# Underwater MEMS Gyrocompassing:
A Virtual Testing Ground

Daniel Engelsman, , and Itzik Klein The authors are with the Charney School of
Marine Sciences, Hatter Department of Marine Technologies, University of
Haifa, Haifa, Israel. E-mails: {dengelsm@campus, <EMAIL_ADDRESS>
###### Abstract

In underwater navigation, accurate heading information is crucial for
accurately and continuously tracking trajectories, especially during extended
missions beneath the waves. In order to determine the initial heading, a
gyrocompassing procedure must be employed. As unmanned underwater vehicles
(UUV) are susceptible to ocean currents and other disturbances, the model-
based gyrocompassing procedure may experience degraded performance. To cope
with such situations, this paper introduces a dedicated learning framework
aimed at mitigating environmental effects and offering precise underwater
gyrocompassing. Through the analysis of the dynamic UUV signature obtained
from inertial measurements, our proposed framework learns to refine disturbed
signals, enabling a focused examination of the earth’s rotation rate vector.
Leveraging recent machine learning advancements, empirical simulations assess
the framework’s adaptability to challenging underwater conditions. Ultimately,
its contribution lies in providing a resilient gyrocompassing solution for
UUVs.

###### Index Terms:

Inertial measurement units, autonomous underwater vehicles, unmanned
underwater vehicles, gyroscopes.

##  I Introduction

Expressing the rotation angles about the principal axes of a three-dimensional
coordinate frame is conveniently achieved using the Euler angles formalism [1,
2]. Determining the initial values of these angles, known as roll, pitch, and
yaw, is crucial for effective navigation. Similar to land and aerial vehicles,
most marine platforms are designed to operate horizontally (leveled),
maintaining their longitudinal axis aligned parallel to the local navigation
plane most of the time. To ascertain such spatial relationship between the x-y
body plane and the north-East plane, accelerometers are employed to sense the
local gravity. Subsequently, the roll and pitch angles can be computed from
the axial components using analytical closed-form relations. Utilizing modern
micro-electro-mechanical systems (MEMS) sensors, successful leveling with an
error of no more than 2 mrad is achievable, thanks to the pronounced gravity
signal ensuring a high signal-to-noise ratio (SNR) [3].
Yet, in the process of orienting the longitudinal axis with respect to true
north, i.e. finding the heading angle, most kinematic mechanisms heavily rely
on external sources to provide the necessary platform information. For
example, consumer-grade magnetometers, priced at just a few US dollars, prove
efficient for tasks such as indoor orientation, pedestrian orientation, and
low-fidelity navigation tasks.
Other strategies include satellite systems, radio frequency (RF)-based
navigation, and, more recently, fusion of visual odometry with the inertial
navigation system (INS) solution. While these methods prove practical for a
wide variety of problems above sea level [4], they fail to operate or provide
accurate initial heading while underwater.
Hence, earth’s rotation rate ($\bm{\omega}^{e}_{ie}$) is employed to compute
the carrier’s initial heading angle in a process known as gyrocompassing.
Relying solely on the gyroscopes sensitivity, this method obviate the need in
external broadcasting and magnetic interference. However, the slow 24 hours
rotation cycle generates an exceedingly subtle signal of 73e-6 [rad/sec],
detectable by only a few high-end sensors [5, 6]. With the advancement of MEMS
technology, off-the-shelf gyroscopes have reached technological maturity,
exhibiting sensitivity to noise levels as low as $\omega_{ie}$. Furthermore,
recent machine learning and deep learning approaches have further advanced
performance in various navigation tasks [7, 8, 9].
So while conventional noise removal methods restrict their operation to
stationary conditions, this paper presents a realistic proof of concept of
underwater gyrocompassing subjected to dynamic effects. From sensor input to
model output, we analyze environmental disturbances and examine our model’s
capability to mitigate them.

##  II Problem Formulation

This section lays the essential groundwork for understanding gyrocompassing
theory, hypothesized UUV dynamics, and how they ultimately manifest through
the inertial sensors.

###  II-A Gyrocompassing

Finding the north direction can be achieved through two distinct methods:
Magnetic north, pointing towards the north magnetic pole, and true Geographic
north, pointing towards earth’s true north pole, around which earth revolves.
Situated far apart, each pair of poles induces distinct rotational axes that
do not coincide, thus resulting in magnetic declination.
Under reasonably sterile conditions, magnetic-based compassing provides a fair
heading estimate while maintaining simplicity. However, when it comes to the
complex UUVs design, their onboard electronics make them susceptible to
electromagnetic interference, rendering magnetic-based compassing unreliable
for underwater tasks.
In contrast, gyro-based navigation offers a standalone solution that remains
robust across a wide variety of UUV tasks, a point further discussed in this
article. To begin with, rigid body orientation is parameterized through an
Euler angle triad, utilizing an orthonormal transformation matrix that
satisfies $\textbf{T}_{i}^{b}=(\textbf{T}_{b}^{i})^{\operatorname{T}}$, and
$\|\textbf{T}_{i}^{b}\|=1$.
By specifying the required rotations between a given body frame ($b$) and a
local inertial frame ($i$), it involves three successive rotations—roll,
pitch, and yaw ($\phi,\theta,\psi$)—about the x-y-z axes, as expressed by:

$\textbf{T}_{i}^{b}=\begin{bmatrix}\text{c}_{\theta}\text{c}_{\psi}&\text{s}_{\phi}\text{s}_{\theta}\text{c}_{\psi}-\text{c}_{\phi}\text{s}_{\psi}&\text{c}_{\phi}\text{s}_{\theta}\text{c}_{\psi}+\text{s}_{\phi}\text{s}_{\psi}\\\
\text{c}_{\theta}\text{s}_{\psi}&\text{s}_{\phi}\text{s}_{\theta}\text{s}_{\psi}+\text{c}_{\phi}\text{c}_{\psi}&\text{c}_{\phi}\text{s}_{\theta}\text{s}_{\psi}-\text{s}_{\phi}\text{c}_{\psi}\\\
-\text{s}_{\theta}&\text{s}_{\phi}\text{c}_{\theta}&\text{c}_{\phi}\text{c}_{\theta}\end{bmatrix}\
,$ (1)

with ’s’, and ’c’, represent the sine and cosine functions, respectively. We
define $\bm{\omega}_{ie}^{e}$ as the earth rotation vector, where $e$ denotes
the earth centered earth-fixed (ECEF) frame. Fig. 1 illustrates the rotation
of $\bm{\omega}_{ie}^{e}$ about the z-axis of the ECEF frame and its
projection onto the body frame by

$\displaystyle\bm{\omega}_{ie}^{b}=\textbf{T}_{n}^{b}\textbf{T}_{e}^{n}\bm{\omega}_{ie}^{e}\
,$ (2)

where $\textbf{T}_{e}^{n}$ represents earth to navigation frame
transformation, followed by $\textbf{T}_{n}^{b}$, which signifies the
navigation to body frame transformation. For simplicity, and without the lose
of generality, we assume that the gyroscopes sensitive axes align with the
body axes such that the gyro output is:

$\displaystyle\bm{\omega}_{ib}^{b}=\begin{bmatrix}\ p&q&r\ \
\end{bmatrix}^{\operatorname{T}}\ .$ (3)

Thus, in stationary conditions when combining (2) and (3), the heading angle
is confined through the following closed-form gyrocompassing expression [10]

$\displaystyle\text{s}_{\psi}$
$\displaystyle=-q\,\text{c}_{\phi}+r\,\text{s}_{\phi}\ ,$ (4)
$\displaystyle\text{c}_{\psi}$ $\displaystyle=\
p\,\text{c}_{\theta}+q\,\text{s}_{\phi}\text{s}_{\theta}+r\,\text{c}_{\phi}\text{s}_{\theta}\
,$ (5)

and can be solved with a four-quadrant arctangent function by

$\displaystyle\underline{\psi}(p,q,r)$
$\displaystyle=\arctan_{2}\big{(}\text{s}_{\psi}\,,\,\text{c}_{\psi}\big{)}\
.$ (6)

Given proper leveling, or equivalently, under horizontal conditions
($\phi=\theta=0$), (6) simplifies into a more streamlined expression

$\displaystyle\underline{\psi}(p,q)=\arctan_{2}\big{(}-q\,,\,p\big{)}\ ,$ (7)

revealing that the predominant component of $\bm{\omega}_{ie}^{e}$ lies only
within the gyroscope x-y plane.

Figure 1: Diagrammatic representation of the transformation between ECEF and
navigation reference frames, using the latitude, $\phi$, and longitude,
$\lambda$ angles.

