# Exploring the 6G Potentials: Immersive, Hyper Reliable, and Low-Latency
Communication

Afsoon Alidadi Shamsabadi, Animesh Yadav, Yasser Gadallah, and Halim
Yanikomeroglu

###### Abstract

The transition towards the sixth-generation (6G) wireless telecommunications
networks introduces significant challenges for researchers and industry
stakeholders. The 6G technology aims to enhance existing usage scenarios,
particularly supporting innovative applications requiring stringent
performance metrics. Among the key performance indicators (KPIs) for 6G,
immersive throughput, hyper-reliability, and hyper-low latency must be
achieved simultaneously in some critical applications to achieve the
application requirements. However, this is challenging due to the conflicting
nature of these KPIs. This article proposes a new service class of 6G as
immersive, hyper reliable, and low-latency communication (IHRLLC), and
introduces a potential network architecture to achieve the associated KPIs.
Specifically, technologies such as ultra-massive multiple-input multiple-
output (umMIMO)-aided terahertz (THz) communications, and reconfigurable
intelligent surfaces (RIS) are viewed as the key enablers for achieving
immersive data rate and hyper reliability. Given the computational complexity
involved in employing these technologies, and the challenges encountered in
designing real-time algorithms for efficient resource allocation and
management strategies as well as dynamic beamforming and tracking techniques,
we also propose the involvement of other potential enabling technologies such
as non-terrestrial networks (NTN), learn-to-optimize (L2O) and generative-AI
(GenAI) technologies, quantum computing, and network digital twin (NDT) for
limiting the latency.

##  I Introduction

Recently, the fifth-generation (5G) networks have become operational in many
countries worldwide, while researchers are working on studies related to the
sixth-generation (6G) wireless networks. These studies are inspired by
potential use cases, requirements, and challenges that are yet to be
addressed. The previous generations of wireless networks have already provided
significant enhancements that enabled a vast range of novel applications
compared to those supported by legacy wireless networks. Nevertheless,
according to the International Telecommunication Union’s (ITU) vision for
international mobile telecommunications 2030 (IMT-2030), 6G usage scenarios
are envisioned as advanced versions of existing 5G scenarios, alongside
entirely new use cases that support the deployment of innovative applications
[1]. Besides, 6G wireless networks are expected to cater to future use cases
and applications that require high mobility, immersive data rates, hyper-
reliability, and low-latency simultaneously. This necessitates the
introduction of significantly more advanced network architectures and enabling
technologies. We refer to this type of service class as immersive, hyper
reliable, and low-latency communication (IHRLLC), which is also the focus of
this article.

The current 5G wireless network already includes ultra-reliable and low
latency communications (URLLC) as one of its service classes. The URLLC class
targets mission-critical applications and a wide range of other applications,
including industry automation, intelligent transportation, health care, and
intelligent societies. Several enabling schemes and technologies have been
developed to meet the stringent requirements of URLLC in terms of reliability
and latency. These include grant-free access, redundant packet transmission,
short packet transmission, edge computing, software-defined network (SDN), and
network function virtualization (NFV) while utilizing both licensed and
unlicensed bands, as well as coordinated multipoint (CoMP) techniques[2].
However, the critical use cases and applications that are envisioned for the
6G wireless networks demand unprecedented data rates combined with hyper-low
latency and reliability simultaneously. In this context, URLLC alone might not
be sufficient to achieve the requirements set for IHRLLC. Furthermore, it is
essential to note that the IHRLLC requirements conflict with each other,
making them challenging to achieve. Therefore, novel network architecture
designs, along with innovative schemes and technologies, must be explored and
investigated.

Recently, a few works have started investigating the role of URLLC in
addressing use cases and applications that require high connectivity,
mobility, and high data rates in 6G wireless networks [3, 4, 5]. In [3], the
potential of terahertz (THz) frequency band is explored for offering high-rate
and highly-reliable low-latency communications in wireless virtual reality
(VR) applications. It is observed that the probability of line-of-sight (LoS)
links in THz play a crucial role in characterizing the end-to-end (E2E) delay.
In [4], three high data rate connectivity scenarios, including ubiquitous,
deep, and holographic connectivity, are proposed, leveraging URLLC. Machine-
learning-based solutions are proposed to achieve the stringent requirements of
the high data rate and URLLC. In [5], a high mobility communication scenario
is considered and the impact of the Doppler spread on URLLC for vehicular
communication is explored. To this end, orthogonal time frequency space (OTFS)
modulation and sensing-aided predictive beamforming schemes are exploited for
reliable and low-latency communications, respectively.

The aforementioned previous works explore individual use cases that require
high data rates while adhering to the URLLC constraints. In this article, we
present several important use cases for 6G wireless networks under the
proposed IHRLLC service class. We then discuss potential enabling technologies
and propose a unified approach to integrate these technologies to support the
IHRLLC services. The main contributions of this paper can be summarized as
follows

  * •

We propose a new type of service class for 6G, namely the IHRLLC class, which
concurrently supports an immersive data rate, hyper reliability, and low-
latency communication. Several important use cases for this type of service
are discussed, along with their specific requirements.

  * •

We discuss a set of enabling technologies to support the IHRLLC class.
Additionally, a network architecture is proposed that combines these
technologies to achieve the goal of IHRLLC. Challenges and open issues
pertaining to the proposed architecture are also discussed.

  * •

We present a numerical simulation-based experiment to validate the high data
rate via ultra-massive multiple-input multiple-output (umMIMO)-aided THz
communication.

##  II Requirements, Use Cases, and Challenges

As 6G wireless networks are expected to encompass a broad spectrum of novel
use cases, applications, and services, more than one KPI must be met
simultaneously to achieve acceptable levels of quality of experience (QoE). In
this paper, we focus on some specific use cases that require the simultaneous
satisfaction of three KPIs, namely, immersive data rates, hyper reliability,
and low latency. The exact values of these KPIs vary depending on service
expectations of the use cases. Certain use cases prioritize specific KPIs or
demand stricter thresholds on a particular KPI to guarantee the desired
performance. In the following section, we discuss a novel service class for 6G
which we term the IHRLLC class, and discuss some key use cases and associated
challenges.

