^{(\infty)}_{\pm}(0)$ being non-vanishing
constants. In the expression (2.20) the leading behaviour of $h(z)$ near
$z=\infty$ is governed by the behaviour of $f(x)$ near $x=1$. Using the
previously-determined behaviour of $X(x)$, and hence $f(x)$, for an unstable
mode we see that the integrand in eqn (2.20) has the behaviour

$\displaystyle\int_{1}^{\infty}e^{2\tilde{\alpha}xz}f(x)dx$ $\displaystyle=$
$\displaystyle
e^{2\tilde{\alpha}z}\int_{1}^{\infty}e^{2\tilde{\alpha}(x-1)z}\,(x-1)^{(\epsilon_{2}-1)\gamma}\Big{(}1+{\cal
O}(x-1)\Big{)}dx.$ (2.40)

Substituting $v=-2\tilde{\alpha}(x-1)z$, we then have:

$\displaystyle\int_{1}^{\infty}e^{2\tilde{\alpha}xz}f(x)dx$
$\displaystyle\propto$ $\displaystyle
e^{2\tilde{\alpha}z}\,z^{-1+\gamma(1-\epsilon_{2})}\int_{0}^{\infty}e^{-v}v^{(\epsilon_{2}-1)\gamma}\Big{(}1+{\cal
O}\left(\frac{v}{z}\right)\Big{)}dv.$ (2.41)

Combining with the pre-factor we then see, provided

$\displaystyle\epsilon_{2}=-1\,,$ (2.42)

that:

$\displaystyle h(z)$ $\displaystyle\propto$ $\displaystyle
e^{+\tilde{\alpha}z}\,y^{\tilde{\gamma}+(1-\epsilon_{2})\gamma}(1+{\cal
O}\left(\frac{1}{z}\right)\Big{)}$ (2.43) $\displaystyle\sim$ $\displaystyle
e^{\tilde{\alpha}z}\,z^{-\tilde{\kappa}}\,.$

Thus for an unstable mode, the plus-sign choice in eqn (2.39) is selected:

$\displaystyle\hbox{Near}\quad z=\infty:\qquad
h_{um}(z)=e^{{{\tilde{\alpha}}}z}\,z^{-{{\tilde{\kappa}}}}\,G^{(\infty)}_{+}\Big{(}{\frac{1}{z}}\Big{)}\,.$
(2.44)

As a check, we note that if $\omega$ has a positive imaginary part,
${{\tilde{\alpha}}}$, given in eqn (2.27), will have a negative real part, and
so $h_{um}(z)$ will be finite as $z$ goes to infinity. (This motivated the
sign choice in the definition of ${{\tilde{\alpha}}}$ in eqn (2.27).)

It is now straightforward to check, using the asymptotic properties of the
radial functions for unstable modes established in this section, that the
boundary term given in eqn (2.22) will vanish for any unstable mode. Thus we
have established that there is a one-to-one mapping between exponentially
unstable modes in the original untransformed radial function $X(x)$ and
exponentially unstable modes in the radial function $h(z)$ obtained by means
of the integral transform (2.20). The final steps in the proof of the mode
stability will be presented in the next section; this will entail establishing
for the transformed radial equation that there cannot exist any exponentially
unstable modes.

##  3 Modes in the Transformed Spacetime

###  3.1 Combining angular and transformed radial equations

The constant $\sigma$ that was introduced in the original process of
separating variables is present in the transformed radial equation (2.25)
through the quantity $\Lambda$ (see eqns (2.14) and (2.26)) and in the angular
equation (2.15) through the quantity $\hat{\Lambda}$ (see eqn (2.16)). It
follows therefore that if we form the combination

$\displaystyle{\frac{z^{2}}{h(z)}}\,{\partial}^{2}_{z}\,h(z)+{\frac{y(1-y)}{Y(y)}}\,{\partial}^{2}_{y}\,Y(y)$
(3.1)

and make use of eqns (2.25) and (2.15) then we shall obtain an equation in
which all the $\sigma$ dependence has cancelled. This equation can in fact be
interpreted as the result of performing a separation of variables in which we
write

$\displaystyle\Psi(t,z,y,\phi,\psi)=h(z)\,Y(y)\,e^{-{{\rm i\,}}\omega
t}\,e^{{{\rm i\,}}m\phi}\,e^{{{\rm i\,}}n\psi}\,.$ (3.2)

We postpone writing the full ‘unseparated” equation for now, and just focus on
the terms proportional to $\omega^{2}$. (That is, the $-{\partial}_{tt}$ terms
in the full five-dimensional wave equation.) Together with the terms involving
the radial and angular derivatives these are

