# Active Sidestick Control Integration for Enhanced Aircraft Flight Envelope
Protection

Çağrı Ege Altunkaya  Fatih Erol  Akın Çatak  Volkan Mert  Pierluigi Capone
Şükrü Akif Ertürk  Emre Koyuncu  Istanbul Technical University, Aerospace
Research Center,
Istanbul, Türkiye.  Zurich University of Applied Sciences, Centre for
Aviation,
Winterthur, Switzerland.  Turkish Aerospace, Air Vehicle Technologies Research
Center,
Istanbul, Türkiye.

###### Abstract

The design of Envelope and Pilot-Induced Oscillation (PIO) Protection
Features, and Failure Cases detection and prevention using Active Control
Sidestick (ACS) is a challenging task. While helping the pilot to respect the
envelope limitations also in failure scenarios and, therefore, increasing
mission effectiveness, these features may have a significant impact on the
aircraft’s agility. ACS characteristics are investigated in an integrated
environment. A set of effective and flexible control laws based on Incremental
Nonlinear Dynamic Inversion have been developed in a state-of-the-art aircraft
simulation model and coupled with a two-ways communication with the selected
ACS. The model can run in real-time in a fixed-based simulator composed of
representative cockpit and out-of-the-window.

###### keywords:

Active Sidestick, Flight Envelope Protection, Pilot-Induced Oscillation
Protection, Incremental Nonlinear Dynamic Inversion, Integrated Aircraft
Simulation Model.

††thanks: This work has been submitted to IFAC for possible publication.

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##  1 Introduction

High performance aircraft are often designed at the leading edge of technology
where mission performance can be heavily impacted by deficiencies. Therefore,
everything which is possible must be done to reduce pilot workload, especially
in situations with rapidly changing flight conditions based on standards
(USDoD (2004)).

In these scenarios the pilot might not be able to monitor the boundaries of
the flight envelope (FE) in an effective way. This results in increased risks
and disadvantage since the available flight performance may not be fully used
(NATO AGARD (2000)). Aircraft limitations can be monitored by Envelope
Protection (EP also called Carefree Handling) functions resulting in increased
mission success due to reduced workload, more aggressive maneuvering, and
reduced risk. Examples for EP functions from the literature are (Lombaerts et
al. (2017)), (Simon et al. (2017)), (Yu et al. (2020)) and,(Moreira et al.
(2022)).

Careful design must be applied to prevent possible disadvantages of EP such as
reduced agility (Gibson and Hess (1997)). Therefore, an integrated approach to
these features where the EP algorithm is coupled with an ACS (SAE (2007) and
(SAE (2018)), able to change its characteristics during the mission, is highly
recommended (Akmenek (2018)). Similar reasoning can be made for PIO detection
and prevention mechanisms. A more integrated solution will allow to cope with
possible limitations of the aircraft system in specific points of FE.
Additionally, possible failure conditions affecting ACS should be considered
and coped with using detection and reconfiguration approaches.

The motivation of this work is to understand the ACS characteristics and to
develop advanced flight protection algorithms within these scope. Therefore,
an integrated framework shall be set up for further demonstrations.

The paper is organized as follows. In Section 2, the motivation of the work
and the problem statement is described. Section 3 details the integrated
framework setup such as simulation models and all the components of the
simulation environment. The architecture of flight envelope protection is
explained in Section 4. Preliminary test results and the initial outcomes are
given in Section 5. Finally, the conclusions are given in Section 6.

##  2 Problem Statement

Many current high-performance aircraft are equipped with Active Control
Sidesticks (ACS). However, the details of these applications are often not
publicly accessible. The design of Envelope Protection (EP), Pilot-Induced
Oscillation (PIO) protection, and failure case detection and prevention using
ACS is a challenging task. These features help pilots adhere to envelope
limitations even in failure scenarios, thereby increasing mission
effectiveness. However, they can also significantly impact the aircraft’s
agility. A thorough understanding of these issues is crucial for designing
current and future aircraft more effectively and efficiently.

To address these challenges, several key research questions must be answered:
What integrated infrastructure is needed to develop and test EP and PIO
prevention algorithms? Which existing EP and PIO prevention algorithms are the
most effective? What are the advantages and disadvantages of ACS? What are the
likely failure scenarios of an ACS, and how can they be mitigated?

The proposed research aims to address these questions through a focus on five
main areas: i) building understanding of ACS features and mathematical models,
ii) integrating flight envelope protection (FEP), PIO detection and prevention
algorithms, and iii) demonstration. This paper details the development of an
integrated framework designed to answer these research questions and presents
preliminary test results.

The framework comprises several key elements: the aircraft simulation model,
flight control laws, active control sidestick, visual system, and cockpit. The
aircraft simulation model is designed to be representative of modern high-
performance aircraft, capable of running in real-time, interfacing with all
other components, supporting detailed linear and non-linear analyses, and
collecting and saving all simulation data. Flight control laws must be
representative of current protection algorithms as benchmarks and should
support advanced protection algorithms. The active control sidestick should
incorporate state-of-the-art ACS features and enable two-way communication
with the simulation model.

The visual system is required to operate in real-time with latency suitable
for task evaluation and to represent dedicated mission task elements
accurately. The cockpit should implement a Hands on Throttle and Stick (HOTAS)
system and support touch-screen instrument panels. By integrating these
components, the framework aims to address the challenges in designing
effective EP and PIO protection features, thus enhancing the safety and
performance of high-performance aircraft.

This integrated approach ensures a comprehensive understanding of ACS
characteristics and the development of advanced flight protection algorithms,
ultimately contributing to more efficient and safer aircraft designs.

##  3 Integrated Active Sidestick Control System

###  3.1 Aircraft Model

In this section, the backbone dynamic, which is a generic F-16 model, are
briefly presented, and these equations establish the foundation for the
nonlinear flight dynamics model. Furthermore, the aerodynamic and actuator
modeling are introduced with relevant references.

####  3.1.1 Equations of Motion

The rigid non-linear body dynamics is given in a compact form in Eq. (1) and
Eq. (2).

$\bm{\dot{V}}=m^{-1}[\bm{F}-\bm{\omega}\times m\bm{V}]$ (1)

$\bm{\dot{\omega}}=J^{-1}[\bm{M}-\bm{\omega}\times J\bm{\omega}]$ (2)

where $\bm{V}\in\mathbb{R}^{3\times 1}$ is the body velocity vector,
$\bm{\omega}\in\mathbb{R}^{3\times 1}$ is the angular rate vector,
$\bm{F}\in\mathbb{R}^{3\times 1}$ is the total body force vector,
$\bm{M}\in\mathbb{R}^{3\times 1}$ is the total moment vector,
$J\in\mathbb{R}^{3\times 3}$ is the inertia tensor, and $m$ is the mass of the
aircraft. Additionally, the rotational kinematics are given by Eq. (3).

$\bm{\dot{\Omega}}=\begin{bmatrix}1&\sin\phi\tan\theta&\cos\phi\tan\theta\\\
0&\cos\phi&-\sin\phi\\\
0&\sin\phi\sec\theta&\cos\phi\sec\theta\end{bmatrix}\bm{\omega}$ (3)

where $\bm{\Omega}$ denotes the Euler angles.

####  3.1.2 Aerodynamics

Aerodynamic modeling is constructed using the data provided in Nguyen (1979),
which consists of wind tunnel test data conducted up to 0.6 Mach. All the data
provided are implemented into interpolation lookup tables. After
interpolation, the force and moment coefficients are calculated according to
the formulation given in the same document Nguyen (1979), and
dimensionalization is carried out using the necessary atmospheric and
geometric properties.