###  II-A Immersive Communication

Immersive communication (IC) will be one of the key service classes of the 6G
wireless networks. It will introduce interactive services such as extended
reality (XR), holographic communication, cloud gaming, high-definition video
(4K and 8K) streaming, and immersive event experiences over cellular networks.
These services demand a large volume of data and require immersive throughput
values. For instance, a live VR concert, streaming to thousands of
participants simultaneously, requires a wireless network capable of handling
substantial data rates without compromising the user’s QoE. According to the
ITU IMT-2030 vision, the envisioned throughput for IC peaks at $1$ Tbps, with
the user-experienced data rate of at least $300$ Mbps[1], which far exceeds
the capabilities of IMT-2020.

###  II-B Hyper reliable and low-latency communication (HRLLC)

The HRLLC class, an extension of URLLC in 5G, will be another core service
class in 6G alongside other classes. Latency in wireless networks is
influenced by various components of the architecture. Key factors include
network management strategies in the core network, signal processing schemes
such as modulation and coding in the E2E parts of the network, and the
propagation delay of the signal within the network. Achieving hyper low-
latency values of $0.1$ to $1$ ms, as specified by IMT-2030, without ensuring
a hyper-reliable network, renders 6G use cases infeasible. Specifically, for
applications such as autonomous vehicles and industrial automation, the
network must simultaneously meet the reliability standard of $10^{-5}\sim
10^{-7}$ while achieving hyper low-latency, thus enabling the HRLLC usage
scenario.

###  II-C Immersive, hyper reliable, low-latency communication (IHRLLC)

The two aforementioned service classes have been studied separately, each with
associated KPIs. Nevertheless, a wide range of use cases require simultaneous
fulfillment of a combined set of three KPIs (i.e., immersive data rate, hyper
reliability and low latency) for an acceptable level of QoE. Hence, we propose
IHRLLC as a new service class for 6G wireless networks. Below are some of the
key applications under this service class.

####  II-C1 Mobile tele-surgery

Mobile tele-surgery in an ambulance racing through the city with a patient
needing an urgent surgical operation is a representative use case for IHRLLC.
The surgeon, located in a hospital far away from the ambulance, is required to
perform the critical procedure remotely to save the patient’s life as they are
being rushed to the hospital. This situation requires not only IC to transmit
substantial data volumes but also necessitates absolute reliability and hyper
low-latency to ensure the safety and efficacy of the medical procedure.

####  II-C2 Multisensory XR

XR refers to various types of reality, including augmented reality (AR), mixed
reality (MR), and virtual reality (VR), encompassing all real-and-virtual
combined environments and human-machine interactions with the assistance of
handheld and wearable end-user devices. Multisensory XR applications have
unique requirements, including bounded low latency and high uplink and
downlink bandwidth. These are challenging requirements due to scale and
mobility at various speeds (e.g., walking, driving, fast trains) as well as
coverage and handovers. The 3rd Generation Partnership Project (3GPP) Release
18 promises to significantly improve support for higher data rates and
capacity while also introducing power and latency reduction techniques to
enable more immersive AR experiences.

####  II-C3 High-frequency trading

In particular, high-frequency trading (HFT) is a trading method that utilizes
powerful computer programs to execute a large number of orders in a very short
period. This approach aims to avoid the high risks associated with fast-
changing trading prices[6]. HFT requires networks that are capable of meeting
the required set of KPIs with a high degree of certainty.

####  II-C4 Cloud gaming

In the gaming industry, cloud gaming refers to use cases where most
computations related to single-player or multiplayer gaming are offloaded from
users’ hardware devices to edge or remote servers. Essentially, games are
hosted on remote servers, with users’ devices serving primarily as peripherals
for interacting with the game. In cloud gaming, the user’s QoE depends on the
network’s capability to deliver IHRLLC service class performance.

####  II-C5 Advance air mobility

Advance air mobility (AAM) is a concept that enables customers to access on-
demand air transportation through multimodal transport networks. One of the
key use cases of AAM is urban air mobility (UAM), which allows vehicles to
either be autonomously controlled or remotely piloted while operating beyond
the visual LoS from the ground. UAM vehicles typically have multiple sensors,
such as light detection and ranging (lidar), radar, and cameras. These sensors
can be utilized for advanced driver assistance systems and can also transmit
real-time immersive data with hyper reliability and low-latency to a ground
pilot for remote operation.

Pursuing a combined set of conflicting KPIs leads to inherent difficulties, as
most existing wireless networks normally focus on achieving a specific KPI at
the expense of others. Balancing these KPIs in 6G wireless networks to meet
the diverse requirements of various applications of IHRLLC remains a
significant challenge, and addressing them necessitates innovative solutions.
In the following section, we overview a set of potential enabling technologies
that we can depend upon and describe how properly aggregating these
technologies overcomes the challenges that arise in simultaneously achieving
conflicting KPIs.

Figure 1: umMIMO and RIS-aided THz communication for 6G mobile tele-surgery
application.

(a) CDF of per user data rate (Gbps). (b) Packet Error Rate vs. SNR.

Figure 2: User data rate and PER in an umMIMO system working at $850$ GHz
carrier frequency with $200$ MHz bandwidth.

##  III Envisioned Enabling Technologies for IHRLLC

Several solutions and technologies are proposed to improve the throughput,
latency, and reliability of wireless networks. These technologies span
different parts of the wireless networks to ensure that the overall
requirements are met. However, techniques used to achieve one specific KPI may
interfere with achieving other KPIs. For example, utilizing higher frequency
bands significantly improves data rates but limits the coverage area due to
severe shadowing, deep fading, and molecular absorption, that result in
decreased reliability. In this section, we provide an overview of our
envisioned IHRLLC enabling technologies and their corresponding advantages and
challenges. We also discuss how the combination of various technologies,
across the different network layers, supports the concurrent fulfilment of
multiple KPIs.

###  III-A umMIMO and RIS-aided THz communication

Migrating to higher frequency bands has been the solution to access wider
bandwidth and, hence, higher data rates for users. The millimeter wave
(mmWave) spectrum, which includes frequencies above $24$ GHz, was utilized in
the 5G wireless networks to provide access to a considerable bandwidth
compared to the previous mobile network generations. However, the huge
capacity demand of 6G applications necessitates the provisioning of extremely
high bandwidths, which can only be achieved through THz communications. The
THz spectrum, in particular, spans the frequencies in the $0.1$ to $3$ THz
range. Extremely high available bandwidth at this frequency range makes it a
promising technology to achieve immersive communication in 6G. Nevertheless,
operating in the THz frequency band exposes new challenges to the wireless
networks [7]. The most significant of these challenges include higher free
space path loss (FSPL) and molecular absorption, channel modeling, and the
design of hardware which includes radio frequency (RF) devices, modulators,
and antennas. Therefore, devising innovative solutions to address the
aforementioned challenges is crucial for leveraging the potential of THz
frequencies in 6G wireless networks. To overcome the short distance
communication range and severe blockage of the THz frequency band, antenna
array architectures, e.g., umMIMO and RIS, can be integrated into the radio
access networks to improve the beam gain toward users and maintain LoS
connection with the base station, respectively, see Fig. 1.

