# A Light Field Front-end for Robust SLAM in Dynamic Environments

Pushyami Kaveti1 and Hanumant Singh2 1Pushyami Kaveti is a student at Khoury
College of Computer Sciences, Northeastern University, Boston
<EMAIL_ADDRESS>Singh is with the Department of Electrical
Engineering,Northeastern University<EMAIL_ADDRESS>
###### Abstract

There is a general expectation that robots should operate in urban
environments often consisting of potentially dynamic entities including
people, furniture and automobiles. Dynamic objects pose challenges to visual
SLAM algorithms by introducing errors into the front-end. This paper presents
a Light Field SLAM front-end which is robust to dynamic environments. A Light
Field captures a bundle of light rays emerging from a single point in space,
allowing us to see through dynamic objects occluding the static background via
Synthetic Aperture Imaging(SAI). We detect apriori dynamic objects using
semantic segmentation and perform semantic guided SAI on the Light Field
acquired from a linear camera array. We simultaneously estimate both the depth
map and the refocused image of the static background in a single step
eliminating the need for static scene initialization. The GPU implementation
of the algorithm facilitates running at close to real time speeds of 4 fps. We
demonstrate that our method results in improved robustness and accuracy of
pose estimation in dynamic environments by comparing it with state of the art
SLAM algorithms.

##  I Introduction

Figure 1: Block diagram of the Light Field Front-end. Frames from the linear
camera array and the segmentation masks are fed into the depth estimation
module. The optimized depth is used to compute the refocused image.

SLAM enables robot navigation in unknown environments by accurately estimating
the robot’s pose and building the map of the world at the same time. It is
essential for collision-free navigation in various indoor and outdoor robotic
applications. Most of the research in SLAM assumes a static world, even though
the robots are expected to function in predominantly dynamic environments. The
static world assumption holds true for some applications during small scale
and short-term runs, but limits the robots from operating in the real-world.
The presence of dynamic entities like people, automobiles and bicycles in the
real world causes errors in various stages of SLAM if these objects are not
detected and dealt with. Feature matches associated with dynamic objects
result in initialization failures, errors in pose estimation and incorrect
maps of the world. If the dynamic features are included in the map they affect
loop closure and re-localization preventing re-usability of maps for long-term
applications. All these problems have motivated researchers to develop
algorithms that target dynamic environments [1][2].

Dynamic environments are usually handled by detecting dynamic content in a
variety of ways [3][4][5][6][7][8] and explicitly omitting the features
associated with the detected dynamic objects while performing SLAM. Discarding
information this way may result in errors if the dynamic content occupies a
significant portion of the image or dominates the scene in terms of feature
space. Therefore, extracting static features is imperative to estimating the
pose of the robot accurately.

In this paper, we introduce a Light Field front-end built on top of
ORBSLAM2[3] that is robust to dynamic environments. The front-end aims at
accurate pose estimation and builds the static map of the world for long term
use. While Light Fields have been researched extensively in the computer
graphics community, their applicability to mobile robotics is far more
recent[9][10][11]. A Light Field captures spatial and angular radiance in the
form of a bundle of rays coming from a point in space as opposed to a single
ray in a monocular camera. This helps us to extract the rays emerging from
partially occluded portions of the scene via Synthetic Aperture Imaging (SAI).

We apply SAI to the Light Field collected from a camera array to synthesize
the refocused image of the static background occluded by dynamic objects which
is then passed on to extract features for tracking. We rely on deep learning
based semantic segmentation for detecting a priori dynamic objects. We use the
dynamic object detections to guide the SAI and recover the static background
image essentially seeing through the dynamic objects as if they were never
present at all. Thus, our method is effective even when the dynamic objects
dominate the scene. Detecting a priori dynamic objects supports long term
mapping when the objects are temporarily static and eliminates the need for
initialization if they are actively moving. Our GPU implementation allows us
to run the front-end processing close to real-time. In the next sections we
discuss the Light Field front-end in detail and evaluate it on real datasets.

##  II Related Work

SLAM in Dynamic Environments A large body of recent research addresses dynamic
scenes by using optical flow [6], RANSAC[3], or residual cost functions
[5][12] to detect and filter dynamic outliers. Outlier rejection strategies
fail when the dynamic content constitutes a significant fraction of the scene.
Other algorithms enforce temporal and structural consistency to detect dynamic
content and omit such content from tracking and mapping. In [13], the
background is reconstructed by accumulating warped depth between consecutive
RGB-D images. [4] track changes by projecting sparse map points into the
current frame. StaticFusion [12] detects moving objects and fuses temporally
consistent data via model alignment. Temporal methods require initialization
of static map in dynamic environments. We use CNN based semantic
segmentation[14][15], which has emerged as an efficient instantaneous approach
to detect apriori dynamic objects in many recent works[8][7][16][17][18].
Unlike other methods, we not only track features belonging to the statically
segmented parts of the image but also extract and track the static features
occluded by dynamic objects from the refocused image of the static background.
Thus, we demonstrate increased accuracy and robustness even when the dynamic
content starts to dominate the scene.
Light Fields in Robotics Light Fields are a popular topic in computer vision
and graphics for refocusing and rendering[19][20][21][22], with applications
that include super-resolution[23] and depth estimation[24][25][26][27][28].
Application of Light Fields in robotics is not as developed but has been
discussed in[9][29][10]. Most of the robotics related work aims to solve the
visual odometry problem [29], and to compensate for challenging light and
weather conditions[30][10][11][31]. None of this work addresses the problem of
dynamic environments. In contrast, our efforts in semantic guided refocusing
and reconstruction specifically deal with dynamic objects in a SLAM framework.
Refocusing[20] is performed by blending rays emerging from a focal surface and
passing through a synthetic aperture without taking into account the semantics
of the rays. The graphics community’s Light Field depth reconstruction
techniques typically are aimed at increasing the accuracy of reconstruction
while trading off speed of computation. Most of the Light Field research is
targeted towards micro lenslet cameras[32] which suffers from small baseline
separation, which are far from ideal in terms of seeing through the dynamic
objects, where wide-baseline arrays are more suitable.

##  III Light Field front-end

At the heart of the Light Field front-end is our method to compute an all-in-
focus synthetic aperture image of the static background from the Light Field
captured with a linear camera array. The processing pipeline is illustrated in
fig. 1.

###  III-A Problem Setup

A calibrated linear array of K cameras forming images $I_{k}$ can be seen as a
single system with synthetic aperture where each camera contributes to a ray
passing through the aperture. Following the two-plane parameterization of
Light Field [20], the camera positions on the array are the (s,t) coordinates
which lie on the entrance plane and the pixels are the (u,v) coordinates on
the image plane. This creates a unique 4D ray (s,t,u,v) as shown in fig. 2.
Given an arbitrary focal surface F and a 3D point (X,Y,Z) on it, we can
combine all the rays of the synthetic aperture which emit from that point and
intersect the cameras into a single refocused ray.

Suppose we have the depth map of the static background. Let $d^{ref}$ and
$x^{ref}$ be the depth and coordinates of a pixel in the reference view, then
the rays $x^{k}$ in the other cameras $k$ can be sampled using:

$x^{k}=\pi_{k}[R_{k}|t_{k}]\pi_{ref}^{-1}(x^{ref},d^{ref})$ (1)

where $\pi_{k}$ is a projective mapping between the a 3D point in space and 2D
pixel coordinates on the image plane, and $R_{k}$ and $t_{k}$ are the rotation
and translation of the $k^{th}$ camera. These rays form our synthetic aperture
and are typically combined by applying an average filter[21, 20] giving equal
weight to all the rays.This helps to generate images with varying focus and
depth of field [19][22]. In free-space, all the camera rays correspond to the
same point on the focal

Figure 2: Light Field refocusing. Given a 3D point on the focal surface the
static rays projected into other cameras are combined to create a single
refocused image.

surface, but, in case of occlusions, some rays get obstructed. Given a large
synthetic aperture, choosing a focal surface behind the foreground object
brings the background into focus and causes the object to blur. This was
applied in [10] to see through rain and snow by manually selecting a focal
plane on the road signs for autonomous navigation. Instead of averaging all
the rays, we design a filter which selects only the rays that map to static
pixels in the respective views obtained from semantic segmentation. This way
the dynamic objects in the foreground are completely eliminated. Denoting
whether a pixel $x^{k}$ belongs to the static or dynamic content with $s^{k}$,
the refocused image $I^{*}$ can be calculated as follows:

$I^{s}=\frac{\sum I(x^{k})*(s^{k}=static)}{\sum(s^{k}=static)}$ (2)

As we can observe in eq. 1, the depth map of the static background is required
to compute the refocused image.

###  III-B Probabilistic Graphical Model

The main challenge in the depth estimation of the static background is to
reconstruct parts of the reference view occluded by dynamic objects. Given
that an array of cameras are acting as the source of image data, we need to
estimate the depth map $D^{*}$ and refocused image $I^{*}$ of static
background for a reference camera view $X^{ref}$. The key to finding correct
depth is to choose the subset of camera views which will exclude the pixels
corresponding to dynamic objects while computing the image correspondences. We
apply MaskRCNN[14] on camera images to compute per-pixel semantic labels. The
CNN model assigns a probability to pixels being segmented
$\\{S_{P}\in\mathbb{R}^{k}|0\leq S_{P}^{k}\leq 1\\}$, where $S_{P}^{k}$ is the
probability in $k^{th}$ view. We introduce a set of binary random variables
$S^{k}$ assigned to each pixel i in camera k which take values either 0 or 1
representing dynamic or static pixels based on a threshold on the probability.
The probabilistic graphical model is shown in the fig. 3. The disparity $d$
can be obtained by maximizing the posterior probability
$p(d|\mathbf{x},x^{ref},T,Sp)$ (MAP estimate) as follows:

$\displaystyle d^{*}=argmax_{d}\;\;p(d|\mathbf{x},T,S_{P},x^{ref}),$
$\displaystyle p(d|\mathbf{x},T,S_{P},x^{ref})\propto
p(d,\mathbf{x}|T,S_{P},x^{ref})$

Figure 3: The Probabilistic Graphical Model. Triangulation $T$ and
segmentation $S$ are pre-computed. $d$ is disparity of a pixel in the
reference view, while $x$ is a set of K reprojections into other cameras.

|  |
---|---|---
|  |

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Various steps associated with the static background disparity
estimation. (a) A sparse set of support points only on the static background
in the reference view. (b) Support points from other views. (c) Reprojected
support points in the reference frame. The support points are colored based on
their disparities (red: close, blue : Far). (d) Final triangulation of the
static background. (e) The coarse disparity map formed by the piece-wise
planar prior. (f) The final refined disparity map of the static background.