$\displaystyle
z^{2}\,{\partial}^{2}_{z}+y(1-y)\,{\partial}^{2}_{y}+{\frac{{\epsilon}_{+}^{2}\,M\,\omega^{2}}{8z^{2}}}+{\frac{{\epsilon}_{-}^{2}\,M\,z^{2}\,\omega^{2}}{8}}+{\frac{[-{\epsilon}_{1}\,({\epsilon}_{-}-{\epsilon}_{+})-{\epsilon}_{2}\,({\epsilon}_{-}+{\epsilon}_{+})]\,M\,\omega^{2}}{4\,{\epsilon}_{-}\,z}}$
$\displaystyle+{\frac{[{\epsilon}_{1}\,({\epsilon}_{-}-{\epsilon}_{+})-{\epsilon}_{2}\,({\epsilon}_{-}+{\epsilon}_{+})]\,M\,z\,\omega^{2}}{4\,{\epsilon}_{+}}}+{\textstyle{\frac{\scriptstyle
3}{\scriptstyle 4}}}M\,\omega^{2}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 8}}}(a^{2}-b^{2})\,(1-2y)\,\omega^{2}+\hbox{rest}\,,$ (3.3)

Since

$\displaystyle{\epsilon}_{+}=\sqrt{1-{\frac{(a+b)^{2}}{2M}}}\,,\qquad{\epsilon}_{-}=\sqrt{1-{\frac{(a-b)^{2}}{2M}}}\,,$
(3.4)

(see eqns (2.10)), and we always assume $a$ and $b$ are non-negative, it
follows that ${\epsilon}_{-}\geq{\epsilon}_{+}$. Consequently, all the
$\omega^{2}$ terms in eqn (3.3) will be non-negative, provided that we choose
the sign of $\epsilon_{2}$ to be

$\displaystyle{\epsilon}_{2}=-1\,.$ (3.5)

Note that the necessity for this choice of sign was already seen in the
previous section, in eqn (2.42). The choice of sign for ${\epsilon}_{1}$ is
undetermined by these considerations. We shall, for definiteness, make the
choice

$\displaystyle{\epsilon}_{1}=-1$ (3.6)

For the remaining $\omega^{2}$ terms, namely with coefficient

$\displaystyle{\textstyle{\frac{\scriptstyle 3}{\scriptstyle
4}}}M-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}(a^{2}-b^{2})(1-2y)\,,$ (3.7)

we note that the range of the angular coordinate is $0\leq y\leq 1$, and so
$-1\leq 1-2y\leq 1$. We also have that

$\displaystyle(a+b)^{2}\leq 2M\,,$ (3.8)

(see eqn (3.4)), and so we have $a^{2}-b^{2}=(a+b)(a-b)\leq(a+b)^{2}\leq 2M$.
Thus the remaining terms (3.7) contributing to $\omega^{2}$ terms in (3.3) are
always positive.

In summary, we have seen that the overall coefficient of $\omega^{2}$ in eqn
(3.3) is always positive, implying that in the transformed metric
$\tilde{g}_{\mu\nu}$ obtained by the process of unseparating variables,
${\frac{{\partial}}{{\partial}t}}$ is always timelike outside the horizon.

We now present the complete result for the combination of the radial and
angular equations. With ${\epsilon}_{1}={\epsilon}_{2}=-1$ as discussed above
we obtain

$\displaystyle\Big{\\{}z^{2}\,{\partial}^{2}_{z}+y(1-y)\,{\partial}^{2}_{y}-\Big{(}{\frac{(a+b)}{4z}}+{\frac{(a-b)\,z}{4}}\Big{)}\,m\omega-\Big{(}{\frac{(a+b)}{4z}}-{\frac{(a-b)\,z}{4}}\Big{)}\,n\omega$
$\displaystyle+\Big{(}{\frac{{\epsilon}_{+}^{2}\,M}{8z^{2}}}+{\frac{{\epsilon}_{-}^{2}\,M\,z^{2}}{8}}+{\frac{M}{2z}}+{\frac{M\,z}{2}}+{\textstyle{\frac{\scriptstyle
3}{\scriptstyle 4}}}M-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}(a^{2}-b^{2})(1-2y)\Big{)}\,\omega^{2}$
$\displaystyle-{\frac{m^{2}}{4(1-y)}}-{\frac{n^{2}}{4y}}+{\frac{1}{4y(1-y)}}\Big{\\}}\Psi=0\,.$
(3.9)

Using the replacements

$\displaystyle\omega\longrightarrow{{\rm i\,}}\,{\partial}_{t}\,,\qquad
m\longrightarrow-{{\rm i\,}}\,{\partial}_{\phi}\,,\qquad
n\longrightarrow-{{\rm i\,}}\,{\partial}_{\psi}$ (3.10)

we can read off the components $\tilde{g}^{\mu\nu}$ of an inverse metric in a
transformed spacetime, such that eqn (3.9) can be written as