####  III-A1 umMIMO beamforming

Higher Frequency bands, such as mmWave and THz, enable the use of antenna
designs that can accommodate a significantly greater number of antenna
elements. Therefore, implementing the umMIMO technology and enhanced
beamforming techniques creates high-gain narrow beams tailored to individual
users. These beams help reduce the substantial FSPL in THz communications[8].
Fig. 2 depicts the cumulative distribution function (CDF) of the data rate and
the packet error rate (PER) performance for varying values of signal-to-noise
ratio (SNR) in a wireless system consisting of one base station, with a cell
radius of $1000$ m, equipped with an umMIMO antenna serving one user. The
system works at the center frequency of $850$ GHz, and the user is allocated a
bandwidth of $200$ MHz. It can be observed that the data rate and PER values
of the user are within the desired range to facilitate a reliable and
immersive connection. Nevertheless, achieving such high data rate and low PER
values necessitates designing efficient advanced beamforming algorithms.

Beamforming via umMIMO architecture in the THz frequency band presents several
challenges. Very narrow pencil beams require accurate knowledge of the channel
state information (CSI), accurate angle of departure of the transmitters, and
the angle of arrival of receivers. Furthermore, since beamforming is
vulnerable to mobility, the difficulty lies in continuous beam realignment in
a short time. Furthermore, a large number of RF power amplifier chains are
needed to perform effective beamforming, thus increasing cost and power
consumption. Alternatively, an energy-efficient holographic MIMO system
appears to be a potential solution.

####  III-A2 RIS for LoS connectivity and diversity

The RIS technology presents a groundbreaking advancement in wireless
communication, offering an evolutionary approach to signal propagation and
control. RISs are artificial surfaces with numerous small, electronically
controlled passive elements capable of altering the phase and amplitude of the
incoming signal, thus enabling the redirection or reshaping of the signal
path[9]. Therefore, an RIS enables a favorable and adaptable wireless
environment for signal propagation, which can enhance the performance of
wireless networks from several perspectives. First, considering the ever-
increasing demand for ubiquitous seamless connectivity, wireless networks
require deploying more base stations to ensure reliable connectivity for
users. However, this will bring a higher energy consumption, higher deployment
and maintenance costs, and a higher level of interference to deal with. RIS
passive elements and their light weight make them capable of being deployed in
many strategic spots in the city at a relatively low cost, which enables them
to overcome the communication distance limitations of the THz frequency band
by offering LoS links, thus paving the way for IHRLLC.

Second, RISs can be strategically positioned to create a scattering channel,
which is desirable for enhancing the performance of umMIMO systems in THz
communications. As a result, RIS systems can add a degree of freedom to umMIMO
systems and improve the reliability of wireless networks.

Figure 3: Role of HAPS and UAVs in enabling UAM use case in 6G.

(a) A comparison between the architecture of classical optimization and
optimization using L2O[10].

(b) Applications of GenAI in wireless networks physical layer[11].

(c) Quantum approximate optimization algorithm (QAOA) for combinatorial
optimization.

Figure 4: L2O, GenAI, and Quantum Computing. The logos in the middle of (b)
and (c) are generated using ChatGPT-4o.

###  III-B Non-terrestrial networks

Non-terrestrial networks (NTN), which include space satellites and aerial
platforms such as high altitude platform stations (HAPS) and unmanned aerial
vehicles (UAVs), present a promising architecture for 6G wireless networks.
Traditionally, NTN have been considered a potential technology for enhancing
wireless networks coverage in rural and remote areas. However, given the ever-
increasing capacity demands of future wireless networks and the need for a
resilient and reliable network architecture, the new paradigm of employing the
NTN is to integrate them into urban terrestrial wireless networks to guarantee
an immersive and hyper reliable connection for the users[12]. HAPS, located at
an altitude of about $20$ Km above the ground, can provide a one-way
propagation delay of the order of microseconds, which is much less than that
of the satellite networks. In addition, as depicted in Fig. 3, HAPS can play
the role of backhauling for the UAVs via enabling airborne integrated access
and backhauling (IAB). In applications such as UAM, multi-connectivity through
aerial platforms, such as UAVs and HAPS, will be a potential solution in case
of losing connection with either the main base station or the RIS (Fig. 3).

In the previous sections, various enabling technologies were discussed to
facilitate IHRLLC. The implementation of such sophisticated technologies, as
well as the proper design of the corresponding parameters, is a critical
challenge. Considering the umMIMO antennas and RISs, with a large number of
elements along with the challenges of THz communication, designing the optimum
parameters through traditional approaches incurs a high computational cost. To
this end, AI algorithms can potentially produce revolutionary changes in
future wireless networks and overcome computational challenges. In the
following discussion, we investigate promising novel computational-related
approaches, including learn-to-optimize (L2O), generative AI (GenAI), and
quantum computing technologies as methods to solve the aforementioned
computational challenges in 6G wireless networks.

###  III-C AI technologies: L2O and GenAI

L2O leverages machine learning (ML) to improve the current optimization
algorithms or develop new optimization approaches [10]. Conventional
optimization techniques often rely on fundamental methods such as gradient
descent, conjugate gradient, and Newton steps, which are theoretically robust
and straightforward. These methods are chosen based on the type of problem and
are expected to provide solutions that meet their theoretical guarantees.
However, L2O presents a different approach in which optimization methods are
developed through training based on their performance with sample problems
(Fig. 4a). These methods may not have a solid theoretical foundation, but are
designed to enhance performance through training, which usually occurs online.
At the same time, this training aims to make the actual application of the
methods more efficient. In situations involving complex problems such as
nonconvex optimization or inverse problems, a well-trained L2O method might
deliver better results than traditional methods.