###  III-C Disparity Estimation

According to the graphical model, the joint probability factorizes into
disparity prior and pixel likelihood. Now, disparity estimation is a task of
picking a value that optimizes:

$\operatorname*{arg\,max}_{d}\;\log\underbrace{p(\mathbf{x}|d,x^{ref},S)}_{\text{pixel
likelihood}}+\log\underbrace{p(d|T,x^{ref})}_{\text{disparity prior}}$ (3)

####  III-C1 Disparity Prior

We use piece-wise planar prior over the disparity space $p(d|T,x^{ref})$ by
forming triangulation on a sparse set of points similar to ELAS [33]. This
helps with poorly textured regions and gives a coarse disparity map, thus
reducing the search space during optimization.

First a sparse set of points are selected on a regular grid in the reference
image and their descriptor vectors are created from the responses of a 5x5
window sobel filter. Since some of these points lie on the dynamic objects we
use segmentation mask to filter them. The remaining points are matched along
the full range of disparities on the epipolar lines of its neighbouring image.
These points together with their matched disparities form the static support
points. We now exploit the Light Field data to extract support points of the
static background occluded by the dynamic objects in reference view, but are
visible in other views. For each image other than the reference view we
extract the support points by matching the sobel descriptors of the static
points against its neighbouring view and choose the points that re-project
onto the parts of the reference image dynamic objects. We then filter out
duplicates and inconsistent support points based on their disparity values.
The final set of support points are used as vertices for the delaunay
triangulation. The prior on disparity $p(d|T,x^{ref})$ is modeled as
combination of a uniform distribution and a sampled Gaussian centered about
the interpolated disparity from the triangulation, for support points in the
neighbourhood[33]. All the above steps are shown in fig. 4 (a-e).

####  III-C2 Pixel Likelihood

The triangulation prior provides a coarse map of the interpolated disparities
and a set of candidate disparities. which we have used to generate accurate
estimates based on the pixel likelihood. Given the coordinate $x^{ref}$ in the
reference frame and a candidate disparity $d$, the corresponding coordinates
of the source images $x^{k}$ can be determined using a warping function
$\mathcal{W}_{k}(x^{ref},d)$. This warping function represents a homography
which can be computed from the reference and $k^{th}$ camera matrices [34].
This allows us to use generalized disparity space suitable for multi-view
camera configuration. We work in the disparity space as opposed to depth space
because it facilitates discrete optimization that results in faster
computation. The pixel likelihood relies on the fact that for correct
disparity value there is a high probability that warped static pixels $x^{k}$
in the source images will have photometric consistency. So, we design our
likelihood function as a Laplace distribution restricted to static pixels such
that the variance between them is minimum.

$p(x^{1}...x^{K}\mid d,x^{ref},S)\\\
\propto\begin{cases}exp\big{(}-\beta\;Var(f(x^{1}),...,f(x^{K}))\big{)}\\\
\;\;\;\;\;for\;\;x^{k}=\mathcal{W}_{k}(x^{ref},d)\;\\\
0\;\;\;\;\;otherwise\end{cases}$

Where the variance is computed on the feature descriptors $f(x^{K})$ of the
pixels that are classified as static:

$Var(...)=\frac{\sum_{k=1}^{K}(S^{k}=1)[f(x^{k})-\hat{f}]^{2}}{\sum(S^{k}=1)}$

with $\hat{f}=\frac{\sum_{k}(S^{k}=1)f(x^{k})}{\sum_{k}(S^{k}=1)}$ as the mean
of the descriptors. We use the descriptors created from 3x3 sobel filter
responses similar to ELAS[33]. The dynamic pixels don’t contribute to the
likelihood as they can have arbitrary intensities. Plugging in the disparity
prior and pixel likelihood into eq. 3 and taking negative logarithm gives the
energy function to be minimized

Figure 5: Top row: Original images in the reference view. Bottom row:
Refocused image of the static background. First column shows the result with
some low textured regions. The second column shows the same in the presence of
a lot of detail. Third column shows the case when dynamic objects dominate the
scene.

###  III-D Refocused Image Synthesis

Once the optimization converges we perform gap interpolation and median filter
on the estimated disparity image to get rid of any speckle noise obtain a
smooth map. The filtered disparity and segmentation masks are used to compute
the refocused image of the static background using eq. 2.

##  IV Experimental Analysis

In this section, we evaluate the Light Field front-end on real-world dynamic
sequences.
Dataset Acquisition and Setup Light Field acquisition is typically done via a
large array of cameras [19] or a micro lenslet camera [32]. The former is
impractical to mount on most mobile robots where as the latter suffers from
limited parallalax due to small baseline separation. Based on these
constraints we use a custom built 5-camera linear array and mount it on a
Clearpath Robotics Husky UGV running ROS for collecting Light Field data. The
array uses 1.6MP Pointgrey BlackflyS global shutter cameras which are hardware
synced for synchronized image capture and calibrated using the Kalibr[35]
package. All our experiments were conducted on an Alienware gaming laptop with
a 8GB GEForce GTX 1080 GPU and Intel core i7 processor. We collected multiple
indoor datasets with people moving in the scene to depict dynamic
environments. The ground truth trajectories were obtained using optitrack
prime x13 cameras [36] which are accurate to a millimeter.

Qualitative results We show the results of our refocused image synthesis on a
few Light Field snapshots from our datasets in fig. 5. The top row shows the
original images and the bottom row features the corresponding refocused
images. As we can see in fig. 5:right, our algorithm does a good job of
reconstructing a complex scene when the people are obstructing our view of the
majority of the background. The left image in fig. 5 presents a case of low
textured regions (blue curtain and floor) and shows that the algorithm can
handle such a situation quite well due to the piece-wise planar prior. In the
center indoor scene we can observe the quality of the reconstruction in the
presence of significant detail. In the reconstructions there are some portions
of the image which are not reconstructed though it is belongs to a dynamic
object eg: some portions of upper body in fig. 5:right. This happens when
there is not enough paralallax and rays cannot reach the background and is a
function of baseline separation. However, we can recover the majority of the
static scene as seen in the images.

| Less Dynamics (A) | Moderate Dynamic (B) | More Dynamics (C)
---|---|---|---
Algorithm |  | ATE[m]
---
mean , median
| RTE
---
[%]
| RRE
---
[∘/m]
| ATE[m]
---
mean , median
| RTE
---
[%]
| RRE
---
[∘/m]
| ATE[m]
---
mean , median
| RTE
---
[%]
| RRE
---
[∘/m]
ORBSLAM2 Mono | 0.021 , 0.019 | 3.474 | 0.404 | 0.399 , 0.019 | 50.987 | 1.238 | 0.987 , 1.000 | 138.24 | 10.798
ORBSLAM2 Stereo | 0.792 , 0.739 | 106.917 | 8.624 | 0.563 , 0.584 | 151.94 | 10.51 | 1.028 , 1.148 | 126.78 | 9.057
DynaSLAM | 0.068 , 0.067 | 5.671 | 0.724 | 0.057 , 0.044 | 6.22 | 0.494 | 0.065 , 0.046 | 5.126 | 0.846
Light Field front-end | 0.022 , 0.021 | 3.492 | 0.439 | 0.023 , 0.018 | 3.66 | 0.57 | 0.021 , 0.020 | 3.943 | 0.733
TABLE I: Comparison of the ATE [m], RTE(%) and RRE(∘/m) of various visual SLAM
algorithms with Light Field front-end in dynamic scenes. The lowest error
values are shown in bold font for convenience

|  |
---|---|---

(a)

(b)

(c)

Figure 6: (a)Example trajectory plot for ORBSLAM2 Mono, Stereo, DynaSLAM and
Light Field front-end compared against the ground truth obtained from
optitrack mocap system. (b) top: The Absolute Trajectory Error (ATE) for the
datasets A,B,C.center: The overall RTE (%) and bottom: RRE (∘/m) for different
sub trajectory lengths. (c) Trajectories of ORBSLAM2 Mono, Stereo and DynaSLAM
compared with that of Light Field front-end for the dataset with loop closure.

Quantitative Analysis The Light Field front-end is compared with other state
of the art SLAM algorithms viz., ORBSLAM2[3] for baseline comparison and
DynaSLAM[7] as it also performs static image synthesis in dynamic environments
via inpainting. However, the inpainted images are not used for tracking. We
demonstrate the effectiveness of our method by comparing the accuracy of pose
estimation using Absolute Trajectory Error (ATE) and Relative Pose Error (RPE)
metrics. We run the algorithms 5 times on each dataset and take the median
errors to account for the non-deterministic nature of system as suggested in
[3]and [7]. The evaluation is conducted on three indoor sequences collected
with less, moderate and significant amounts of dynamic objects by roughly
moving along the same path in each sequence.

The ATE and RPE for translational and rotational components for the three
cases are computed using the RPG trajectory evaluation package[37] and are
shown in table I. In fig. 6 (a) the estimated trajectories are aligned and
scaled with respect to the ground truth and overlaid on a plot. We can see
that Light Field front-end estimates the trajectory closest to the ground
truth amongst other algorithms. We observe that ORBSLAM2 Monocular version
shows slightly better accuracy than the Light Field front-end in the sequence
with less dynamic content due to its robust initialization and outlier
rejection strategy. Even though the initialization phase helps with accurate
tracking it also often delays bootstrapping in the presence of dynamic
objects. This results in missing parts of the estimated trajectory. The
performance deteriorates as the proportion of dynamic content increases
because the outlier rejection is no longer effective. ORBSLAM2 stereo shows
highest errors across all the datasets because the system is initialized from
the first frame causing the dynamic keypoints to be tracked.

Both DynaSLAM and Light Field front-end outperform ORBSLAM2(Mono and Stereo)
as the dynamic content increases. Our Light Field front-end achieves better
accuracy than DynaSLAM in all the datasets except for the relative rotational
error (RRE) in dataset b. However we achieve a lower overall RTE and RRE per
meter computed for all the datasets when compared to DynaSLAM as shown in fig.
6 (b). While DynaSLAM maintains consistent ATE and RTE values across the three
datasets, we can see from fig. 6 (b) that the spread of ATE increases with the
dynamic content. This is because in DynaSLAM the features corresponding to the
dynamic objects are discarded and only the remaining features are used in the
low-cost tracking module for pose estimation. This can introduce errors in
tracking when the dynamic features dominate the scene. The Light Field front-
end on the other hand demonstrates consistent performance across datasets
because instead of discarding dynamic key points associated with segmented
pixels we recover the static background, thus gaining the support of static
features.

We further prove that our method is effective in loop closures in the presence
of dynamic objects as shown in the fig. 6(d). This sequence is a rectangular
path recorded with people present in the view at the start of the loop. These
people, ”dyamic objects”, move as the trajectory progresses and finally go out
of view when we come back to finish the loop. Dynamic objects introduce errors
in pose estimation at different points along the trajectory. Monocular
ORBSLAM2 experiences extreme errors due to bad initialization. Stereo ORBSLAM2
shows errors initially and DynaSLAM loses tracking at the middle of the
trajectory; Both of them recover at a later point but none of the other
algorithms achieve loop closure when we come back to the starting point
because the feature distribution has changed.