####  3.1.3 Actuators

The dynamics of actuators are treated as a first-order system with the time-
constant, rate, and position saturation constraints, as specified in Nguyen
(1979). Each control surface has a time constant of $0.0495s$ (that gives
circa $20$ rad/s), whereas rate saturations are $60^{\circ}/s$,
$80^{\circ}/s$, and $120^{\circ}/s$ for the horizontal tail, ailerons, and
rudder respectively. Moreover, the position saturation are $\pm 25^{\circ}$,
$\pm 21.5^{\circ}$, and $\pm 30^{\circ}$ for the horizontal tail, aileron, and
rudder, respectively.

###  3.2 Flight Control

The control augmentation system consists of a single-loop angular rate control
law using the incremental nonlinear dynamic inversion (INDI). The derivation
of the control law is quite straightforward due to the control-affine form of
Euler’s equations of motion, given by Eq. (4) in a decomposed form.

$\bm{\dot{\omega}}=-J^{-1}(\bm{\omega}\times
J\bm{\omega})+\underbrace{J^{-1}\bar{q}_{\infty}S\begin{bmatrix}b&&\\\
&\bar{c}&\\\ &&b\\\
\end{bmatrix}\Phi}_{\begin{subarray}{c}\bm{g}(\bm{x})\end{subarray}}\underbrace{\bm{\delta}}_{\begin{subarray}{c}\bm{u}\end{subarray}}$
(4)

where $\bar{q}_{\infty}$, $S$, $b$, and $\bar{c}$ are the dynamic pressure,
wing area, wing span, and mean aerodynamic chord, respectively. Additionally,
$\Phi\in\mathbb{R}^{3\times n}$ represents the control effectivity matrix,
which consists of the moment coefficient derivatives with respect to control
surface deflections at the current instant, with $n$ denoting the number of
control surfaces. The INDI control law for the control of the angular rates is
derived in the form given in Eq. (5) (for further details, please see Ludeña
Cervantes et al. (2020)),

$\bm{u}=\bm{g}(\bm{x}_{0})^{-1}[\bm{\dot{\omega}}_{c}-\bm{\dot{\omega}}_{0}]+\bm{u}_{0}$
(5)

where the subscript $"0"$ denotes the current state and
$\bm{\dot{\omega}}_{c}\in\mathbb{R}^{3}$ is the virtual input to be designed,
the finalized form of the control law is given in Eq. (6).

$\bm{\delta}=\Bigg{\\{}J^{-1}\bar{q}_{\infty}S\begin{bmatrix}b&&\\\
&\bar{c}&\\\ &&b\\\
\end{bmatrix}\Phi\Bigg{\\}}^{-1}[\bm{\dot{\omega}}_{c}-\bm{\dot{\omega}}_{0}]+\bm{\delta}_{0}$
(6)

where $\bm{\delta}\in\mathbb{R}^{3}$ signifies the control surface
deflections, i.e. horizontal tail, aileron, and rudder, respectively.
Moreover, since there are also three control surfaces ($n=3$), the control
effectivity matrix is a square matrix, that is, it is invertible unless it is
rank deficient. Furthermore, the virtual input $\bm{\dot{\omega}}_{c}$ is
provided by Eq. (7).

$\bm{\dot{\omega}}_{c}=\begin{bmatrix}K_{p}&&\\\ &K_{q}&\\\ &&K_{r}\\\
\end{bmatrix}\begin{bmatrix}p_{c}-p\\\ q_{c}-q\\\ r_{c}-r\\\ \end{bmatrix}$
(7)

where $K_{p}$, $K_{q}$, and $K_{r}$ represent the gains for the roll, pitch,
and yaw channels, respectively. The expressions presented provide the required
generation of control surface deflections with respect to the pilot inputs of
$p_{c}$, $q_{c}$, and $r_{c}$.

Consequently, the aforementioned relations complete the flight control law for
the angular rates. The constructed flight control architecture is
straightforward and flexible, allowing for seamless integration with other
components and subsystems, e.g. it can accommodate various flight envelope
protection algorithms and active sidestick features, facilitating the
realization of a comprehensive pilot-in-the-loop environment.

###  3.3 Active Sidestick Control

This section explains the characterization of the active side-stick control
system, which plays a crucial role in improving pilot-aircraft interactions
and enabling flight envelope protection features.

STIRLING DYNAMICS® Compact Stick Active Inceptor is a state-of-the-art low-
force development ACS and uses a mass-spring-damper model with adjustable
parameters. The Inceptor Control Module (ICM) is an integrated microcomputer
and brushless DC motor drive designed to be used as a common control module
for a variety of compatible active control inceptors. The ICM runs the control
law for the force feedback system and interfaces with the host computer in the
simulator system.

The ACS grip has four switches: S1 (left), S2 (right), S3 (trim), and S4
(trigger). The default power-up mode is Disabled Mode. After the Initial
Built-In Test (IBIT), the system zeroes force measurements. In Enabled Mode,
the system can be commanded to provide artificial force feedback. In Jammed
Mode, the motors can be commanded to lock the motion of the stick. Message
transmission via UDP on Ethernet occurs only with command changes at 0-200 Hz
for all axes. ACS features are sent through ID1-12s & 50 and read over
ID20-24s as identification number, (ID) messages.

Figure 1: Force Feel Characteristics Curve and Enable Flags

Control message (ID2) manages features including trim, shaker, and damping
coefficients. The Force Feel Characteristics Curve (FFC) requires increasing
force/position values. Characteristic Control (ID5) sets up the fade time when
transitioning between different FFC. Trim Control (ID6) adjusts axis trim
positions. Shaker Control (ID7) specifies shaker amplitude and frequency.
Damping Control (ID8) sets the damping ratio (0.2-6). Cueing Force Control
(ID9) adjusts cueing force characteristics. Inertia Control (ID10) defines
perceived inertia (0.01-10 lbf/in²). Two ACS units can be linked for two-
pilots configuration.

Rotary Status Message (ID20) reads axes status at set rates, including mode
and grip switch states. Limited Characteristic Message (ID22), ’limited,’
restricts control characteristics due to motor capabilities. IP Address Change
(ID50) modifies network addresses.

Both pitch and roll axes have 24° position and a 27 lbf force limits, as this
is a Low Force ACS. An example pitch axis configuration is shown in Fig.1.

The system’s dynamics are modeled through a second-order transfer function
with parameters including FFC, Damping Ratio, Moment of Inertia (MOI), and
friction. The governing equation of motion is

$m\ddot{\theta}+c\dot{\theta}=F_{char}+F_{friction}+F_{grip}$ (8)

where $m$ denotes the MOI adjusted by 0.01, and $c$ is calculated from the
damping ratio $\zeta$ and force gradient $k$. Assuming no friction and fixed
$k$, the model simulates the sidestick’s response to grip force $F_{grip}(s)$.

$\mathbf{\theta(s)}=\frac{1}{ms^{2}+(2\zeta\sqrt{mk})s+k}\mathbf{F_{grip}(s)}$
(9)

where $m=0.6$, $\zeta=0.35$, $k=2$ and $F_{grip}=11lbf$.