Although conventional AI/ML algorithms have already been utilized in
developing algorithms for wireless networks, GenAI has recently been
considered a potential tool for future wireless networks. GenAI can be
employed in wireless networks to overcome the challenges of the conventional
AI algorithms[11]. Specifically, GenAI models can be utilized in various
physical layer aspects of the networks, such as IRS deployment design, joint
source-channel coding, beamforming, etc., as illustrated in Fig. 4b. In
addition, the GenAI models can generate data for training purposes and can be
implemented near the training locations.

###  III-D Quantum Computing

Future wireless networks will experience massive communications among users in
various usage scenarios. This makes it increasingly difficult to decide the
order in which resource allocation should be made. These challenges can be
defined as combinatorial problems which involve determining the service order
for different users and the routes for traffic transfer to ensure everyone
receives the promised QoE. Additionally, networks are becoming software-
oriented through the use of the SDN and NFV technologies. Next-generation
networks will employ AI and cognitive technologies for self-healing, self-
management, and decision making, which require massive computing capabilities,
leading to high latency [13]. Traditionally, combinatorial problems have so
far been largely solved using heuristic methods that simplify the problem.
However, these methods typically yield approximate answers that, in certain
situations, may lead to inferior performance. Optimization methods based on
statistics and data, on the other hand, consume a significant amount of energy
and may not be suitable for dynamic situations that require quick reactions.
To address these issues, hybrid algorithms combining quantum and classical
computing, such as the quantum approximate optimization algorithm (QAOA), are
used. These algorithms can be deployed at the core network, in various scales
according to the hardware capabilities, offering considerable performance
improvements in many combinatorial optimization problems [14]. Hybrid
algorithms optimize the strengths of both computing approaches, ultimately
increasing the network capacity. This ensures that demanding applications
receive the necessary service quality, even in a busy network, while also
saving energy.

Figure 5: NDT-empowered network for IHRLLC in mobile tele-surgery and UAM use
cases.

###  III-E Network Digital Twin (NDT)

Digital twin (DT) is an emerging technology that virtually represents a
physical system. It accurately models complex systems by replicating real
physical entities and functionalities of a physical system into the digital
world in real-time. Combining the DT technology with AI/ML tools, real-time
data can be used to improve the operation and performance of the physical
system in terms of precise decision-making and efficient resource utilization.
The network DT (NDT) has recently been proposed as a key enabler for efficient
control and management of modern communication networks [15]. Through the NDT
dynamic state of the network, the status of the edge and cloud servers can be
captured in real-time. Furthermore, combining 3D map services with NDT can
determine the location and health of network infrastructure elements such as
base stations and RISs. The NDT does therefore improve network performance.
However, it also suffers from problems of data asynchronization and deviation
between the NDT and the real network. Such issues can be addressed by
appropriately applying the tools from the AI/ML framework.

##  IV Proposed Network Architecture for IHRLLC

We now propose a network architecture that integrates the aforementioned
technologies to support and achieve the requirement for the IHRLLC service
class. We use NDT as the underlying technology that helps the IHRLLC use cases
to make more informed and swift decisions. In particular, NDT is used to gain
information about the infrastructure locations and the status of edge and
cloud computing servers in the entire network. As an example, the proposed
NDT-empowered network for IHRLLC in mobile tele-surgery and UAM use cases is
shown in Fig. 5. Essentially, the network relies on the THz band to support
immersive communication. To reliably transmit data at high data rates, the
communication relies on numerous RISs deployed at different locations
throughout the city. After obtaining the relative location information via
NDT, and with the help of the umMIMO, the ambulance continuously forms pencil
beams toward multiple RISs in LoS with an ambulance. The use or multiple RISs
not only facilitates LoS communications but also enhances diversity reception.
Both the LoS link and diversity reception significantly improve reliability.
Each RIS is tuned such that all reflected signals are directed to the nearest
base station and subsequently to the core network and then eventually to the
hospital. Furthermore, multi-connectivity transmission is available. If the
ambulance can not find an RIS in the range, it then starts communication via a
dedicated UAV that is associated with this ambulance. For the UAM use case,
UAVs and HAPS will be part of the IHRLLC network, which ensures seamless
connectivity for the user.

AI enablers (L2O and GenAI) and quantum computing play a crucial role in the
proposed network by reducing computational delays for algorithms related to
efficient resource allocation, beamforming design, beam-tracking, AL/ML-based
operations, and various functions for implementing SDN and NFV. The L2O and
GenAI models can be utilized at the base stations and the core network, in
different scales, to support designing efficient algorithms. Accordingly,
quantum computing resources are distributed across the core network
components. In this way, tasks requiring heavy computation are delegated to
the core network due to the higher processing capabilities. On the other hand,
tasks requiring real-time decision making will be handled at the base stations
using L2O and GenAI models. Task allocation is based on the status of all the
servers, monitored via NDT. Additionally, HAPS serves as a computing platform
for specific use cases such as UAM.

##  V Challenges and Open Issues

In the previous sections, we discussed the 6G use cases that require IHRLLC to
achieve an acceptable QoE, and we proposed a combination of technologies to
achieve IHRLLC. However, designing an E2E wireless network, equipped with the
proposed technologies, and is capable of dealing with a large volume of data
in a hyper reliable manner within a hyper low-latency, is a significant
challenge. Considering the THz frequency band, combined with umMIMO and RIS
technologies, the development of hardware devices such as antenna elements and
electronic or photonic devices that can generate high frequency
electromagnetic waves is an open area of research among wireless researchers.
The THz-specific air interface techniques and multiple access protocols need
further exploration to ensure network reliability. Furthermore, the human
health and safety issues associated with the THz technology should be
thoroughly analyzed and evaluated.

Although AI has been introduced as a game changer for future wireless
networks, specific characteristics of these networks make the implementation
of the AI algorithms challenging. One of the key challenges is the randomness
of the wireless network environments inherited from the random channel models.
This is considerably important in urban areas where wireless channels
experience significant small-scale fading. Consequently, developing AI-driven
algorithms to support the E2E network functionality in a robust, resilient,
dynamic, and efficient manner is a significant challenge and, hence, an
important research area. This is in addition to the privacy and
standardization concerns related to the implementation of AI algorithms.
Finally, integrating a combination of all potential technologies into wireless
networks necessitates interdisciplinary research among experts in wireless
networks, AI, and quantum computing to ensure the efficiency and scalability
of the developed algorithms.