$\displaystyle\tilde{g}^{\mu\nu}\,{\partial}_{\mu}\,{\partial}_{\nu}\Psi+{\frac{1}{4y(1-y)}}\,\Psi=0\,,$
(3.11)

with

$\displaystyle\tilde{g}^{zz}$ $\displaystyle=$ $\displaystyle
z^{2}\,,\qquad\tilde{g}^{yy}=y(1-y)\,,\quad\tilde{g}^{\phi\phi}={\frac{1}{4(1-y)}}\,,\quad\tilde{g}^{\psi\psi}={\frac{1}{4y}}\,,$
$\displaystyle\tilde{g}^{t\phi}$ $\displaystyle=$
$\displaystyle-{\frac{(a+b)}{8z}}-{\frac{(a-b)\,z}{8}}\,,\quad\tilde{g}^{t\psi}=-{\frac{(a+b)}{8z}}+{\frac{(a-b)\,z}{8}}\,,$
$\displaystyle\tilde{g}^{tt}$ $\displaystyle=$
$\displaystyle-\Big{(}{\frac{{\epsilon}_{+}^{2}\,M}{8z^{2}}}+{\frac{{\epsilon}_{-}^{2}\,M\,z^{2}}{8}}+{\frac{M}{2z}}+{\frac{M\,z}{2}}+{\textstyle{\frac{\scriptstyle
3}{\scriptstyle 4}}}M-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}(a^{2}-b^{2})(1-2y)\Big{)}\,.$ (3.12)

We may find a suitable conformal factor $\Omega^{2}$ and a redefined wave
function $\Phi$ such that the D’Alembertian of $\Phi$ in a rescaled metric
$\hat{g}_{\mu\nu}=\Omega^{2}\,\tilde{g}_{\mu\nu}$ gives rise to eqn (3.11). We
define

$\displaystyle\Phi(t,z,y,\phi,\psi)={\frac{1}{z\,\sqrt{y(1-y)}}}\,\Psi(t,z,y,\phi,\psi)\,,$
(3.13)

and, noting from (3.12) that we have

$\displaystyle\sqrt{-\hat{g}}={\frac{8\sqrt{2}\,\Omega^{5}}{\sqrt{M}\,(1+z)^{2}}}\,,$
(3.14)

it can be seen that if we choose $\Omega$ so that

$\displaystyle\Omega^{3}={\frac{(1+z)^{2}\,\sqrt{M}}{8\sqrt{2}}}\,,$ (3.15)

then the transformed equation (3.11) is equivalent to the following equation
for $\Phi$ in the $\hat{g}_{\mu\nu}$ metric:

$\displaystyle{\partial}_{\mu}\big{(}\sqrt{-\hat{g}}\,\hat{g}^{\mu\nu}\,{\partial}_{\nu}\Phi\big{)}=0\,.$
(3.16)

This can be derived from the Lagrangian

$\displaystyle{\cal L}=-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\sqrt{-\hat{g}}\,\hat{g}^{\mu\nu}\,{\partial}_{\mu}\bar{\Phi}\,{\partial}_{\nu}\Phi\,.$
(3.17)

From the resulting energy-momentum tensor

$\displaystyle
T_{\mu\nu}={\partial}_{(\mu}\bar{\Phi}\,{\partial}_{\nu)}\Phi-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\hat{g}_{\mu\nu}\,\hat{g}^{\rho\sigma}\,{\partial}_{\rho}\bar{\Phi}\,{\partial}_{\sigma}\Phi$
(3.18)

we may construct a conserved current $J^{\mu}=-K^{\nu}\,T^{\mu}{}_{\nu}$,
where $K={\frac{{\partial}}{{\partial}t}}$ is the time-translation Killing
vector. This gives rise to a conserved energy

$\displaystyle{\cal E}=\int\sqrt{-\hat{g}}\,J^{0}\,d^{4}x\,,$ (3.19)

with

$\displaystyle J^{0}$ $\displaystyle=$
$\displaystyle-\hat{g}^{t\rho}\,{\partial}_{(\rho}\bar{\Phi}\,{\partial}_{t)}\Phi+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\hat{g}^{\rho\sigma}\,{\partial}_{\rho}\bar{\Phi}\,{\partial}_{\sigma}\Phi$
(3.20) $\displaystyle=$ $\displaystyle-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\hat{g}^{tt}\,|{\partial}_{t}\Phi|^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\hat{g}^{zz}\,|{\partial}_{z}\Phi|^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\hat{g}^{yy}\,|{\partial}_{y}\Phi|^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\hat{g}^{\phi\phi}\,|{\partial}_{\phi}\Phi|^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\hat{g}^{\psi\psi}\,|{\partial}_{\psi}\Phi|^{2}\,.\quad{}$