Although the ACS is similar to a mass-spring-damper system, it is a combined
system with complex electrical and magnetic features. A set of different
fitting methods have been used to investigate this highly nonlinear behaviour.
First, the stick was moved to the deflection of 10° and then released. The
recorded data was fitted with a 2nd order transfer function with 4 different
approaches. Noise and system nonlinearity cause discrepancies between
optimized and calculated damping and frequency values, indicating practical
deviations from the theoretical model as illustrated in Fig. 2.

Figure 2: The ACS Parameter Identification

##  4 Flight Envelope Protection

To integrate with the ASC, a couple of conventional FEP algorithms are
developed and implemented. The protection algorithms for angular rates, angle
of attack, and bank angle are developed based on a common and simple
rationale: applying a counter-action as a restorative measure to prevent FEP
violations.

####  4.0.1 Pitch, Roll, and Yaw Rate Protection

Since the commanded pilot inputs are also the angular rates, it is simply
possible to restrict the pilot commands with a dynamic saturation, which is
fed by the scheduled flight envelope database, to prevent a possible excess of
the flight envelope. Subsequent to subjecting the pilot commands of
$\omega_{p}$ to the maximum and minimum allowable values of the flight
envelope, the resulting commanded angular rates of $\omega_{c}$ are then
applied to the control augmentation system.

####  4.0.2 Angle of Attack and Load Factor Protection

The violation of the angle of attack and load factor boundaries can be avoided
by applying a ’restorative’ pitch rate command. Prior to reaching the
predetermined limits of the angle of attack and load factor, the pilot
commands are faded out in order to prevent the aircraft from entering into an
upset condition. Furthermore, both the angle of attack and load factor
protection should be considered holistically to generate a single and unified
restorative pitch action; therefore, the load factor boundary is also
converted to the angle of attack as given by Eq. (10).

$\alpha_{max}^{n_{z}}=\dfrac{Wn_{z_{max}}}{\bar{q}_{\infty}SC_{z_{\alpha}}}$
(10)

where $\alpha_{max}^{n_{z}}$ denotes the angle of attack equivalent to the
maximum allowable load factor $n_{z_{max}}$, $W$ is the weight of the
aircraft, $C_{z_{\alpha}}$ is the force coefficient in z-direction derivative
with the angle of attack. Afterward, the minimum of the main angle of attack
limits, $\alpha_{max}^{\alpha}$, and equivalent angle of attack limit,
$\alpha_{max}^{n_{z}}$, is used to generate the restorative action. The
general framework is depicted in Fig. 3

Figure 3: The angle of attack and load factor protection scheme: four main
components, S-I/II/III/IV.

S-I is the pitch rate protection component, which has been discussed in the
preceding section. S-II is the logic of the selection of the restrictive angle
of attack; furthermore, the S-III and IV consist of designed restorative
actions. To ensure the generality of the generated action in terms of
magnitude, normalization is performed in S-II. The normalized angle of attack
indicator, $\bar{\alpha}$, reflects the current $\alpha$ as the percentage of
$\alpha_{max}$. This normalization enables the generation of a restorative
pitch rate based on the proximity to the limit. Consequently, in S-IV, a
damping component is incorporated to reduce the magnitude of oscillation
during protection, leveraging the current pitch rate, $q$. Note that this
structure is highly adaptable for design, and a reverse rationale can be
applied to protect against violations of the minimum angle of attack and load
factor limits as well.

####  4.0.3 Bank Angle Protection

The bank angle protection is established with the same philosophy as the
previous section. Simply, a restorative roll action is applied to not exceed
the predetermined maximum allowable bank angle limit. The general framework is
illustrated in Fig. 4.

Figure 4: The bank angle protection scheme: four main components,
S-I/II/III/IV.

S-I is the roll rate protection component, while S-II is the normalization
component considering the absolute of the current bank angle, $\phi$.
Moreover, the S-III and IV, again, consists of designed restorative actions.
In S-III, a counter-action roll rate is generated based on the current bank
angle as the percentage of the allowable limit. In addition, in S-IV, two
distinct damping components are incorporated to fade the magnitude of
oscillation out during protection, using both the current roll rate, $p$, and
the yaw rate, $r$. However, the S-IV component not only addresses the roll
motion-induced bank angle build-up but also accounts for the possibility of
yaw motion-induced bank angle build-up. Therefore, measures are taken to
protect the bank angle envelope in both scenarios.

##  5 Simulation Setup and Results

The Visualization System runs in UNITY® which is a cross-platform game engine
developed by Unity Technologies™ that is primarily used to develop video games
and simulations for computers, consoles, and mobile devices. The F-16 3D model
has been drawn in BLENDER® in Fig. 5. All control surfaces have been
individually separated and rigged with bones to enable independent rotation.

Figure 5: F-16 3D CAD Model used in the Simulation Environment and Terrain Map
Model in UNITY®

Istanbul is selected for terrain map and Istanbul Technical University is
selected for reference axes. Marmara region digital elevation data is used for
recreating terrain in UNITY® as illustrated in Fig. 5. Terrain information
from Digital Terrain Elevation Data (DTED) contains only point clouds.
Triangle meshes are created between these point clouds for visualization.

###  5.1 Aircraft Simulator Setup

The aircraft cockpit shell is equipped with an 8x20 touch screen large area
display (LAD) as instrument panel, a tablet used as control panel, and a
THRUSTMASTER® Hotas Wartog Hands-on-Throttle-and-Stick (HOTAS) system, which
includes a sidestick, a dual rotary throttle, and two pedals. The general view
of the integrated simulation environment is shown in Fig. 6.

Figure 6: General view of Simulator Setup

###  5.2 ACS Component Interfaces

For system identification purpose of ACS, input and output signals were
decoded, and tested using a network packet analyzer software that captures,
aggregates, and reads data packets between local and target devices, such as
Ethernet networks or recorded files as in Fig. 7. The preliminary phase also
involved comprehensive diagnostics on the Low Force Inceptor, utilizing
STIRLING DYNAMICS®’ IBIT software to confirm system integrity and operational
safety.

Figure 7: Example Signal Acquisition for System Identification (pitch & roll
sweeps)

A key aspect of the testing was the load calibration, which achieved a
precision level of %2 as shown in Fig. 8. This was conducted using a load cell
positioned at the detent button, facilitating accurate force measurements
across control positions. Additionally, the control stick’s response was
evaluated from a constrained negative, down-leftmost position to a positive,
upright-most position, emphasizing the robustness of the high force feedback
mechanism.

Figure 8: Calibration Tests with Load Cell Setup

Ad hoc tests were instrumental in pinpointing uncertainties, thereby
establishing definitive operational parameters, including minimum and maximum
force limits and positional thresholds.

The aforementioned tests not only demonstrated the reliability of the ACS
control but also revealed the identification of the axes through the binary
analysis of the message header bits. This allowed for the verification of any
bits that may have been overlooked between the pitch and roll axis readings.
The observed discrepancies led to the implementation of alternative
methodologies in subsequent development stages.

Latency was evaluated by monitoring UDP messages with scripts, which indicated
a comparative decrease in efficiency against SIMULINK® Real-Time Blocks. Thus,
it is observed that the SIMULINK® Real-Time Blocks are working faster. The
integration of ’Matlab Functions’ within SIMULINK® was also used to improve
performance.

The commanded angular rates and their responses with respect to time are shown
in Fig. 9. The performance of the flight control law is satisfactory in both
decoupled and coupled pilot inputs. The affect of envelope annihilation and
the effect of flight envelope protection structure is given in Fig. 10 for
longitudinal axis.