##  VI Conclusion

In this paper, we discussed the 6G use cases that require achieving immersive,
hyper reliable, low-latency communication (IHRLLC) simultaneously.
Accordingly, we proposed a new class, termed the IHRLLC class, to be
considered as one of the main 6G service classes. We also proposed potential
network solutions to meet the KPI requirements and overcome the challenges of
achieving multiple KPIs. To this end, THz communication, combined with umMIMO
and RIS, plays an important role in making IHRLLC feasible. An NDT-empowered
network architecture was proposed to facilitate the IHRLLC applications.
Moreover, novel computation and communication techniques, including L2O,
GenAI, quantum computing, and NDT, were proposed to design the parameters of
various parts of the architecture.

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## Biography

Afsoon Alidadi Shamsabadi [M]<EMAIL_ADDRESS>is a PhD
Candidate in the Non-Terrestrial Networks (NTN) Lab in the Department of SCE
at Carleton University, Canada.
---
Animesh Yadav [SM]<EMAIL_ADDRESS>is an assistant professor in the School
of EECS at Ohio University, Athens, OH, USA.
---
Yasser Gadallah [SM]<EMAIL_ADDRESS>is a Professor at the Department
of ECE at The American University in Cairo (AUC), Egypt. His research
interests include IoT, wireless communication and smart systems. He has
extensive telecommunications industrial experience, having worked at several
high-profile industry-leading companies in this area.
---
Halim Yanikomeroglu [F]<EMAIL_ADDRESS>is a Chancellor’s Professor of
Systems and Computer Engineering in the Non-Terrestrial Networks Lab., at
Carleton University, Canada. He is a Fellow of EIC (Engineering Institute of
Canada), CAE (Canadian Academy of Engineering), and AAIA (Asia-Pacific
Artificial Intelligence Association); and a Distinguished Speaker for IEEE
Communications Society and IEEE Vehicular Technology Society.
---


# Free-falling motion of an elastic rigid rod towards a Schwarzschild black
hole

Luís Machado, José Natário and Jorge Drumond Silva  Center for Mathematical
Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto
Superior Técnico, Universidade de Lisboa, Portugal

###### Abstract.

We study the motion of an elastic rigid rod which is radially free-falling
towards a Schwarzschild black hole. This is accomplished by reducing the
corresponding free-boundary PDE problem to a sequence of ODEs, which we
integrate numerically. Starting with a rod at rest, we show that it is
possible to choose its initial compression profile so that its midpoint falls
substantially faster, or slower, than a free-falling particle with the same
initial conditions. This seems to be a purely kinematic effect, since on
average there is no net transfer of elastic to mechanical energy.

## Introduction

It is well known that the trajectories of free-falling test particles in
General Relativity are given by timelike geodesics. The motion of extended
bodies, however, is far more complicated [7, 8, 9], and the geodesic
approximation is only valid in the asymptotic limit when the size of the body
is much smaller than the local radius of curvature [11, 10, 21]. In fact,
extended bodies often exhibit surprising motions (sometimes dubbed “swimming”
or “gliding” motions), examples of which have been studied in [25, 12, 14, 13,
6, 22, 17, 23, 15, 24].

In Classical Mechanics, the simplest model to describe the motion of an
extended body is to assume that it is rigid (meaning undeformable). In General
Relativity, however, causality considerations prevent the use of such a
simplistic assumption. Therefore, relativistic elasticity has become the
standard tool to study simple extended bodies [2, 1, 5, 3, 19, 4, 20, 16], as
it offers a fully coherent framework within General Relativity which not only
models the bulk motion but also accounts for the inevitable deformations that
any real object is subject to.

In this work, we focus on the motion of a rigid rod, that is, a
$1$-dimensional elastic object whose speed of sound is equal to the speed of
light111Note that “rigid” is taken to mean “as rigid as possible”, implying
the largest admissible speed of sound. In Classical Mechanics this means
“undeformable”, with an infinite speed of sound; in General Relativity it
implies that the speed of sound is equal to the speed of light., falling
radially towards a Schwarzschild black hole. This leads to a free boundary
problem for the wave equation on the $2$-dimensional Lorentzian manifold
corresponding to the radial direction. Using harmonic coordinates (essentially
d’Alembert’s method for solving the wave equation along null directions), we
turn this problem into a sequence of ODEs, which we then integrate numerically
for a few illustrative examples, concentrating on the strong field region near
the black hole. Specifically, we analyze the motion of a rod starting at rest,
for three different choices of the initial compression factor: completely
relaxed (for comparison purposes), and two cases where the two halves of the
rod are uniformly compressed by different amounts. The soundness of the
numerical integration is checked by computing the conserved energy of the rod.
We also compute the (non-conserved) elastic energy, whose value seems to
oscillate without any discernible trend.

The organization of the paper is as follows: in Section 1 we briefly review
the general theory of one-dimensional rigid elastic bodies moving in two-
dimensional spacetimes. In Section 2 we set up the algorithm for solving the
equation of motion of a rigid rod moving radially on a general static
spherically symmetric spacetime. Section 3 contains a derivation of the
expression for the rod’s conserved energy, as well as for its elastic energy.
Finally, we numerically integrate the equation of motion for the three
examples mentioned above in Section 4. We conclude in section 5 by discussing
the results and suggesting further research problems.

We follow the conventions of [18]; in particular, we use a geometrized system
of units, for which $c=G=1$.

##  1\. Elastic rod

We are interested in the motion of an elastic rod, that is, a one-dimensional
elastic body, along a fixed direction (namely radial motion in a spherically
symmetric spacetime). Therefore, it suffices to consider two-dimensional
spacetimes $(M,g)$. The elastic rod can be modeled by a scalar field
$\lambda:M\to\mathbb{R}$ with nonvanishing spacelike gradient, whose level
sets correspond to the worldlines of the particles that constitute the rod
(see [19]). The compression factor of the rod is then the positive function
$n$ such that

(1)
$n^{2}={|\text{grad}\>\lambda|}^{2}=({\nabla}^{\alpha}\lambda)({\nabla}_{\alpha}\lambda)\,.$

Since we are considering only one spatial dimension, the energy-momentum
tensor of the elastic rod is that of a perfect fluid. As mentioned in the
introduction, we will assume that the rod is rigid, that is, its speed of
sound equals the speed of light. Under these assumptions, as shown in [19],
the scalar field $\lambda$ satisfies the wave equation,

(2) $\Box\lambda=0\,.$

Generically, the motion of a finite rod leads to a free boundary problem with
nonlinear boundary conditions for this equation. Such problems are highly
nontrivial, and a careful choice of comoving coordinates, for which the
boundary remains fixed, is typically the best approach to study them. In this
simple $(1+1)$-dimensional setting, an obvious choice is the function
$\lambda$ itself, since the free boundary, corresponding to the endpoints of
the rod, are level sets of this function. As shown in [19], one can always
choose a time function $\tau:M\to\mathbb{R}$, also satisfying the wave
equation, such that the metric in the (harmonic) coordinates $(\tau,\lambda)$
is written as

(3)
$ds^{2}=\frac{1}{n^{2}(\tau,\lambda)}\left(-d\tau^{2}+d\lambda^{2}\right)\,.$

Conversely, if the metric can be written in the form (3) then $\lambda$ is a
solution of the wave equation. Therefore, a method for solving the problem of
the rod’s motion is simply to find a harmonic coordinate system.