###  II-B Gyroscope Measurement Model

In general, instrumental errors are typically categorized into stochastic and
deterministic sources. The former is unmodelable due to an unknown origin,
fluctuates randomly between time steps, and maintains a constant variance upon
remeasurement. The latter exhibits constancy between time steps but may vary
run-to-run. Conventional linearization schemes establish a relationship
between the gyro outputs $\bm{\tilde{\omega}}_{b}$ and the ground truth (GT)
quantities $\bm{\omega}_{b}$, characterized by

$\displaystyle\bm{\tilde{\omega}}_{b}=\bm{M}\bm{\omega}_{b}+\textbf{{b}}_{g}+\textit{{w}}_{g}\
.$ (8)

Here, $\bm{\tilde{\omega}}_{b}$ is directly influenced by a scale-factor and
misalignment matrix $\bm{M}$, affecting its intensity. Concurrently,
$\textbf{{b}}_{g}$ represents the uncompensated bias, and $\textit{{w}}_{g}$
introduces additional signal obscuration, modeled as a zero-mean white
Gaussian noise [11, 12]. Evidently, deterministic errors manifest as linear
mappings from noisy inputs to their respective outputs.

###  II-C Rigid Body Kinematics

A six degrees of freedom (DoF) motion model is defined by one set of 3D
coordinates addressing linear translations and another set addressing the
associated rotations. Let the following state vector denote the position and
orientation in the inertial frame

$\displaystyle\bm{\eta}_{b}=\begin{bmatrix}\ \bm{\eta}_{1}^{\operatorname{T}}\
|\ \bm{\eta}_{2}^{\operatorname{T}}\ \
\end{bmatrix}^{\operatorname{T}}=\begin{bmatrix}\ x&y&z\ \ |\ \
\phi&\theta&\psi\ \end{bmatrix}^{\operatorname{T}}\,,$ (9)

and similarly, the 6-DoF body velocities are given by

$\displaystyle\bm{\nu}_{i}=\begin{bmatrix}\ \bm{\nu}_{1}^{\operatorname{T}}\
|\ \bm{\nu}_{2}^{\operatorname{T}}\ \
\end{bmatrix}^{\operatorname{T}}=\begin{bmatrix}\ u&v&w\ \ |\ \ p&q&r\
\,\end{bmatrix}^{\operatorname{T}}\,.$ (10)

|  DoF |  Forces & moments |  Notation |  Inertial frame |  Body frame
---|---|---|---|---|---
Translational | 1 | Heave | $X$ | x | $u$
2 | Sway | $Y$ | y | $v$
3 | Surge | $Z$ | z | $w$
Rotational | 4 | Rolling | $L$ | $\phi$ | $p$
5 | Pitching | $M$ | $\theta$ | $q$
6 | Yawing | $N$ | $\psi$ | $r$
TABLE I: Key notations in UUV motion variables.

Unlike the orthonormal transformation (1) which relates translational
velocities between inertial and body frames, the mapping of body-axis rates to
Euler angle rates ($\dot{\phi},\dot{\theta},\dot{\psi}$) is carried out
through

$\textbf{T}_{\dot{\Omega}}^{\,b}=\begin{bmatrix}1&\text{s}_{\phi}\text{tan}_{\theta}&\text{c}_{\phi}\text{tan}_{\theta}\\\
0&\text{c}_{\phi}&-\text{s}_{\phi}\\\
0&\text{s}_{\phi}/\text{c}_{\theta}&\text{c}_{\phi}/\text{c}_{\theta}\end{bmatrix}\
.$ (11)

Therefore, by jointly combining (1) and (11)) into

$\displaystyle\textbf{T}=\begin{bmatrix}\textbf{T}_{i}^{b}&\bm{0}_{3\times 3}\
\ \\\ \ \bm{0}_{3\times 3}&\textbf{T}_{\dot{\Omega}}^{\,b}\end{bmatrix}\ ,$
(12)

the kinematic model is given in vectorial representation by

$\displaystyle\bm{\dot{\eta}}=\textbf{T}\,\bm{{\nu}}\ .$ (13)

Table I compiles the relevant nomenclature, assuming that the sensor frame is
aligned with the UUV center of mass. This arrangement expects the
accelerometers to sense the induced forces ($X,Y,Z$) along the longitudinal
($\hat{{x}}_{b}$), transverse ($\hat{{y}}_{b}$), and vertical axes
($\hat{{z}}_{b}$), while the gyros will detect the angular velocities
resulting from the 3D moments ($L,M,N$) around these axes. To visualize this,
Fig. 2 provides simplified schematics of the dynamics acting along the UUV
body axes and the local navigation frame.

Figure 2: Relationship between the navigation frame and UUV body reference
frames.

Despite extensive literature coverage on control aspects, this work focuses
solely on observing vehicle states under external interruptions, such as
characteristic underwater forces and moments [13, 14, 15]. Hence, control
inputs are disregarded to maintain focus solely on gyrocompassing.

###  II-D Rigid Body Dynamics

While kinematics studies the object’s motion without considering the driving
causes, dynamics, on the other hand, explores the cause-and-effect
relationships underlying the observed motion. To begin with, Newton’s second
law states that the net force exerted on an object of mass ($m$) is directly
proportional to its linear acceleration ($\dot{\bm{\nu}}_{1}$), as given in

$\displaystyle\bm{\tau}_{1}=m\ \dot{\bm{\nu}}_{1}\ .$ (14)

The rotational analog is provided by Euler’s second law, which relates changes
in angular momentum to external torques [16]

$\displaystyle\bm{\tau}_{2}=\bm{I}_{0}\,\dot{\bm{\omega}}_{b}+\bm{\omega}_{b}\times(\bm{I}_{0}\,\bm{\omega}_{b})\approx\bm{I}_{0}\,\dot{\bm{\nu}}_{2}\
,$ (15)

where $\bm{I}_{0}$ represents a 3D inertia tensor, and $\bm{I}_{3}$ is a
$3\times 3$ identity matrix. After augmentation with (14), the external
factors acting on the UUV are collectively determined by

$\displaystyle\bm{\tau}=\begin{bmatrix}\ \bm{\tau}_{1}^{\operatorname{T}}\ |\
\bm{\tau}_{2}^{\operatorname{T}}\ \
\end{bmatrix}^{\operatorname{T}}=\begin{bmatrix}\ X&Y&Z\ \ |\ \ L&M&N\ \
\end{bmatrix}^{\operatorname{T}},$ (16)

resulting in the following 6-DoF rigid body model:

$\displaystyle\bm{M}\,\dot{\bm{\nu}}+\bm{C}(\bm{\nu})\,\bm{\nu}+\bm{D}(\bm{\nu})\,\bm{\nu}+\bm{g}(\bm{\eta})=\bm{{\tau}}\
.$ (17)

The mass-inertia matrix, $\bm{M}$, maps the state vector $\dot{\bm{\nu}}$ to
the associated mass and inertia terms (Appendix -A), and the skew-symmetric
operator is given by
$[\,\cdot\,]_{\times}:\mathbb{R}^{3}\rightarrow\mathfrak{so}(3)$.
In a bottom-heavy configuration, the center of gravity is positioned below the
body-frame origin, i.e.
$\bm{r}_{W}=\begin{bmatrix}0&0&z_{W}\end{bmatrix}^{\operatorname{T}}$,
yielding the following $6\times 6$ structure

$\displaystyle\bm{M}=\begin{bmatrix}m\,\bm{I}_{3}&-m\,[\bm{r}_{W}]_{\times}\\\
m\,[\bm{r}_{W}]_{\times}&\bm{I}_{0}\end{bmatrix}\ .$ (18)

At the water-air interface, surface vessels experience frequent wave
disturbances—a dynamic that involves cross-coupling between $\bm{\nu}$ and
$\dot{\bm{\nu}}$—expressed in the Coriolis matrix $\bm{C}$. In contrast,
subsea vehicles are exposed to smaller amplitudes and slower oscillations
induced by ocean currents, leading to an approximation of $\bm{C}\approx 0$.
This facilitates a critical underwater capability known as station keeping—a
disturbance-free period that enables a stable hovering while executing
intricate tasks such as localization and mapping.
The third term in (17) is the damping matrix ($\bm{D}$), which captures the
body’s resistive forces countering its motion in the medium. Assuming
uncoupled motion, its approximate coefficients suggest the following diagonal
structure

$\displaystyle\bm{D}({\bm{\nu}})=-\operatorname{diag}\\{\,X_{u},\,Y_{v},\,Z_{w},\,L_{p},\,M_{q},\,N_{r}\,\\}\
.$ (19)

The concluding term, $\bm{g}(\bm{\eta})$, signifies the restoring forces and
moments that actively counteract disruptions in the delicate weight ($\bm{W}$)
buoyancy ($\bm{B}$) equilibrium [17], formulated as

$\displaystyle\bm{f}_{W}(\bm{\eta}_{2})$
$\displaystyle=+\textbf{T}_{b}^{i}\,(\bm{\eta}_{2})\,(\bm{W}\cdot\hat{z}_{b})\
,$ (20) $\displaystyle\bm{f}_{B}(\bm{\eta}_{2})$
$\displaystyle=-\textbf{T}_{b}^{i}\,(\bm{\eta}_{2})\,(\bm{B}\,\cdot\hat{z}_{b})\
.$ (21)