Dataset | depth map [ms] | refocusing[ms] | Total Avg Time[ms]
---|---|---|---
Dataset A | 153.354 | 13.297 | 208.717
Dataset B | 155.482 | 14.357 | 233.918
Dataset C | 159.569 | 14.518 | 246.831
TABLE II: Average time taken by various processing steps.

Finally, we report the average computational time for carrying out the
different stages of the front-end processing for various sequences in table
II. To speed up processing we refocus only the dynamic pixels as the other
pixels already have the static background intensities. Thus, the computational
complexity scales with respect to the amount of dynamic content. Due to the
GPU implementation we can achieve speed of 4.35 fps facilitating close to
real-time performance.

##  V Conclusion

We presented a Light Field SLAM front-end where we compute the depth map and
synthetic aperture image of the static background using semantic segmentation
of apriori dynamic objects. The synthesized image allows us to extract static
features occluded by dynamic objects resulting in better tracking accuracy and
robustness. This approach performs well even when the scene is dominated by
dynamic objects, eliminates the need for an initialization phase, and enables
parallel processing resulting in close to real-time executions at 3 fps on a
laptop with a GPU. We show promising results by evaluating our algorithm on
real-world datasets collected with dynamic content. In the future, we would
like to tightly couple the apriori semantic knowledge with the dynamics of the
scene to detect actively moving objects and develop a pose graph back-end for
dynamic Light Field SLAM.

## ACKNOWLEDGMENTS

This work was funded in part by ONR Grant N00014-19-1-2131. We would like to
thank S Ramalingam, A Techet and A Bajpayee for their guidance and NU Field
Robotics lab for helping with data collection.

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aainstitutetext: Institut de Physique Théorique, Université Paris Saclay, CEA,
CNRS,
F-91191 Gif-sur-Yvette, Francebbinstitutetext: Dipartimento di Fisica,
Università di Roma “Tor Vergata” & INFN Roma 2,
Via della Ricerca Scientifica 1, 00133, Roma, Italy

# Non-spinning tops are stable

Iosif Bena a, b Giorgio Di Russo b Jose Francisco Morales b Alejandro Ruipérez
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider coupled gravitational and electromagnetic perturbations of a
family of five-dimensional Einstein-Maxwell solutions that describes both
magnetized black strings and horizonless topological stars. We find that the
odd perturbations of this background lead to a master equation with five
Fuchsian singularities and compute its quasinormal mode spectrum using three
independent methods: Leaver, WKB and numerical integration. Our analysis
confirms that odd perturbations always decay in time, while spherically
symmetric even perturbations may exhibit for certain ranges of the magnetic
fluxes instabilities of Gregory-Laflamme type for black strings and of Gross-
Perry-Yaffe type for topological stars. This constitutes evidence that
topological stars and black strings are classically stable in a finite domain
of their parameter space.

##  1 Introduction

The physics of black holes is responsible for some of the deepest puzzles in
our quest to formulate a unified quantum theory of gravity. On one hand, black
holes have an entropy proportional to the area of the event horizon in Planck
units, and hence, according to Statistical Mechanics, an enormous number of
states ($e^{10^{90}}$ for the Sgr.A black hole at the center of the Milky
Way). On the other hand, uniqueness theorems indicate that General Relativity
is not able to distinguish any of these states and this, in turn, leads to
violations of quantum unitarity. It is also possible to argue Mathur:2009hf ;
Almheiri:2012rt ; Guo:2021blh  that the only way to avoid such violations is
if these states give different physics from that of the classical General-
Relativity black-hole solution at the scale of the event horizon.

However, constructing such states is no easy feat: as the black-hole horizon
is null, any horizon-sized object we can construct using normal four-
dimensional matter will immediately fall in. The only mechanism to avoid
gravitational collapse in a classical theory is to use (or mimic) higher-
dimensional theories with nontrivial fluxes wrapping topologically-nontrivial
cycles Gibbons:2013tqa ; deLange:2015gca , of the type one naturally finds in
String Theory.

Furthermore, the black hole horizon is perhaps the only thing in our universe
that grows when gravity becomes stronger. Hence, if one wants to construct
horizon-sized extreme-compact-objects (ECO’s) that replace the black hole,
these objects must contain non-perturbative solitonic degrees of freedom,
which become lighter and larger when gravity becomes stronger.

For supersymmetric and extremal-non-supersymmetric black holes there is an
almost-20-year history of string-theory and supergravity constructions of such
horizon-sized ECO’s111See Bena:2022rna  and references therein, which are also
known as microstate geometries, fuzzballs geometries, or “topological stars"
Bena:2013dka . The string-theory ingredients that enter their construction
have exactly the properties needed to avoid collapse and grow when gravity
becomes stronger. However, constructing and analyzing topological stars for
non-supersymmetric black holes is much more challenging, requiring in general
solving non-linear PDE’s that, absent supersymmetry, do not factorize.

Besides some artisanal solutions Jejjala:2005yu ; Bena:2009qv , there are now
three systematic routes to build such solutions. The oldest is the floating-
JMaRT factorization Bossard:2014ola ; Bossard:2014yta , which can in principle
produce non-extremal rotating solutions, but which unfortunately does not seem
to produce solutions that have the same charges and angular momenta as non-
extremal black holes with a macroscopic horizon Bena:2015drs ; Bena:2016dbw ;
Bossard:2017vii . The second is to use numerics in a consistent truncation to
three-dimensional supergravity Mayerson:2020tcl  to produce microstrata
Ganchev:2021pgs ; Ganchev:2021ewa ; Ganchev:2023sth ; Houppe:2024hyj  that
have the same charges as non-extremal asymtptotically-AdS rotating black
holes. The third is to use the Bah-Heidmann factorization Bah:2020pdz  to
produce non-extremal non-rotating multi-bubble topological stars, which can
have both flat-space and AdS asymptotics Bah:2020ogh ; Bah:2021owp ;
Bah:2021rki ; Bah:2021irr ; Heidmann:2021cms ; Bah:2022pdn ; Bah:2022yji ;
Bah:2023ows . Absence of rotation notwithstanding, this later method appears
to be the most prolific at generating solitonic solutions with non-extremal
black-hole charges and mass, including Schwarzschild microstate Bah:2022yji ;
Bah:2023ows .

The main question about these non-extremal microstate geometries is whether
they are stable or unstable, and what are the physical implications of their
stability or absence thereof. Since the asymptotically-AdS3 solutions are dual
to coherent CFT states that have both right movers and left movers, one can
argue that generic solutions with a large number of identical CFT strands
should be unstable Chowdhury:2008bd , much like the JMaRT solution is
Cardoso:2005gj ; Chowdhury:2007jx ; Bianchi:2023rlt . However, this argument
does not extend to solutions without an AdS near-horizon region, such as
Schwarzschild microstates.

The purpose of this paper is to investigate the stability under coupled
gravitational and electromagnetic perturbations of the simplest spherically-
symmetric topological star, which can be constructed as four-dimensional
Euclidean Schwarzschild (ES) solution times time, to which one adds magnetic
flux on the ES bolt. The solution is specified by two parameters $r_{s}$ and
$r_{b}$ that interpolate between a magnetic black string solution with a
finite-area event horizon when $r_{s}>r_{b}$ and a smooth horizonless solution
one when $r_{s}<r_{b}$. Throughout this paper we will call the former the
black string and the later the top star. The top star solution is one of the
key ingredients in the building of the more general multi-bubble Bah-Heidmann
solutions, and hence our investigation is the first step in a programme to
determine the stability or instability of these solutions and the physical
consequences thereof222The stability of top stars under scalar perturbations
was established in Bianchi:2023sfs ; Heidmann:2023ojf ; Cipriani:2024ygw ..

The equations governing the perturbation around the black string and top star
are the same, and couple the electromagnetic fluctuations to the gravitational
ones. They separate into a rather intractable parity-even sector and a more
tractable parity-odd sector, on which we focus in this paper. We find that in
this sector the perturbations separate into two coupled systems of ODE’s. We
show that one of these systems boils down to two independent second-order
ODE’s for the so-called _master variables_. These ODE’s have five Fuchsian
singularities, and cannot be mapped to a Heun equation or confluent version
thereof. Hence, they are harder to solve then the scalar perturbations of
black holes Regge:1957td ; Teukolsky:1973ha , of D-brane bound states
Bianchi:2021mft ; Bianchi:2021mft , and of topological stars Bianchi:2023sfs ;
Heidmann:2023ojf ; Cipriani:2024ygw , where the linear dynamics is described
by confluent forms of the Heun equation.

We primarily rely on the Leaver method to find the quasinormal (QNM)
frequencies of topological stars and magnetic black strings. The then verify
the results against those obtained through direct numerical integration and
WKB analysis. The WKB approximation works well when the orbital number of the
perturbation is large. Direct numerical integration of the differential
equation produces reliable results for frequencies with a small imaginary
part. Our calculations show that QNM frequencies of odd perturbations have
always negative imaginary part, so they represent modes decaying in time.

For even perturbations, we are able to separate the equations only for
spherically symmetric perturbations and find stable solutions for all
topological stars with $r_{s}<2r_{b}$ and black strings with $r_{b}<2r_{s}$ in
agreement with Bah:2021irr ; Stotyn:2011tv . Our analysis provide a unifying
picture that interpolates between the Gross-Perry-Yaffe-type instability of
top stars with zero or small magnetic fields Gross:1982cv ; Witten:1981gj ;
Bah:2021irr ; Stotyn:2011tv  and the Gregory-Laflamme instability
Gregory:1993vy ; Gregory:1994bj  of magnetized black strings with zero or
small magnetic charge Miyamoto:2007mh . Both these instabilities manifest
themselves through the existence of QNMs that blow up in time.

The key result of our calculation is that in the regimes of parameters where
these instabilities are absent, all other perturbations decay in time. Hence,
in these regimes, both the top star and the magnetized black strings are
stable!

In Section 2 we derive the system of ordinary differential equations governing
the odd sector of the coupled gravitational and electromagnetic perturbations.
In Section 3 we compute the spectrum of QNM frequencies for odd perturbations.
In Section 4 we determine the regimes of Gregory-Laflamme instability.
Finally, in Section 5 we present some conclusions and future directions.

Note added: When this work was nearly complete, we were informed that another
group was working independently on the same problem Paolo-new . Although there
is significant overlap between our results, our analysis and that of Paolo-new
also focus on different aspects and numerical methods and are therefore
complementary to each other. We have compared some of our numerical results to
those of Paolo-new , finding very good agreement.

