# Against Scale: Provocations and Resistances to Scale Thinking

Alex Hanna<EMAIL_ADDRESS>Google Research  0000-0002-8957-0813 and
Tina M. Park<EMAIL_ADDRESS>Independent Researcher

††copyright: rightsretained††conference: CSCW ’20: Reconsidering Scale and
Scaling in CSCW Research; October 17–18, 2020; Virtual

##  1\. Introduction

At the heart of what drives the bulk of innovation and activity in Silicon
Valley and elsewhere is scalability. Scalability is that most-important,
desirable attribute that “connotes the ability of a system to accommodate an
increasing number of elements or objects, to process growing volumes of work
gracefully, and/or be susceptible to enlargement” (])p. 195]Bondi2000. It
means that a small start-up is capable of growing into a multinational
corporation, serving a global audience in a short period of time. This
unwavering commitment to scalability - to identify strategies for efficient
growth - is at the heart of what we refer to as “scale thinking.” Whether
people are aware of it or not, scale thinking is all-encompassing. It is not
just an attribute of one’s product, service, or company, but frames how one
thinks about the world (what constitutes it and how it can be observed and
measured), its problems (what is a problem worth solving versus not), and the
possible technological fixes for those problems.

We argue that scale thinking represents a very particular form of growth, a
form that has become the default and predominant approach within the current
technology sector. Scale thinking presumes that everything can be made more
efficient – that products and services can be supplied and consumed at faster
and faster speeds, for more and more people. Such efficiency is possible when
production of the good or service can be expanded without significantly
modifying the inputs (e.g., the type of raw material used, the type of labor
required) or infrastructure necessary. Such growth is indisputably a good
thing. For developers and investors, it means capturing greater market shares
with lower levels of investment. Even those committed to social good over
financial gain tend to believe that the efficient use of resources and the
rapid expansion of potential solutions would help deal with the world’s ills
faster and more effectively – they are able to do more good with fewer
resources.

This paper examines different facets of scale thinking and its implication on
how we view technology and collaborative work. We argue that technological
solutions grounded in scale thinking are unlikely to be as liberatory or
effective at deep, systemic change as their purveyors imagine. As we will
discuss, the logics that drive scale thinking are antithetical to constructing
solutions which can produce systemic, equity-driven social change. Rather,
solutions which resist scale thinking are necessary to undo the social
structures which lie at the heart of social inequality. We draw on recent work
on mutual aid networks and propose questions to ask of collaborative work
systems as a means to evaluate technological solutions and identify
resistances to scale thinking.

##  2\. Prior Literature on Scale & What is Scale Thinking?

“Scale thinking” is an approach that centers on and prioritizes
scalability111Our intervention borrows heavily from Lilly Irani’s (Irani,
2018) critique of ”design thinking,” which characterizes the concept as a type
of racialized Western expertise that resists the imaginary mechanistic
reasoning of a looming Asian labor threat. It also borrows from Ruha
Benjamin’s critique of design as a ”colonizing project” which ”submerg[es] so
much heterogeneity” under its name (Benjamin, 2019, pp. 175-76).. Scalability
refers to the ability of a system to expand without having to change itself in
substantive ways or rethinking its constitutive elements (Tsing, 2012).
Scalability goes hand-in-hand with other macro-level processes of making
modernity legible, including standardization (Timmermans and Epstein, 2010),
classification (Bowker and Star, 2000), and colonial trade (Tsing, 2012).
Investors seek technological start-ups which are designed to grow quickly –
businesses that are structured to meet any level of demand at moment’s notice.
In an essay titled ”Startup = Growth,” Y Combinator founder Paul Graham
discusses how the distinguishing feature of startups is its ability to grow,
stating ”not every newly founded company is a startup” and most of those are
in service. The startup is an instantiation of scale thinking which requires
immediate growth. ”A barbershop doesn’t scale,” he quips. In scale thinking,
this feature of any given system is assumed to be the most relevant for
success and longevity. As Werner Vogels, chief technology officer of Amazon,
wrote, “scalability cannot be an after-thought. It requires applications and
platforms to be designed with scaling in mind…” (O’Hanlon, 2006).

By centering scalability in the design and development process, scale thinking
revolves around three key tenets: (1) Scalability is a morally good quality of
a system; (2) Quantification is a necessary part of designing scalable
systems; and (3) Scalability is achieved by identifying and manipulating
quantifiable, core elements or attributes of a system.

First, in scale thinking, scalability is taken from being a desirable trait to
a morally good trait. Insofar as capitalism values the maximization of
production and the minimization of waste – whether it is unused labor or raw
materials and goods – scalability is imbued with a moral goodness because it
centers on designing a system that is able to serve a greater number of people
with fewer resources over time. When a system is scalable, growth – which
itself is fundamentally a positive and good thing – can take place without the
loss of time, resources, or effort. The absence of scalability is equated with
wastefulness: opportunities to meet the needs of an ever-growing body of
people are lost and precious resources (money, time, raw materials) are
utilized ineffectively. Especially at a time when natural resources are
dwindling, while populations are growing, wastefulness is marked as evil and
morally corrupt. It becomes taken-for-granted that to be a “good” company or
person, one should obviously seek scalability.

Moreover, scalability is a prerequisite to technical implementation;
accordingly, solutions which don’t scale are morally abject. Large tech firms
spend much of their time hiring developers who can envision solutions which
can be implemented algorithmically. Code and algorithms which scale poorly are
seen as undesirable and inefficient. Many of the most groundbreaking
infrastructural developments in big tech have been those which increase
scalability, such as Google File System (and subsequently the MapReduce
computing schema) and distributed and federated machine learning models.
Large-scale technical systems operate much like the markets that preceded
them, and are considered by liberals to be in and of themselves a moralizing
and virtuous force (Fourcade and Healy, 2007).

Given the importance of growth to scalability, the measurement of growth and
efficiency is paramount to measuring the worth and value of the scalable
system. The greater the growth, the more efficient the use of inputs and
resources, and the greater the capture of demand, the better and more valuable
the system. In scale thinking, developing tactics for measuring such growth is
as important as designing for scalability itself (because if no one sees the
growth, did it really happen?). Quantification, or the production and act of
transforming human observations and experiences into quantities based on a
common metric (Espeland and Stevens, 2008) is an important procedural aspect
of scalability.

Scalability is achieved when a system is able to expand without rethinking
basic elements (Tsing, 2012). The system is designed in such a way that can
accommodate new inputs without changing its fundamental framework. To create
such a system, thus, requires an understanding of what those core elements or
attributes of the system are – of what aspects can remain unchanged and what
elements can be changed to accommodate a growing system. Improving efficiency
also means eliminating excess and removing non-essential elements to ensure
the system runs smoothly. To engage in scale thinking, then, is to try and
reduce a complex process or interaction to its most elemental and simplistic
exchange of input to output.

##  3\. Implications of scale thinking

The implications of scale thinking are three fold. In the first, scale
thinking requires units of work to be interchangeable, abstract, and
universal. Units themselves can be servers, offices, databases, individual
workers. Next, scale thinking requires users to be of the same kind, or within
a narrow set of predefined bounds, which is most often to the detriment of
people ”at the margins,” marked by systems of racism, transphobia, and
ableism. Lastly, scale thinking has implications for the legibility of users,
with a natural endpoint in data gathering and datafication of individuals.