Maintaining a stable upright position depends on the body’s resistance to
externally induced reactive moments. With the center of buoyancy positioned in
the upper $xy$ half-plane,
$\bm{r}_{B}=\begin{bmatrix}0&0&z_{B}\end{bmatrix}^{\mathrm{T}}$, the restoring
force vector can be written in a compact form as

$\displaystyle\bm{g}(\bm{\eta})=-\begin{bmatrix}\bm{f}_{W}(\bm{\eta}_{2})+\bm{f}_{B}(\bm{\eta}_{2})\\\
\bm{r}_{W}\times\bm{f}_{W}(\bm{\eta}_{2})+\bm{r}_{B}\times\bm{f}_{B}(\bm{\eta}_{2})\end{bmatrix}\
.$ (22)

Ultimately, our Newton-Euler equations in body coordinates take the form of

$\displaystyle\bm{M}\,\dot{\bm{\nu}}+\bm{D}(\bm{\nu})\,\bm{\nu}+\bm{g}(\bm{\eta})=\bm{{\tau}}\
.$ (23)

For an inertial-fixed frame, $\bm{{\nu}}$ can be replaced by inverting (13)
and taking its time derivative approximation, namely

$\displaystyle\bm{{\nu}}=$ $\displaystyle\ \textbf{T}^{-1}\,\bm{\dot{\eta}}\
,$ (24) $\displaystyle\bm{\dot{{\nu}}}=$ $\displaystyle\
\cancelto{0}{\dot{\textbf{T}}^{-1}}\bm{\dot{\eta}}+\
{\textbf{T}}^{-1}\,\bm{\ddot{\eta}}\approx\
{\textbf{T}}^{-1}\,\bm{\ddot{\eta}}\ .$ (25)

##  III Proposed Approach

This section introduces our approach to combining the established theory for
understanding both gyros errors and simplified UUV dynamics, aiming to
investigate the feasibility of underwater MEMS gyrocompassing.

###  III-A The Big Picture

In our earlier study [18], a commercially available Emcore device111Sensor
specs @ https://Emcore.com/products/SDN500 was employed to measure angular
rates on a stationary unit that can be freely rotated while leveled. With the
primary objective of discerning how $\bm{\omega}_{ie}$ is affected by heading
angle variations, a deep learning (DL) model222Data & code @
https://GitHub.com/ansfl/LB-Gyrocompassing was proposed as means of tackling
the weak rotation signal, thus enabling a direct heading estimate. Here, we
propose a superposition ($\oplus$) between real-world stationary measurements
($\bm{\tilde{\omega}}_{b}$) and our self-generated dynamics
($\bm{\breve{\omega}}_{b}$), resulting in the following synthesis

$\displaystyle\bm{\breve{\omega}}_{b}=\bm{\tilde{\omega}}_{b}\,\oplus\,\bm{\check{\omega}}_{b}\
.$ (26)

Building upon the prior success of our DL model in learning from stationary
samples, the analysis of synthetic dynamics can now provide valuable insights
into the upper limits of environmental disturbances under which gyrocompassing
can be sustained. Fig. 3 conceptualizes this integration architecture,
resulting in a synthesized trainable dataset.

Figure 3: Conceptual flow of the proposed setup.

###  III-B Simulated Dynamics

The complete 6-DoF dynamic model (23) establishes the analytical relationship
between linear accelerations and angular velocities, correlating them with
external forces and moments, respectively. Upon further examination of the
gyrocompassing equation (7), it becomes evident that accelerations
($\dot{\bm{\nu}}_{1}$) have no impact on it, rendering it translation-
invariant.
Conversely, angular velocities ($\dot{\bm{\nu}}_{2}$) do indeed affect the
heading estimate, establishing its rotation-variance property. Mathematically,
this eliminates the ${\bm{\nu}}_{1}$ equations, reducing the system to be
solely subjected to external moments $\bm{{\tau}}_{2}$, which act as the
forcing term in the right hand side of

$\displaystyle\bm{I}_{0}\,\dot{\bm{\nu}}_{2}+\bm{D}_{2}(\bm{\nu}_{2})\,\bm{\nu}_{2}+\bm{g}_{2}(\bm{\eta}_{2})=\bm{{\tau}}_{2}(t)\
,$ (27)

where the subscript ’$2$’ denotes rotational coordinates. Various disturbance
models are discussed in details in [16] (page 71), and [19]. However, to
ensure the practicality of our simulation scheme, we confine our analysis to
three distinct modes:

  1. (i)

Impulse $\bm{{\tau}}_{2}(t)=\delta(t)$: Signifies an instantaneous
perturbation at $t_{0}$, resulting in short-term system responses.

  2. (ii)

Step $\bm{{\tau}}_{2}(t)=\mathbf{1}_{[0,t_{ss})}(t)$: Marking a sudden change
at $t_{0}$, characterized by a transient behavior, subsequently transitioning
to a decay towards a steady-state.

  3. (iii)

Sinusoidal $\bm{{\tau}}_{2}(t)=\mathcal{W}\cos(\omega_{0}t+\varphi_{0})$:
Featuring periodic wave-like patterns with amplitude $\mathcal{W}$ and phase
$\varphi_{0}$. The resulting oscillatory response decays in proportion to
$\bm{D}$.

Upon normalizing (27), the system manifests as a set of linear ordinary
differential equations (ODE), solvable either by integrating factors or the
Laplace transform (Appendix -B). While the system parameters naturally remain
constant, the parameters of $\bm{{\tau}}_{2}$ vary randomly between
iterations.
This strategy enhances the model’s generalizability by closely simulating
underwater conditions, thereby bridging the reality gap during inference.

###  III-C A Learning-Based Approach

Though the analytical structure presented thus far might suggest the
possibility of a closed-form solution, two main reasons defer such an
assumption. Firstly, the initial conditions, i.e., body Euler angles, are not
known a priori, particularly at depths of tens of meters. Secondly, the
presence of time-varying dynamics renders conventional tools such as the
Fourier transform or wavelet analysis suboptimal. To that end, learning
approaches excel in adapting and generalizing from data, especially in our
case where the meaningful signal $\bm{\omega}_{ie}^{e}$, is ’buried’ under
both instrumental errors $\bm{\tilde{\omega}}_{b}$ and complex underwater
dynamics $\bm{\check{\omega}}_{b}$.
Within a supervised learning paradigm, the problem formulation differentiates
the input sample space $\bm{\Omega}$ form the corresponding label space
$\bm{\psi}$, linked by a statistical probability measure $\mathcal{P}$. A
learner in the hypothesis space $f\in\mathcal{F}$ is tasked with performing an
inference mapping $f:\bm{\Omega}\rightarrow\bm{\psi}$, aiming to minimize
generalization error $\mathcal{L}_{\mathcal{P}}$ as follows

$\displaystyle\bm{\theta}^{*}=\arg\min_{\bm{\theta}}\mathcal{L}_{\mathcal{P}}=\mathbb{E}_{(\omega,\psi)\sim\mathcal{P}}\left\\{\ell\left(\hat{\psi},\psi\right)\right\\}\
,$ (28)

where $\bm{\theta}$ denotes the tuned parameters,
$\hat{\psi}=f_{\theta}(\bm{\breve{\omega}}_{b})$ expresses the model
predictions (heading angles), and the inference is evaluated using quadratic
loss

$\displaystyle\ell:\mathcal{F}\times\bm{\Omega}\times\bm{\psi}\rightarrow\mathbb{R}_{+}\
.$ (29)

Input : Labeled sensor data ( $\bm{\tilde{\Omega}},\bm{\psi}$ )

$\bm{\breve{\Omega}}\leftarrow$ [ ] ;

// Initialization

for  _$(i\,,\,\bm{\tilde{\omega}}_{i})$ in $\bm{\tilde{\Omega}}$_ do

$\bm{\tau}_{2}(t)\leftarrow\texttt{generate}(\mathcal{W},\,\omega_{0},\,\varphi_{0})$

$\bm{\check{\omega}}_{i}\leftarrow\texttt{solve}(\ref{eq:simplified})$

$\bm{\breve{\omega}}_{i}\leftarrow\bm{\tilde{\omega}}_{i}\,\oplus\,\bm{\check{\omega}}_{i}$

$\bm{\breve{\Omega}}\texttt{.add}$( $\bm{\breve{\omega}}_{i}$ )

end for

return tensor( $\bm{\breve{\Omega}},\bm{\psi}$ )

Algorithm 1 The augmented dynamics algorithm

Algorithm 1 outlines the additive process by which the stationary dataset
$\bm{\tilde{\Omega}}$ incorporates artificial dynamics $\bm{\check{\Omega}}$.
During the $i$-th iteration, simulation parameters are randomly drawn
(generate) to allow for a wide range of excitation profiles
$\bm{\tau}_{2}(t)$. With all variables assigned, further substitution in the
system equation facilitates the synthesis of disturbances (solve), resulting
in a new mixed-mode volume $\bm{\breve{\Omega}}$.