##  2 Gravitational and electromagnetic perturbations

We consider solutions of Einstein-Maxwell theory in five dimensions describing
magnetically-charged topological stars and black strings. Our goal is to study
linear perturbations of the metric and gauge field around these backgrounds.
The study of perturbations around magnetically charged backgrounds poses a
technical problem well known in the literature on black-hole perturbation
theory Gerlach:1980tx ; Kodama:2003kk ; Pereniguez:2023wxf : the usual
decoupling between even and odd perturbations does not occur. Here we
circumvent this problem by dualizing the vector field into a two-form
potential $C_{\mu\nu}$ with field strength
$F_{\mu\nu\rho}=3\partial_{[\mu}C_{\nu\rho]}$. In terms of this field, the
equations of motion are

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(F_{\mu\alpha\beta}F_{\nu}{}^{\alpha\beta}-\frac{1}{6}g_{\mu\nu}F_{\alpha\beta\gamma}F^{\alpha\beta\gamma}\right)\,,$
(2.1) $\displaystyle\nabla_{\mu}F^{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle 0\,.$ (2.2)

We study linear perturbations around solutions of (2.1) and (2.2) with metric
${\bar{g}}_{\mu\nu}$ and two-form potential ${\bar{C}}_{\mu\nu}$ given by
Horowitz:1991cd ; Bah:2020ogh ; Bah:2020pdz

$\displaystyle\mathrm{d}{\bar{s}}^{2}$ $\displaystyle=$ $\displaystyle-
f_{s}(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f_{s}(r)f_{b}(r)}+r^{2}\left(\mathrm{d}\theta^{2}+\sin\theta^{2}\mathrm{d}\phi^{2}\right)+f_{b}(r)\mathrm{d}y^{2}\,,$
(2.3) $\displaystyle{\bar{C}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3r_{s}r_{b}}}{r}\,\mathrm{d}t\wedge\mathrm{d}y\,,$
(2.4)

where $r_{s}$, $r_{b}$ are some real positive numbers and

$f_{s}(r)=1-\frac{r_{s}}{r}\,,\hskip 28.45274ptf_{b}(r)=1-\frac{r_{b}}{r}\,.$
(2.5)

There are two regimes of interest:

$\displaystyle{\rm Black\leavevmode\nobreak\ string}:$ $\displaystyle\qquad
r_{b}<r_{s}$ $\displaystyle{\rm Topological\leavevmode\nobreak\ star}:$
$\displaystyle\qquad r_{s}<r_{b}$ (2.6)

The first is called the black string regime Horowitz:1991cd , in which the
solution has an event horizon at $r=r_{s}$. In the second, the solution
describes a topological star Bah:2020ogh ; Bah:2020pdz , which is a smooth
horizonless solution provided the coordinate $y$ is periodically identified
with period

$y\sim y+\frac{4\pi r_{b}^{3/2}}{\sqrt{r_{b}-r_{s}}}\,.$ (2.7)

For this periodicity the spacetime smoothly ends at the hypersurface
$r=r_{b}$. The geometry near $r=r_{b}$ is
${\mathbb{R}}_{t}\times{\mathbb{R}}^{2}\times{\mathbb{S}}^{2}$.

###  2.1 Linear perturbations

We consider perturbations of the background metric and two-form potentials,

$g_{\mu\nu}={\bar{g}}_{\mu\nu}+h_{\mu\nu}\,,\hskip
28.45274ptC_{\mu\nu}={\bar{C}}_{\mu\nu}+c_{\mu\nu}\,.$ (2.8)

Following Regge:1957td , we separate the perturbations according to their
transformations under parity in even- and odd-types. For the background
solutions we are considering, even and odd perturbations do not couple.
Therefore, we can study them separately. Here we focus on odd perturbations,
which are considerably simpler. These are given by

$\displaystyle h_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}\,$
$\displaystyle=$ $\displaystyle\,2\,e^{-{\rm i}\omega
t}\sum_{a=t,r,y}h_{a}(r)\mathrm{d}x^{a}\left[-\frac{1}{\sin\theta}\frac{\partial
Y_{\ell m}}{\partial\phi}\mathrm{d}\theta+\sin\theta\frac{\partial Y_{\ell
m}}{\partial\theta}\mathrm{d}\phi\right]\,,$ (2.9) $\displaystyle
c_{\mu\nu}\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}\,$ $\displaystyle=$
$\displaystyle\,2\,e^{-{\rm i}\omega
t}\sum_{a=t,r,y}c_{a}(r)\mathrm{d}x^{a}\wedge\left[-\frac{1}{\sin\theta}\frac{\partial
Y_{\ell m}}{\partial\phi}\mathrm{d}\theta+\sin\theta\frac{\partial Y_{\ell
m}}{\partial\theta}\mathrm{d}\phi\right]\,,$ (2.10)

where $Y_{\ell m}(\theta,\phi)$ are the spherical harmonics satisfying

$\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}+\ell(\ell+1)\right]Y_{\ell
m}=0\,.$ (2.11)

Since the background is spherically symmetric there is no dependence on $m$,
so from now on we set $m=0$. In addition, we consider no dependence on the
coordinate $y$ for the time being. Plugging (2.9) and (2.10) into the field
equations, (2.1) and (2.2), and expanding to linear order in the
perturbations, one finds two decoupled systems of ordinary differential
equations: one for the $(h_{t},h_{r},c_{y})$ perturbations

$\displaystyle
h_{r}^{\prime}(r)+\left[\frac{r(r_{s}+r_{b})-2r_{s}r_{b}}{r(r-r_{s})(r-r_{b})}\right]h_{r}(r)+\frac{ir^{3}\omega
h_{t}(r)}{\left(r-r_{b}\right)\left(r-r_{s}\right){}^{2}}=0\,,$ $\displaystyle
h_{t}^{\prime}(r)-\frac{2h_{t}(r)}{r}+{\rm
i}\left[\frac{\omega^{2}r^{3}{-}\left(\ell{+}2\right)\left(\ell{-}1\right)\left(r{-}r_{s}\right)}{\omega
r^{3}}\right]h_{r}(r)+\frac{\sqrt{3r_{b}r_{s}}c_{y}(r)}{r(r-r_{b})}=0\,,$
(2.12) $\displaystyle
c_{y}^{\prime\prime}(r){+}\frac{r_{s}c_{y}^{\prime}(r)}{r\left(r{-}r_{s}\right)}{+}\left[\frac{\left(r^{5}\omega^{2}-\ell(\ell{+}1)r^{2}\left(r{-}r_{s}\right){+}3r_{b}r_{s}\left(r_{s}{-}r\right)\right)}{r^{2}\left(r{-}r_{b}\right)\left(r{-}r_{s}\right){}^{2}}\right]c_{y}(r)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad{+}\frac{i\sqrt{3}(\ell+2)\left(\ell-1\right)\sqrt{r_{b}r_{s}}h_{r}(r)}{r^{4}\omega}=0\,,$

and one for $(c_{t},c_{r},h_{y})$ perturbations

$\displaystyle
c_{r}^{\prime}(r)+\left[\frac{r\left(r_{b}+r_{s}\right)-2r_{b}r_{s}}{r\left(r-r_{b}\right)\left(r-r_{s}\right)}\right]c_{r}(r)+\frac{ir^{3}\omega
c_{t}(r)}{\left(r-r_{b}\right)\left(r-r_{s}\right){}^{2}}=0\,,$ $\displaystyle
c_{t}^{\prime}(r)+\frac{\left[\ell(\ell+1)(r-r_{s})-r^{3}\omega^{2}\right]c_{r}(r)}{ir^{3}\omega}-\frac{\sqrt{3}\ell(\ell+1)\sqrt{r_{b}r_{s}}h_{y}(r)}{r\left(r-r_{b}\right)}=0\,,$
(2.13) $\displaystyle
h_{y}^{\prime\prime}(r){+}\frac{r_{s}h_{y}^{\prime}(r)}{r(r{-}r_{s})}{+}\left[\frac{\left(r^{5}\omega^{2}{-}\left(r{-}r_{s}\right)\left(r\left(\ell(\ell{+}1)r{-}2r_{b}\right){+}3r_{b}r_{s}\right)\right)}{r^{2}\left(r{-}r_{b}\right)\left(r{-}r_{s}\right){}^{2}}\right]h_{y}(r){-}\frac{i\sqrt{3}c_{r}(r)\sqrt{r_{b}r_{s}}}{r^{4}\omega}=0\,.$

Before studying the complete system, we discuss two limits of interest.

###  2.2 Schwarzschild black string and soliton limits

The limits are the $r_{b}\to 0$ and the $r_{s}\to 0$ limits. In the first the
background solution corresponds to the “Schwarzschild black string” (meaning a
four-dimensional Schwarzschild solution times a transverse direction). The
second limit corresponds to a horizonless solution which we are going to call
the Schwarzschild soliton. These solutions are mapped one into another via a
double Wick rotation along the coordinates $t$ and $y$ and an exchange of
$r_{s}\leftrightarrow r_{b}$. This does not imply, as we are going to see,
that the equations for the perturbations around these backgrounds coincide (up
to an exchange of $r_{s}\leftrightarrow r_{b}$). The reason is that here we
are considering perturbations with no dependence in $y$. Therefore, the
perturbations are not mapped one into each other by the double Wick rotation,
even if the background solutions are.

  * •

Schwarzschild black string: $r_{b}\to 0$. In this limit the full system
reduces to four independent second-order differential equations for
$h_{r},h_{y},c_{r},c_{y}$. When recast into Schrödinger form, they are given
by

$\Psi_{s}^{\prime\prime}(r)+Q_{s}(r)\Psi_{s}(r)=0\,,$ (2.14)

where $Q_{s}$ is the spin-$s$ Regge-Wheeler potential Regge:1957td

$Q_{s}(r)=\frac{r^{4}\omega^{2}-\ell(\ell+1)r(r-r_{s})+s^{2}r_{s}\left(r-r_{s}\right)+\frac{r_{s}^{2}}{4}}{r^{2}(r-r_{s})^{2}}\,,$
(2.15)

and

$\Psi_{s}\sim\left\\{\begin{array}[]{ccc}c_{r}&\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\hfil&s=0\\\
h_{y},c_{y}&&s=1\\\ h_{r}&&s=2\end{array}\right.\,.$ (2.16)

  * •

Schwarzschild soliton: $r_{s}\to 0$. The four differential equations can be
cast in Schrödinger form,

$\Psi_{\epsilon}^{\prime\prime}+Q_{\epsilon}(r)\Psi_{\epsilon}=0\,,$ (2.17)

with

$Q_{\epsilon}=\frac{r^{3}\omega^{2}-\ell(\ell+1)r+2\epsilon
r_{b}}{r^{2}\left(r-r_{b}\right)}\,,$ (2.18)

and

$\Psi_{s}\sim\left\\{\begin{array}[]{ccc}c_{r},c_{y}&\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\hfil&\epsilon=0\\\
h_{r},h_{y}&&\epsilon=1\\\ \end{array}\right.\,.$ (2.19)

####  2.2.1 The $(h_{r},c_{y},h_{t})$ system

We focus on the system (2.1) describing the perturbations
$(h_{t},h_{r},c_{y})$. The first equation can be solved for $h_{t}$, leading
to a coupled system of two differential equations for $h_{r}$, $c_{y}$. The
system can be decoupled by the linear transformation,