Scale thinking demands a standardization of inputs and outputs. Global supply
chains demand modularity: shipping containers require only a pickup point and
destination (Posner, 2018). As The Atlantic contributor Alexis Madrigal has
suggested, containers are the ”embodiment…of global capitalism.” (Madrigal,
2017). What’s significant is how readily that embodiment has translated to
technological work.

The container metaphor readily extends to the realm of software development
and deployment. Docker, the first widely used ”container” service, allows
developers to develop ”standard unit[s] of software that package up code” and
run them in different environments. Docker’s logo is a whale with a set of
shipping containers set on top of it. Kubernetes, likewise, is a software
package used to manage ”deployment, scaling, and management of containerized
applications.” Accordingly, its logo is a ship’s wheel, steering containers to
where they need to go.

Figure 1. The Docker and Kubernetes logos, respectively.

The ubiquity of the container metaphor highlights the embeddedness of scale
thinking in modern computing and infrastructural development; the metaphor
extends to how developers think about work units, software teams, and
technological organizations. If modern software companies are supposed to be
”flat,” then the work teams within them operate as standardized containers
that take as input design and requirement documents, and output code,
processes, and product. These organizations scale with the addition of more
containerized work teams. At the base unit of this operation, the individual
tech worker needs to have such standardized inputs and outputs. As such,
scaling has a fractal quality within tech development.

While scale thinking permeates the organization of the production side of tech
development, it sharply determines its consumptive side as well. Scale
thinking requires a sameness of user, or a user which falls within a tightly
bound constraint of imagination. In a containerized world, interchangeability
is critical to the operation working, and users operate in a standardized
manner. The desire of the startup is to ensure that users fall within the
bounds of the universal. Heterogeneity becomes antithetical to scalability,
because the same product/service can no longer be duplicated to sufficiently
serve a suffuse audience. A varied user base means that many different
solutions are needed, rather than a scalable solution. Despite Graham’s cry to
”do things that don’t scale,” (Graham, 2012) a startup’s outputs need to be
constrained insofar as they do just that.

The failures for users that fall outside of universality of scaled solutions
abound. Media scholar Safiya Noble accounts in stark detail the racist and
sexist failures of web search for queries for Black, Latina, and Asian women
and girls (Noble, 2018). In the book’s conclusion, she details how Kandis, the
owner of a Black-serving salon in a college town of a predominantly white
institution, had to contend with Yelp and the erasure of her business: ”I
quickly realized that Internet/Yelp told people that I did not exist” (])p.
175]Noble2018. Kandis goes on to detail the extensive hoops she has to go
through to get listed on the service, and the hesitance of her primarily Black
clientele to ”check-in” using Yelp’s functionality, because of their sense of
already being over surveilled.

Datafication, then, becomes the endpoint of scale thinking. Much has been made
of the impulse of datafication of the individual (Van Dijck, 2014; Cheney-
Lippold, 2017) and its anti-Black, carceral dimensions (Browne, 2015;
Benjamin, 2019). Datafication’s move – and therefore scale thinking’s move –
is to find ways to rationalize the individual into legible data points. A
single data point is rarely useful on its own – data points only matter
insofar as they accumulate and move through the world as a new form of capital
(Sadowski, 2019). This can only be made possible via creation of systems
undergirded by scale thinking, and the building of systems which operate as
massive accrual machines.

##  4\. Resistances to Scale Thinking

In April 2020, at the height of the Coronavirus pandemic in New York City,
George Farcasiu, the ”lead automation engineer” of the mutual aid network Bed-
Stuy Strong told VICE’s technology-focused publication Motherboard that “We’re
making sure we’re building tools that are about organizing people to interact
with neighbors, not treating volunteers as boxes that do work.” He continued,
“We’re not looking for a system that scales. We’re looking at the social
system we have and augmenting and enabling that” (Rose, 2020).

On the flip side of this, tech companies continue to struggle with diversity
and inclusion among their employees, despite years of investment and highly
publicized efforts. One early strategy adopted by many tech companies was the
installation of “people analytics” teams with human resources divisions to
provide “data-driven,” scalable solutions (Bersin, 2017). Efforts focused on
scalable tactics – focusing on hiring percentages and inclusion metrics,
implementing bias workshops, developing employee resource groups – with little
to no impact on the composition of the workforce (Rooney and Khorram, 2020) or
the experiences of marginalized people within the company (Ozoma and Tiku,
2020).

These two examples illustrate two opposite, but active resistances to scale
thinking. In the former, the resistance is centered on the hopeful and local
in mutual aid networks. In the latter, however, we find resistance anchored in
the pessimistic and anti-social. To conclude this piece, we draw on recent
writings on mutual aid to discuss the ways in which scale limits – and
actively inhibits – participation in tech and society. Indeed, scale thinking
forces particular types of participation to operate as extractive or
exploitative labor (Sloane et al., 2020). We ask readers to consider what
potential resistances to scale thinking may look like, and invite them to
think through what kinds of technology encourage collaborative work in ways
which don’t replicate its logics.

One such framework is that of mutual aid. Mutual aid allows a possible way to
think of collaborative work more fruitfully. Critical legal scholar Dean Spade
defines mutual aid as:

> a form of political participation in which people take responsibility for
> caring for one another and changing political conditions, not just through
> symbolic acts or putting pressure on their representatives in government but
> by actually building new social relations that are more survivable. (Spade,
> 2020, p. 136)

The concept of mutual aid has been around for years (many trace the concept to
the anarchist theorist Peter Kropotkin) and the practice for much longer. It
is a particular tradition of relationship building and has many local
instantiations, including Black mutual aid traditions in the diaspora and the
Black Panthers breakfast program in Oakland to name a few (Harris, 1979;
Heynen, 2009; Mochama and Blackpowerbarbie, [n.d.]). Unlike other social
assistance programs, mutual aid is oriented around collective problem-solving
to develop strategies and obtain resources to help a community of people meet
one another’s needs while simultaneously organizing towards different social
and political relations, both among community members and between the
community and other structures of power. It is the latter that distinguishes
mutual aid from forms of charity or assistance: assistance provided between
group members is not only about sharing the resources on hand, but changing
what resources are made available and how they are distributed in general; it
is about shifting dynamics of power.

For example, for the Black Panther Party, the Breakfast Program was an
important mutual aid program that was integral to their overall political
strategy to transform U.S. politics. In addition to providing necessary
support and aid for the communities they served, it also functioned as a model
that the federal government could use to establish a national school breakfast
program (Heynen, 2009). The urgency of the COVID pandemic has revived interest
and networks around mutual aid. In the US, with the lack of federal and state
intervention resulting in mass infections and death, mutual aid groups have
emerged as a means to provide material relief for vulnerable populations,
including Black and brown, elderly, low-income, disabled, and queer and
transgender people.

Mutual aid networks, by their nature, are not intended to “scale”. While scale
thinking emphasizes abstraction and modularity, mutual aid networks encourage
concretization and connection. Mutual aid is intended to operate as a mode of
radical collective care (Spade, 2020, p. 131) in which individuals in the
network have their direct material needs met, regardless of considerations of
those receiving aid falling into a set of datatified categories, such as
”deserving or undeserving.” While scale thinking encourages top-down
coordination, mutual aid considers building skills for ”collaboration,
participation, and decision making” (Spade, 2020, p. 137). Mutual aid focuses
on ”how to organize human activity without coercion” (Spade, 2020). And mutual
aid encourages building of solidarities between people with different needs.
While mutual aid is not the only framework through which we can consider a
move away from scale thinking-based collaborative work arrangements, we find
it to be a fruitful one to theorize and pursue.