The integrand in the energy integral (3.19) is therefore given by

$\displaystyle\sqrt{-\hat{g}}\,J^{0}$ $\displaystyle=$
$\displaystyle{\frac{1}{2z^{2}}}\,\Big{\\{}P\,|{\partial}_{t}\Phi|^{2}+z^{2}\,|{\partial}_{z}\Phi|^{2}+y(1-y)\,|{\partial}_{y}\Phi|^{2}$
(3.21)
$\displaystyle\qquad\qquad\qquad+{\frac{1}{4(1-y)}}\,|{\partial}_{\phi}\Phi|^{2}+{\frac{1}{4y}}\,|{\partial}_{\psi}\Phi|^{2}\Big{\\}}\,,$

where

$\displaystyle
P={\frac{{\epsilon}_{+}^{2}\,M}{8z^{2}}}+{\frac{{\epsilon}_{-}^{2}\,M\,z^{2}}{8}}+{\frac{M}{2z}}+{\frac{M\,z}{2}}+{\textstyle{\frac{\scriptstyle
3}{\scriptstyle 4}}}M-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}(a^{2}-b^{2})(1-2y)\,.$ (3.22)

The four-dimensional integration in eqn (3.19) is over the coordinates of the
3-sphere (with the ranges $0\leq y\leq 1$, $0\leq\phi<2\pi$,
$0\leq\psi<2\pi$), and over the transformed radial variable $z$. This ranges
over $0\leq z\leq\infty$, and as we discussed in section 2.4 the transformed
radial function $h(z)$ for any putative unstable mode goes rapidly to zero at
$z=0$ (see eqn (2.37)), and it goes rapidly to zero at $z=\infty$ (see eqn
(2.44)), ensuring the convergence of the integrals of all the terms in eqn
(3.19).

From eqns (3.4) we have $|a^{2}-b^{2}|\leq 2M$, and since $y$ lies in the
interval $0\leq y\leq 1$, it follows that the quantity $P$ satisfies $P\geq
0$. Since every term in the energy integral (3.19) is integrable for any
putative unstable mode, and each contribution is non-negative, it follows in
particular that the integral of the
${\frac{1}{2z^{2}}}\,P\,|{\partial}_{t}\Phi|^{2}$ term is bounded from above
by the conserved energy ${\cal E}$. Thus $\Phi$ cannot grow exponentially in
time, and therefore there cannot in fact exist any exponentially unstable
modes.

##  4 3-Charge Five-Dimensional STU Supergravity Black Holes

The 3-charge rotating black-hole solution in five-dimensional STU supergravity
was obtained in [12], by using a solution-generating procedure. A convenient
form of the solution was given in [16]. With minor change of notation, to
achieve consistency with our present conventions, the metric is given by

$\displaystyle
ds^{2}=(H_{1}H_{2}H_{3})^{1/3}\,({{\tilde{x}}}+{{\tilde{y}}})\,d\hat{s}^{2}\,,$
(4.1)

where

$\displaystyle d\hat{s}^{2}=-\Phi(dt+{{\cal A}})^{2}+ds_{4}^{2}\,,$ (4.2)

with

$\displaystyle
ds_{4}^{2}={\frac{d{{\tilde{x}}}^{2}}{4X}}+{\frac{d{{\tilde{y}}}^{2}}{4Y}}+{\frac{U}{G}}\,\Big{(}d\chi-{\frac{Z}{U}}\,d\sigma\Big{)}^{2}+{\frac{XY}{U}}\,d\sigma^{2}\,.$
(4.3)

The various functions above are given by333There was one typo in [16]: a
missing factor of $({{\tilde{x}}}+{{\tilde{y}}})$ in the first of the two
terms in the expression for the 1-form ${{\cal A}}$. This is corrected here.

$\displaystyle X$ $\displaystyle=$
$\displaystyle({{\tilde{x}}}+a^{2})({{\tilde{x}}}+b^{2})-2M\,{{\tilde{x}}}\,,\qquad
Y=-(a^{2}-{{\tilde{y}}})(b^{2}-{{\tilde{y}}})\,,$ $\displaystyle G$
$\displaystyle=$
$\displaystyle({{\tilde{x}}}+{{\tilde{y}}})({{\tilde{x}}}+{{\tilde{y}}}-2M)\,,\qquad
U={{\tilde{y}}}\,X-{{\tilde{x}}}\,Y\,,\qquad Z=ab\,(X+Y)\,,$
$\displaystyle{{\cal A}}$ $\displaystyle=$
$\displaystyle{\frac{2Mc_{1}c_{2}c_{3}\,({{\tilde{x}}}+{{\tilde{y}}})}{G}}\,[(a^{2}+b^{2}-{{\tilde{y}}})d\sigma-
abd\chi]-{\frac{2Ms_{1}s_{2}s_{3}}{{{\tilde{x}}}+{{\tilde{y}}}}}\,(abd\sigma-{{\tilde{y}}}d\chi)\,,$
$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle{\frac{G}{({{\tilde{x}}}+{{\tilde{y}}})^{3}\,H_{1}H_{2}H_{3}}}\,,\qquad
H_{i}=1+{\frac{2Ms_{i}^{2}}{{{\tilde{x}}}+{{\tilde{y}}}}}\,.\qquad i=1,2,3.$
(4.4)

Here $s_{i}=\sinh\delta_{i}$ and $c_{i}=\cosh\delta_{i}$, where $\delta_{i}$
are the boost parameters that correspond to turning on the three electric
charges. When $\delta_{i}=0$, the metric reduces to the five-dimensional
Myers-Perry black hole.