Figure 9: Controller Outputs Integrated with Active Sidestick Control

Although the performance of the INDI controller is satisfactory it may lead to
envelope exceedance if certain pitch rate command is demanded from pilot. The
red lines in the Fig. 10 shows the limits in pitch, angle of attack and load
factor. Even the pitch command is within the limits, it may result a in angle
of attack or load factor exceedance. The results show that the proposed
envelope protection algorithm prevents the exceedance before it happens.

Figure 10: Envelope Protection in Longitudinal Axis

The corresponding response of the active sidestick for the results shown in
Fig.10 is given in Fig.11 to give the force displacement curve.

Figure 11: ACS Displacement-Force Curve

##  6 Conclusion & Outlook

This work looked on how ACS technologies can be effectively integrated in a
modern Flight Control System. The focus has been on FEP functionalities and on
PIO protection. Another important aspect has been to verify how ACS works with
pilot inputs. This is important for making aircraft safer and more efficient.
The limitations of current ACS can be identified and the corresponding
pitfalls in the development of effective integrated solutions should,
therefore, be avoided.

An integrated environment that takes into accounts all the different aspects
and components of the problem has been built and is currently under testing.
In particular, a state-of-the-art simulation model of a high-performance
aircraft has been coupled with effective and versatile control laws that will
allow to develop ”standard” and modern/advanced EP and PIO protection
mechanisms. The model has been interfaced with 2-ways connections with a
configurable ACS and also with a representative cockpit and out-of-windows.

Preliminary testing give encouraging results and confidence in the capability
of the integrated framework to face the challenges of designing more easier to
fly, better controllable and, ultimately, more safe aircrafts.

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Overcharging problem and thermodynamics in five-dimensional black holes

Deyou Chen and Shan Zeng

School of Science, Xihua University, Chengdu 610039, China

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract: In this paper, the overcharging problem and thermodynamics in the
extended phase spaces of the five-dimensional spherically symmetric
topological black holes are investigated by absorptions of scalar particles
and fermions. The cosmological constant is regarded as a variable related to
pressure and its conjugate quantity is a thermodynamic volume. The first law
of thermodynamics is recovered. The second law is violated in the extended
phase space of the extremal and near-extremal black holes. The overcharging
problem is tested by the existence of the event horizons. The event horizon is
determined by the metric component $f(r)$. The minimal values of the metric
component at the final stage show that the extremal and near-extremal black
holes can not be overcharged.

##  1 Introduction

It is well known that a spacetime singularity can be formed at the end of
gravitational collapse. Near the singularity, physical laws fail. To avoid the
physical damage caused by the singularity, Penrose pointed out that under
normal material properties and initial conditions, naked singularities cannot
be formed in the real physical process [2]. This is the weak form of the
cosmic censorship conjecture. It shows that singularities are not naked and
hide behind event horizons. This cosmic censorship conjecture plays an
important role in black hole physics, and its validity is a necessary
condition to ensure the predictability of physical laws. Although its
correctness is widely accepted, there is no complete evidence to prove it.

An effective method to test the weak cosmic censorship conjecture (WCCC) is
the Gedanken experiment designed by Wald [3]. In this experiment, a particle
with energy, large enough angular momentum and charge is thrown into a charged
rotating black hole to test the evolution of the black hole. The evolution
result is described by the new spacetime metric. If the final state of the
black hole evolution is no longer a black hole, there is no event horizon and
the singularity is naked. The solution of the event horizon is determined by
the metric components. Therefore, one only needs to check the existence of the
horizon to judge whether the singularity is naked. The energy, charge and
angular momentum of the black hole can also be gotten by the interaction
between the black hole and a field. The field has a finite energy (i.e. it
contains a wave packet). At first, the field does not exist, there is only the
black hole. Then, the field comes from infinity and interacts with the black
hole. Due to the interaction, energy, charge and angular momentum are
transferred between the black hole and field. Part of the field is absorbed by
the black hole and the other part is reflected back to infinity. Finally, the
field decays, and the energy, charge and angular momentum of the black hole
change. Using this field view, Semiz first adopted a complex scalar field to
test the WCCC and found that the dyonic Kerr-Newman black hole could not be
overcharged [4]. This result was supported by the work of Toth [5]. Based on
the seed work of Wald, the validity of the WCCC in the various spacetimes has
been tested [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29]. In the relevant discussions, the self-force
effect, tunnelling effect, radiation reaction were taken into account [30, 31,
32, 33]. However, there is no consensus. In these researches, the black holes
are assumed to exist in advance. The formation of the naked singularity was
also discussed by the numerical method [34]. In the discussion, it was assumed
that a celestial body before particles being absorbed is not a black hole, but
a general dense star.

The cosmological constant used to be a fixed constant. Recently, it was seen
as a variable in the work of Kastor and others [35, 36, 37, 38]. One of the
reasons for this consideration is that the cosmological constant, as a
variable, can reconcile the inconsistency between the first law of
thermodynamics of black holes and the Smarr relation derived from the scaling
method. The second reason is that if people consider ”more fundamental”
theories, physical constants, such as the gauge coupling constant, Yukawa
coupling constant, Newtonian constant or cosmological constant arise as vacuum
expectation values are not fixed in advance and can vary [39]. The
cosmological constant was seen as a pressure, and its conjugate quantity is a
thermodynamic volume. Using the standard thermodynamics, Kubiznak and Mann
studied the critical behavior of the charged anti-de Sitter black hole. They
found that its critical index was consistent with that of van der Waals system
[38]. The original first law of black hole thermodynamics was modified to

$\displaystyle dM=TdS+\phi dQ+VdP+\Omega dJ.$ (1)

The corresponding Smarr relation is

$\displaystyle(n-3)M=(n-2)TS+(n-2)\Omega J+(n-3)\phi Q-2VP.$ (2)

where the mass $M$ is expressed as the enthalpy and $n$ is the dimension of
the spacetime. Introducing the thermodynamic pressure and volume into the
black hole system, one can use the standard thermodynamic formulae to
calculate various thermodynamic quantities and discuss various thermodynamic
phenomena. In the extended phase space, many interesting thermodynamic
phenomena have been found, including the reentant phase transition, repulsive
interaction in the high temperature black holes, multiphase transition in the
quasi topological gravity and Lovelock gravity, and other thermodynamic
phenomena [40, 41, 42, 43, 44, 45, 46].

In the recent work, Gwak studied the WCCC in the extended phase space of the
rotating black hole, where the pressure and volume were introduced [47]. He
found that the black hole could not be overspun by throwing a particle with
energy and large enough angular momentum into the hole. This result is in
consistence with that gotten in the new version Gedanken experiment [48, 49,
50, 51, 52, 53]. In the extremal and near-extremal cases, the entropies
decreased. This means the violation of the second law of thermodynamics in the
extended phase space. The same result was gotten in the subsequent work [54,
55, 56, 57, 58, 59, 60, 61]. In these work, an important relation between the
momentum, energy and angular momentum or charge of the particle was gotten by
the Hamilton-Jacobi equation.

In this paper, we investigate the WCCC and thermodynamics in extended phase
spaces of five-dimensional charged topological black holes by absorptions of
scalar particles and fermions. These black holes are a spacial case of a
$(n+1)$-dimensional Einstein-Maxwell-dilaton gravity solution [62], where the
dilaton $\alpha=0$. The test particles in this paper are scalar particles and
fermions, which satisfy the Klein-Gordon equation and Dirac equation in curved
spacetime, respectively. The first law of thermodynamics is recovered by the
absorptions of the particles. In the extremal and near-extremal cases, the
second law is violated in the extended phase spaces. The overcharging problem
is tested by the existence of the event horizon. The minimal values of the
metric component $f(r)$ at the final stages show that the extremal and near-
extremal black holes can not be overcharged.