$\displaystyle h_{r}=$ $\displaystyle
g_{h}(r)\left[\Psi_{+}(r)+\Psi_{-}(r)\right]\,,$ (2.20) $\displaystyle c_{y}=$
$\displaystyle
g_{c}(r)\left[(1+\gamma)\Psi_{+}(r)+(1-\gamma)\Psi_{-}(r)\right]\,,$

with

$g_{h}(r)={r^{7\over 2}\over(r-r_{b})(r-r_{s})^{3/2}}\,,\hskip
28.45274ptg_{c}(r)=-{{\rm i}(2r_{b}+3r_{s})\over
2\omega\sqrt{3r_{b}r_{s}}}{r^{1\over 2}\over(r-r_{s})^{1\over 2}}\,,$ (2.21)

and

$\gamma\equiv\sqrt{1+\frac{12(\ell+2)(\ell-1)r_{b}r_{s}}{(2r_{b}+3r_{s})^{2}}}\,.$
(2.22)

In terms of these variables the system reduces to two decoupled ordinary
differential equations of the Schrödinger form,

$\Psi^{\prime\prime}_{\pm}+Q_{\pm}\Psi_{\pm}=0\,,$ (2.23)

with

$\displaystyle Q_{\pm}\,=$
$\displaystyle\frac{r^{3}}{\left(r-r_{s}\right)^{2}\left(r-r_{b}\right)}\left[\omega^{2}-\frac{\ell(\ell+1)}{r^{2}}+\frac{\left(2r_{b}+3r_{s}\right)\left(1\pm\gamma\right)+2r_{s}\left(\ell^{2}+\ell+1\right)}{2r^{3}}\right.$
(2.24)
$\displaystyle\left.-\frac{r_{s}\left[\left(2r_{b}+3r_{s}\right)\left(1\pm\gamma\right)+8r_{b}+\frac{3}{2}r_{s}\right]}{2r^{4}}+\frac{15r_{b}r_{s}^{2}}{4r^{5}}\right]\,.$

Let us remark that (2.23) is a differential equation with five Fuchsian
singularities: three regular ones at $r=0,r_{s},r_{b}$, and two colliding at
$r=\infty$. Crucially, this radial equation cannot be mapped to a Heun
equation or any of its confluent versions, as is the case for the scalar
perturbations in the topological star geometry, which are described by a
Confluent Heun equation Bianchi:2023sfs , similar to all black holes in the
Kerr-Newman family Bianchi:2022wku .

####  2.2.2 The $(h_{y},c_{t},c_{r})$ system

We can proceed analogously for the system (2.13). We solve the first equation
for $c_{t}$, which yields a system of two coupled second-order ODEs. Then we
look for a change of variables of the form

$\begin{bmatrix}h_{y}\\\ c_{r}\\\
\end{bmatrix}=\begin{bmatrix}g_{1+}(r)&g_{1-}(r)\\\ g_{2+}(r)&g_{2-}(r)\\\
\end{bmatrix}\begin{bmatrix}\Phi_{+}\\\ \Phi_{-}\\\ \end{bmatrix}\,,$ (2.25)

and fix the functions $g_{1+},g_{2-},g_{2+}$ and $g_{2-}$ such that the system
decouples in two independent second-order ODEs for $\Phi_{+}$ and $\Phi_{-}$.
We find that for this system this not possible. However, for the choice of
functions

$g_{1+}=g_{2+}=1\,,\hskip
28.45274ptg_{1-}=-g_{2-}=\frac{r^{3}}{\left(r-r_{b}\right)\left(r-r_{s}\right)}\,,$
(2.26)

we find that the second system (2.13) reduces to the following differential
equations

$\displaystyle\Phi^{\prime\prime}_{\pm}+\frac{r_{s}}{r\left(r-r_{s}\right)}\Phi^{\prime}_{\pm}+\frac{[-4r_{b}r_{s}+r\left(2r_{b}+r_{s}\right)]\omega\pm{\rm{i}}\sqrt{3r_{b}r_{s}}\,r\left[1+\ell\left(\ell+1\right)\omega^{2}\right]}{2r^{2}\left(r-r_{b}\right)\left(r-r_{s}\right)}\Phi_{\mp}$
(2.27)
$\displaystyle+\left[\frac{r^{3}\omega^{2}}{\left(r-r_{b}\right)\left(r-r_{s}\right)^{2}}\pm\frac{{\rm{i}}\sqrt{3r_{b}r_{s}}\,}{2r\left(r-r_{b}\right)\left(r-r_{s}\right)}\left(\ell(\ell+1)\,\omega+\frac{1}{\omega}\right)\right.$
$\displaystyle\left.+\frac{2r\left(r_{b}-\ell(1+\ell)r\right)-r_{s}\left(r+2r_{b}\right)}{2r^{2}\left(r-r_{b}\right)\left(r-r_{s}\right)}\right]\Phi_{\pm}=0\,.$

Then we can use the equation which does not contain derivatives of $\Phi_{-}$
(equivalently, $\Phi_{+}$) and solve it (for $\Phi_{-}$). After plugging the
resulting expression in the remaining equation, we get a fourth-order ODE for
$\Phi_{+}$. Given its complexity, we omit the details and the study of the
QNMs.

##  3 QNMs of topological stars and black strings

In this section we compute the QNM spectrum associated to the
$(h_{t},h_{r},c_{y})$ perturbations, whose dynamics is described by the master
equations (2.23). We will mainly use a semi-analytical method developed by
Leaver Leaver:1985ax , which has been recently applied to the study of scalar
perturbations of topological stars, Bianchi:2023sfs ; Heidmann:2023ojf ;
Cipriani:2024ygw . Nevertheless, we will compare the results obtained using
Leaver’s method against those obtained using the WKB approximation and via a
direct numerical integration of the differential equation.

QNMs correspond to solutions of the Schrodinger-type equation (2.23)
satisfying the boundary conditions333We omit the $\pm$ subindex in the master
variables for the sake of clarity.

$\Psi(r)\underset{r\to r_{0}}{\sim}(r-r_{0})^{\lambda_{0}}\,,\hskip
28.45274pt\Psi(r)\underset{r\to\infty}{\sim}r^{\lambda_{\infty}}e^{i\omega
r}\,.$ (3.1)

with

$r_{0}=\left\\{\begin{array}[]{ccc}r_{b}&\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\hfil&{\rm(top\leavevmode\nobreak\
star)}\\\ r_{s}&&{\rm(black\leavevmode\nobreak\ string)}\\\
\end{array}\right.\,,\hskip
56.9055pt\lambda_{0}\in\left\\{\begin{array}[]{ccc}\mathbb{R}_{+}&\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\hfil&{\rm(top\leavevmode\nobreak\
star)}\\\ {\rm i}\,\mathbb{R}_{-}&&{\rm(black\leavevmode\nobreak\ string)}\\\
\end{array}\right.\,.$ (3.2)

Namely, we are imposing outgoing boundary conditions at infinity for both
background solutions. The difference lies in the boundary conditions imposed
at $r_{0}$. For top stars we just demand regularity at the _cap_ : $r=r_{b}$.
Instead, for the black string we impose incoming boundary conditions at the
horizon: $r=r_{s}$. The specific values of $\lambda_{0}$ and
$\lambda_{\infty}$ can be obtained by solving the differential equation around
$r=r_{0}$ and $r=\infty$. The details are given in sections 3.3 and 3.4.

Solutions satisfying these boundary conditions exist only for a discrete
choice of complex frequencies $\omega_{n}$: the QNMs. The rest of the section
is organized as follows. First, in sections 3.1 and 3.2 we briefly describe
the WKB approximation and the numerical integration methods we are going to
use to further confirm the results obtained using Leaver’s method. The latter
will be discussed in sections 3.3 (top star) and 3.4 (black string).

###  3.1 WKB approximation

A rough estimate of the QNM frequencies, $\omega$, can be obtained from a WKB
semiclassical approximation of the wave solution Schutz:1985km ;
Cardoso:2008bp ; Bianchi:2020des ; Bianchi:2020yzr ; Bianchi:2021mft  around
the “light rings”: extrema of the effective potential $-Q(r)$ where both $Q$
and its first derivative vanish:

$Q(r_{c};\omega_{c})=Q^{\prime}(r_{c};\omega_{c})=0\,.$ (3.3)

To estimate the QNM frequencies, we promote $\omega_{c}$ to a complex number
(adding a small imaginary part) and demand that the integral between two zeros
$r_{\pm}$ of $Q(r)$ satisfies the Bohr-Sommerfeld quantization condition:

$\int_{r_{-}}^{r_{+}}\sqrt{Q(r;\omega)}dr=\pi\left(n+\frac{1}{2}\right)\,.$
(3.4)

The solution to linear order in the imaginary part of $\omega$ is given by the
WKB formula, namely

$\omega_{n}^{\rm WKB}=\omega_{c}-{\rm
i}\left(n+\frac{1}{2}\right)\frac{\sqrt{\partial_{r}^{2}Q}}{\partial_{\omega}Q}\Big{|}_{\omega=\omega_{c},r=r_{c}}\,.$
(3.5)

###  3.2 Direct numerical integration

QNM frequencies can be alternatively obtained via a direct numerical
integration of the differential equation in the domain $r\in[r_{0},\infty]$
with boundary conditions (3.1). Since both $r=r_{0}$ and $r=\infty$ are
singular points, boundary conditions cannot be imposed exactly at these
points, but one can choose some arbitrary points nearby. The procedure has
several steps: First, one determines the solution (with the appropriate
boundary conditions) as a Taylor expansion around $r_{0}$. Then, one evaluates
the solution at a point $r_{0}+\epsilon$ with $\epsilon$ small, and uses the
result as a boundary condition to find a numerical solution, $\Psi_{0}$. One
then proceeds in the same way to compute a numerical solution $\Psi_{\infty}$
satisfying outgoing boundary conditions at infinity.

The QNM frequencies are then obtained by requiring the matching of the two
functions at an intermediate point. This is equivalent to demanding the
vanishing of the Wronskian:

$\Psi_{0}^{\prime}(r;\omega)\Psi_{\infty}(r;\omega)-\Psi_{0}(r;\omega)\Psi_{\infty}^{\prime}(r;\omega)=0\,,$
(3.6)

at some point $r$ in the middle. We notice that the Wronskian is independent
of the choice of this point. Thus, (3.6) becomes an equation for $\omega$
whose solutions are the QNMs. When comparing this method against the Leaver
method, we will bear in mind that this numerical method is known to be
numerically stable only for frequencies with a small imaginary part
ae16f484-8e04-30db-a6a3-902e6f6fd388 ; Berti:2009wx ; Cardoso:2014sna ;
Bianchi:2023sfs .