The coordinates $\sigma$ and $\chi$ are related to the standard azimuthal
angular coordinates $\phi$ and $\psi$ (each with period $2\pi$) by

$\displaystyle\sigma={\frac{a\phi-b\psi}{a^{2}-b^{2}}}\,,\qquad\chi={\frac{b\phi-a\psi}{a^{2}-b^{2}}}\,,$
(4.5)

as can be seen from eqn (15) in [16] after turning off the gauge coupling
constant $g$. The standard radial and angular coordinates $r$ and $\theta$ are
related to ${{\tilde{x}}}$ and ${{\tilde{y}}}$ by

$\displaystyle{{\tilde{x}}}=r^{2}\,,\qquad{{\tilde{y}}}=a^{2}\,\cos^{2}\theta+b^{2}\,\sin^{2}\theta\,.$
(4.6)

Thus ${{\tilde{x}}}$ here is the same as ${{\tilde{x}}}$ in eqn (2.6) of the
uncharged case. The coordinate ${{\tilde{y}}}$ is related to our coordinate
$y=\cos^{2}\theta$ by

$\displaystyle{{\tilde{y}}}=(a^{2}-b^{2})y+b^{2}\,.$ (4.7)

Proceeding as in the earlier uncharged case, we may separate variables and
write the radial equation in the same form as eqn (2.13), and the angular
equation in the same form as eqn (2.15). Only the expressions for the various
$\kappa$, $\Lambda$, $\beta$ and $\gamma$ coefficients will change when the
charges are turned on.

The coefficients $\kappa$, $\Lambda$, $\beta$ and $\gamma$ in the potential
for the radial equation, generalising those in (2.14) for Myers-Perry, are now
given by

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}M{\epsilon}_{+}\,{\epsilon}_{-}\,\omega^{2}\,,$ $\displaystyle\beta$
$\displaystyle=$ $\displaystyle{\frac{{{\rm
i\,}}}{4\sqrt{2M}}}\,\Big{[}{\frac{2(\Pi_{c}+\Pi_{s})M\omega-(a+b)(m+n)}{{\epsilon}_{+}}}-{\frac{2(\Pi_{c}-\Pi_{s})M\omega-(a-b)(m-n)}{{\epsilon}_{-}}}\Big{]}\,,$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle{\frac{{{\rm
i\,}}}{4\sqrt{2M}}}\,\Big{[}{\frac{2(\Pi_{c}+\Pi_{s})M\omega-(a+b)(m+n)}{{\epsilon}_{+}}}+{\frac{2(\Pi_{c}-\Pi_{s})M\omega-(a-b)(m-n)}{{\epsilon}_{-}}}\Big{]}\,,$
(4.8) $\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle\sigma+{\frac{1}{2}}-{\frac{1}{4}}M(3+2s_{1}^{2}+2s_{2}^{2}+2s_{3}^{2})\omega^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 8}}}(a^{2}-b^{2})\omega^{2}$
$\displaystyle+{\frac{[2(\Pi_{c}+\Pi_{s})M\omega-(a+b)(m+n)]^{2}}{16M{\epsilon}_{+}^{2}}}+{\frac{[2(\Pi_{c}-\Pi_{s})M\omega-(a-b)(m-n)]^{2}}{16M{\epsilon}_{-}^{2}}}\,.$

Crucial properties that held previously in the uncharged case continue to hold
here. In particular, since

$\displaystyle\Pi_{c}+\Pi_{s}\geq\Pi_{c}-\Pi_{s}\geq 1\,,$ (4.9)

together with the usual inequalities ${\epsilon}_{+}\leq{\epsilon}_{-}\leq 1$,
it follows that when $\omega$ has a positive imaginary part, the real parts of
$\beta$ and $\gamma$ will be negative.