The Bed-Stuy neighborhood in New York City’s Brooklyn borough reflects
gentrification trends. It is a mix of newly arrived residents who are white,
more financially-secure, technologically-oriented, both professionally and
personally, and long-time residents who are Black or Latinx, older, and more
likely to be negatively harmed by COVID-19 and its related economic impacts
(of the State Deputy Comptroller for the City of New York, 2017).222It is also
a neighborhood with a rich history of localized Black activism that opposed
its designation as the “largest ghetto” (Woodsworth, 2016). And, given the
disparate impacts of COVID-19 on Black and Brown communities and lower-income
communities, the desire to make the aid distribution system nimble enough to
grow quickly and across more neighborhoods, if not entire cities and regions,
is a well-intentioned one. Like many other “tech for social good” projects, it
is not unreasonable to fixate on the end goal of the project - in this case,
distributing money and food to as many people as possible - in the hopes of
assisting the most number of people.

Yet, the developers who volunteered to build out the tech-enabled aspect of
the work resisted building a system focused on scalability, explicitly
focusing on developing a system that centers both the recipients of the aid
and the volunteers who distribute the aid. Instead, they looked to using
technology to augment the relationships formed between volunteers and
recipients. As noted by Alyssa Dizon, a civic technologist and a volunteer
with Bed-Stuy Strong:

> The most important piece of our operation is our relationship to the
> neighborhood, our sensitivity to the community, our approach, our values,
> our culture as an organization. The tech is a layer of it, and it’s not the
> most important layer of it at all.” (Mckenzie, 2020)

The priority for developing the technological infrastructure for Bed-Stuy
Strong is focused less on building a system that can expand quickly to serve
the most people possible. Instead, it is focused on easing and strengthening
social connections between the people involved in doing the mutual aid: the
volunteers, the donors, and the recipients. The technology layer is built with
the understanding that those involved, rather than being easily replaceable
units, are not homogenous. Organizers presume that the people and their
relationships with one another will change, including their roles within the
mutual aid network. A volunteer may be a donor one week, and due to changes in
circumstances, may become a recipient the next as well. People who receive aid
may also volunteer their time to provide aid to others.

Furthermore, there is an expectation that the users of the mutual aid system
are different and have different needs. Rather than building for a homogenous
base of users or clients, the tech-oriented developers within Bed-Stuy Strong
oriented themselves towards identifying technology that could help them
identify differences between users, such as those who are elderly, immuno-
compromised, living with children, or in need of immediate assistance. By
doing so, they can render assistance in a way that is most functional and
appropriate for the recipient. Rather than trying to identify users to who
fit, or could be made to fit, into the system, they adapted the work of
volunteers and donors to meet the needs of a vast array of recipients.

We conclude this section with a few considerations around how to organize
collaborative work in technological systems. Spade proposes four questions in
assessing reforms and tactics which we use as our jump-off point:

> Does [the reform or tactic] provide material relief? Does it leave out an
> especially marginalized part of the affected group (e.g., people with
> criminal records, people without immigration status)? Does it legitimize or
> expand a system we are trying to dismantle? Does it mobilize people,
> especially those most directly impacted, for ongoing struggle? (Spade, 2020,
> p. 133)

We add three more, related questions for technical collaborative work systems:
Does the technological system centralize power (either through coordination,
data extraction, or authority) or distribute it between developers and users?
Does the technological system treat the contributions and experiences of
individuals as interchangeable or as uniquely essential? Lastly, does it open
up avenues for participation, and are those avenues of participation
mobilizing or demobilizing?

##  5\. Conclusion

In this piece, we identify ”scale thinking” as a mode of thinking prevalent in
Silicon Valley and other technological development centers. We discuss the
dimensions and implications of scale thinking, and how this type of thinking
is universally valorized within technological development. We then introduce
resistances to scale thinking, namely mutual aid, as an alternative mode of
arranging social relations, and offer several provocations and questions for
designers and theorists of cooperative and collaborative work systems.

Pandemics and disasters are sites of immense loss of life, property, and
livelihood. They also starkly highlight what elements, at their root, are not
working for most people and contribute to interlocking systems of inequality.
Failures under disasters are often failures of technological scale and the
attendant scale thinking which undergirds them. We hope that this set of
provocations introduces a mode of evaluating whether systems created by
designers and other tech workers are working for those who need them most.

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# Electromagnetic Field Exposure Avoidance thanks to Non-Intended User
Equipment and RIS

Hao Guo1, Dinh-Thuy Phan-Huy2, and Tommy Svensson1  1Chalmers University of
Technology, 41296 Gothenburg, Sweden, {hao.guo<EMAIL_ADDRESS>2Orange Innovation/Networks, 92326 Chatillon, France,
<EMAIL_ADDRESS>
###### Abstract

5G and beyond systems call for improved service qualities. To this end,
advanced communication technologies such as massive multiple-input multiple-
output (MIMO) and mm-wave beamforming have been developed in recent years to
use radio resources more effectively. However, efficient beamforming
potentially also increases the electromagnetic field exposure (EMFE) for
certain users. Thus, EMFE is currently gaining increased attention from both
regulatory authorities and the public and should be part of the design
criteria for evolved 5G and 6G mobile communications. Reconfigurable
Intelligent Surface (RIS) has emerged as a promising future technical
candidate as it can program the radio environment and obtain the desired
propagation paths, which potentially could also help to control EMFE. To this
end, recent research studies have addressed the beamforming design at both the
base station (BS) and the RIS with limited beam power requirements as EMF
constraints. In this work, treating the EMF requirement as an opportunity
instead of just a limitation, we consider different cases where the non-
intended user (NIU), either in between the BS and the intended user (IU) or
alongside the IU, could help the network design. Specifically, we investigate
the potential of EMF-constrained RIS-assisted networks with power allocation
between the direct BS-IU and the in-direct BS-RIS-IU paths and efficient
beamforming methods. As we presented in the analytical and numerical results,
proper power allocation and beam optimization can boost the performance of the
system remarkably with limited complexity and channel information.

###### Index Terms:

Reconfigurable intelligent surface (RIS), massive multiple-input-multiple-
output (M-MIMO), electromagnetic field exposure (EMFE), beamforming,
millimeter wave (mmWave), power allocation.

##  I Introduction

The 5th Generation (5G) and most likely coming 6th Generation (6G) use massive
multiple input multiple output (MIMO) beamforming and millimeter wave (mmWave)
[1] to boost the spectral efficiency and the energy efficiency of networks.
Beyond 5G or 6G networks are expected to keep on using such technologies.
However, in some countries or some cities, the electromagnetic field exposure
(EMFE) limits set by the local regulation can reach a very low level,
sometimes ten times lower than the levels recommended by International
Commission on non-ionizing radiation protection (ICNIRP) [2]. Such arbitrarily
low levels slow down the deployment of new networks, as it has been observed
as early as with 4th Generation (4G) [3]. For these reasons, in [4, 5], it has
been proposed to take the opportunity when designing the new generation of
networks, the future 6G, to offer a new type of service: arbitrarily low EMFE
as a service.