##  IV Empirical Assessment

Assuming the intrinsic properties $\bm{I}_{0}$ and $\bm{D}_{2}$ to be time-
invariant, the system response seems to be dominated by the torque-to-inertia
ratio $\left(\frac{\bm{\tau}(t)}{\bm{I}_{0}}\right)$, or $\gamma$ for brevity.
This normalization term ensures geometry-independence, allowing a focus on the
order of magnitude (OoM) regardless of specific configurations. Furthermore,
it facilitates the integration of varying disturbance intensities, enhancing
the model’s ability to learn and adapt to characteristic responses.
Figure 4 illustrates the UUV’s dynamic response, showcasing three distinct
solutions derived from (27) and unveiling the system’s angular response to
diverse inputs. When subjected to a wave input, the UUV exhibits an initial
angle tilt, revealing an oscillatory relationship intricately linked to its
specific geometry and mass properties.

Figure 4: Simulation of the characteristic excitation profiles: impulse (top),
step (middle), and sinusoidal (bottom) functions.

The top two plots capture the system’s reaction to impulse and step inputs,
demonstrating transient behavior originating from the damping effect.
Observably, the angular states exhibit a gradual relaxation process before
reaching a steady-state. On the contrary, the bottom plot illustrates the
system’s response to a sinusoidal input, underscoring the preservation of
input periodicity. Notably, variations in scaling and phase shift in the
sinusoidal input are attributed to the inherent properties of the system.
Subsequently, a detailed analysis of $\bm{\eta}_{2}$ (see (39) in Appendix -B)
is conducted to discern the spectral response of the system, shedding light on
the frequencies experiencing amplification and attenuation. In marine
environments, ocean currents typically manifest at relatively low frequencies.
Fig. 5 investigates the spectral content of the angular response to
characteristic inputs. Ocean currents manifest a broad spectrum of
frequencies, ranging from the gradual rhythms of slow tidal waves to the more
frequent patterns associated with stormy seas. In this context, the damping
effect is notably pronounced at higher frequencies, wherein the opposing force
of water resistance effectively counters oscillations, resulting in a gradual
attenuation.

Figure 5: Spectral response to unit input $\bm{\tau}(t)$.

As observed, lower values of $\gamma$ correspond to decreased amplitudes in
the system’s response. This is a predictable outcome, as less wave energy is
considered in such scenarios. Nevertheless, these lower values provide
valuable insights into the interplay between the UUV and its surrounding
environment. Finally, to comprehend how everything converges in the
gyrocompassing procedure, let us examine the dynamics’ influence on signal
clarity in terms of SNR, given by

$\displaystyle\text{SNR}_{\text{dB}}=\frac{P_{\text{signal,dB}}}{P_{\text{noise,dB}}}=20\log_{10}\left(\frac{A_{\text{signal}}}{A_{\text{noise}}}\right)\
,$ (30)

where $P$ stands for power and $A$ stands for amplitude, and expressed in
decibels (dB). To visualize that, Fig. 6 depicts the gyroscopes SNR versus
averaging time. The computation of their amplitudes involves dividing the
square of $\bm{\omega}_{ie}^{e}$ by the square norm of the gyros’ standard
deviation. In addition to the four $\gamma$ values observed previously, the
thick blue line represents a non-dynamic scenario (N/D), effectively serving
as a control group. Being stationary, it exhibits an expected linear growth of
SNR by 20 decibels (1 OoM) per decade, on a logarithmic time scale.

Figure 6: SNR comparison with varying averaging time and $\gamma$’s.

The challenging task of signal enhancement becomes evident: the stronger the
waves impacting the UUV, the more undesirable frequencies propagate within the
signal, compromising $\bm{\omega}_{ie}^{e}$ clarity, and thereby reducing the
SNR. A notable instance is observed in the bottom plot ($\gamma=10$), where
the intense wave input results in a negative SNR for at least 10 seconds of
sampling before it can even be identified (0 dB).
To that end, our proposed DL algorithm comes to the rescue, enabling a
learning-based extraction of the rotation signal from the obscuring dynamics,
accounting for varying intensities. Fig. 7 depicts the gyrocompassing error
through the model validation loss, comparing performance against the number of
epochs and increasing $\gamma$ values.
Firstly, a noticeable correlation emerges between low $\gamma$ values and a
low root mean squared error (RMSE), likely attributed to the heightened
frequency content that corrupts the GT signal. Secondly, the presence of
frequency contents introduce learning instability, rendering the
generalization task more challenging as the GT pattern becomes less
recognizable.

Figure 7: Validation curve against number of epochs. TABLE II: Analyzing gyrocompassing errors across benchmark models and varying intensities. | RMSE [deg]
---|---
|  |  N/D | $\gamma=0.1$ | $\gamma=0.5$ | $\gamma=1$ | $\gamma=10$
Model | Wavelet | 4.83 | 7.23 | 9.72 | 14.18 | 31.59
Wiener | 4.90 | 7.77 | 10.44 | 14.76 | 35.64
SG | 4.56 | 6.33 | 8.52 | 12.91 | 28.96
FIR | 4.87 | 7.43 | 9.97 | 13.64 | 35.64
Ours | 3.26 | 5.13 | 5.88 | 7.23 | 13.91

Table II summarizes the data from these plots and provides a comparison with
four state-of-the-art filters: wavelet denoising, the Wiener filter, the
Savitzky-Golay (SG) filter, and the finite impulse response (FIR) filter [20,
21, 22].
While they are regarded as reliable benchmarks for signal enhancement, their
lower error bounds fall short compared to our model, which exhibits
improvement rates ranging from $30\%$ to $60\%$, depending on $\gamma$ values.
This underperformance is attributed to the inherent model-based limitations,
which rely on simplistic cause-and-effect relationships. When complex dynamics
are introduced, these approaches struggle to refine noisy inputs, rendering
them unsuitable for operational platforms with minimal error tolerance.

##  V Conclusions

In this paper, we address the challenging task of enabling gyrocompassing in
an underwater environment where external ocean currents may interact with the
UUV body. Through the design of a simplified vehicle structure and a dynamic
motion model mimicking wave disturbances, we have successfully established an
integrated pipeline capable of simulating realistic hovering scenarios.
Exhibiting noise in both instrumentation and mechanical vibration aspects, the
synthesized signals demonstrate low SNR, significantly complicating the
gyrocompassing task. To overcome this challenge, we propose and train a
dedicated learning model, which is then compared with conventional filtering
techniques. The results indicate a substantial improvement, often in tens of
percentages, as our model learns to selectively process meaningful
information, effectively ’ignoring’ additive dynamics. These findings
underscore the potential of learning-based approaches to redefine error
limits, making off-the-shelf gyroscopes competent for underwater
gyrocompassing.

## Acknowledgement

D.E. is supported by the Maurice Hatter foundation and the Bloom School
Institutional Excellence Scholarship for outstanding doctoral students at the
University of Haifa.

###  -A Second-Order Ordinary Differential Equation

Presented below are the simplified coefficient matrices of the ODE, utilized
for solving the system dynamics.

####  -A1 Mass-Inertia Matrix

To begin with, let us assume a simplified solid cylinder, with mass $m$,
radius $r$, length $l$, and centered about the ${z}$-axis such that its 3D
inertia tensor is

$\displaystyle\bm{I}_{0}=\begin{bmatrix}\frac{1}{2}mr^{2}&0&0\\\
0&\frac{1}{12}m(3r^{2}+l^{2})&0\\\
0&0&\frac{1}{12}m(3r^{2}+l^{2})\end{bmatrix}\ .$ (31)

Ideally, the mass-inertia matrix of an ideal 3D axisymmetric object, whose
center of gravity coincides with its body-frame origin
($\bm{r}_{W}=\bm{0}_{3}$), degenerates into

$\displaystyle\bm{M}=\operatorname{diag}\\{\,m,\,m,\,m,\,I_{xx},\,I_{yy},\,I_{zz}\,\\}\
.$ (32)

But since mass is concentrated in the lower half $xy$ plane,

$\displaystyle\bm{r}_{W}=\begin{bmatrix}0&0&-z^{\prime}\end{bmatrix}^{\operatorname{T}}\
,$ (33)

along port/starboard and fore/aft symmetries, about the $xz$ and the $yx$
planes, respectively, turns $\bm{M}$ into

$\displaystyle\bm{M}=\begin{bmatrix}m&0&0&0&mz_{G}^{\prime}&0\\\
0&m&0&-mz_{G}^{\prime}&0&0\\\ 0&0&m&0&0&0\\\
0&-mz_{G}^{\prime}&0&I_{xx}&0&0\\\ mz_{G}^{\prime}&0&0&0&I_{yy}&0\\\
0&0&0&0&0&I_{zz}\end{bmatrix}\ .$ (34)

####  -A2 Restoring forces and moments

In a frictionless medium, an initially set-in-motion pendulum persists in
oscillating indefinitely, thanks to mechanical energy conservation. However,
real-world scenarios introduce non-conservative forces, including hydrodynamic
drag, turbulence, and sea currents. These external factors gradually dissipate
the initial energy input, ultimately leading the pendulum to return to
equilibrium.
Similarly, UUVs adhere to analogous principles, wherein their longitudinal and
lateral planes respond to external torques by revolving around the $\phi$ and
$\theta$ angles, tilting both gravity and buoyancy components. Being bottom-
heavy, the center of gravity is below the body-fixed origin,
$\bm{r}_{W}=\begin{bmatrix}0&0&z_{W}\end{bmatrix}^{\operatorname{T}}$, while
the center of buoyancy is above at
$\bm{r}_{B}=\begin{bmatrix}0&0&z_{B}\end{bmatrix}^{\operatorname{T}}$,
resulting in the restoring force being