###  3.3 QNMs: top stars

In this section we compute the QNM spectrum of top stars using Leaver’s
method. As explained in the previous section, top stars correspond to smooth
horizonless geometries ending a cap at $r=r_{b}$ with $r_{b}>r_{s}$. Hence,
QNMs in this geometry correspond to solutions of (2.23) satisfying regular
boundary conditions at the cap and behaving as an outgoing wave at infinity.
Given this, we consider the following ansatz

$\Psi(r)=e^{i\omega r}r^{-{3\over
2}}(r-r_{s})^{\lambda_{s}}\sum_{n=0}^{\infty}c_{n}\left(\frac{r-r_{b}}{r-r_{s}}\right)^{n}\,,$
(3.7)

with

$\lambda_{s}=\frac{1}{2}+{{\rm i}\omega\over 2}(r_{b}+2r_{s})\,.$ (3.8)

The expansion (3.7) satisfies the required boundary conditions at the cap and
infinity, and solves (2.23) near $r=r_{b}$ and near $r\to\infty$. Plugging
(3.7) into (2.23) yields a four-term recursion relation,

$\alpha_{n}c_{n+1}+\beta_{n}c_{n}+\gamma_{n}c_{n-1}+\delta_{n}c_{n-2}=0\,,\hskip
14.22636pt\quad n\geq 0\,.$ (3.9)

The explicit expressions of the coefficients $\alpha_{n},\beta_{n},\gamma_{n}$
and $\delta_{n}$ for $\Psi_{+}$ are

$\displaystyle\alpha_{n}\,=$ $\displaystyle-(n+1)(n+2)\,,$ (3.10)
$\displaystyle\beta_{n}\,=$
$\displaystyle\ell(\ell{+}1){+}2n(n{+}2){+}1{-}\gamma+\frac{r_{s}(3(1+\gamma)-2n(n-2))}{2r_{b}}{-}3{\rm
i}(n{+}1)\omega r_{b}{-}\frac{\omega^{2}r_{b}^{3}}{r_{b}{-}r_{s}}\,,$
$\displaystyle\gamma_{n}\,=$
$\displaystyle{-}n(n{+}1){+}1{+}\gamma{+}\frac{r_{s}(3\gamma{-}4(n{-}3)n{-}2\ell(\ell{+}1){-}5)}{2r_{b}}+\frac{1}{2}i\omega\left(2nr_{b}{+}r_{b}{+}(10n{-}7)r_{s}\right){+}\frac{\omega^{2}r_{b}^{2}\left(r_{b}{+}7r_{s}\right)}{4\left(r_{b}{-}r_{s}\right)}\,,$
$\displaystyle\delta_{n}\,=$ $\displaystyle\frac{(n-2)r_{s}\left(n-2i\omega
r_{s}-2\right)}{r_{b}}-\frac{1}{4}\omega r_{s}\left(\omega
r_{b}+4i(n-2)+4\omega r_{s}\right)-\frac{\omega^{2}r_{s}^{3}}{r_{b}-r_{s}}\,.$

The coefficients of the recursion for $\Psi_{-}$ are obtained by sending
$\gamma\to-\gamma$.

Figure 1:  Left Panel) QNMs for top star with $r_{b}=1$, $r_{s}=0.8$ and
$\ell=2$. Right Panels) Convergence of the continuous-fraction method.

$\ell$ | Leaver($n=0$) | Dir. Int.
---|---|---
$2$ | $0.9102-0.0774i$ | $0.9102-0.0774i$
$3$ | $1.3939-0.0477i$ | $1.3939-0.0477i$
$4$ | $1.8618-0.0286i$ | $1.8618-0.0286i$
$5$ | $2.3241-0.0164i$ | $2.3241-0.0164i$
$8$ | $3.6952-0.0019i$ | $3.6952-0.0019i$
$10$ | $4.6003-0.0003i$ | $4.6003-0.0003i$

$n$ | Leaver ($\ell=2$)
---|---
$0$ | $0.9102-0.0774i$
$1$ | $1.1201-0.3126i$
$2$ | $1.3775-0.6018i$
$3$ | $1.6711-0.9105i$
$4$ | $1.9868-1.2224i$
$5$ | $2.3415-1.5428i$

Table 1: QNMs for top star with $r_{b}=1$, $r_{s}=0.8$.

Truncating $n$ to some large number $N$, the recursion relation (3.9) becomes
an $N$-dimensional matrix equation $M\cdot c=0$, where $c$ is a vector
containing the $c_{n}$ coefficients entering in the ansatz (3.7). Therefore, a
non-trivial solution exists only if the determinant of $M$ vanishes, which
provides the equation satisfied by the QNM frequencies. However, we are not
going to use the vanishing of the determinant to find the QNMs. Instead, we
find more convenient to first tridiagonalize $M$,

$\displaystyle\alpha^{\prime}_{0}\,=$
$\displaystyle\alpha_{0},\quad\beta^{\prime}_{0}=\beta_{0}\,,$
$\displaystyle\alpha^{\prime}_{1}\,=$
$\displaystyle\alpha_{1},\quad\beta^{\prime}_{1}=\beta_{1},\quad\gamma^{\prime}_{1}=\gamma_{1}$
$\displaystyle\alpha^{\prime}_{n}\,=$
$\displaystyle\alpha_{n},\quad\beta^{\prime}_{n}=\beta_{n}-\frac{\delta_{n}}{\gamma^{\prime}_{n-1}}\alpha^{\prime}_{n},\quad\gamma^{\prime}_{n}\,=\,$
$\displaystyle\gamma_{n}-\frac{\delta_{n}}{\gamma^{\prime}_{n-1}}\beta^{\prime}_{n},\quad\delta^{\prime}_{n}=0\,,\quad
n\geq 2$

and then use the fact that the integrability of the three-term recursion boils
down to the continuous fraction equation:

$\beta^{\prime}_{n}+\frac{\alpha^{\prime}_{n}\gamma^{\prime}_{n}}{\beta^{\prime}_{n-1}-\frac{\alpha^{\prime}_{n-2}\gamma^{\prime}_{n-1}}{\beta^{\prime}_{n-2}-\dots}}+\frac{\alpha^{\prime}_{n}\gamma^{\prime}_{n+1}}{\beta^{\prime}_{n+1}-\frac{\alpha^{\prime}_{n+1}\gamma^{\prime}_{n+2}}{\beta^{\prime}_{n+2}-\dots}}=0\,.$
(3.11)

The QNM frequencies can be obtained by solving the above equation for a
particular $n$, let us say $n=0$. In practice, the continuous fraction is
truncated by keeping $N$ terms, with $N$ a large enough number. In Figure 1
(left) the result for the QNM frequencies is displayed for the $\ell=2$ mode
in a top star with $r_{s}=0.8$, $r_{b}=1$ with $N=50$. The plots on the right
show the convergence of the method, which typically occurs for $N\geq 30$.

In Table 1 we display the results for QNM frequencies for the fundamental mode
$n=0$ as $\ell$ varies from 2 to 10. The results are compared against those
obtained from a direct numerical integration of the differential equation,
showing an excellent agreement. We omit the results based on a WKB
approximation of the solution as the latter fails in reproducing the imaginary
part of the QNM frequencies for the low $n$ modes.

Finally, in appendix A we collect more results for various representative
choices of $r_{s}$, $r_{b}$, $\ell$. In all the solutions we have checked, QNM
frequencies have negative imaginary parts, suggesting the stability of top
stars against metric and electromagnetic odd perturbations.

###  3.4 QNM: black string

Figure 2:  $R_{+}$ QNM’s for black string with $r_{b}=0.8$, $r_{s}=1$,
$\ell=2$

$\ell$ | WKB | Leaver | Dir. Int.
---|---|---|---
$2$ | $0.6323-0.1239i$ | $0.6648-0.1225i$ | $0.6648-0.1225i$
$3$ | $1.0608-0.1267i$ | $1.0826-0.1263i$ | $1.083-0.1263i$
$4$ | $1.4678-0.1282i$ | $1.4839-0.1279i$ | $1.4839-0.1279i$
$5$ | $1.8659-0.129i$ | $1.8787-0.1288i$ | $1.8787-0.1288i$
$8$ | $3.0402-0.1301i$ | $3.0482-0.13i$ | $3.0482-0.13i$
$10$ | $3.8166-0.1304i$ | $3.8229-0.1303i$ | $3.8229-0.1303i$

Table 2: The fundamental $n=0$ QNM for a black string with $r_{b}=0.8$,
$r_{s}=1$,

.

The analysis for the black string $r_{b}<r_{s}$ follows mutatis mutandis the
same steps than that for topological stars, but now incoming boundary
conditions have to be imposed at the horizon $r=r_{s}$. Therefore, the ansatz
now becomes

$\Psi(r)=e^{i\omega r}r^{-{3\over
2}}(r-r_{s})^{\lambda_{s}}(r-r_{b})^{\lambda_{b}}\sum_{n=0}^{\infty}c_{n}\left(\frac{r-r_{s}}{r-r_{b}}\right)^{n}\,,$
(3.12)

with

$\lambda_{s}={1\over 2}+{\omega r_{s}^{3\over
2}\over\sqrt{r_{b}-r_{s}}}\,,\hskip 28.45274pt\lambda_{b}=1+{{\rm
i}\omega\over
2(r_{s}-r_{b})}\left(2r_{s}^{2}-r_{b}^{2}-r_{b}r_{s}+2\sqrt{r_{s}^{3}(r_{s}-r_{b})}\right)\,.$
(3.13)

Plugging this into (2.23) yields again a four-term recursion relation as in
(3.9), but this time with coefficients given by:

$\displaystyle\alpha_{n}$ $\displaystyle=-(n+1)^{2}r_{s}+\frac{2i(n+1)\omega
r_{s}^{2}}{\sigma}\,,$ $\displaystyle\beta_{n}$
$\displaystyle=-\frac{1}{2}r_{s}\left(-2\gamma\sigma^{2}+5\gamma-2\ell^{2}-2\ell+2n^{2}\sigma^{2}-6n^{2}-6n\sigma^{2}+2n+2\sigma^{2}+1\right)$
$\displaystyle-\frac{i(\sigma+1)\omega\left(2n\sigma^{2}-6n\sigma+12n+\sigma^{2}+5\sigma-2\right)r_{s}^{2}}{2\sigma}-\frac{(\sigma+1)^{3}\omega^{2}r_{s}^{3}}{\sigma^{2}}\,,$
$\displaystyle\gamma_{n}$
$\displaystyle=\frac{1}{2}r_{s}\left(-2\gamma\sigma^{2}+5\gamma+2\ell^{2}\sigma^{2}-2\ell^{2}+2\ell\sigma^{2}-2\ell+4n^{2}\sigma^{2}-6n^{2}-16n\sigma^{2}+16n+14\sigma^{2}-9\right)\,$
$\displaystyle-\frac{i(\sigma+1)^{2}\omega\left(n\sigma^{3}-2n\sigma^{2}+6n\sigma-6n-2\sigma^{3}+4\sigma^{2}-10\sigma+8\right)r_{s}^{2}}{\sigma}+\frac{\left(\sigma^{2}-8\sigma+8\right)(\sigma+1)^{4}\omega^{2}r_{s}^{3}}{4\sigma^{2}}\,,$
$\displaystyle\delta_{n}$
$\displaystyle=-\frac{i(2n-5)(\sigma-2)(\sigma-1)(\sigma+1)^{3}\omega
r_{s}^{2}}{2\sigma}-(n-3)(n-2)(\sigma-1)(\sigma+1)r_{s}$
$\displaystyle+\frac{(\sigma-2)^{2}(\sigma-1)(\sigma+1)^{5}\omega^{2}r_{s}^{3}}{4\sigma^{2}}\,,$

with

$\sigma=\sqrt{1-{r_{b}\over r_{s}}}\,.$ (3.14)

Figure 2 shows some QNM frequencies (different overtones) $r_{s}=1$,
$r_{b}=0.8$ and $\ell=2$, again for the $\Psi_{+}$ perturbation. Those for
$n=0$ with $\ell$ varying from $2$ to $10$ are displayed in Table 2, together
with a comparison against the WKB and direct integration methods. As we can
see, we find an pretty good agreement. Finally, other representative examples
are collected in the Appendix A. For $r_{b}=0$ we reproduce the results for
Schwarzschild black holes, as expected.