##  2\. Motion on static spherically symmetric spacetimes

We now focus on the case of radial motion of the rod in a general static
spherically symmetric spacetime. The metric on the totally geodesic
$2$-dimensional submanifold $M$ corresponding to a fixed radial direction can
be written as

(4) $ds^{2}=-F(r)dt^{2}+\frac{1}{G(r)}dr^{2}\,,$

where $r$ is the usual areal radius coordinate and $F$ and $G$ are smooth
positive functions depending on $r$. Changing to the null coordinates

(5) $\begin{cases}u=t-x\\\ v=t+x\end{cases}\,,\quad\text{where}\quad
x(r)=\int\frac{dr}{\sqrt{F(r)G(r)}}\,,$

the metric of the $2$-dimensional spacetime takes the form

(6) $ds^{2}=-F\left(r\left(\frac{v-u}{2}\right)\right)du\>dv\,,$

where $r(x)$ is the inverse function of $x(r)$ in equation (5).

On the other hand, changing to null coordinates in the harmonic coordinate
system associated to the rod,

(7) $\begin{cases}U=\tau-\lambda\\\ V=\tau+\lambda\end{cases}\,,$

the metric (3) is represented as

(8) $ds^{2}=-\frac{1}{n^{2}(U,V)}\,dU\,dV\,.$

Since $(u,v)$ and $(U,V)$ are both future-directed null coordinate systems for
our $2$-dimensional spacetime222We are assuming that the time functions $t$
and $\tau$ induce the same time orientation., they can be rescaled into each
other, in the sense that there must exist smooth increasing functions $f$ and
$g$ such that

(9) $\begin{cases}u=f(U)\\\ v=g(V)\end{cases}\,.$

Inserting this in equation (6) yields

(10)
$ds^{2}=-F\left(r\left(\frac{g(V)-f(U)}{2}\right)\right)f^{\prime}(U)\,g^{\prime}(V)\,dU\,dV\,.$

From (8) and (10) we obtain the following equation, which fully determines the
motion of the elastic rod:

(11)
$\frac{1}{n^{2}(U,V)}=F\left(r\left(\frac{g(V)-f(U)}{2}\right)\right)f^{\prime}(U)\,g^{\prime}(V)\,.$

Given initial and boundary conditions, our task is then to determine the
functions $f$ and $g$ by solving equation (11) in the region corresponding to
$\tau>0$ and $0<\lambda<L$, where $L$ is the proper length of the rod.

As the first initial condition, we will impose that the rod is initially at
rest:

(12) $\frac{\partial x}{\partial\tau}(0,\lambda)=0\,,\qquad 0<\lambda<L\,,$

which is equivalent to

(13) $g^{\prime}(V)=f^{\prime}(-V)\,,\qquad 0<V<L\,.$

Integrating this equation, we obtain

(14) $g(V)+f(-V)=0\,,\qquad 0<V<L\,,$

where we set the integration constant to zero by fixing $t=0$ when $\tau=0$.
Additionally, we have to impose a second initial condition to define the
initial state of the elastic rod: the values of the compression factor at
$\tau=0$, that is, $n(-V,V)$ for $0<V<L$.

As boundary conditions, we will impose that the rod is not being pulled or
pushed at its ends. This corresponds to setting the values of the compression
factor at both ends of the rod equal to one for all time (see [19]), that is,
$n(V,V)=1$ for $V>0$ and $n(V-2L,V)=1$ for $V>L$.

Let us suppose that the endpoint of the rod corresponding to $\lambda=0$ is
initially at $r=r_{0}$. Mathematically, this can be expressed by

(15) $\frac{g(0)-f(0)}{2}=x(r_{0})\,,$

which, together with equation (14), implies

(16) $\begin{cases}g(0)=x(r_{0})\\\ f(0)=-x(r_{0})\end{cases}\,.$

In the segment corresponding to $\tau=0$, the equation of motion (11) becomes

(17)
$\frac{1}{n^{2}(-V,V)}=F\left(r\left(\frac{g(V)-f(-V)}{2}\right)\right)f^{\prime}(-V)\,g^{\prime}(V)\,,\qquad
0<V<L\,.$

Combining this result with equation (14), we obtain

(18) $\frac{1}{n^{2}(-V,V)}=F(r(g(V)))(g^{\prime}(V))^{2}\,,\qquad 0<V<L\,,$

which, using the with initial condition (16), can be solved to determine
$g(V)$ for $0<V<L$, since the values of $n(-V,V)$ in this interval are known.
The values of $f(U)$ for $-L<U<0$, in turn, can be obtained from equation
(14).

Setting $U=V-2L$, the equation of motion (11) becomes

(19)
$\frac{1}{n^{2}(V-2L,V)}=F\left(r\left(\frac{g(V)-f(V-2L)}{2}\right)\right)f^{\prime}(V-2L)\,g^{\prime}(V)\,.$

This equation can be solved to determine $g(V)$ for $L<V<2L$, since the values
of $n(V-2L,V)$ in this interval are known, as are the values of $f(U)$ for
$-L<U<0$. Similarly, by setting $V=U$, the equation of motion (11) becomes

(20)
$\frac{1}{n^{2}(U,U)}=F\left(r\left(\frac{g(U)-f(U)}{2}\right)\right)f^{\prime}(U)\,g^{\prime}(U)\,.$

Using the known values of $g(V)$ for $0<V<L$ and $n(U,U)$ for $U>0$, we can
determine $f(U)$ for $0<U<L$.