To do so, [4, 5] proposes to use reconfigurable intelligent surfaces, to shape
the radio propagation environment [6, 7]. RIS are new network nodes reflecting
waves in the desired direction and can be seen as intelligent mirrors or
passive relays [8]. In [9, 10, 11, 12], RIS-aided schemes have been proposed
to reduce the self-EMFE of a customer to his/her own user equipment (UE)
transmissions in the uplink (UL) direction. In [13, 14, 15, 16], RIS-aided
downlink (DL) beamforming schemes have been proposed to confine the over-
exposed area, due to the transmission of the base station (BS) in the DL
direction, inside a predefined circle. To our best knowledge, up to now, no
study has been performed about the full potential of an EMFE-constrained RIS-
assisted system.

In this paper, for the first time, we introduce the notation of non-intended
user (NIU). An NIU is a customer of a mobile network operator. When the NIU is
not using his/her smartphone for data transmission, it agrees to let the
network use some control signaling and pilot exchanges between his/her
smartphone and the network. In this case, the smartphone is not considered by
the network as usual UE to serve with the best data rate possible to an
intended user (IU). Instead, the smartphone is considered by the network as a
non-intended user equipment (NIUE). The NIUE has no data needs to be connected
to the network, but being connected it has the opportunity to impose an EMFE
constraint at its location. By being connected it could also help the network
to decide how much power could still be transmitted in its path with a
limitation, while at the same time, preventing itself from unnecessary radio
exposures.

In order to realize this purpose, we propose different methods to maximize the
throughput of an IU and at the same time meet an EMFE requirement (i.e. a
threshold) at an NIU side (this EMFE being caused by the DL transmission
between a BS and an IU), thanks to the help of the NIUE, the UE, and an RIS.
Our contributions can be summarized as follows:

  * •

Two propagation scenarios are investigated: 1) the NIUE is located on the
direct propagation path between the BS and the IU, and 2) the NIUE is located
out of this path alongside the IU.

  * •

We first formulate a joint beamforming and transmit power optimization (JBPO)
problem at the BS side, with the objective to provide the best data rate at
the IU under the NIU EMFE constraint.

  * •

We then propose two schemes to solve the JBPO problem with different
performance and different assumptions on channel state information (CSI)
availability: alternating optimization (AO)-, and discrete Fourier transform
(DFT)-based optimization. They are later combined with five power allocation
methods between the direct path and the RIS-assisted path. Finally, we provide
analytical insights on 1) performance upper bound with transmitting different
codewords as well as 2) a simple power filling rule with priority on the
direct-link transmission.

The remainder of the paper is organized as follows. Section II introduces the
system model and the problem formulation of JBPO. In Section II, the AO\- and
DFT-based optimization schemes are proposed and analysed. Section IV presents
simulation results to validate the performance of the proposed methods while
Section V concludes the paper.

##  II System Model

Consider an RIS-assisted MIMO DL system with one BS deployed with
$N_{\text{T}}$ transmit antennas, one RIS with $N$ elements, and a pair of
single-antenna IU and NIU. Here, IU is the user who wants to receive service
from the BS while the NIU does not want to connect to the BS. More
importantly, NIU prefers to have limited EMFE given that it requires no
communication service. We assume that the NIU does not decode the message or
absorb the energy from the BS and acts as one probe in space to detect the
signal strength. Different from recent studies, e.g., [13, 11], as shown in
Figs. 1-2, two cases are considered in this work:

  * •

Case 1: The NIU moves between the BS and the IU. Depending on the location of
the NIU and the beam width from the BS, at some points/areas, the NIU and IU
could receive the same transmission beam from the BS.

  * •

Case 2: The NIU moves alongside the IU and it could receive a different
beam/side beam from the BS.

Figure 1: EMFE-constrained RIS system, Case 1: The NIU moves between the BS
and IU and it receives the same beam as the IU.
Figure 2: EMFE-constrained RIS system, Case 2: The NIU is alongside the IU and
it receives a different beam/side beam compared to the IU.

In this way, the received message for the IU at time $t$ can be written as

$\displaystyle y(t)=\sqrt{P}\bm{h}(t)\bm{x}(t)+n(t),$ (1)

where $P$ is the transmit power at the BS and $\bm{x}$ is the transmitted
message with a unit power. $n$ represents the additive Gaussian noise at the
receiver side. To simplify the presentation, we omit $t$ in the following. The
channel $\bm{h}\in\mathcal{C}^{1\times N_{\text{T}}}$ between BS-IU includes
both the direct link (BS-IU) as well as the RIS-assisted link (BS-RIS-IU),
i.e.,

$\displaystyle\bm{h}=\bm{h}_{\text{D}}+\bm{h}_{\text{RI}}\bm{\Theta}\bm{h}_{\text{BR}}.$
(2)

Here, $\bm{h}_{\text{D}}\in\mathcal{C}^{1\times N_{\text{T}}}$ is the direct
link, while $\bm{h}_{\text{BR}}\in\mathcal{C}^{N\times N_{\text{T}}}$ and
$\bm{h}_{\text{RI}}\in\mathcal{C}^{1\times N}$ are the channel between BS-RIS
and RIS-IU, respectively. We assume that $\bm{h}_{\text{D}}$ is independent of
$\bm{h}_{\text{BS}}$ and $\bm{h}_{\text{RI}}$. Moreover,

$\displaystyle\bm{\Theta}=\text{diag}(e^{j\phi_{1}},...,e^{j\phi_{N}})$ (3)

is the reflection coefficient matrix of the RIS. With BS-active and RIS-
passive beamforming, the data rate at the IU can be expressed as

$\displaystyle
R=B\log_{2}\left(1+\frac{P_{\text{D}}\left|\bm{h}_{\text{D}}\bm{w}_{\text{D}}\right|^{2}+P_{\text{R}}\left|\bm{h}_{\text{RI}}\bm{\Theta}\bm{h}_{\text{BR}}\bm{w}_{\text{R}}\right|^{2}}{BN_{0}}\right),$
(4)

where $P_{\text{D}}$, $P_{\text{R}}$,
$\bm{w}_{\text{D}}\in\mathcal{C}^{N_{\text{T}}\times 1}$,
$\bm{w}_{\text{R}}\in\mathcal{C}^{N_{\text{T}}\times 1}$ are the power of the
direct link, the power of the RIS link, the precoder of the direct link, and
the precoder of the RIS link, respectively. Both $\bm{w}_{\text{D}}$ and
$\bm{w}_{\text{R}}$ have unit power. Also, $B$ and $N_{0}$ represent the
system bandwidth and the noise power spectral density, respectively.

Assuming the NIU occupies a unit area, the EMFE constraint in terms of
received power can be written as [17]

$\displaystyle P_{\text{N, Rx}}=\frac{4\pi}{\lambda}P_{\text{N,
Tx}}\left|\bm{h}_{\text{N}}\bm{w}_{\text{N}}\right|^{2},$ (5)

with $P_{\text{N, Tx}}$ ($P_{\text{N, Rx}}$) being the transmit (receive)
power of the NIU, and $\bm{h}_{\text{N}}$ and $\bm{w}_{\text{N}}$ are the
channel and the precoder of BS-NIU link, respectively. Both direct and
reflected paths can be considered using such a model.