For the angular equation, the hatted quantities $\hat{\kappa}$,
$\hat{\Lambda}$, $\hat{\beta}$ and $\hat{\gamma}$ are given by

$\displaystyle\hat{\kappa}$ $\displaystyle=$
$\displaystyle-{\frac{1}{8}}(a^{2}-b^{2})\omega^{2}\,,\qquad\hat{\beta}={\frac{n}{2}}\,,\qquad\hat{\gamma}={\frac{m}{2}}$
$\displaystyle\hat{\Lambda}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}+\sigma-{\frac{(m^{2}+n^{2})}{4}}+{\frac{(a^{2}-b^{2})\omega^{2}}{8}}\,,$
(4.10)

unchanged from the results (2.16) for the uncharged black holes.

Following the same steps as we did previously for the uncharged Myers-Perry
black hole, we find that after implementing the same integral transformation
of the radial equation as before, we again arrive at an “unseparated” equation
of the form (3.9), with the only difference being in the coefficient of
$\omega^{2}$:

$\displaystyle\Big{\\{}z^{2}\,{\partial}^{2}_{z}+y(1-y)\,{\partial}^{2}_{y}-\Big{(}{\frac{(a+b)}{4z}}+{\frac{(a-b)\,z}{4}}\Big{)}\,m\omega-\Big{(}{\frac{(a+b)}{4z}}-{\frac{(a-b)\,z}{4}}\Big{)}\,n\omega$
$\displaystyle+\Big{(}{\frac{{\epsilon}_{+}^{2}\,M}{8z^{2}}}+{\frac{{\epsilon}_{-}^{2}\,M\,z^{2}}{8}}+{\frac{(\Pi_{c}+\Pi_{s})M}{2z}}+{\frac{(\Pi_{c}-\Pi_{s})M\,z}{2}}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}M(3+2\sum_{i}s_{i}^{2})$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}(a^{2}-b^{2})(1-2y)\Big{)}\,\omega^{2}-{\frac{m^{2}}{4(1-y)}}-{\frac{n^{2}}{4y}}+{\frac{1}{4y(1-y)}}\Big{\\}}\Psi=0\,.$
(4.11)

This correspondingly implies that the components of the tilded inverse metric
$\tilde{g}^{\mu\nu}$ are unchanged except for $\tilde{g}^{tt}$, which becomes

$\displaystyle\tilde{g}^{tt}$ $\displaystyle=$
$\displaystyle-\Big{(}{\frac{{\epsilon}_{+}^{2}\,M}{8z^{2}}}+{\frac{{\epsilon}_{-}^{2}\,M\,z^{2}}{8}}+{\frac{(\Pi_{c}+\Pi_{s})M}{2z}}+{\frac{(\Pi_{c}-\Pi_{s})M\,z}{2}}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}M\big{(}3+2\sum_{i}s_{i}^{2}\big{)}$ (4.12)
$\displaystyle\quad\quad-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}(a^{2}-b^{2})(1-2y)\Big{)}\,.$

Calculating the determinant, we now find that instead of eqn (3.14) we have

$\displaystyle\sqrt{-\hat{g}}$ $\displaystyle=$
$\displaystyle{\frac{8\sqrt{2}\,e^{\delta_{1}+\delta_{2}+\delta_{3}}\,\Omega^{5}}{\sqrt{M}}}\Big{[}(e^{\delta_{1}+\delta_{2}}+e^{\delta_{3}}z)(e^{\delta_{2}+\delta_{3}}+e^{\delta_{1}}z)(e^{\delta_{1}+\delta_{3}}+e^{\delta_{2}}z)$
(4.13)
$\displaystyle\qquad\qquad\qquad\qquad(1+e^{\delta_{1}+\delta_{2}+\delta_{3}}\,z)\Big{]}^{-{\frac{1}{2}}}\,\,,$

Following the remaining steps of the previous discussion for the uncharged
case, we find that the conserved energy is given by integrating
$\sqrt{-\hat{g}}\,J^{0}$ as in equation (3.21), with the function $P$ now
given not by eqn (3.22) but instead

$\displaystyle P$ $\displaystyle=$
$\displaystyle{\frac{{\epsilon}_{+}^{2}\,M}{8z^{2}}}+{\frac{{\epsilon}_{-}^{2}\,M\,z^{2}}{8}}+{\frac{(\Pi_{c}+\Pi_{s})M}{2z}}+{\frac{(\Pi_{c}-\Pi_{s})M\,z}{2}}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}M(3+\sum_{i}2s_{i}^{2})$ (4.14)
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}(a^{2}-b^{2})(1-2y)\,.$

The same arguments that established that $P$ was non-negative in the uncharged
case show that $P\geq 0$ here also, and hence again there cannot exist any
unstable exponentially-growing modes.