By iterating the previous steps, we can determine $f(U)$ and $g(V)$ for $U>-L$
and $V>0$, thus obtaining the motion of the rod: the compression factor can be
computed at any point $(U,V)$ using equation (11), and the trajectory of any
particle in the rod, located at $\lambda=\lambda_{0}$ , is given by the curve

(21)
$\gamma(V)=\left(\frac{f(V-2\lambda_{0})+g(V)}{2},\,r\left(\frac{g(V)-f(V-2\lambda_{0})}{2}\right)\right),\qquad
V>\lambda_{0}\,,$

where the first coordinate is the time $t$ and the second coordinate is the
radius $r$.

##  3\. Energy

Since $\frac{\partial}{\partial t}$ is a Killing vector field, there is a
conserved energy for the motion of the rod, given by

(22) $E(\tau)=\int_{0}^{L}T\left(\frac{\partial}{\partial
t},n\frac{\partial}{\partial\tau}\right)\frac{d\lambda}{n}=\int_{0}^{L}T\left(\frac{\partial}{\partial
t},\frac{\partial}{\partial\tau}\right)d\lambda\,,$

where

(23) $T=d\lambda^{2}-\frac{1}{2}\left(n^{2}+1\right)ds^{2}$

is the rod’s energy-momentum tensor in units where the rest density is
$\rho_{0}=1$ (see [19, 20]). Using equations (5), (7) and (9), we can write
this expression as

(24)
$E(\tau)=\int_{0}^{L}\frac{n^{2}(\tau,\lambda)+1}{4n^{2}(\tau,\lambda)}\left(\frac{1}{f^{\prime}(\tau-\lambda)}+\frac{1}{g^{\prime}(\tau+\lambda)}\right)d\lambda\,.$

We can use the conservation of $E(\tau)$ to double check the soundness of the
numerical integrations. We can also compute the (non-conserved) sum of the
energies of the individual particles making up the rod, given by

(25) $K(\tau)=-\int_{0}^{L}ds^{2}\left(\frac{\partial}{\partial
t},n\frac{\partial}{\partial\tau}\right)d\lambda=\int_{0}^{L}\frac{1}{2n(\tau,\lambda)}\left(\frac{1}{f^{\prime}(\tau-\lambda)}+\frac{1}{g^{\prime}(\tau+\lambda)}\right)d\lambda\,.$

The difference $E(\tau)-K(\tau)$ can be interpreted as the elastic energy of
the rod. Monitoring the evolution of $K(\tau)$ will signal whether elastic
energy is being converted to mechanical energy (meaning kinetic and
gravitational potential energy) or vice-versa.

##  4\. Falling into a Schwarzschild black hole

We will now study the radial motion of an elastic rod in the Schwarzschild
spacetime. The metric of the $2$-dimensional totally geodesic submanifold
corresponding to a fixed radial direction is

(26)
$ds^{2}=-\left(1-\frac{1}{r}\right)dt^{2}+\left(1-\frac{1}{r}\right)^{-1}dr^{2}\,,$

where by simplicity we take the Schwarzschild radius to be our length unit,
$2M=1$. In this case we have $F(r)=G(r)=1-\frac{1}{r}$, and so

(27) $x(r)=\int\left(1-\frac{1}{r}\right)^{-1}dr=r+\ln{(r-1)}\,,$

frequently referred to as the tortoise coordinate. Note that we assume $r>1$,
implying that the rod is totally outside the black hole.

We will concentrate on the strong field region near the black hole, and
present three illustrative examples. Specifically, we will discuss the motion
of a rod starting at rest, for three different choices of the initial
compression factor: completely relaxed (for comparison purposes), and two
cases where the two halves of the rod are uniformly compressed by different
amounts. Other examples, such as the uniformly compressed or stretched rod,
yield comparable but less dramatic results.

###  4.1. Initially relaxed free-falling rod

As a first example, we will explore the scenario in which the rod is initially
completely relaxed and falls freely into the black hole. Mathematically, this
corresponds to setting $n=1$ at $\tau=0$. Since equation (11) cannot be solved
explicitly, one needs to resort to numerical integration. We use the Runge-
Kutta method, repeating the integration for successively smaller time steps in
order to check the convergence of the solution. The results are depicted in
Figure 1, where the geodesics representing test particles starting at rest
from the same points as the rod’s endpoints are also presented for comparison.
We can see that the endpoint nearer the black hole falls slower than the
particle starting from the same position, whereas the opposite endpoint falls
faster. This is due to the rod’s elastic nature, which exerts an elastic force
on its endpoints, forcing them to move slightly inwards.

Figure 1. Initially relaxed free-falling rod in the Schwarzschild spacetime.
The worldlines of the left and right endpoints are the blue curves. The
geodesics corresponding to particles starting at rest from the same positions
are the orange dashed curves.

We can also plot the trajectory of the midpoint of the rod, that is, the point
corresponding to $\lambda=L/2$, and compare it with the geodesic corresponding
to the free-falling test particle starting at rest from the same position. As
shown in Figure 2, the midpoint’s worldline is significantly closer to the
test particle’s geodesic than the endpoints, which may be due to the action of
elastic forces in both directions rather than just one.

Figure 2. Worldline of the midpoint of the rod, initially relaxed and falling
freely in the Schwarzschild spacetime. The worldline of the midpoint is the
blue curve. The geodesic corresponding to a particle starting at rest from the
same position is the orange dashed curve.

As an aside, we can study the initial behavior of the compression factor — in
particular, whether it is initially increasing or decreasing. To do that, we
use equations (11) and (14) to calculate the derivative of the quantity
$\bar{n}\coloneqq 1/n^{2}$ with respect to $\tau$ at $\tau=0$, that is, at the
points satisfying $U=-V$, for $0<V<L$:

(28)
$\frac{\partial\bar{n}}{\partial\tau}(-V,V)=\frac{\partial\bar{n}}{\partial
U}(-V,V)+\frac{\partial\bar{n}}{\partial V}(-V,V)\,,\qquad 0<V<L\,.$

Using (11), we obtain:

$\displaystyle\frac{\partial\bar{n}}{\partial\tau}(U,V)=$
$\displaystyle\,\,\frac{1}{2}F^{\prime}\left(r\left(\frac{g(V)-f(U)}{2}\right)\right)r^{\prime}\left(\frac{g(V)-f(U)}{2}\right)\left(f^{\prime}(U)\left(g^{\prime}(V)\right)^{2}-\left(f^{\prime}(U)\right)^{2}g^{\prime}(V)\right)$
(29)
$\displaystyle+F\left(r\left(\frac{g(V)-f(U)}{2}\right)\right)\left(f^{\prime\prime}(U)g^{\prime}(V)+f^{\prime}(U)g^{\prime\prime}(V)\right)\,.$