As in the analysis of the topological star, we find that for all choices of
$r_{s}$, $r_{b}$, $\ell$ we have checked the imaginary parts of the QNM
frequencies of black strings are always negative, strongly suggesting the
stability of these solutions against this type of perturbations.

##  4 Gregory-Laflamme instabilities

Figure 3: Black string unstable modes for various parameters; the frequency of
the unstable mode is at $\omega=0.0332837i$ (Top left), $p=0.435943$ (Bottom
left).

Black strings and branes in dimensions higher than four typically exhibit a
classical instability, known as the Gregory-Laflamme (GL) instability
Gregory:1994bj . A characteristic feature of the GL perturbations is that they
have a momentum, $p$, along the internal directions. Typically, the
instability appears when $p$ is smaller than a certain threshold value,
$p_{\star}$, while the system is stable under perturbations with
$p>p_{\star}$. This suggests that the mode with $p=p_{\star}$ (which is
referred to as the _threshold unstable mode_) is time independent.

Based on the existence of a threshold time-independent mode, the domain of GL
stability of black strings was determined in Miyamoto:2007mh  to be
$r_{s}<2r_{b}$. Furthermore, Bah:2021irr ; Stotyn:2011tv  used the double Wick
rotation symmetry that exchanges black strings with topological stars,
$r_{s}\leftrightarrow r_{b}$ and $\omega\leftrightarrow{\rm i}p$ to argue that
top stars are stable when $r_{b}<2r_{s}$.444In the analysis of Bah:2021irr ,
the role of the threshold mode is played by a mode with $p=0$ and $\omega={\rm
i}p_{\star}$. We also know that when $r_{b}$ is fixed and $r_{s}=0$, the top
star becomes Euclidean Schwarzschild times time, and this solution suffers
from a Gross-Perry-Yaffe instability Gross:1982cv ; Witten:1981gj . Hence, we
expect the instability of top stars with $0<r_{s}<r_{b}/2$ to be of the same
type Bah:2021irr ; Stotyn:2011tv .

Here we review the arguments of Miyamoto:2007mh ; Bah:2021irr , adapting them
to the more general perturbations where both $p$ and $\omega$ are non-
vanishing. We then apply the Leaver method to compute the QNM frequencies and
show directly the existence of modes with positive imaginary parts outside of
the stability domains.

Let us consider the following even perturbation of the metric and two-form
with $\ell=0$ (hence, spherically-symmetric),

$\displaystyle h_{\mu\nu}\mathrm{d}{x}^{\mu}\mathrm{d}{x}^{\nu}\,=$
$\displaystyle e^{{\rm i}py-{\rm i}\omega
t}\left[h_{1}(r)\mathrm{d}r^{2}{+}r^{2}k(r)\left(\mathrm{d}\theta^{2}{+}\sin^{2}\theta\,\mathrm{d}\phi^{2}\right){+}2\,h_{2}(r)\,\mathrm{d}r\,\mathrm{d}y{+}2\,h_{3}(r)\,\mathrm{d}r\,\mathrm{d}t\right]\,,$
(4.1) $\displaystyle c_{\mu\nu}\mathrm{d}{x}^{\mu}\mathrm{d}{x}^{\nu}\,=$
$\displaystyle 2e^{{\rm i}py-{\rm i}\omega
t}c(r)\,\mathrm{d}t\wedge\mathrm{d}y\,.$

One can check that for any choice of $p$ and $\omega$ this is not pure gauge:
$h_{\mu\nu}$ cannot be written as $\nabla_{(\mu}\zeta_{\nu)}$. The field
equations (2.1) and (2.2) allow us to solve for $c$, $h_{1}$, $h_{2}$ and
$h_{3}$ in terms of $k$, and we find that the latter satisfies the second-
order differential equation

${d\over dr}\left[A(r)k^{\prime}(r)\right]-B(r)k(r)=0\,,$ (4.2)

with

$\displaystyle A(r)$
$\displaystyle=\frac{\left(r{-}r_{b}\right)\left(r{-}r_{s}\right)}{{\cal
W}(r)^{2}}\,,$ (4.3) $\displaystyle B(r)$ $\displaystyle=\frac{r^{3}{\cal
W}(r)\left[p^{2}\left(r{-}r_{s}\right){-}\omega^{2}\left(r{-}r_{b}\right)\right]+2\left(r{-}r_{b}\right)\left(r{-}r_{s}\right)\left[p^{2}\left(2r_{b}{-}r_{s}\right){-}\omega^{2}\left(2r_{s}{-}r_{b}\right)\right]}{{\cal
W}(r)^{3}\left(r{-}r_{b}\right)\left(r{-}r_{s}\right)}\,,$

and

${\cal W}(r)=p^{2}\left(4r-3r_{s}\right)-\omega^{2}\left(4r-3r_{b}\right)\,.$
(4.4)

Figure 4: GL unstable modes for the topological star for various parameters:
the frequencies are $\omega=0.435929i$ (top left) and
$\omega=0.260928-0.166396i$ (top right).

We look for solutions to (4.2) satisfying the boundary conditions,

$k(r)\underset{r\to r_{0}}{\sim}(r-r_{0})^{\lambda_{0}}\,,\hskip
56.9055ptk(r)\underset{r\to\infty}{\sim}e^{{\rm
i}r\sqrt{\omega^{2}-p^{2}}}\,,$ (4.5)

where $r_{0}={\rm{max}}\left[r_{s},r_{b}\right]$ and $\lambda_{0}$ is
determined by solving the equation near $r=r_{0}$ and imposing incoming (for
the black string) or regular (for the top star) boundary conditions at
$r=r_{0}$. The GL modes correspond to solutions of (4.2) satisfying (4.5), and
exhibiting an exponential decay at infinity Gregory:1994bj :

$\displaystyle{\rm Re}\left({\rm i}\sqrt{\omega^{2}-p^{2}}\right)<0\,.$ (4.6)

In order to review the arguments presented in Miyamoto:2007mh ; Bah:2021irr
in a unified way, let us first consider perturbations where $p$ is real and
$\omega$ is purely imaginary and (4.6) is automatically satisfied. Multiplying
(4.2) by $k(r)$, integrating it over the domain $[r_{0},\infty]$, and
integrating by parts, we obtain:

$\int^{\infty}_{r_{0}}\mathrm{d}r\left[A(r)\left(k^{\prime}(r)\right)^{2}+B(r)k(r)^{2}\right]=\int^{\infty}_{r_{0}}d\left[A(r)k(r)k^{\prime}(r)\right]=0\,.$
(4.7)

The vanishing of the right hand side follows from the fact that $A(r)$
vanishes at $r_{0}$ and $k(r)$ vanishes at infinity. In addition if $p$ is
real and $\omega$ purely imaginary, both $A(r)$ and $B(r)$ are real and
positive in the domain:

${\rm{top\leavevmode\nobreak\ star:}}\,\,\,r_{s}<r_{b}<2r_{s}\,,\hskip
28.45274pt{\rm{black\leavevmode\nobreak\ string:}}\,\,\,r_{b}<r_{s}<2r_{b}\,.$
(4.8)

Thus, the only (real) solution of (4.7) is the trivial one, $k=0$. The
geometry is therefore stable in this domain.

Given this, we provide further evidence applying again Leaver’s method. Since
the procedure is analogous to the one described in the previous section, we
shall ignore the technical details and present directly the results in Figures
3 and 4, where we plot the argument of the QNM eigenvalue equation for various
choices of $r_{s}$ and $r_{b}$. The QNM frequencies show up as peaks on these
plots.

A summary of our findings is:

  * •

Black string (Figure 3): We find unstable modes only in the regime
$r_{s}>2r_{b}$ for $p<p_{\star}\approx 0.435943$ in agreement with
expectations in Miyamoto:2007mh .

  * •

Top star (Figure 4): We find unstable modes for $r_{b}>2r_{s}$ and $p$ small
enough. However when $p$ is finite, it cannot be arbitrarily small since
regularity of the geometry requires the quantization condition:

$p=\frac{n_{y}}{R_{y}}=n_{y}\,\frac{{\sqrt{r_{b}-r_{s}}}}{2r_{b}^{3/2}}\,,\hskip
28.45274ptn_{y}\in{\mathbb{Z}}\,,$ (4.9)

We find that all modes with $n_{y}\geq 1$ are stable, therefore only the
$n_{y}=0$ mode leads to a potential instability.

##  5 Conclusions

We have calculated the QNM frequencies associated to odd perturbations of
topological stars and magnetized black strings in five-dimensional Einstein-
Maxwell theory. Our analysis of the QNM spectrum mainly relies on Leaver’s
method Leaver:1985ax , which has been suitably generalized in order to apply
it to an ODE with five Fuchsian singularities, reducing the 4-term recursion
relation to a 3-term one via a tri-diagonalization of the eigenvalue matrix.

We have confirmed the expectations of Bah:2021irr ; Stotyn:2011tv  which
suggest classical stability inside the domain

${\rm{top\leavevmode\nobreak\ star:}}\,\,\,r_{s}<r_{b}<2r_{s}\,,\hskip
28.45274pt{\rm{black\leavevmode\nobreak\ string:}}\,\,\,r_{b}<r_{s}<2r_{b}\,,$
(5.1)

as we have found that all the perturbations we have considered inside the
above domains decay exponentially in time. Outside of these domains we have
verified that topological stars suffer from a Gross-Perry-Yaffe-type
instability Gross:1982cv , whereas magnetic black strings suffer from the
usual Gregory-Laflamme instability Miyamoto:2007mh . Such instabilities
reflect themselves on the existence of QNMs with positive imaginary part which
are associated to even perturbations with $\ell=0$.

There are several obvious next steps to our investigation. The first is to
study the possible gravitational-wave signatures that may distinguish top
stars from black holes: multipolar structureMayerson:2020tpn ; Bianchi:2020des
, echoes Ikeda:2021uvc  and tidal effects during the inspiral phase of black-
hole merger Fucito:2023afe .

The next is to consider the even perturbations, and try to solve the
underlying system of ODE’s numerically. This will involve a shooting problem
in several functions of one variable. Systems of similar complexity have been
solved by shooting in other circumstances Ganchev:2021pgs ; Buchel:2000ch , so
we believe this problem is within reach.