$\displaystyle\bm{g}(\bm{\eta})=\begin{bmatrix}\vrule\\\ \ \bm{g}_{1}\quad\\\
\rule[-4.30554pt]{0.5pt}{10.76385pt}\vspace{1mm}\\\
\hline\cr\rule[-4.30554pt]{0.5pt}{10.76385pt}\\\ \bm{g}_{2}\\\
\rule[-4.30554pt]{0.5pt}{10.76385pt}\end{bmatrix}=\begin{bmatrix}(\bm{W}-\bm{B})\text{s}_{\theta}\\\
-(\bm{W}-\bm{B})\text{c}_{\theta}\text{s}_{\phi}\\\
-(\bm{W}-\bm{B})\text{c}_{\theta}\text{c}_{\phi}\\\
(z_{W}\bm{W}-z_{B}\bm{B})\text{c}_{\theta}\text{s}_{\phi}\\\
(z_{W}\bm{W}-z_{B}\bm{B})\text{s}_{\theta}\\\ 0\end{bmatrix}\ .$ (35)

As observed, an upright vehicle ($\phi=\theta=0$) would experience all
reactive moments as zeros, except for the downward gravity force. Upon
perturbation, all reactive forces and moments (except the yawing moment)
manifest as non-homogeneous solutions, causing both roll and pitch angles to
successfully revert to their initial states, while the heading angle will
consistently drift away. To preserve the linearity of the ODE, a small-angle
approximation is applied to (35) and substituted into (17), resulting in a
second-order differential relation.

###  -B Simulated Solutions

Generating practical solutions for the system involves assuming zero initial
conditions and applying the small-angle approximation. This is followed by
substituting (24) into (23) to transition to an inertial frame, resulting in

$\displaystyle\bm{I}_{0}\,\ddot{\bm{\eta}}_{2}(t)+\bm{D}_{2}\,\dot{\bm{\eta}}_{2}(t)+\textbf{g}_{2}\,\bm{\eta}_{2}(t)=\bm{{\tau}}_{2}(t)\
,$ (36)

where $\bm{W}-\bm{B}$ takes the matrix form of
$\textbf{G}_{2}=\operatorname{diag}(\bm{g}_{2})$.
Given that all moment inputs $\bm{{\tau}}_{2}(t)$ are time-deterministic, the
Laplace transform of the system is given by

$\displaystyle
s^{2}\bm{I}_{0}\,{\bm{\eta}}_{2}(s)+s\bm{D}_{2}\,{\bm{\eta}}_{2}(s)+\textbf{G}_{2}\,\bm{\eta}_{2}(s)=\bm{{\tau}}_{2}(s)\
,$ (37)

resulting in the transfer function of the orientational response

$\displaystyle\bm{H}(s)=\frac{\bm{\eta}_{2}(s)}{\bm{\tau}_{2}(s)}=\big{(}s^{2}\bm{I}_{0}+s\bm{D}_{2}+\textbf{G}_{2}\big{)}^{-1}\
.$ (38)

In the frequency domain ($s=j\omega$), each of the angular states
($\phi,\theta,\psi$) assumes a standard oscillatory form, comprising a scaling
factor multiplied by the response term

$\displaystyle\bm{\eta}_{i}(\omega)=\left(\frac{\bm{\tau}_{i}(\omega)}{\bm{I}_{0,ii}}\right)\cdot\frac{1}{(\omega_{0}^{2}-\omega^{2})-(2\omega\zeta)j}\
,$ (39)

where $i\in\\{\text{x, y, z}\\}$, the system frequency is
$\omega_{0}=\sqrt{\frac{G_{ii}}{I_{ij}}}$, and its damping ratio is
$\zeta=\frac{D_{ij}}{2\sqrt{G_{ij}I_{ij}}}$. Note that both terms are
functions of the inertia tensor, restoring forces, and torsional friction.
Given the diverse nature of ocean currents, input waves $\bm{\tau}_{2}(t)$ can
be modeled as either an impulse function or a sinusoidal wave, leading to
distinct dynamic responses. For instance, a pitching moment ($\tau_{2,x}$)
characterized as a step function would produce an oscillatory roll angle, as
follows

$\displaystyle\phi(t)=1-e^{-\zeta\omega_{0}t}\frac{\sin\left(\sqrt{1-\zeta^{2}}\omega_{0}t+\phi_{0}\right)}{\sin(\phi_{0})}$
(40)

where the initial phase angle is given by $\phi_{0}=\cos^{-1}(\zeta)$.

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Strong Cosmic Censorship in light of Weak Gravity Conjecture for Charged Black
Holes
Jafar Sadeghi ⋆111Email<EMAIL_ADDRESS>Mohammad Reza Alipour ⋆222Email:
<EMAIL_ADDRESS>Saeed Noori Gashti⋆,‡333Email:
<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
‡School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran

###### Abstract

In this paper, we investigate the strong cosmic censorship conjecture (SCC)
for charged black holes in the de Sitter space by considering the weak gravity
conjecture (WGC). Using analytical methods, we find that the SCC is preserved
for dS-charged black holes with respect to some restriction $qQ\gg 1$ and
$r_{+}\geq Q$ with the help of the WGC condition viz $\frac{q}{m}\geq 1$ for
scalar fields. Where q, m are the charge and mass of the scalar field, and
$r_{+}$, Q determine the radius of the outer event horizon and the charge of
the black hole, respectively. In that case, when the (WGC) is valid, SCC will
definitely be satisfied for the dS-charged black holes. On the other hand, the
SCC is violated when the WGC is not satisfied. Also, we examined the RN-dS
charged black hole in the extremality state and found that SCC can be violated
with the condition $\Lambda r_{+}^{2}=1$.

Keywords: Strong cosmic censorship conjecture; Weak gravity conjecture; RN-dS
charged black hole

###### Contents

  1. 1 Introduction
  2. 2 Weak Gravity Conjecture
  3. 3 Charged Black Holes in dS Space
  4. 4 The Quasinormal Resonant Frequency Spectrum
  5. 5 Discussion and Result
  6. 6 Acknowledgments

##  1 Introduction

One of the studies with a long history in general relativity is the study of
the collapse of small perturbations. We need more information on how these
oscillations decay to understand better the gravity concept, use gravitational
wave data, and study and investigate the valuable features of general
relativity. One of the signs of the failure of determinism in general
relativity can be the emergence of an interesting phenomenon known as Cauchy
horizons that appear in the astrophysical solutions of Einstein’s equations.
These horizons are such that it is impossible to specify the history of the
future of an observer that passes through such horizons using Einstein’s
equations and initial data. With these descriptions in the black holes’ space-
time background, it is an expected possibility that the perturbations of the
outer region are infinitely amplified by a mechanism known as the blue shift.
They lead to a singularity boundary beyond the Cauchy horizon in the interior
of black holes, where field equations cease to make sense. The Penrose strong
cosmic censorship (SCC) confirms such an expectation. Of course, another point
is that astrophysical black holes are stable due to a special mechanism called
the perturbation-damping mechanism, which is applied in the outer region.
Therefore, considering whether SCC retains real hinges or not depends on the
very subtle competition between the collapse of perturbations in the outer
region and their amplification (blue shift) in the inner space-time of black
holes. In general, the fate of Cauchy horizons is related to the collapse of
small perturbations outside the event horizon. Hence, the validity of SCC is
tied to the extent of external damps fluctuation. In connection with various
structures and conditions, SCC and its satisfaction and violation have been
investigated in various theories. The violation of this conjecture near the
extremal region studied in the investigation of higher curvature gravity [1].
Also, this conjecture has been challenged in investigating many charged black
holes. In [2, 3], this conjecture was checked for a charged AdS black hole. It
was shown that for a specific interval for the parameter ($\beta$), this
conjecture is satisfied and violated in other areas as well. The strong cosmic
censorship conjecture has also been investigated in two dimensions. There have
been interesting outcomes regarding the violation of this conjecture near the
extremal region at specific points [4]. The study of this conjecture in the
structure of three-dimensional black strings has also carried interesting
results, which you can see [5] for a deeper study. Also, you can see [6, 7, 8,
9] for further study. The effectiveness of mass-inflation systems, which are
involved in the transformations of the inner Cauchy horizon associated with
the space-time of black holes that are approximately flat, which is
pathological in the estimation of SCC, into a series of hypersurfaces which is
singular non-extendable. Those that are in an indivisible form are related to
two different types of physical mechanisms[10, 11, 12, 13, 14, 15, 16]. First,
the events in the exterior space-time regions of dynamic black holes formed
viz the collapse of the remnant perturbation fields and second amplification
of exponential blue shift related to the fields falling into the inside of
black holes. We can manage these two introduced different systems through
parameters such as ($g$) and ($k_{-}$). It can be stated that the
dimensionless physical ratio with the help of these two parameters can
determine the fate of the inner Cauchy horizons inside such space-times of
non-asymptotic flat black holes[8, 17, 18],