##  III EMFE-aware Joint Beamforming and Transmit Power Optimization

The target of this work is to jointly maximize the rate at the IU with proper
beamforming design ($\bm{w}_{\text{D}}$, $\bm{w}_{\text{R}}$, $\bm{\Theta}$)
and power allocation ($P_{\text{D}}$, $P_{\text{R}}$), in the presence of the
NIU with an EMFE constraint. It can be mathematically stated as

$\displaystyle\max_{P_{\text{D}},P_{\text{R}},\bm{w}_{\text{D}},\bm{w}_{\text{R}},\bm{\Theta}}$
$\displaystyle R$ (6) s.t. $\displaystyle P_{\text{N, Rx}}\leq\bar{P},$
$\displaystyle 0\leq P_{\text{D}}\leq P,0\leq P_{\text{R}}\leq P,$
$\displaystyle P_{\text{D}}+P_{\text{R}}=P,$
$\displaystyle\|\bm{w}_{\text{D}}\|^{2}=1,\|\bm{w}_{\text{R}}\|^{2}=1,$
$\displaystyle\phi_{n}\in[0,2\pi],n=1,...,N,\forall{n}.$

Here, $\bar{P}$ is a constant which represents the EMFE constraint in terms of
the received power at the NIU and can be modeled by, e.g., (5) as a function
of $\bm{h}_{\text{N}}$. Since the beamforming optimization is independent of
the transmit power, in the following, we first jointly optimize the beam of
the BS and the RIS using two schemes with different CSI requirements. Then, we
propose different methods to allocate the transmit power at the BS considering
the EMFE requirement of NIU.

###  III-A Joint Beam Optimization for the BS and RIS

With the assumption of independent paths of the direct link and the RIS-
assisted link, the beamforming pattern of the direct link can be obtained by,
e.g., maximum-ratio transmission (MRT), as
$\bm{w}_{\text{D}}=\bm{h}_{\text{D}}^{H}/\left\|\bm{h}_{\text{D}}\right\|$,
with $(\cdot)^{H}$ representing the Hermitian transpose.

For the RIS-aided indirect link, we need to jointly optimize the active
beamforming vector $\bm{w}_{\text{R}}$ at the BS, as well as the passive
beamforming matrix $\bm{\Theta}$ at the RIS. This problem has been widely
studied in the literature under different setups/assumptions. For instance,
with perfect knowledge of CSI, i.e., $\bm{h}_{\text{BR}}$ and
$\bm{h}_{\text{RI}}$, AO has been shown to approach the optimal solution of
(6) [18, Sec. III-B] [19, Sec. III][20, Algorithm I]. In this work, we use AO
in EMFE-aware RIS systems as an upper bound to compare to, as presented in
Algorithm. 1. Here, maximizing $R$ is equivalent to optimize
$|\bm{h}_{\text{RI}}\bm{\Theta}\bm{h}_{\text{BR}}\bm{w}_{\text{R}}|$. Using
AO, for a fixed $\bm{w}_{\text{R}}$, the optimal $\phi_{n}$ resulting in
$-\angle\left\\{\bm{h}_{\text{RI}}^{*}(n)\bm{h}_{\text{w}}(n),\forall{n}\right\\}$
with $\bm{h}_{\text{w}}=\bm{h}_{\text{BR}}\bm{w}_{\text{R}}$. Then, for fixed
$\Theta$, the optimal $\bm{w}_{\text{R}}$ can be obtained by calculating the
dominant right eigenvector of
$\bm{g}=\bm{h}_{\text{RI}}\bm{\Theta}\bm{h}_{\text{BR}}$. Once the optimal
beams of the BS and RIS are settled, various power allocation methods can be
applied to reach the maximum rate (4), which are presented in Sec. III-B.

Algorithm 1 Joint beam optimization using AO in EMFE-aware RIS systems

0: $\bm{h}_{\text{BR}}$, $\bm{h}_{\text{RI}}$, $\bm{h}_{\text{N}}$, and
$\bar{P}$1\. Initialize $\bm{w}_{\text{R}}$ to some feasible values at the BS.

repeat

2\. Calculate $\bm{h}_{\text{w}}=\bm{h}_{\text{BR}}\bm{w}_{\text{R}}$;

3\. Set
$\phi_{n}=-\angle\left\\{\bm{h}_{\text{RI}}^{*}(n)\bm{h}_{\text{w}}(n),\forall{n}\right\\}$;

4\. Compute $\bm{g}=\bm{h}_{\text{RI}}\bm{\Theta}\bm{h}_{\text{BR}}$;

5\. Set $\bm{w}_{\text{R}}$ as the right dominant eigenvector of $\bm{g}$;

until Convergence6. Compute power allocation $P_{\text{D}}$ and $P_{\text{R}}$
use methods in Sec. III-B, considering EMFE constraint $\bar{P}$ under channel
$\bm{h}_{\text{N}}$.

return Optimal rate $R$ for IU (4).

In some scenarios, especially with large number of RIS elements, optimizing
$\phi_{n}$ and $\bm{w}_{\text{R}}$ with explicit CSI may not be practical.
Inspired by the precoding scheme designed in [21, Algorithm 1], we propose to
use a DFT codebook-based beam optimization where the RIS beam is selected from
the pre-defined beam patterns while only the concatenated channel
$\bm{h}_{\text{RI}}\bm{\Theta}\bm{h}_{\text{BR}}\bm{w}_{\text{R}}$ is needed
to optimize $\bm{w}_{\text{R}}$, as presented in Algorithm 2.

Complexity: The computational complexity of Algorithm. 1 is
$\mathcal{O}\left(I_{\text{AO}}\left(N^{2}+2NN_{\text{T}}+N+N_{\text{T}}\right)\right)$,
where $I_{\text{AO}}$ represents the number of iterations. For Algorithm. 2
the complexity is
$\mathcal{O}\left(\left(N+1\right)\left(N^{2}+NN_{\text{T}}\right)\right)$.

Algorithm 2 Joint beam optimization using DFT codebook-based beamforming in
EMFE-aware RIS systems

0: Concatenate channel
$\bm{h}_{\text{RI}}\bm{\Theta}\bm{h}_{\text{BR}}\bm{w}_{\text{R}}$ with
selected $\bm{\Theta}$, pre-defined DFT codebook
$\bm{V}\in\mathcal{C}^{N\times N}$, $\bm{h}_{\text{N}}$, and $\bar{P}$

1\. The BS send training and control message to RIS;

for $i$ = 1:$N$ do

2\. The RIS pick $i$-th beam $\bm{v}_{i}$ from the predefined codebook
$\bm{V}$ and form the reflection matrix as
$\Theta=\text{diag}\left(\bm{v}_{i}\right)$;

3\. The IU calculate the received power for the selected RIS beam
$\left\|\bm{h}_{\text{RI}}\text{diag}\left(\bm{v}_{i}\right)\bm{h}_{\text{BR}}\right\|^{2}$;

end for

5\. The IU feeds back the best beam index $i_{\text{best}}$ in terms of
received power to the BS;

6\. The BS calculate the precoder as
$\bm{w}_{\text{DFT}}=\frac{\bm{h}_{\text{DFT}}^{H}}{\left\|\bm{h}_{\text{DFT}}\right\|}$,
where
$\bm{h}_{\text{DFT}}=\bm{h}_{\text{RI}}\text{diag}\left(\bm{v}_{i_{\text{best}}}\right)\bm{h}_{\text{BR}}$
;

7\. The RIS generate Compute power allocation $P_{\text{D}}$ and
$P_{\text{R}}$ use methods in Sec. III-B, considering EMFE constraint
$\bar{P}$ under channel $\bm{h}_{\text{N}}$.

return Optimal rate $R$ for IU (4).