Equation (13) implies that this derivative vanishes when evaluated at points
of the form $(-V,V)$, with $0<V<L$. Therefore, we proceed to calculate the
second derivative of $\bar{n}$ with respect to $\tau$ at $\tau=0$. By making
use of equation (13) to relate the derivatives of $f$ and $g$, we obtain:

$\displaystyle\frac{\partial^{2}\bar{n}}{\partial\tau^{2}}(-V,V)=$
$\displaystyle\,\,F^{\prime}(r(g(V)))r^{\prime}(g(V))\left(g^{\prime}(V)\right)^{2}g^{\prime\prime}(V)-2F(r(g(V)))\left(g^{\prime\prime}(V)\right)^{2}$
(30) $\displaystyle+2F(r(g(V)))g^{\prime}(V)g^{\prime\prime\prime}(V)\,.$

We can calculate the derivatives of the function $g$ from equation (18). After
a lengthy computation, we arrive at

(31)
$\frac{\partial^{2}\bar{n}}{\partial\tau^{2}}(-V,V)=\frac{2}{\left(r(g(V))\right)^{3}}\,.$

Therefore, the second derivative of $\bar{n}$ is positive, indicating that the
rod starts out by stretching. This can be understood as the effect of the
tidal forces produced by the black hole.

###  4.2. Gliding motion in the Schwarzschild spacetime

Finally, we will study the motion of a rod whose initial configuration is
given by

(32) $n(-V,V)=n_{1}(1-H(V,\varepsilon))+n_{2}H(V,\varepsilon)\,,\qquad
0<V<L\,,$

where $n_{1}$ and $n_{2}$ are two positive constants and $H$ is the function

(33)
$H(V,\varepsilon)=\frac{1}{2}+\frac{1}{2}\tanh\left(\frac{V-\frac{L}{2}}{\varepsilon}\right)\,.$

Note that if we choose $\varepsilon>0$ sufficiently small then $H$ is a smooth
approximation of a Heaviside step function that jumps from $0$ to $1$ at
$V=L/2$, the midpoint of the rod. In our simulation we considered
$\varepsilon=10^{-3}$. With this definition, the initial configuration
corresponds to a rod whose compression factor is $n_{1}$ on the left half of
the rod and $n_{2}$ on the right half. For instance, by choosing the
parameters $n_{1}=0.03$ and $n_{2}=6$ we obtain the result depicted in Figure
3.

(a)

(b)

Figure 3. Worldlines of (a) the midpoint, and (b) the endpoints of the
initially compressed free-falling rod in the Schwarzschild spacetime, with
initial parameters $n_{1}=0.03$ and $n_{2}=6$. The worldlines of points in the
rod are the blue curves, while the geodesics corresponding to particles
starting at rest from the same points are the orange dashed curves.

(a)

(b)

Figure 4. Worldlines of (a) the midpoint, and (b) the endpoints of the
initially compressed free-falling rod in the Schwarzschild spacetime, with
initial parameters $n_{1}=0.9$ and $n_{2}=0.03$. The worldlines of points in
the rod are the blue curves, while the geodesics corresponding to particles
starting at rest from the same points are the orange dashed curves.

For this choice of parameters, we can see that the rod’s midpoint falls
significantly faster than a free-falling particle starting at rest from the
same point. On the other hand, if we choose the parameters appropriately then
we can set up an initial configuration such that the midpoint of the rod falls
much slower than the particle. For example, the case with $n_{1}=0.9$ and
$n_{2}=0.03$ is shown in Figure 4, where we can clearly see that the rod’s
midpoint falls slower than the free-falling particle starting at rest from the
same initial position: not only does it not cross the geodesic, but the
distance between the midpoint and the particle after each complete oscillation
increases over time.

In order to check energy conservation and other energy considerations, the
quantities $E$ and $K$ are presented as a function of $\tau$ in Figure 5 for
the two cases above (the integrals were computed using Simpson’s rule). We can
see that in both cases the total energy of the rod, $E$, is constant,
corroborating the soundness of the numerical integration of the equations of
motion. The results depicted in Figure 5 also show that there is a periodic
conversion of elastic energy to mechanical energy and vice-versa. However,
there does not seem to be a net transfer from one form of energy to the other
on average; the faster or slower falling motion of the rod when compared to
geodesics cannot be attributed to an accumulation or depletion of elastic
energy.

(a)

(b)

Figure 5. Quantities $E$ (blue curve) and $K$ (orange curve) as a function of
$\tau$ for (a) $n_{1}=0.03$ and $n_{2}=6$, and (b) $n_{1}=0.9$ and
$n_{2}=0.03$.

##  5\. Conclusion

In this paper we considered, for the first time, the free-falling motion of an
extended elastic body in the Schwarzschild gravitational field. For an
initially relaxed rod, we saw that the motion of the midpoint is very close to
pure geodesic motion, although the endpoints do deviate from the corresponding
geodesics (as expected for an elastic body). What is interesting about the
other two examples we presented, however, is the fact that the speed of the
rod’s midpoint can be significantly affected by the initial compression, when
compared to pure geodesic motion.

In all these examples, only the initial compression profile can be chosen.
Naturally, an interesting question would be to study the possibility of
controlling the motion by some internal mechanism that would allow for
changing the rod’s shape as it falls. This would necessitate a more
complicated mathematical model, in which energy flow along the rod would have
to be included.

This $(1+1)$-dimensional formalism can only be used to study radial fall. A
more complete description of the free-falling motion of a relativistic elastic
body on the Schwarszchild field would require the full $(1+3)$-dimensional
theory of relativistic elasticity, whose implementation can be considerably
more difficult.

## Acknowledgements

This work was partially supported by FCT/Portugal through CAMGSD, IST-ID
(projects UIDB/04459/2020 and UIDP/04459/2020) and project NoDES (PTDC/ MAT-
PUR/1788/2020), and also by the H2020-MSCA-2022-SE project EinsteinWaves, GA
no. 101131233. LM gratefully acknowledges the Calouste Gulbenkian Foundation
for the scholarship program _Novos Talentos em Matemática_.

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