The rest is organized as follows. The five-dimensional topological black hole
is given and the thermodynamics are discussed in the next section. In section
3, we investigate the absorptions of the scalar particle and fermion by the
black hole. The relation between the momentum, energy and charge of the
particle is gotten. In section 4, the thermodynamics in the extended phase
space are investigated by the absorptions of the particles. In section 5, the
overcharging problem is tested by throwing a particle in the near-extremal and
extremal black holes. The last section is devoted to our discussion and
conclusion.

##  2 The black hole solution

The action described the $(n+1)$-dimensional Einstein-Maxwell-dilaton gravity
is [63]

$\displaystyle\mathcal{S}=\frac{1}{16\pi}\int{dx^{n+1}\sqrt{-g}\left[\mathcal{R}-\frac{4}{n-1}\left(\nabla\Phi\right)^{2}-V(\Phi)-e^{-\frac{4\alpha\Phi}{n-1}}F_{\mu\nu}F^{\mu\nu}\right]},$
(3)

where $n\geq 3$, $\mathcal{R}$ is the Ricci scalar curvature, $\Phi$ is a
dilaton field, and $V(\Phi)$ is a potential for $\Phi$,

$\displaystyle V(\Phi)=2\Lambda
e^{\frac{4\alpha\Phi}{n-1}}+\frac{k(n-1)(n-2)\alpha^{2}}{b^{2}(\alpha^{2}-1)}e^{\frac{4\Phi}{(n-1)\alpha}}.$
(4)

$\Lambda$ is a negative cosmological constant. $\alpha$ is a constant
determining the coupling strength of the scalar and electromagnetic fields.
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor and $A_{\mu}$ denotes the electromagnetic
potential. The black hole solutions in the Einstein-Maxwell-dilaton gravity
with the dilaton coupling parameter $\alpha$ are given by [62]

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{{f(r)}}+r^{2}R^{2}(r)d\Omega^{2}_{k,n-1},$
(5)

where

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{k(n-2)(\alpha^{2}+1)^{2}b^{-2\gamma}r^{2\gamma}}{(\alpha^{2}-1)(\alpha^{2}+n-2)}-\frac{m}{r^{(n-1)(1-\gamma)-1}}$
(6)
$\displaystyle+\frac{2q^{2}(\alpha^{2}+1)^{2}b^{-2(n-2)\gamma}r^{2(n-2)(\gamma-1)}}{(\alpha^{2}-1)(\alpha^{2}+n-2)}+\frac{2\Lambda(\alpha^{2}+1)^{2}b^{2\gamma}r^{2(1-\gamma)}}{(n-1)(\alpha^{2}-n)},$

$R(r)=e^{\frac{2\alpha\Phi}{n-1}}$,
$\Phi(r)=\frac{(n-1)\alpha}{2(1+\alpha^{2})}ln\left(\frac{b}{r}\right)$,
$\gamma=\frac{\alpha^{2}}{\alpha^{2}+1}$ and $b$ is an arbitrary constant.
$d\Omega^{2}_{k,n-1}$ describes a $(n-1)$-dimensional subspace with $n(n-1)k$
constant curvature and volume $\omega_{n-1}$. $k$ is a constant and
characterizes the hypersurface. Without loss of generality, the value of $k$
is $-1,0,1$, which correspond to a negative (hyperbolic), zero (flat),
positive (elliptic) constant curvature hypersurface in the metric (5),
respectively. $m$ and $q$ are integration constants. According to the
definition of mass due to Abbott and Deser [64, 65], the mass takes the form

$\displaystyle
M=\frac{(n-1)b^{(n-1)\gamma}m\omega_{n-1}}{16\pi(\alpha^{2}+1)}.$ (7)

The charge is

$\displaystyle Q=\frac{q\omega_{n-1}}{4\pi}.$ (8)

The electric potential is $A_{t}=-\frac{qb^{(3-n)\gamma}}{\lambda
r^{\lambda}}$, where $\lambda=(n-3)(1-\gamma)+1$. The thermodynamic properties
in the extended phase space of the $(n+1)$-dimensional dilaton black holes
were deeply discussed in [66, 67, 68, 69, 70, 71].

In this paper, our interest is focused on the overcharging problem and
thermodynamics in the extended phase spaces of the five-dimensional black
holes. Therefore, we let $n=4$ in the metric (5). Considering that the
cosmological constant is a variable related to pressure,
$P=-\frac{\Lambda}{8\pi}=\frac{n(n-1)}{16\pi l^{2}}$, we replace the
cosmological constant with the pressure in the following discussions. In the
absence of the non-trivial dilaton ($\alpha=0$), the solution (6) is reduced
to

$\displaystyle
f(r)=k-\frac{M}{r^{2}}\frac{16\pi}{3\omega_{3}}+\frac{Q^{2}}{r^{4}}\frac{16\pi^{2}}{3\omega_{3}^{2}}+\frac{4}{3}\pi
r^{2}P,$ (9)

Thus the metric (5) describes a five-dimensional anti-de Sitter topological
spacetime where $R(r)=0$. Now the electric potential is given by

$\displaystyle A_{t}=-\frac{2\pi Q}{\omega_{3}r^{2}}.$ (10)

The Hawking temperature and entropy are

$\displaystyle T=\frac{f^{\prime}(r_{+})}{4\pi},\quad\quad
S=\frac{\omega_{3}r_{+}^{3}}{4},$ (11)

respectively. $r_{+}$ is the event horizon radius derived by $f(r)=0$. The
mass is expressed by the entropy, charge and pressure

$\displaystyle
M=\frac{3k\omega_{3}}{16\pi}\left(\frac{4S}{\omega_{3}}\right)^{\frac{2}{3}}+\frac{\pi
Q^{2}}{\omega_{3}}\left(\frac{\omega_{3}}{4S}\right)^{\frac{2}{3}}+\frac{\omega_{3}P}{4}\left(\frac{4S}{\omega_{3}}\right)^{\frac{4}{3}}.$
(12)

Here, $S$, $Q$ and $P$ are regarded as the extensive parameters for the mass
$M$. Their conjugate quantities are the temperature, electric potential and
volume, respectively.

$\displaystyle T$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial
S}\right)_{Q,P}=\frac{k\omega_{3}}{8\pi}\left(\frac{4}{\omega_{3}}\right)^{\frac{2}{3}}S^{-\frac{1}{3}}-\frac{2\pi
Q^{2}}{3\omega_{3}}\left(\frac{\omega_{3}}{4}\right)S^{-\frac{5}{3}}+\frac{\omega_{3}P}{3}\left(\frac{4}{\omega_{3}}\right)^{\frac{4}{3}}S^{\frac{1}{3}},$
$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial Q}\right)_{P,S}=\frac{2\pi Q}{\omega_{3}r_{+}^{2}},\quad\quad
V=\left(\frac{\partial M}{\partial
P}\right)_{S,Q}=\frac{\omega_{3}r_{+}^{4}}{4}.$ (13)

After a straightforward calculation, it is easily found that these
thermodynamic quantities obey the first law of thermodynamics

$\displaystyle dM=TdS+\phi dQ+VdP,$ (14)

and the Smarr relation

$\displaystyle 2M=3TS+2\phi Q-2VP.$ (15)

In what follows the thermodynamics are investigated by the absorptions of the
scalar particle and fermion and the first law of thermodynamics is recovered.