Another extension of our calculation is to the running-Bolt solutions
Bena:2009qv , which are also obtained by magnetizing the bolt of Euclidean
Schwarzschild times time in five dimensions. For these solutions we also
expect a Gross-Perry-Yaffe-type instability for small magnetic fluxes, and
possibly a stable solution for larger fluxes Avery . Another cohomogeneity-one
solution with fluxes that can be studied using our method is the magnetized
Atiyah-Hitchin solution Atiyah:1985dv ; Bena:2007ju , which is the M-theory
uplift of Type-IIA Orientifold 6-planes with fluxes.

Our calculation serves as a first step towards and a benchmark in the
determination of the stability or lack thereof for more generic non-extremal
topological stars that have a fluxed bolt. The most generic such solutions are
cohomogeneity-two Bah:2021owp , and hence determining the quasinormal
frequencies will require solving PDE’s, using methods similar to those of
Dias:2009iu ; Dias:2014eua ; Santos:2015iua . These calculations should reduce
in certain limits to the calculations we do here, and hence our calculations
should serve as a useful benchmark for these more complicated calculations.

## Acknowledgments:

We would like to thank Massimo Bianchi, Pablo A. Cano, Giuseppe Dibitetto,
Alexandru Dima, Francesco Fucito, Pierre Heidmann, Marco Melis, Paolo Pani and
David Pereniguez for stimulating discussions and exchanges. The work of IB is
supported in part by the ERC Grants 787320 - QBH Structure and 772408 -
Stringlandscape. The work of GDR, JFM and AR is supported by the MIUR-PRIN
contract 2020KR4KN2 - String Theory as a bridge between Gauge Theories and
Quantum Gravity and by the INFN Section of Rome “Tor Vergata”.

##  Appendix A QNMs

In this appendix we collect some tables displaying QNM frequencies
corresponding to the perturbations $\Psi_{+}$ (left) and $\Psi_{-}$ (right)
for various choices of $r_{s}$, $r_{b}$ and $\ell$.

  * •

$r_{s}=0.8$, $r_{b}=1$:

$\ell$ | Leaver ($n=0$) | Dir. Int. ($n=0$)
---|---|---
$2$ | $0.9102-0.0774i$ | $0.9102-0.0774i$
$3$ | $1.3939-0.0477i$ | $1.3939-0.0477i$
$4$ | $1.8618-0.0286i$ | $1.8618-0.0286i$
$5$ | $2.3241-0.0164i$ | $2.3241-0.0164i$
$6$ | $2.7833-0.0087i$ | $2.7833-0.0087i$
$7$ | $3.2403-0.0043i$ | $3.2403-0.0043i$
$8$ | $3.6952-0.0019i$ | $3.6952-0.0019i$
$9$ | $4.1484-0.0008i$ | $4.1484-0.0008i$
$10$ | $4.6003-0.0003i$ | $4.6003-0.0003i$

$\ell$ | Leaver ($n=0$) | Dir. Int ($n=0$)
---|---|---
$2$ | $1.5803-0.1064i$ | $1.5803-0.1064i$
$3$ | $2.0518-0.0694i$ | $2.0518-0.0694i$
$4$ | $2.5181-0.0454i$ | $2.5181-0.0454i$
$5$ | $2.9815-0.0288i$ | $2.9815-0.0288i$
$6$ | $3.4429-0.0174i$ | $3.4429-0.0174i$
$7$ | $3.9025-0.0098i$ | $3.9025-0.0098i$
$8$ | $4.3604-0.0051i$ | $4.3604-0.0051i$
$9$ | $4.8166-0.0024i$ | $4.8166-0.0024i$
$10$ | $5.2711-0.001i$ | $5.2711-0.001i$

$n$ | Leaver ($\ell=2$)
---|---
$0$ | $0.9102-0.0774i$
$1$ | $1.1201-0.3126i$
$2$ | $1.3775-0.6018i$
$3$ | $1.6711-0.9105i$
$4$ | $1.9868-1.2224i$
$5$ | $2.3415-1.5428i$

$n$ | Leaver ($\ell=2$)
---|---
$0$ | $1.5803-0.1064i$
$1$ | $1.7748-0.345i$
$2$ | $2.0085-0.6302i$
$3$ | $2.274-0.9405i$
$4$ | $2.5645-1.2634i$
$5$ | $2.8598-1.5919i$

$n$ | Leaver ($\ell=10$) | Dir. Int. ($\ell=10$)
---|---|---
$0$ | $4.6003-0.0003i$ | $4.6003-0.0003i$
$1$ | $4.7872-0.0321i$ | $4.7872-0.0321i$
$2$ | $4.9538-0.1546i$ | $4.9538-0.1546i$
$3$ | $5.139-0.3201i$ | $-$
$4$ | $5.3355-0.5135i$ | $-$
$5$ | $5.5417-0.729i$ | $-$

$n$ | Leaver ($\ell=10$) | Dir. Int. ($\ell=10$)
---|---|---
$0$ | $5.2711-0.001i$ | $5.2711-0.001i$
$1$ | $5.4367-0.0486i$ | $5.4368-0.0486i$
$2$ | $5.5997-0.1799i$ | $-$
$3$ | $5.7793-0.348i$ | $-$
$4$ | $5.9699-0.5423i$ | $-$
$5$ | $6.1702-0.7575i$ | $-$

  * •

$r_{s}=0$, $r_{b}=1$:

$\ell$ | Leaver ($n=0$)
---|---
$2$ | $1.8517-1.2691i$
$3$ | $3.0513-1.3549i$
$4$ | $4.157-1.3815i$
$5$ | $5.222-1.3933i$
$6$ | $6.266-1.3996i$
$7$ | $7.2979-1.4034i$
$8$ | $8.3221-1.4059i$
$9$ | $9.3411-1.4076i$
$10$ | $10.3564-1.4088i$

$\ell$ | Leaver ($n=0$)
---|---
$2$ | $2.2995-1.4362i$
$3$ | $3.3569-1.425i$
$4$ | $4.3888-1.4206i$
$5$ | $5.409-1.4185i$
$6$ | $6.423-1.4173i$
$7$ | $7.4333-1.4165i$
$8$ | $8.4412-1.416i$
$9$ | $9.4474-1.4156i$
$10$ | $10.4524-1.4154i$

$n$ | Leaver ($\ell=2$)
---|---
$0$ | $1.8517-1.2691i$
$1$ | $1.4597-2.7725i$
$2$ | $1.0743-4.6179i$
$3$ | $0.8437-6.6137i$
$4$ | $0.7107-8.6372i$
$5$ | $0.6269-10.6623i$

$n$ | Leaver ($\ell=2$)
---|---
$0$ | $2.2995-1.4362i$
$1$ | $1.9111-3.0645i$
$2$ | $1.5047-4.9713i$
$3$ | $1.2273-7.0143i$
$4$ | $1.0492-9.0811i$
$5$ | $0.927-11.1432i$

$n$ | Leaver ($\ell=10$)
---|---
$0$ | $10.3564-1.4088i$
$1$ | $10.2491-2.828i$
$2$ | $10.0721-4.2681i$
$3$ | $9.8284-5.7402i$
$4$ | $9.5235-7.2556i$
$5$ | $9.1654-8.8256i$

$n$ | Leaver ($\ell=10$)
---|---
$0$ | $10.4524-1.4154i$
$1$ | $10.3457-2.841i$
$2$ | $10.1698-4.2876i$
$3$ | $9.9277-5.766i$
$4$ | $9.6249-7.2876i$
$5$ | $9.2694-8.8632i$

  * •

$r_{b}=0.8$, $r_{s}=1$:

$\ell$ | WKB ($n=0$) | Leaver ($n=0$) | Dir. Int. ($n=0$)
---|---|---|---
$2$ | $0.6323-0.1239i$ | $0.6648-0.1225i$ | $0.6648-0.1225i$
$3$ | $1.0608-0.1267i$ | $1.0826-0.1263i$ | $1.083-0.1263i$
$4$ | $1.4678-0.1282i$ | $1.4839-0.1279i$ | $1.4839-0.1279i$
$5$ | $1.8659-0.129i$ | $1.8787-0.1288i$ | $1.8787-0.1288i$
$6$ | $2.2595-0.1295i$ | $2.2702-0.1294i$ | $2.2702-0.1294i$
$7$ | $2.6507-0.1299i$ | $2.6598-0.1297i$ | $2.6598-0.1297i$
$8$ | $3.0402-0.1301i$ | $3.0482-0.13i$ | $3.0482-0.13i$
$9$ | $3.4288-0.1303i$ | $3.4358-0.1302i$ | $3.4358-0.1302i$
$10$ | $3.8166-0.1304i$ | $3.8229-0.1303i$ | $3.8229-0.1303i$

$\ell$ | WKB ($n=0$) | Leaver ($n=0$) | Dir. Int. ($n=0$)
---|---|---|---
$2$ | $1.0695-0.1349i$ | $1.0925-0.1335i$ | $1.0925-0.1335i$
$3$ | $1.4785-0.1338i$ | $1.4952-0.1332i$ | $1.4952-0.1332i$
$4$ | $1.8773-0.1333i$ | $1.8905-0.1329i$ | $1.8905-0.1329i$
$5$ | $2.2714-0.133i$ | $2.2822-0.1327i$ | $2.2822-0.1327i$
$6$ | $2.6627-0.1328i$ | $2.672-0.1326i$ | $2.672-0.1326i$
$7$ | $3.0524-0.1326i$ | $3.0605-0.1325i$ | $3.0605-0.1325i$
$8$ | $3.441-0.1325i$ | $3.4482-0.1324i$ | $3.4482-0.1324i$
$9$ | $3.8289-0.1324i$ | $3.8353-0.1323i$ | $3.8353-0.1323i$
$10$ | $4.2162-0.1323i$ | $4.2221-0.1322i$ | $4.2221-0.1322i$

$n$ | WKB ($\ell=2$) | Leaver ($\ell=2$)
---|---|---
$0$ | $0.6323-0.1239i$ | $0.6648-0.1225i$
$1$ | $0.6323-0.3716i$ | $0.6369-0.372i$
$2$ | $0.6323-0.6194i$ | $0.5891-0.6385i$
$3$ | $0.6323-0.8671i$ | $0.5083-0.9566i$
$4$ | $0.6323-1.1149i$ | $0.4527-1.2544i$
$5$ | $0.6323-1.3626i$ | $0.3972-1.603i$

$n$ | WKB ($\ell=2$) | Leaver ($\ell=2$)
---|---|---
$0$ | $1.0695-0.1349i$ | $1.0925-0.1335i$
$1$ | $1.0695-0.4047i$ | $1.0785-0.4024i$
$2$ | $1.0695-0.6745i$ | $1.0518-0.6752i$
$3$ | $1.0695-0.9443i$ | $1.0226-0.9586i$
$4$ | $1.0695-1.2141i$ | $0.9858-1.2474i$
$5$ | $1.0695-1.484i$ | $0.9531-1.531i$

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