##  5 Discussion

It has become apparent that many of the equations governing massless fields on
black hole spacetimes are of Heun type (in cases where the cosmological
constant is non-zero), or one of its many confluent variants, such as for non-
extreme Kerr [1], or the extreme case [4]. The differential equations we find
for massless scalar fields in the five-dimensional Myers-Perry black hole
spacetime (see (2.13) and (2.15)) are of yet another confluent Heun type. That
observation has allowed us to extend to this (and the related STU) case the
analysis originally applied to massless fields of all spin in the Kerr
spacetime [1]. Before our present work, that analysis had also been extended:
i) by using a different integration contour, to rule out unstable modes on the
real axis for the Kerr spacetime [6], ii) by considering a modified integral
transform, to deal with the extreme ($|a|=M$) Kerr black hole [4], and iii),
by looking carefully at more complicated examples, to establish the absence of
unstable modes for massless scalar fields in STU spacetimes and all more
specialized sub-cases [7]. Remarkably, the integral transform we have used
here is, effectively, an inverse of that developed for the extreme Kerr
spacetime [4]. In this context, it is also worth noting that quite different
techniques, stemming from Seiberg-Witten theory (see, for example [17]), and
based on the spectral properties of the operators involved, have been used to
discuss both Kerr quasi-normal modes[18] and Kerr-de Sitter stability [19].
The relevance of such an approach to the spacetimes we consider here is yet to
be determined.

##  6 Conclusion

We have shown that a massless scalar field has no exponentially unstable modes
in the five-dimensional Myers-Perry black hole spacetime. We have also shown
that the same is true in the five-dimensional supergravity-motivated STU
spacetimes and, previously [7], that this holds, too, for the four-dimensional
STU spacetimes. Together, these encompass a number of other special cases
which arise from restricting the parameters in these more general examples.
Although these results may serve as suggestive for the behaviour for fields of
higher spin – in particular, Maxwell fields and gravitational perturbations –
it would be useful to have some more direct indication, perhaps by writing
down (at least) the analogue of the Teukolsky equation in these more general
cases. That task currently remains for future work.

## Acknowledgements

We are grateful to Mihalis Dafermos, Harvey Reall and Jorge Santos for helpful
discussions. The authors acknowledge support from the Mitchell Institute,
Texas A&M University, during the early stages of this work. M.C. is supported
in part by DOE Grant Award de-sc0013528 and the Fay R. and Eugene L. Langberg
Endowed Chair. C.N.P. is supported in part by DOE grant DE-FG02-13ER42020.
B.F.W. is supported in part by NSF grant PHY 1607323, and thanks the Newton
Institute, Cambridge, and the Institut d’Astrophysique de Paris, for
hospitality while this manuscript was being written.

##  Appendix A Global Structure of the Myers-Perry Black Hole

Defining a coordinate $r^{*}$ by the relation

$\displaystyle
dr_{*}={\frac{(r^{2}+a^{2})(r^{2}+b^{2})\,dr}{r^{2}\,\Delta}}\,,$ (A.1)

we may introduce retarded coordinates $(u,\phi_{-},\psi_{-})$, where

$\displaystyle du=dt-dr_{*}\,,\qquad
d\phi_{-}=d\phi-{\frac{a\,(r^{2}+b^{2})\,dr}{r^{2}\,\Delta}}\,,\qquad
d\psi_{-}=d\psi-{\frac{b\,(r^{2}+a^{2})\,dr}{r^{2}\,\Delta}}\,.$ (A.2)

In terms of these, the metric (2.1) then becomes

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
du^{2}-2dr\,(du-a\sin^{2}\theta\,d\phi_{-}-b\cos^{2}\theta\,d\psi_{-})+\rho^{2}\,d\theta^{2}$
$\displaystyle+{\frac{2M}{\rho^{2}}}\,(du-a\sin^{2}\theta\,d\phi_{-}-b\cos^{2}\theta\,d\psi_{-})^{2}+(r^{2}+a^{2})\,\sin^{2}\theta\,d\phi_{-}^{2}+(r^{2}+b^{2})\,\cos^{2}\theta\,d\psi_{-}^{2}\,.$

This form of the metric is regular in the neighbourhood of future null
infinity.

We may also introduce advanced coordinates $(v,\phi_{+},\psi_{+})$ by

$\displaystyle dv=dt+dr_{*}\,,\qquad
d\phi_{+}=d\phi+{\frac{a\,(r^{2}+b^{2})\,dr}{r^{2}\,\Delta}}\,,\qquad
d\psi_{+}=d\psi+{\frac{b\,(r^{2}+a^{2})\,dr}{r^{2}\,\Delta}}\,,$ (A.4)

with respect to which the metric (2.1) becomes

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
dv^{2}+2dr\,(dv-a\sin^{2}\theta\,d\phi_{+}-b\cos^{2}\theta\,d\psi_{+})+\rho^{2}\,d\theta^{2}$
$\displaystyle+{\frac{2M}{\rho^{2}}}\,(dv-a\sin^{2}\theta\,d\phi_{+}-b\cos^{2}\theta\,d\psi_{+})^{2}+(r^{2}+a^{2})\,\sin^{2}\theta\,d\phi_{+}^{2}+(r^{2}+b^{2})\,\cos^{2}\theta\,d\psi_{+}^{2}\,.$

It can be seen from this form of the metric that it is regular as one crosses
the future horizon.