###  III-B Power Allocation Schemes for the Direct and RIS-assisted Links with
EMFE Constraints

Depending on the channel condition as well as the EMFE constraint, the power
division between the direct link $P_{\text{D}}$ and the RIS-aided link
$P_{\text{R}}$ may vary. In this subsection, various power allocation methods
with different performance-complexity trade-offs are presented. Specifically,
we consider the following:

  * •

Method 1: All power is allocated to the BS-RIS-IU link, i.e.,
$P_{\text{R}}=P$.

  * •

Method 2: All power is allocated to the direct link, i.e., $P_{\text{D}}=P$.

  * •

Method 3: Based on the information from the NIU, i.e., $P_{\text{N, Tx}}$ in
(5), the BS fills the direct link with $P_{\text{D}}=P_{\text{N, Tx}}$ and
allocate the remaining power to $P_{\text{R}}=P-P_{\text{D}}$.

  * •

Method 4: As one upper bound of Method 3, the power allocation factor
$\alpha$, i.e.,

$\displaystyle P_{\text{D}}=\alpha P,\alpha\leq\frac{P_{\text{N, Tx}}}{P},$
(7)

can be adaptively and exhaustively optimized with the knowledge of the
optimized beams from the corresponding steps in Algorithms 1-2.

  * •

Method 5: As another upper bound of Method 3, we consider transmitting
different codewords in the direct link and the RIS link, i.e., the IU rate can
be upper bounded as

$\displaystyle R_{\text{upper}}=$ $\displaystyle
B\Bigg{(}\log_{2}\left(1+\frac{P_{\text{D}}\left|\bm{h}_{\text{D}}\bm{w}_{\text{D}}\right|^{2}}{BN_{0}}\right)+$
$\displaystyle\log_{2}\left(1+\frac{P_{\text{R}}\left|\bm{h}_{\text{RI}}\bm{\Theta}\bm{h}_{\text{BR}}\bm{w}_{\text{R}}\right|^{2}}{BN_{0}}\right)\Bigg{)}.$
(8)

In the following Lemma 1, we show that $\alpha$ can be analytically determined
when considering Method 5.

######  Lemma 1.

Define

$\displaystyle
c_{1}=\frac{\mathbb{E}\left[\left|\bm{h}_{\text{RI}}\bm{\Theta}\bm{h}_{\text{BR}}\bm{w}_{\text{R}}\right|^{2}\right]}{BN_{0}},$
(9)

and

$\displaystyle
c_{2}=\frac{\mathbb{E}\left[\left|\bm{h}_{\text{D}}\bm{w}_{\text{D}}\right|^{2}\right]}{BN_{0}}$
(10)

as the average received signal-to-noise ratio (SNR) with optimized BS and RIS
beamforming. The optimal power allocation can be calculated as

$\displaystyle\min\left\\{\frac{P_{\text{N,
Tx}}}{P},\frac{Pc_{2}-Pc_{1}+P^{2}c_{1}c_{2}}{2\left(P^{2}c_{1}c_{2}\right)}\right\\}.$
(11)

###### Proof.

The proof of Lemma 1 can be found in Appendix A. ∎

Lemma 1 is useful for evaluating the upper bounded performance in RIS-assisted
networks with EMFE constraint. With the knowledge of optimized beam pattern
leading to average received SNR, the optimal power allocation for the direct
link and the RIS-assisted link can be analytically derived.

######  Lemma 2.

To obtain the best IU rate (6), the considered EMFE-constrained RIS system
should always fill in the direct link with the maximum possible power if the
direct link is better in the sense that $c_{2}>c_{1}$.

###### Proof.

The proof of Lemma 2 can be found in Appendix B. ∎

##  IV Simulation Results

In this section, we present numerical results for the proposed EMFE-aware
power allocation methods for RIS-assisted networks. The simulation setup is
presented in Fig. 3. Consider one BS with $N_{\text{T}}=32$ antennas, one RIS
with $N=100$ elements, one IU and one NIU both having one antenna. The carrier
frequency $f_{\text{c}}$ is set to 28 GHz with 100 MHz channel bandwidth $B$.
The noise power is set as -174 dBm/Hz with 10 dB noise figure. The antenna
gain of the BS, RIS, and the users are set to 18 dBi, 18 dBi, and 0 dBi,
respectively. The total transmit power is $P=$ 43 dBm in Fig. 4. As shown in
Fig. 3, the BS is located at [-80m, 0] while the IU is fixed at [80m, 0] for
Case 1 and [-70m, 0] for Case 2. Except for otherwise stated, the RIS is
located at [0, 50m] for Case 1 and [-70m, 10m] for Case 2. Finally, for the
NIU, it moves along the x-axis for the Case 1 (Fig. 1), while it stands close
or far from the IU (Case 2, Fig. 2) with the same distance to the BS. For
simplification, we omit the EMFE from the RIS-NIU link since the RIS beam is
supposed to be optimized towards IU. Nevertheless, we verify this assumption
in one of the curves in Fig. 4.

Figure 3: The simulation setup for the two considered cases.

Our proposed methods are generic to different channel models. We assume that
we have all line-of-sight (LoS) paths and the path loss at distance $d$ can be
obtained by the mmMAGIC model [22, Table IV]

$\displaystyle\text{PL}=19.2\log_{10}(d)+32.9+20.8\log_{10}(f_{\text{c}})$
(12)

with 2 dB shadowing. Also, to fully evaluate the performance, for Case 1 (Fig.
1) we consider Rayleigh fading with a unit variance while Case 2 (Fig. 2) uses
a more generic mmWave channel with angle-of-departure (AoD) and multipath:

$\displaystyle\bm{h}=\sqrt{\frac{\text{PL}}{L}}\sum_{l=1}^{L}\beta_{l}\bm{a}(\psi_{l}).$
(13)

Here, $\beta\sim\mathcal{CN}(0,1)$ and $L$ is the number of paths (set to 3 in
the simulation). Furthermore,
$\bm{a}(\psi_{l})=\left\\{e^{jkd_{a}\left(n_{\text{T}}-1\right)\sin\left(\psi_{l}\right)}\right\\}_{n_{\text{T}}=1}^{N_{\text{T}}}$
is the antenna steering vector with $k=2\pi/\lambda$ and
$d_{\text{a}}=\lambda/2$ when $\lambda$ is the wavelength. Moreover, the
codebook-based beamforming proposed in Algorithm 2 can be applied with
different codebooks. Here, we present results with [23]

$\displaystyle\bm{V}=\left\\{\bm{\mu}_{i}\right\\}_{i=1}^{\sqrt{N}}\otimes\left\\{\bm{\nu}_{j}\right\\}_{j=1}^{\sqrt{N}},$
(14)

where

$\displaystyle\bm{\mu}_{i}=\left[1,e^{\frac{2\pi(i-1)}{\sqrt{N}}},...,e^{\frac{2\pi(i-1)(\sqrt{N}-1)}{\sqrt{N}}}\right],$
(15)
$\displaystyle\bm{\nu}_{j}=\left[1,e^{\frac{2\pi(j-1)}{\sqrt{N}}},...,e^{\frac{2\pi(j-1)(\sqrt{N}-1)}{\sqrt{N}}}\right],$
(16)

and $\otimes$ represents Kronecker product.