##  3 Particles’ absorption

###  3.1 Scalar particle’s absorption

In this subsection, we discuss the absorption of the scalar particle in the
five-dimensional spacetime. In the curved spacetime, the motion of the charged
scalar particle obeys the Klein-Gordon equation

$\displaystyle\frac{1}{\sqrt{-g}}\left(\frac{\partial}{\partial
x^{\mu}}-\frac{iq_{e}}{\hbar}A_{\mu}\right)\left[\sqrt{-g}g^{\mu\nu}\left(\frac{\partial}{\partial
x^{\nu}}-\frac{iq_{e}}{\hbar}A_{\nu}\right)\right]\Psi_{S}-\frac{m_{e}^{2}}{\hbar^{2}}\Psi_{S}=0,$
(16)

where $m_{e}$ and $q_{e}$ are the mass and charge of the particle,
respectively. $\Psi_{S}$ is the scalar field. Following the ansatz [72, 73,
74], we use the $WKB$ approximation and adopt the following expression of the
wave function

$\displaystyle\Psi_{S}=exp(\frac{i}{\hbar}I_{s}+I_{1}+\mathcal{O}(\hbar)).$
(17)

Inserting the above ansatz and the contravariant metric components of the
five-dimensional black hole into the Klein-Gordon equation yields an equation
about the action $I_{s}$. Dividing by the exponential term and taking the
leading contribution of $\hbar$, we get

$\displaystyle
f^{-1}(\partial_{t}I_{s}-q_{e}A_{t})^{2}-f(\partial_{r}I_{s})^{2}-g^{22}(\partial_{\theta_{2}}I_{s})^{2}-g^{33}(\partial_{\theta_{3}}I_{s})^{2}-g^{44}(\partial_{\theta_{4}}I_{s})^{2}+m_{e}^{2}=0.$
(18)

Here we use $\theta_{2}$, $\theta_{3}$, $\theta_{4}$ to express the
coordinates $x^{2}$, $x^{3}$, $x^{4}$, respectively. And $g^{22}$, $g^{33}$
and $g^{44}$ are the corresponding contravariant metric components in the
five-dimensional metric. Their expressions are not specified here, since they
do not affect the later calculation result. This is different from the
previous discussion of tunneling radiation with consideration of quantum
gravity effects [75]. Taking into account the symmetries of the spacetime, we
carry out the separation of variables to the action,

$\displaystyle I_{s}=-\omega t+W(r)+\Theta(\theta_{2},\theta_{3},\theta_{4}).$
(19)

In the above equation, $\omega$ is the energy of the absorbed scalar particle.
Substituting the separated action into Eq. (18) and solving it yield

$\displaystyle\partial_{r}W=\pm\frac{\sqrt{\left(\omega-q_{e}\frac{2\pi
Q}{\omega_{3}r^{2}}\right)^{2}+\left[m_{e}^{2}-g^{22}(\partial_{\theta_{2}}\Theta)^{2}-g^{33}(\partial_{\theta_{3}}\Theta)^{2}-g^{44}(\partial_{\theta_{4}}\Theta)^{2}\right]f}}{f},$
(20)

where $+(-)$ denote the cases of the ingoing (outgoing) particles. We assume
that the particle is completely absorbed by the black hole. Then the negative
sign in the above equation is neglected. Defining $p_{r}=\partial_{r}I_{s}$,
we get

$\displaystyle p^{r}$ $\displaystyle=$ $\displaystyle g^{rr}p_{r}$ (21)
$\displaystyle=$ $\displaystyle\sqrt{\left(\omega-q_{e}\frac{2\pi
Q}{\omega_{3}r^{2}}\right)^{2}+\left[m_{e}^{2}-{g^{22}}(\partial_{\theta_{2}}\Theta)^{2}-{g^{33}}(\partial_{\theta_{3}}\Theta)^{2}-{g^{44}}(\partial_{\theta_{4}}\Theta)^{2}\right]f},$

Near the event horizon, we have $f(r)\to 0$ and the above equation is
simplified to

$\displaystyle p^{r}=\omega-q_{e}\phi,$ (22)

where $\phi=\frac{2\pi Q}{\omega_{3}r_{+}^{2}}$ represents the electric
potential at the event horizon. The condition of the superradiation is that
the boundary condition of the scalar field should be in the asymptotic region
and $\omega<q_{e}\phi$. In this paper, the effects of superradiation are not
taken into account, which leads to $\omega>q_{e}\phi$ and $p^{r}>0$. The
relation (22) plays an important role in the derivation of thermodynamics and
is recovered by the absorption of the fermion in the next subsection.

###  3.2 Fermion’s absorption

In this subsection, we use a charged fermion with spin-1/2 to discuss its
absorption at the event horizon. The motion of the fermion obeys the Dirac
equation in curved spacetime

$i\gamma^{\mu}\left(\partial_{\mu}+\Omega_{\mu}+\frac{i}{\hbar}q_{e}A_{\mu}\right)\Psi_{F}+\frac{m_{e}}{\hbar}\Psi_{F}=0,$
(23)

where $m_{e}$ and $q_{e}$ denote the mass and charge of the fermion,
respectively. $\Psi_{F}$ is the wave function.
$\Omega_{\mu}\equiv\frac{i}{2}\omega_{\mu}\,^{ab}\Sigma_{ab}$,
$\omega_{\mu}\,^{ab}$ is the spin connection defined by the ordinary
connection and the tetrad $e^{\lambda}\,_{b}$, namely,
$\omega_{\mu}\,^{a}\,{}_{b}=e_{\nu}\,^{a}e^{\lambda}\,_{b}\Gamma^{\nu}_{\mu\lambda}-e^{\lambda}\,_{b}\partial_{\mu}e_{\lambda}\,^{a}$.
The Greek indices are raised and lowered by the curved metric $g_{\mu\nu}$.
The Latin indices are governed by the flat metric $\eta_{ab}$. The
construction of the tetrad needs to use the following definitions

$g_{\mu\nu}=e_{\mu}\,^{a}e_{\nu}\,^{b}\eta_{ab},\hskip
14.22636pt\eta_{ab}=g_{\mu\nu}e^{\mu}\,_{a}e^{\nu}\,_{b},\hskip
14.22636pte^{\mu}\,_{a}e_{\nu}\,^{a}=\delta^{\mu}_{\nu},\hskip
14.22636pte^{\mu}\,_{a}e_{\mu}\,^{b}=\delta_{a}^{b}.$ (24)

The Lorentz spinor generators are defined by
$\Sigma_{ab}=\frac{i}{4}\left[{\gamma^{a},\gamma^{b}}\right]$ and $\gamma^{a}$
satisfy $\\{\gamma^{a},\gamma^{b}\\}=2\eta^{ab}$. We can readily construct the
$\gamma^{\mu}$’s in curved spacetime as

$\displaystyle\gamma^{\mu}=e^{\mu}\,_{a}\gamma^{a},\hskip
19.91692pt\left\\{{\gamma^{\mu},\gamma^{\nu}}\right\\}=2g^{\mu\nu}.$ (25)

For a spin-$1/2$ fermion, there are two states corresponding to spin up and
spin down. First of all, we discuss the spin up state. The wave function takes
the form