$\begin{split}\beta\equiv\frac{g}{k_{-}}.\end{split}$

Of course, a certain range of parameters of black holes, such as mass and
charge, etc., as indicated in [8, 17, 18],

$\begin{split}\beta>\frac{1}{2}.\end{split}$

So, space-time of the corresponding black holes can be physically expanded
beyond their Cauchy horizon which includes a pathological fact and a sign of
algebraic failure or a violation of the Penrose SCC in classical general
relativity. For the dynamics of Einstein’s equations as well as the destiny of
the observers, the explosive structure of the curvature that is related to
($\beta<1$) does not have per se much physical significance: it indicates two
theorems, not the failure of the field equations mentioned in [19] and of
course not the destruction of macroscopic observers which is discussed in
[13]. Therefore, the physical and mathematical formulation of the conjecture
of a SCC in such conditions leads to ignoring physical phenomena such as
impulsive gravitational waves or the formation of shocks in relativistic
fluids. Due to the aforementioned reasons, the modern form of the SCC
conjecture was introduced that requires a stronger constraint
($\beta<\frac{1}{2}$ ) and many works have been done to fit such constraints.
In [8], it was found that when there are _neutral_ scalar fields in the
presence of the R-N dS black hole, it leads to the violation of SCC. But in
continuation, Hod has shown in [34] that the presence of _charged_ massive
scalar fields near _charged_ black holes is inevitable. By considering
_charged_ scalar fields near a _charged_ black hole and using WKB techniques,
he has shown that the SCC will be meted. In this article, by using an
analytical method and WKB approximation, we show that _charged_ scalar fields
play an essential role in satisfying the SCC in light of WGC for charged black
holes. Therefore, we find that the numerical results presented in [8] have no
physical relevance to the question of the (in) validity of the SCC in
_charged_ black-hole spacetimes. In particular, the SCC conjecture in the
context of _charged_ black hole spacetimes must be tested in the presence of
_charged_ matter fields, whereas the numerical results presented in [8] are
based only on the presence of _neutral_ scalar fields in the _charged_ black
hole spacetime. Therefore, in this article, we are going to study different
structure of this conjecture. According to the above description, we consider
the general configuration of _charged_ black holes in the presence of massive
_charged_ scalar fields. Then, using the weak gravity conjecture, we will
prove that SCC is valid for specific values for all _charged_ black holes. We
will use the weak gravity conjecture to prove a general relation with respect
to SCC for all _charged_ black holes. According to the above explanations, we
organize the article in the following form.
In section 2, we will give basic explanations about the weak gravity
conjecture and also the motivation to use it. In section 3, we introduce
charged black holes in dS space, and then we show the quasinormal resonant
frequency spectrum in section 4. We check the conditions of compatibility and
violation of (SCC) with respect to (WGC) for RN-dS charged black holes.
Finally, we describe the results in section 5.

##  2 Weak Gravity Conjecture

As it is known in the literature, a new idea has been put forward as a
swampland program to check theories coupled to gravity, the consistency of
quantum gravity and finally, a proof for string theory. Recently, ones have
done lots of work on this field [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].
Due to the special conditions of string theory and the fact that its testing
and experimental investigations seem a bit difficult, this idea has been
proposed to test and investigate various concepts of cosmology. The swampland
program is challenged from two sides. From an up-bottom view for introducing
principles and limitations to introduce conjectures, as well as mathematical
formulations to examine cosmological concepts. A second look from the bottom-
up in order to test each of these conjectures with various concepts of
cosmology including inflation and matching with observable data, which is both
a proof for this new idea and a proof for string theory. So far, many
conjectures have been proposed from this theory, and now, according to the
structure and further investigations, new conjectures will be added to this
program. Some of these conjectures face challenges and as a result,
corrections are made to the conjectures. We face some limitations in quantum
gravity (QG). At the point when gravity is considered at the quantum level,
the hypothesis will be incompatible. Generally having a reliable quantum
hypothesis of gravity isn’t really straightforward and can in any case hold
many surprises and can be interesting for physical science at low energies.
The objective of the swampland program is to decide the limitations that an
effective field theory(EFT) should fulfill to be viable with the consideration
of ultraviolet completion(UV) in QG. They are called swampland limitations,
and different suggestions are figured out as far as swampland conjectures(SC).
The objective is to recognize these limitations, accumulate proof to
demonstrate or refute them inside the structure of QG, give reasoning to make
sense of them in a model-free manner, and comprehend their phenomenological
suggestions for low-energy EFTs. Albeit the swampland idea isn’t restricted to
string theory on a fundamental level, SC are frequently examined by string
theory backdrops. Without a doubt, the string theory gives an ideal structure
to thorough quantitative testing of conjectures and works on how we might
interpret potential compressions of string theory. Strangely, it has as of
late been uncovered that a large number of these conjectures are to be sure
related, recommending that they may essentially be various countenances of
some yet-to-be-found crucial standard of QG. As far as possible have
significant ramifications for cosmology and particle physics. They can give
new core values to building conjectures past the standard models in high-
energy physics. They may likewise prompt $UV/IR$ blending, which breaks the
assumption for scale detachment and possibly gives new bits of knowledge into
the regular issues seen in our universe. Consequently, the presence of
swampland is extraordinary information for phenomenology. For a total rundown
of references connected with swampland that might be valuable, we allude in
[20] the swampland program (SP) has likewise been surveyed and presented. The
shortfall of global symmetry (GS) and the completeness of charge spectra are
at the center of the SP. Nonetheless, they need phenomenological suggestions
except if we can restrict the global symmetries [21, 22] and whether there is
any limitations point on the mass of charged states. In any case, they just
bound the complete hypothesis but not the low-energy EFTs. Specifically, it is
phenomenologically important whether all charged particles can be really super
heavy and even compare to black holes(BHs), or whether there is some thought
of completeness of the range that gets by at low energies. A large portion of
the SCs examined address exactly these inquiries. They want to profoundly
explore these assertions and measure them so we can draw nearer to the
recuperation of a few global symmetries. For instance, we can deduce
recuperate a global symmetry (GS) $U(1)$ by sending the gauge coupling(GC) to
nothing, which ought not to be permitted in $QG$. Attempting to comprehend
string theory for the study of this issue, it might turn out that if one
somehow managed to try to do such work, can give data about the imperatives
that an EFT can fulfill to be viable with QG. Likewise, WGC forbids this cycle
by flagging the presence of new charged states that denies the depiction of
the EFTs. Thusly, it gives an upper bound on the mass of these charged states.
The WGC comprises of some parts: the electric and the magnetic electric-WGC:
As indicated by a quantum hypothesis, we have the following condition [20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30],

$\begin{split}\frac{Q}{m}\geq\frac{\mathcal{Q}}{M}|_{ext}=\mathcal{O}(1),\end{split}$
(1)

and

$\begin{split}Q=gq,\end{split}$ (2)

where, $g$ and $q$ are the gauge coupling and the quantized charge. The
electric-WGC needs the presence of an electrically charged condition of a
higher charge-to-mass proportion than extremal BH in that hypothesis, which is
regularly a variable of the order one. One more understanding of this
conjecture is that the limitations region shows that scale force determines
stronger than the gravity on this mode — so subsequently is called WGC. This
is an identical equation since it expects that electromagnetic force is
stronger gravitational force [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30],

$F_{Grav}\leq F_{EM}$ (3)

It implies that the charge is more prominent than the mass, so we get a
similar condition as above. This is as of now false within the sight of
massless scalar fields. The motivations of $WGC$ are twofold. To begin with,
it makes a $QG$ boundary to reestablish the GS of $U(1)$ by sending
$g\rightarrow 0$. If a GC goes to zero as indicated by $WGC$, this conducts
new light particles and the cutoff the hypothesis arrives at nothing and
nullifies the EFT. Because of the littleness of the scale coupling, it relies
upon how much energy the interaction with which you need to portray the viable
EFT. The smallness of the cycle energy leads to the smallness of the scale
coupling. On the other hand, if you need to keep the EFT substantial up to an
extremely high cut-off, the GC can’t be excessively small. This is an
illustration of swampland limitations that becomes more grounded for higher
energies. Obviously, a hypothesis with disappearing measure coupling i.e., GS
is incompatible because the cutoff of the viable EFT is likewise zero. One
more fundamental inspiration for $WGC$ is that a kinematic prerequisite
permits extremality $BH$ to have decomposed. Charged BHs should fulfill an
extremality breaking point to stay away from the presence of exposed
singularities, as shown by the weak cosmic censorship $(WCC)$. For a given
charge $\mathcal{Q}$, this super bound shows that the this super bound shows
that the mass $M$ of the BHs should be more noteworthy than the charge [20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30],

$M\geq\mathcal{Q}$ (4)

For the BHs to have a regular horizon. Here, we set the extremal factor
$\mathcal{O}(1)$ to one for simplicity. The primary condition for starting the
decay to the small black hole and particle is the existence of the extremal
BHs ($M=\mathcal{Q}$). So, one can consider the decay of an extremal BHs which
one of the rot items has a charge more modest than its mass as far as
possible, so $M_{1}\geq\mathcal{Q}_{1}$. Then the rot item can never again
have a charge more modest than the mass, that is $m_{2}\leq Q_{2}$. It is just
a kinematic necessity. Since the second rot item violates the WCC, it can’t be
a BH, so it should be a particle. The above kinematic necessity can be
acquired by applying preservation of mass/energy and protection of charge as
follows. The initial mass of the BH should be more prominent than the amount
of the mass of the rot items $M_{i}$ and the charge of the initial BH.