Figure 4: IU throughput as a function of the position of the NIU for Case 1.
The EMFE constraint $\bar{P}$ is set to 0.005 mW.
Figure 5: IU throughput as a function of the position of the NIU for Case 1.
The EMFE constraint $\bar{P}$ is set to 0.5 mW.

Figure 4 presents the IU throughput as a function of the position of NIU for
Case 1, with the EMFE constraint $\bar{P}$ is set to 0.005 mW. Here, we
consider the cases with Method 1 (RIS only), Method 2 (direct link only),
Method 3 with both AO and DFT-based beam optimization (corresponding to
Algorithm 1 and 2, respectively), Method 4 with exhaustive power allocation
using AO, Method 5 as an upper bound from both simulation and Lemma 1, and the
case without EMFE constraint and transmit with the BS-IU direct link. The
interference from RIS to NIU is considered in the case ”Method 3-DFT, RIS-
NIU”. Then, in Fig. 5, with a more relaxed EMFE constraint ($\bar{P}=$ 0.5
mW), we focus on the comparison of Methods 1-4 as well as AO and DFT-based
beam optimization.

Figure 6: IU throughput as a function of transmit power $P$ for Case 2. The
EMFE constraint $\bar{P}$ is set to 0.1 mW and the AoD for the close and far
NIU are set to $\pi/16$ and $\pi/8$, respectively. The coordinates for the
close and far RIS are (-70, 10)m, and (-30, 10)m, respectively.

To study the performance of Case 2 where the NIU is alongside the IU, in Fig.
6, we plot the IU throughput versus the transmit power. Here, the EMFE
constraint $\bar{P}$ set to 0.1 mW and the AoD for the close and far NIU are
set to $\pi/16$ and $\pi/8$, respectively. We consider the cases with Method 1
(RIS only), Method 2 (direct link only), and Method 3 with DFT-based
optimization (Algorithm 2). Finally, to evaluate the impact of the deployment
of the RIS, both close and far RIS with respect to the IU are considered, and
they have coordinates as (-70, 10)m, and (-30, 10)m, respectively.

According to these simulation results, the following conclusions can be drawn:

  * •

The EMFE constraint from, e.g., regulation and standardization, could
drastically affect the system performance, according to the no constraint case
in Figs. 4-5 and saturation of Method 2 in Fig. 6, and thus needs to be
carefully taking into account with network design.

  * •

For the considered two cases, i.e., the NIU between the BS and the IU (Case 1)
and alongside the IU (Case 2), the system performance is affected by various
parameters such as the strength of the EMFE constraint, the position of RIS,
the total transmit power, as well as the relative distance between the NIU and
IU.

  * •

With the EMFE constraint being considered, it is still beneficial to fully
exploit the potential of the direct BS-IU path (which normally has better
propagation conditions in terms of, e.g., path loss compared to the RIS path),
with the help of NIU. As shown in a power-limited system in Figs. 4-5, the
performance of only using the direct path increases with smaller EMFE
constraints, i.e., further NIU. On the other hand, with an increased transmit
power budget in Fig. 6, RIS-assisted link could eventually overperform even
with a far-deployed RIS.

  * •

If the direct path has better propagation condition, it is preferable to fill
the link first before considering the RIS-assisted link (Lemma 2), which is
supported by the similar performance of Methods 3-4 in Figs. 4-5.

  * •

The analytical results of the upper bound (• ‣ III-B) in Lemma 11 agrees well
with the simulations, and it reveals the system performance potential with
transmitting different codewords for the direct and the RIS-assisted link.

  * •

As can be seen in Figs. 4-5, the proposed DFT-based scheme (Algorithm. 2)
could reach a close performance compared to the AO method, especially with a
more relaxed EMFE constraint, and it does not require explicit CSI when
optimizing the beams.

##  V Conclusions

In this paper, we studied the EMFE-limited RIS-assisted system with two new
types of constrained positions. Specifically, we reveal that with the help of
NIUE detection, it is beneficial in terms of the system throughput to fully
utilize the better propagation path, which is normally the direct BS-IU link.
Also, we evaluated the system performance with the proposed AO\- and DFT-based
scheme with various power allocation methods. The DFT-based method is shown to
be efficient with limited channel information. In addition, the analytical
contributions on the upper bound of the system as well as the preference of
the direction in power allocation provide solid insights on the considered
setups.

## Acknowledgement

This work was supported by the EU H2020 RISE-6G project under grant number
101017011.

## Appendix A

With power allocation factor $\alpha$, the throughput of IU (• ‣ III-B) can be
written as

$\displaystyle
R_{\text{upper}}=B\left(\log_{2}\left(1+(1-\alpha)Pc_{1}\right)+\log_{2}\left(1+\alpha
Pc_{2}\right)\right).$ (17)

Then, the derivative of $R_{\text{upper}}$ with respect to $\alpha$ is

$\displaystyle\frac{\text{d}R_{\text{upper}}}{\text{d}\alpha}=B\log_{2}\left(\left(1+Pc_{1}-\alpha
Pc_{2}\right)\left(1+\alpha Pc_{2}\right)\right)$
$\displaystyle=B\log_{2}\left(1+Pc_{1}+\alpha\left(Pc_{2}+P^{2}c_{1}c_{2}-Pc1\right)-\alpha^{2}P^{2}c_{1}c_{2}\right).$
(18)

Setting (Appendix A) equal to zero the optimal power allocation for the upper
bounded IU rate (• ‣ III-B) can be obtained as 11.

## Appendix B

(6) with power allocation factor $\alpha$ can be simplified as

$\displaystyle R=B\log_{2}\left(1+\left(1-\alpha\right)Pc_{1}+\alpha
Pc_{2}\right).$ (19)

Its derivative with respect to $\alpha$ is

$\displaystyle\frac{\text{d}R}{\text{d}\alpha}=\frac{BP\cdot\left(c_{2}-c_{1}\right)}{\left(Pc_{2}-Pc_{1}\right)x+Pc_{1}+1}.$
(20)

When $c_{2}>c_{1}$, i.e., the direct link is better, the derivative is always
positive. As a result, the maximum allowed power ($P_{\text{N, Tx}}$) should
be allocated to the direct path.

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  *[5G]: 5th Generation
  *[6G]: 6th Generation
  *[MIMO]: multiple input multiple output
  *[mmWave]: millimeter wave
  *[EMFE]: electromagnetic field exposure
  *[ICNIRP]: International Commission on non-ionizing radiation protection
  *[4G]: 4th Generation
  *[RIS]: reconfigurable intelligent surface
  *[UE]: user equipment
  *[UL]: uplink
  *[DL]: downlink
  *[BS]: base station
  *[NIU]: non-intended user
  *[IU]: intended user
  *[NIUE]: non-intended user equipment
  *[JBPO]: joint beamforming and transmit power optimization
  *[CSI]: channel state information
  *[AO]: alternating optimization
  *[DFT]: discrete Fourier transform
  *[MRT]: maximum-ratio transmission
  *[SNR]: signal-to-noise ratio
  *[LoS]: line-of-sight
  *[AoD]: angle-of-departure


# Gödel-type solutions in hybrid metric-Palatini gravity

J. S. Gonçalves<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal de Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil  A. F. Santos
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal de
Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil

###### Abstract

In this paper, the hybrid metric-Palatini gravity is an approach to modified
gravity in which is added to the usual Einstein-Hilbert action a supplementary
term containing a Palatini-type correction of the form $f(\mathcal{R},T)$.
Here, $\mathcal{R}$ is the Palatini curvature scalar, which is constructed
from an independent connection and $T$ is the trace of the energy-momentum
tensor. This theory describes a non-minimal coupling between matter and
geometry. The modified Einstein field equations in this hybrid metric-Palatini
approach are obtained. Then, it is investigated whether this modified theory
of gravity and its field equations allow Gödel-type solutions, which
essentially lead to violation of causality. Considering physically well-
motivated matter sources, causal and non-causal solutions are explored.