$\displaystyle\Psi_{F\uparrow}=\left(\begin{array}[]{c}A\\\ 0\\\ B\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I_{\uparrow}\right),$ (30)

where $A$, $B$ and $I_{\uparrow}$ are the functions of
$(t,r,\theta_{2},\theta_{3},\theta_{4})$. To solve the wave function, we
should construct the $\gamma^{\mu}$ matrices. There are many choices to
construct them. For the five-dimensional metric, we first choose the tetrad

$\displaystyle
e_{\mu}\,^{a}=\rm{diag}\left(\sqrt{f},1/\sqrt{f},\sqrt{g^{22}},\sqrt{g^{33}},\sqrt{g^{44}}\right).$
(31)

Then the $\gamma^{\mu}$ matrices are

$\displaystyle\gamma^{t}=\frac{1}{\sqrt{f\left(r\right)}}\left(\begin{array}[]{cc}0&I\\\
-I&0\end{array}\right),$
$\displaystyle\gamma^{\theta_{2}}=\sqrt{g^{22}}\left(\begin{array}[]{cc}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),$ (36)
$\displaystyle\gamma^{r}=\sqrt{f\left(r\right)}\left(\begin{array}[]{cc}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right),$
$\displaystyle\gamma^{\theta_{3}}=\sqrt{g^{33}}\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right),$ (41)
$\displaystyle\gamma^{\theta_{4}}=\sqrt{g^{44}}\left(\begin{array}[]{cc}-I&0\\\
0&I\end{array}\right).$ (44)

$\sigma^{1}$, $\sigma^{2}$ and $\sigma^{3}$ are the Pauli matrices and $I$ is
a identity matrix. Inserting the spin connection, wave function and gamma
matrices into the Dirac equation, dividing the exponential term and
multiplying by $\hbar$ yield four equations to the leading order in $\hbar$.
They are

$\displaystyle-B\frac{1}{\sqrt{f}}\left({\partial_{t}I_{\uparrow}-q_{e}A_{t}}\right)-B\sqrt{f}\partial_{r}I_{\uparrow}+A\sqrt{g^{44}}\partial_{\theta_{4}}I_{\uparrow}+Am_{e}=0,$
(45)

$\displaystyle
A\frac{1}{\sqrt{f}}\left({\partial_{t}I_{\uparrow}-q_{e}A_{t}}\right)-A\sqrt{f}\partial_{r}I_{\uparrow}-B\sqrt{g^{44}}\partial_{\theta_{4}}I_{\uparrow}+Bm_{e}=0,$
(46)

$\displaystyle-B\left[{\sqrt{g^{22}}\partial_{\theta_{2}}I_{\uparrow}+i\sqrt{g^{33}}\partial_{\theta_{3}}I_{\uparrow}}\right]=0,$
(47)

$\displaystyle-A\left[{\sqrt{g^{22}}\partial_{\theta_{2}}I_{\uparrow}+i\sqrt{g^{33}}\partial_{\theta_{3}}I_{\uparrow}}\right]=0.$
(48)

The last two equations are simplified to an equation and yield
$g^{22}(\partial_{\theta_{2}}I_{\uparrow})^{2}+g^{33}(\partial_{\theta_{3}}I_{\uparrow})^{2}=0$.
Previous discussions have shown that the contribution of the angular part does
not affect the results of the tunneling radiation when quantum gravity effects
are not taken into account [75]. Therefore, we focus on the first two
equations. Due to the same symmetry in the black hole spacetime, we still
adopt the separation of variables in the above subsection

$\displaystyle I_{\uparrow}=-\omega
t+W(r)+\Theta(\theta_{2},\theta_{3},\theta_{4}).$ (49)

Now $\omega$ denotes the energy of the absorbed fermion. Combining these two
equations and eliminating $A$ and $B$, we get an equation about the action
$I_{\uparrow}$. Substituting the separated variables into this equation and
solving it, we get

$\displaystyle\partial_{r}W=\pm\frac{\sqrt{\left(\omega-q_{e}\frac{2\pi
Q}{\omega_{3}r^{2}}\right)^{2}+\left[m_{e}^{2}-g^{44}\partial_{\theta_{4}}I_{\uparrow}\right]f}}{f},$
(50)

where $+(-)$ denote the cases of the ingoing (outgoing) fermion. As addressed
in the above subsection, the particle is assumed to be completely absorbed.
Therefore, there is no outgoing fermion. Using the definition
$p_{r}=\partial_{r}I_{\uparrow}$, we get
$p^{r}=g^{rr}\partial_{r}I_{\uparrow}$. Near the event horizon,

$\displaystyle p^{r}=\omega-q_{e}\phi,$ (51)

where $\phi$ is the electric potential at the event horizon given by Eq. (13).
It recovers the relation gotten by the absorption of the scalar particle. For
the spin down, the calculation is parrel and the same relation (51) is gotten.

##  4 Thermodynamics and particles’ absorption

Thermodynamics of black holes were deeply discussed through particles’
tunnelling across event horizons. In this section, it is shown that the
thermodynamics in the extended phase space of the five-dimensional black hole
can be recovered by the absorptions of the scalar particle and fermion. The
relation $p^{r}=\omega-q_{e}\phi$ is used in the derivation. Therefore, the
particle can be either a scalar particle or a fermion. When a particle with
energy and charge falls into the black hole, the energy and charge of the
black hole increase. Let $f(M,Q,P,r_{+})=f(r_{+})$ represent the metric
component of the original state of the black hole and
$f(M+dM,Q+dQ,P+dP,r_{+}+dr_{+})$ be the metric component of the final state
after the particle being absorbed. These metric components satisfy

$\displaystyle f(M+dM,Q+dQ,P+dP,r_{+}+dr_{+})-f(M,Q,P,r_{+})=$
$\displaystyle\left.\frac{\partial f(M,Q,P,r)}{\partial
M}\right|_{r=r_{+}}dM+\left.\frac{\partial f(M,Q,P,r)}{\partial
Q}\right|_{r=r_{+}}dQ+\left.\frac{\partial f(M,Q,P,r)}{\partial
P}\right|_{r=r_{+}}dP$ $\displaystyle+\left.\frac{\partial
f(M,Q,P,r)}{\partial r}\right|_{r=r_{+}}dr_{+},$ (52)

where

$\displaystyle\left.\frac{\partial f(M,Q,P,r)}{\partial M}\right|_{r=r_{+}}$
$\displaystyle=$
$\displaystyle-\frac{16\pi}{3\omega_{3}r_{+}^{2}},\quad\quad\left.\frac{\partial
f(M,Q,P,r)}{\partial
Q}\right|_{r=r_{+}}=\frac{32\pi^{2}Q}{3\omega_{3}^{2}r_{+}^{4}},$
$\displaystyle\left.\frac{\partial f(M,Q,P,r)}{\partial P}\right|_{r=r_{+}}$
$\displaystyle=$ $\displaystyle\frac{4\pi
r_{+}^{2}}{3},\quad\quad\quad\left.\frac{\partial f(M,Q,P,r)}{\partial
r}\right|_{r=r_{+}}=4\pi T.$ (53)

Following Gwak’s work, we assume that the final state is still a black hole.
This implies $f(M+dM,Q+dQ,P+dP,r_{+}+dr_{+})=f(M,Q,P,r_{+})=0$. From Eqs. (52)
and (53), the increase of the horizon radius takes the form

$\displaystyle
dr_{+}=\frac{4p^{r}}{3\omega_{3}r_{+}^{2}T-4\omega_{3}r_{+}^{3}P}.$ (54)

Using the entropy relation $S=\frac{1}{4}\omega_{3}r_{+}^{3}$ yields

$\displaystyle dS=\frac{3p^{r}}{3T-4r_{+}P},$ (55)

which shows that the increase of the entropy is related to the Hawking
temperature, pressure and $p^{r}$. For the non-extremal black hole or the
fixed cosmological constant, we readily get $dS>0$, which implies that the
entropy increase when the particle is absorbed by the black hole. In the
extremal and near-extremal cases, $dS<0$. This implies that the entropy
decreases when the particle falls into the black hole. It violates the second
law of thermodynamics which shows the entropy never decreases in the clockwise
direction.