##  3 Charged Black Holes in dS Space

The metric of charged black hole in spherical symmetric space is defined as
follows,

$dS^{2}=f(r)dt^{2}-f^{-1}(r)dr^{2}-r^{2}d\Omega^{2},\qquad
d\Omega^{2}=(d\theta^{2}+\sin^{2}(\theta)d\varphi^{2}).$ (5)

Here, we consider $f(r)=H(M,Q)-\frac{\Lambda r^{2}}{3}$ in general; where Q,
M, $\Lambda>0$ are electric charge, the mass of the black hole and the
cosmological constant respectively. In this case, we can obtain its event
horizons as follows,

$f(r_{\star})=0\qquad\rightarrow\qquad\star\in(-,+,...,c).$ (6)

Considering the metric in general terms, we have different event horizons,
where ($r_{-}$) is the Cauchy horizon, ($r_{+}$) is the outer event horizon,
and ($r_{c}$) is the cosmological horizons. Using Klein-Gordon’s differential
equation, we can determine the dynamics of a massive charged particle near a
charged black hole [31, 32, 33, 34],

$\frac{1}{\sqrt{-g}}\partial_{\mu}(g^{\mu\nu}\sqrt{-g}\partial_{\nu}\Phi)-2iqg^{\mu\nu}A_{\mu}\partial_{\nu}\Phi-q^{2}g^{\mu\nu}A_{\mu}A_{\nu}\Phi-m^{2}\Phi=0,$
(7)

where $m$ and $q$ are the mass and charge of the particle, respectively also,
$A_{\mu}=\left(\frac{Q}{r},0,0,0\right)$. We can define the scalar field
$\Phi$ according to relation (7) as follows [36],

$\Phi(t,r,\theta,\phi)=\sum_{m}\sum_{\ell}e^{-i\omega t}Y_{\ell
m}(\theta,\varphi)\Phi(r).$ (8)

The integer parameters $\ell$ and $m$ are the spherical and the azimuthal
harmonic indices of the resonant eigenmodes which characterize the charged
massive scalar fields in the charged black-hole spacetime. By putting Eq.(8)
in Eq.(7) and using $dx=\frac{dr}{f(r)}$, we get the Schrödinger-like
differential equation ,

$\frac{d^{2}\phi(r)}{dx^{2}}+V(r)\phi(r)=0.$ (9)

The effective radial potential due to a massive charged particle near a
charged black hole is defined as [8],

$V(r)=\frac{qm}{r^{2}}\left[\frac{q}{m}\alpha(r)-\frac{m}{q}\beta(r)\right],$
(10)

where

$\alpha(r)=Q^{2}\left(1-\frac{\omega
r}{qQ}\right)^{2},\qquad\beta(r)=r^{2}f(r)H(r),\qquad
H(r)=\left(\frac{\ell(\ell+1)}{m^{2}r^{2}}+\frac{f^{\prime}(r)}{m^{2}r}+1\right).$
(11)

Also, we can consider the boundary conditions for the special radial function
near the outer event horizon as an incoming wave and at the largest event
horizon as an outgoing wave [34, 35]:

$\displaystyle\phi(x)\sim\left\\{\begin{array}[]{ll}e^{-i(\omega-\frac{qQ}{r_{+}})x},&\hbox{for
\hskip 5.69046pt $r\rightarrow r_{+}\hskip
2.84544pt(x\rightarrow-\infty)$;}\\\
e^{-i(\omega-\frac{qQ}{r_{c}})x},&\hbox{for \hskip 5.69046pt $r\rightarrow
r_{c}\hskip 2.84544pt(x\rightarrow\infty)$.}\end{array}\right.$ (14)

According to the above boundary conditions, we can obtain the discrete
spectrum of $\omega$, defined as the resonance frequency of the imaginary
quasi-normal state.

##  4 The Quasinormal Resonant Frequency Spectrum

In this section, we need to obtain the imaginary part of the resonance
frequency to investigate the linear dynamics of a massive charged particle
near a general charged black hole. Also, we need to do this in a dimensionless
regime to do this analytically. Since, the
$\frac{q^{2}}{\hbar}\simeq\frac{1}{137}$ relationship exists in our universe,
we can consider it for black holes, even slightly charged, and get $qQ\gg 1$.
In addition, the mechanism of the Schwinger-type pair-production in space-time
of charged black hole creates a limit to the black hole electric field with
the $\frac{Q}{r_{+}^{2}}\ll\frac{m^{2}}{q}$ relationship [37, 38, 39, 40].
Therefore, according to the above statement, we can consider SCC and define
our constraint regime following ansans,

$m^{2}r_{+}^{2}\gg\ell(\ell+1)\qquad and\qquad m^{2}r_{+}^{2}\gg 2k_{+}r_{+},$
(15)

where $k_{+}=f^{\prime}(r_{+})/2$ is the gravitational acceleration of the
black hole at the outer event horizon. In this area, we try to obtain the
imaginary part of the resonance frequency in the background of the general
charged black hole near the event horizon. Now, we use radial potential (10)
to determine the linear dynamics of the massive charged particle near the
event horizon of the black hole. We can consider this potential in region (15)
as an effective potential and obtain the quasinormal resonance modes
analytically using standard WKB techniques [41, 42]. In this region, we
consider the maximum effective potential near the event horizon of the black
hole at point $r=r_{0}$. In the following, we use the relationship (10), (11),
and $V^{\prime}(r_{0})=0$ to obtain the point where the effective potential is
maximum as follows,

$r_{0}=\frac{q^{2}Q^{2}}{qQ\omega-m^{2}r_{+}^{2}k_{+}}$ (16)

According to the Schrödinger-like differential equation (9) and [41, 42, 43],
we use the $WKB$ method to obtain the quasinormal mode frequencies through the
following,

$iK-(n+\frac{1}{2})-\Lambda(n)=\Omega(n)$ (17)

where

$\begin{split}&K=\frac{V_{0}}{\sqrt{2V_{0}^{(2)}}}\\\
&\Lambda(n)=\frac{1}{\sqrt{2V_{0}^{(2)}}}\left[\frac{\left(\alpha^{2}+\frac{1}{4}\right)}{8}\frac{V_{0}^{(4)}}{V_{0}^{(2)}}-\frac{\left(60\alpha^{2}+7\right)}{288}\left(\frac{V_{0}^{(3)}}{V_{0}^{(2)}}\right)^{2}\right]\\\
&\Omega(n)=\frac{n+\frac{1}{2}}{2V_{0}^{(2)}}\left[\frac{5\left(188\alpha^{2}+77\right)}{6912}\left(\frac{V_{0}^{(3)}}{V_{0}^{(2)}}\right)^{4}-\frac{\left(100\alpha^{2}+51\right)}{384}\frac{\left(V_{0}^{(3)}\right)^{2}V_{0}^{(4)}}{\left(V_{0}^{(2)}\right)^{3}}\right]\\\
&+\frac{n+\frac{1}{2}}{2V_{0}^{(2)}}\left[\frac{\left(68\alpha^{2}+67\right)}{2304}\left(\frac{V_{0}^{(4)}}{V_{0}^{(2)}}\right)^{2}+\frac{\left(28\alpha^{2}+19\right)}{288}\frac{\left(V_{0}^{(3)}V_{0}^{(5)}\right)}{\left(V_{0}^{(2)}\right)^{2}}-\frac{\left(4\alpha^{2}+5\right)}{288}\frac{V_{0}^{(6)}}{V_{0}^{(2)}}\right]\end{split}$
(18)

Here, $V_{0}^{(k)}\equiv\frac{d^{k}V}{dx^{k}}|_{r=r_{0}}$ is the spatial
derivative of the effective potential of equation (10), and its scattering
peak is evaluated at the point $r=r_{0}$. Using relations (10), (11), (16) and
(18), we will have the following relation in the region of (15),