## References

  * [1] B.F. Whiting, Mode stability of the Kerr black hole, J. Math. Phys. 30, 1301 (1989) doi:10.1063/1.528308
  * [2] M. Dafermos, G. Holzegel, I. Rodnianski and M. Taylor, The non-linear stability of the Schwarzschild family of black holes, [arXiv:2104.08222 [gr-qc]].
  * [3] Y. Shlapentokh-Rothman and R. Teixeira da Costa, Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range $|a|<M$: frequency space analysis, [arXiv:2007.07211 [gr-qc]].
  * [4] R. Teixeira da Costa, Mode stability for the Teukolsky equation on extremal and subextremal Kerr spacetimes, Commun. Math. Phys. 378 (2020) no.1, 705-781 doi:10.1007/s00220-020-03796-z [arXiv:1910.02854 [gr-qc]].
  * [5] Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, Annales Henri Poincare 16 (2015), 289-345 doi:10.1007/s00023-014-0315-7 [arXiv:1302.6902 [gr-qc]].
  * [6] L. Andersson, S. Ma, C. Paganini and B. F. Whiting, Mode stability on the real axis, J. Math. Phys. 58 (2017) no.7, 072501 doi:10.1063/1.4991656 [arXiv:1607.02759 [gr-qc]].
  * [7] M. Cvetič, G.W. Gibbons, C.N. Pope and B.F. Whiting, Positive energy functional for massless scalars in rotating black hole backgrounds of maximal ungauged supergravity, Phys. Rev. Lett. 124, no.23, 231102 (2020) doi:10.1103/PhysRevLett.124.231102 [arXiv:1912.08988 [gr-qc]].
  * [8] M. Cvetič and D. Youm, Entropy of nonextreme charged rotating black holes in string theory, Phys. Rev. D 54, 2612-2620 (1996) doi:10.1103/PhysRevD.54.2612 [arXiv:hep-th/9603147 [hep-th]]
  * [9] Z. W. Chong, M. Cvetič, H. Lu and C. N. Pope, Charged rotating black holes in four-dimensional gauged and ungauged supergravities, Nucl. Phys. B 717, 246-271 (2005) doi:10.1016/j.nuclphysb.2005.03.034 [arXiv:hep-th/0411045 [hep-th]].
  * [10] A. G. P. Gomez-Lobo and J. M. Martin-Garcia, Spinor calculus on 5-dimensional spacetimes, J. Math. Phys. 50 (2009), 122504 doi:10.1063/1.3256124 [arXiv:0905.2846 [gr-qc]].
  * [11] R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172, 304 (1986) doi:10.1016/0003-4916(86)90186-7
  * [12] M. Cvetič and D. Youm, General rotating five-dimensional black holes of toroidally compactified heterotic string, Nucl. Phys. B 476, 118-132 (1996) doi:10.1016/0550-3213(96)00355-0 [arXiv:hep-th/9603100 [hep-th]].
  * [13] R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237-238 (1963) doi:10.1103/PhysRevLett.11.237
  * [14] M. Cvetič and F. Larsen, General rotating black holes in string theory: Grey body factors and event horizons, Phys. Rev. D 56, 4994-5007 (1997) doi:10.1103/PhysRevD.56.4994 [arXiv:hep-th/9705192 [hep-th]].
  * [15] K. D. Kokkotas and B. G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Rel. 2 (1999), 2 doi:10.12942/lrr-1999-2 [arXiv:gr-qc/9909058 [gr-qc]].
  * [16] Z.W. Chong, M. Cvetič, H. Lü and C.N. Pope, Non-extremal rotating black holes in five-dimensional gauged supergravity, Phys. Lett. B 644, 192-197 (2007) doi:10.1016/j.physletb.2006.11.012 [arXiv:hep-th/0606213 [hep-th]].
  * [17] K. Ito, S. Kanno and T. Okubo, Quantum periods and prepotential in $\mathcal{N}=2$ SU(2) SQCD, JHEP 08 (2017), 065 doi:10.1007/JHEP08(2017)065 [arXiv:1705.09120 [hep-th]].
  * [18] G. Aminov, A. Grassi and Y. Hatsuda, Black Hole Quasinormal Modes and Seiberg–Witten Theory, Annales Henri Poincare 23 (2022) no.6, 1951-1977 doi:10.1007/s00023-021-01137-x [arXiv:2006.06111 [hep-th]].
  * [19] M. Casals and R. T. da Costa, Hidden spectral symmetries and mode stability of subextremal Kerr(-dS) black holes, [arXiv:2105.13329 [gr-qc]].