We associate the particle’s energy and charge with the increases of the
internal energy and charge of the black hole,

$\displaystyle\omega=dU,\quad q_{e}=dQ.$ (56)

Since the cosmological constant is regarded as the thermodynamic pressure, the
black hole mass is seen as the enthalpy. The relationship between the mass and
internal energy is

$\displaystyle U=M-PV.$ (57)

Combining Eqs. (55), (56) and (57) yields

$\displaystyle dM=TdS+\phi dQ+VdP,$ (58)

which is the first law of thermodynamics recovered by the absorptions of the
scalar particle and fermion. This derivation is inevitable, since the final
state of the black hole after the particle being absorbed was assumed to be
still a black hole.

##  5 Overcharging problem in the extremal and near-extremal black holes

In this section, we test the WCCC in the extremal and near-extremal five-
dimensional black holes by the absorptions of the scalar particle and fermion.
Here the final states of the black holes are unknown. An effective way to test
the WCCC is to check whether a event horizon exists when a particle with
energy and large enough charge or angular momentum falls into a black hole.
The event horizon is determined by the metric component $f(r)$. If the minimal
value of $f(r)$ is negative or zero, the horizon exists. Otherwise, the
horizon does not exist.

We use $f(M,Q,P,r_{0})$ to represent the metric component of the original
state and $f(M+dM,Q+dQ,P+dP,r_{0}+dr_{0})$ to denote the metric component of
the final state, where $r_{0}$ is the location of the minimal value. The
relation between these two metric components is

$\displaystyle
f(M+dM,Q+dQ,P+dP,r_{0}+dr_{0})=f(M,Q,P,r_{0})+\left.\frac{\partial
f(M,Q,P,r)}{\partial r}\right|_{r=r_{0}}dr_{0}$
$\displaystyle\left.\frac{\partial f(M,Q,P,r)}{\partial
M}\right|_{r=r_{0}}dM+\left.\frac{\partial f(M,Q,P,r)}{\partial
Q}\right|_{r=r_{0}}dQ+\left.\frac{\partial f(M,Q,P,r)}{\partial
P}\right|_{r=r_{0}}dP,$ (59)

where

$\displaystyle f(M,Q,P,r_{0})$ $\displaystyle=$ $\displaystyle
k-\frac{M}{r_{0}^{2}}\frac{16\pi}{3\omega_{3}}+\frac{Q^{2}}{r_{0}^{4}}\frac{16\pi^{2}}{3\omega_{3}^{2}}+\frac{4}{3}\pi
r_{0}^{2}P,$ $\displaystyle\left.\frac{\partial f(M,Q,P,r)}{\partial
r}\right|_{r=r_{0}}$ $\displaystyle=$ $\displaystyle
0,\quad\quad\quad\quad\left.\frac{\partial f(M,Q,P,r)}{\partial
P}\right|_{r=r_{0}}=\frac{4\pi r_{0}^{2}}{3},$
$\displaystyle\left.\frac{\partial f(M,Q,P,r)}{\partial M}\right|_{r=r_{0}}$
$\displaystyle=$
$\displaystyle-\frac{16\pi}{3\omega_{3}r_{0}^{2}},\quad\left.\frac{\partial
f(M,Q,P,r)}{\partial
Q}\right|_{r=r_{0}}=\frac{32\pi^{2}Q}{3\omega_{3}^{2}r_{0}^{4}},$ (60)

For the extremal black hole, its Hawking temperature is zero. The location of
the event horizon coincides with that of the minimal value, namely,
$r_{+}=r_{0}$. Thus,

$\displaystyle f(M+dM,Q+dQ,P+dP,r_{0}+dr_{0})$
$\displaystyle=f(M,Q,P,r_{+})+\frac{16\pi}{3\omega_{3}r_{+}^{2}}\left[dM-
VdP-\phi dQ\right]$ $\displaystyle=0,$ (61)

where $f(M,Q,P,r_{+})=0$ and the first law of thermodynamics for the extremal
black hole were used to derive the second equal sign. It shows that the final
state of the extremal black hole is still a extremal one with new mass and
charge. Therefore, the extremal black hole can not be overcharged.

In the near-extremal case, $f(M,P,Q,r_{0})<0$. We order
$r_{0}=r_{+}-\epsilon$, where $0<\epsilon\ll 1$. Eq. (59) is reduced to

$\displaystyle
f(M+dM,Q+dQ,P+dP,r_{0}+dr_{0})=f(M,Q,P,r_{0})+\mathcal{O}(\epsilon)$
$\displaystyle-\frac{16\pi}{3\omega_{3}r_{+}^{2}}\left(1+\frac{4\epsilon}{r_{+}}\right)\frac{3Tp^{r}}{3T-4r_{+}P}+\frac{2\epsilon}{r_{+}}\left(\frac{16\pi}{3\omega_{3}r_{+}^{2}}dM-4\pi
r_{+}^{2}dP\right).$ (62)

The near-extremity leads to $T\to 0$. Considering $r_{+}\sim l\gg
1\gg\epsilon$, we neglect the second term of the second line in the above
equation. Thus we get $f(M+dM,Q+dQ,P+dP,r_{0}+dr_{0})<0$. It shows that the
final state of the near-extremal black hole is still a near-extremal one with
the new mass and charge.

Therefore, the extremal and near-extremal five-dimensional black holes can not
be overcharged by absorbing the scalar particle or fermion with mass and large
enough charge.

##  6 Discussion and Conclusion

In this paper, we investigated the overcharging problem and thermodynamics in
the extended phase spaces of the five-dimensional spherically symmetric
topological black holes by the absorptions of the scalar particle and fermion.
In the investigation, the cosmological constant was regarded as a variable
related to pressure and its conjugate quantity is the thermodynamic volume.
The calculations of the absorptions of the scalar particle and fermion reduced
to the same relation $p^{r}=\omega-q_{e}\phi$. This relation is full in
consistence with that gotten by the Hamilton-Jacobi equation [47]. Maybe the
reason is that the Hamilton-Jacobi equation can be derived by the Klein-Gordon
equation and Dirac equation. This derivation is referred to Ref. [76]. We used
this relation to discuss the thermodynamics and recovered the first law of
thermodynamics. This result is inevitable because the final state of the black
hole after the particles being absorbed was assumed to be still a black hole.
But the second law of thermodynamics was violated in the near-extremal and
extremal black holes. The event horizon is determined by the metric component
$f(r)$. To check the existence of the horizons at the final stages of the
extremal and near-extremal black holes, we estimate the minimal values of the
metric component. For the extremal black hole, its minimal value is zero,
which shows that the final state is still a extremal black hole. For the near-
extremal black hole, its minimal value is a negative value, which corresponds
to a near-extremal black hole with new mass and charge. Therefore, the
extremal and near-extremal black holes can not be overcharged.

##  7 Acknowledgments

This work is supported by the NSFC (Grant No. 11205125).

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