$\begin{split}&K\simeq\frac{k_{+}^{2}m^{4}r_{+}^{4}qQ}{2f_{0}\left(k_{+}m^{2}r_{+}^{2}-qQ\omega\right)^{2}}\\\
&\Lambda(n)\simeq\frac{k_{+}^{2}m^{4}\left[17-60\left(n+\frac{1}{2}\right)^{2}\right]r_{+}^{4}+2k_{+}m^{2}\left[36\left(n+\frac{1}{2}\right)^{2}-7\right]qQr_{+}^{2}\omega}{16qQ\left(qQ\omega-3k_{+}m^{2}r_{+}^{2}\right)^{2}}\times
f_{0}\\\
&\mathcal{A}=15k_{+}^{4}m^{8}\left[148(n+\frac{1}{2})^{2}-41\right]r_{+}^{8}+12k_{+}^{3}m^{6}\left[121-420(n+\frac{1}{2})^{2}\right]qQr_{+}^{6}\omega\\\
&\mathcal{B}=64q^{5}Q^{5}\left(k_{+}m^{2}r_{+}^{2}-qQ\omega\right)^{4}\\\
&\Omega(n)\simeq-(n+\frac{1}{2})q^{3}Q^{3}f_{0}^{2}\times\frac{\mathcal{A}}{\mathcal{B}}\end{split}$
(19)

where $f_{0}=f(r_{0})$. In the next step, to determine the study of SCC, we
need to obtain the minimum value of the fundamental imaginary resonance mode
of the system. For this purpose, using equations (17) and (19), we can
calculate the $Im(\omega_{0})$,

$\begin{split}&\omega\simeq\frac{qQ}{r_{+}}-\frac{2k_{+}m^{2}r_{+}^{2}}{qQ}\left[1-\frac{14400}{11644}\left(\frac{(n+1/2)f_{0}}{qQ}\right)^{4}\right]\\\
&-i\left\\{4f_{0}k_{+}(n+\frac{1}{2})\frac{m^{2}r_{+}^{2}}{q^{2}Q^{2}}\left[1-\frac{34qQf_{0}^{4}}{11664}\right]+\mathcal{O}(f_{0}^{2})\right\\}\end{split}$
(20)

Since we consider $r_{0}$ near the event horizon ($r_{+}$), we have $f_{0}\ll
1$. For investigation the SCC, it is necessary to find the minimum value of
the resonance mode and evaluate its ratio to the surface gravity of the event
horizon,

$\beta=\frac{-Im(\omega_{0})}{k_{+}}\simeq
2f_{0}\frac{m^{2}r_{+}^{2}}{q^{2}Q^{2}}\left[1-\frac{34qQf_{0}^{4}}{11664}\right].$
(21)

Since it is $f_{0}\ll 1$, it is sufficient to have the conditions
$q^{2}Q^{2}>m^{2}r_{+}^{2}$ in the relation above concepts so that
$\frac{-Im(\omega_{0})}{k_{+}}<\frac{1}{2}$ is established. Therefore, we have
the following condition for the study of SCC,

$\frac{q}{m}\geq\frac{r_{+}}{Q}.$ (22)

from equation (22) determine that when $r_{+}\geq Q$, we have the weak gravity
conjecture condition. We know that $k_{-}>k_{+}$, so the relationship of (21)
and (22) is also established for
$\beta=\frac{-Im(\omega_{0})}{k_{-}}<\frac{1}{2}$. Also, according to relation
(21), when $qQ<2\sqrt{f_{0}}mr_{+}$, SCC can be violated. Since $qQ\gg 1$ and
$f_{0}\ll 1$, the mass of the scalar field and the radius of the event horizon
must be very massive and very large respectively. In the following, we obtain
the extremality state of the RN-dS black hole. We will have the following
relation for the RN-dS black hole with respect to equation(5),

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda r^{2}}{3}.$ (23)

When $k_{+}=k_{-}=0$, we can obtain the black hole extremality state,

$Q_{exe}=r_{+}\sqrt{1-\Lambda r_{+}^{2}},\qquad
M_{exe}=r_{+}(1-\frac{2}{3}\Lambda r_{+}^{2}).$ (24)

We substitute Eq.(24) in Eq.(21) to obtain $\beta$ in the extremality state of
the RN black hole,

$\beta\simeq 2f_{0}\frac{m^{2}}{q^{2}(1-\Lambda
r_{+}^{2})}\left[1-\frac{34qf_{0}^{4}\sqrt{1-\Lambda
r_{+}^{2}}r_{+}}{11664}\right]$ (25)

According the above relationship, when the condition
$\frac{q}{m}\geq\frac{1}{1-\Lambda r_{+}^{2}}$ is satisfied, the SCC will
definitely be preserved, and since $\Lambda r_{+}^{2}<1$, the weak gravity
conjecture will also be satisfied. On the other hand, when $\Lambda
r_{+}^{2}\ll 1$, we will have the $SCC$ condition in light of the $WGC$,

$\frac{q}{m}\geq 1+\Lambda r_{+}^{2},$ (26)

from the above relation WGC is clearly obtained. In relation (25) when
$\Lambda r_{+}^{2}=1$, we have $\beta\rightarrow\infty$ and the SCC is
violated. Also, these result and conditions are completely compatible with
[44, 45].

##  5 Discussion and Result

One of the indications of the failure of determinism GR can be the rise of a
fascinating peculiarity known as the Cauchy horizon that shows up in the
astrophysical solutions of Einstein’s equations. These horizons are such that
it is difficult to indicate the history of the future of an observer that
passes come of such horizons utilizing Einstein’s conditions and initial
information. With these descriptions in the black holes’ background space-
time, it is a predicted possibility that the perturbations of the external
area are infinitely enhanced by a system known as the blue shift. They lead to
a singularity beyond the Cauchy horizon the inside of BHs, where field
conditions fail to seem good. The Penrose cosmic censorship conjecture (SCC)
affirms such an assumption. Obviously, another point is that astrophysical BHs
are stable because of an exceptional component called the perturbation-damping
system, which is applied in the outer region. Also, the SCC resolves the issue
of the idea of the singularities tracked down in many answers to Einstein’s
gravitational field equations: Are such singularities conventionally described
by unbounded curvature? Is the presence of a Cauchy horizon, an unsteady
characteristic element of answers of Einstein’s equations? Recently
researchers, remarking on the historical backdrop of the SCC conjecture,
overview a portion of the headway made in research coordinated either toward
satisfying SCC or toward revealing a portion of its shortcomings. They
specifically around model adaptations of SCC which have been demonstrated for
constrained groups of spacetimes viz the Gowdy spacetimes and the role played
by the conventional presence of Asymptotically speed term dominated conduct in
these answers. Also additionally note some work on spacetimes containing weak
null singularities, and their importance for the SCC [44, 45, 47]. SCC
conjecture has been one of the main acts of pure confidence with regard to GR,
confirming the deterministic idea of the related field relations. However, it
holds well for asymptotically level spacetimes, an expected disappointment of
the SCC conjecture could emerge for spacetimes acquiring Cauchy horizon
alongside a positive cosmological constant viz its potential failure about
this issue. Researchers have unequivocally exhibited that infringement of the
restriction SCC turns out as expected within the sight of a Maxwell field even
with the presence of higher spacetime aspects. Specifically, for higher
dimensions of the RN black holes, the infringement of SCC is at a bigger scope
compared with the 4D case, for specific of the cosmological constant. Then
again, for a brane world BH, the impact of an additional dimension is to make
the infringement of cosmic censorship weaker. For rotating BHs, intriguingly,
the SCC is constantly holding even in the presence of higher dimensions. A
comparable situation is likewise noticed for rotating BHs on the brane [47].
In this paper, we investigated dynamically formed charged black holes. Also,
to satisfy the SCC, the inner Cauchy horizons of the black hole must be
unstable. Here, to check the SCC, it is necessary to get two $-Im(\omega_{0})$
and $k_{-}$ parameters to demonstrate the decay rate of the remaining
perturbation fields in the outer regions of the black hole and the blue-shift
growth rate of the in-falling fields of the black hole, respectively.
Therefore, if $\beta=\frac{-Im(\omega_{0})}{k_{-}}<1/2$, SCC will be
maintained. We found that for the dS charged black hole with respect to
$r_{+}\geq Q$ in light of the WGC, viz $q/m\geq 1$, SCC will definitely be
satisfied. We also found that there will be a possibility of violation of SCC
for the massive scalar field as well as when the radius of the event horizon
of the charged black hole is very large. We also found SCC will be violated in
the extremality state for the charged RN-dS black hole when $\Lambda
r_{+}^{2}=1$ which is also mentioned in [44, 45]. Also, these results and
conditions are completely compatible with [44, 45]. On the other hand, in [8,
46], when the scalar field is uncharged, the SCC is violated, which is
consistent with (21) in this paper. Because can be obtained $\beta>1/2$ if
assume the charge of the scalar field is zero viz $q=0$. The above study also
raises some questions as follows.
Is the relationship researched in this article also valid for black holes in
higher dimensions? Do other black holes in different frames satisfy the SCC
and WGC simultaneously? Is it possible to consider the SCC relation with WGC
monitoring for all black holes? Is it may such a structure also be established
for black holes on the brane? We leave these questions for future work.

##  6 Acknowledgments

The authors would like to thank the referee for the fruitful comments to
improve the introduction section.

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