##  I Introduction

Cosmological observations indicate that the universe is currently expanding at
an accelerated rate Riess ; Per ; Adam ; Cole ; Anderson . To explain the
accelerated expansion of the universe, a large number of modified theories of
gravity have been proposed (for a review, see Clifton ). These approaches
consider more general Lagrangian functions, extra dimensions, geometric
quantities, among others. These gravitational theories can be obtained from
two different theoretical approaches, called the metric and the Palatini
formalism. In the first formalism, the dynamic variable is just the metric
tensor, while in the latter, in the Palatini variational approach, the metric
tensor and the affine connection are taken as independent field variables.
Both formalisms, when applied to general relativity, lead to the same
gravitational field equations. However, in modified theories of gravity, this
is no longer the case. For a review of modified theories of gravity in the
Palatini approach, see palatini_motivation1 . Although these approaches have
led to possible ways to explain the observed accelerated expansion of the
universe, they do have some problems. For example, the $f(R)$ gravity has
difficulties explaining the evolution of cosmological perturbations and local
gravity constraints palatini_motivation1 . Then, a third approach can be
considered to study the modified theories of gravity.

An alternative way to explain the current accelerated expansion of the
universe is the hybrid metric-Palatini gravity first_hybrid_paper . This
theory consists of the superposition of the metric (Einstein–Hilbert action)
and Palatini formalisms. Several studies have been developed with this
approach, such as, classical gravitational tests in the solar system have been
investigated hybrid_motivation1 , the dark energy problem has been analyzed
Capo , models of the accelerating universe without a cosmological constant
have been presented Capo2 , Noether symmetries have been studied Boro , axial
gravitational perturbations and the quasi-normal modes frequencies of black
holes have been studied hybrid_motivation2 , Kerr black hole solutions and
their stability have been explored Rosa , static and spherically symmetric
solutions have been discussed hybrid_motivation3 ; Bron ; esf ; esf1 , among
others. In addition, pathologies such as Ostrogradsky instability have been
analyzed Adria . It is observed that the hybrid formalism does not, in
general, guarantee the absence of ghosts. Furthermore, to avoid these
pathologies, it is necessary to impose additional constraints on the gravity
sector. For a review of hybrid metric-Palatini gravity, see Rev ; Rev2 ;
hybrid_motivation4 .

The present paper continues the study of the hybrid metric-Palatini gravity.
Here, the $f(R,T)$ gravity is analyzed. This gravitational model was first
proposed in first . This model is characterized by the non-minimal coupling
between matter, described by the trace of the energy-momentum tensor $T$, and
geometry, defined by the Ricci scalar $R$. In the hybrid formalism, the
function $f(\mathcal{R},T)+R$ is introduced, where $R$ represents the
Einstein–Hilbert part of the action and $\mathcal{R}=R(g,\tilde{\Gamma})$ is
Ricci-Palatini scalar. As an application of this gravitational theory, the
causality problem is investigated. Then the Gödel-type solution tipo_godel1
is considered. The motivation is based on the fact that, if gravity is
described by the hybrid metric-Palatini $f(\mathcal{R},T)$ gravity theory,
there are a number of issues that must be re-examined in its context, such as
the question of whether this gravity theory allows Gödel-type solutions, which
are allowed in general relativity and in various modified gravity theories.

The Gödel-type metric is a generalization of the Gödel solution proposed by
Kurt Gödel in 1949 godel . The main feature of the Gödel solution is the
possibility of Closed Timelike Curves (CTCs) that lead to causality violation.
This causality violation does not occur locally, since in general relativity
the space-times locally have the same causal structure of the special
relativity. However, on a non-local scale significant differences may arise.
Then, globally, a violation of causality is allowed beyond a region limited by
a critical radius. These CTCs appear in other cosmological models that have
hyperbolic or spherical symmetry, such as Kerr black hole, Van-Stockum model,
cosmic string, among others ctc1 ; ctc2 . The causality problem has been
verified in many gravitational theories, such as, $f(R)$ theory fr_and_godel ,
$k$-essence theory kessence_and_godel , Chern-Simons gravity
chersimon_and_godel1 ; chersimon_and_godel2 , $f(T)$ gravity ft_and_godel ,
$f(R,T)$ gravity frt_and_godel , bumblebee gravity bumblebeee_and_godel ,
Horava-Lifshitz gravity horava_and_godel , Brans-Dicke theory brans_and_godel
, $f(R,Q)$ gravity frq_and_godel , $f(R,\phi,X)$ gravity fr_phi_X , $f(R,T)$
Palatini gravity frt_palatini , among others.

The present paper is organized as follows. In section II, an introduction to
the hybrid metric-Palatini $f(\mathcal{R},T)$ gravity is made. In section III,
the Gödel-type metric is introduced and the field equations are solved for
this metric. Then, the causality problem is discussed. Different matter
contents, such as perfect fluid, a combination of perfect fluid and scalar
field and only a scalar field, are considered. Causal and non-causal solutions
are obtained. The critical radius for non-causal solution is calculated. In
section IV, remarks and conclusions are presented.

##  II Hybrid Metric-Palatini $f(\mathcal{R},T)$ gravity

In this section, the field equations for the hybrid metric-Palatini
$f(\mathcal{R},T)$ gravity are calculated. The action that describes this
theory is given as

$S=\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}\left[\ R+\
f(\mathcal{R},T)+\mathcal{L}_{m}\right],$ (1)

where $\kappa^{2}=8\pi G$, $g$ is the determinant of the metric tensor
$g_{\mu\nu}$, $R$ is the Ricci scalar associated with the metric $g_{\mu\nu}$,
$f(\mathcal{R},T)$ is a function explicitly dependent on the Ricci-Palatini
scalar $\mathcal{R}=R\left(g,\tilde{\Gamma}\right)$ and the trace of the
energy-momentum tensor $T$ and $\mathcal{L}_{m}$ is the matter Lagrangian that
is a function of $g$ and physical fields $\psi$. The Ricci-Palatini scalar
that depends on both the metric and an independent connection $\tilde{\Gamma}$
is written as

$\mathcal{R}=g^{\mu\nu}\tilde{R}_{\mu\nu}\left(\tilde{\Gamma}\right),$ (2)

where the Ricci tensor associated with the connection $\tilde{\Gamma}$ is

$\tilde{R}_{\mu\nu}\left(\tilde{\Gamma}\right)=\partial_{\lambda}\tilde{\Gamma}_{\mu\nu}^{\lambda}-\partial_{\nu}\tilde{\Gamma}_{\mu\lambda}^{\lambda}+\tilde{\Gamma}_{\mu\nu}^{\lambda}\tilde{\Gamma}_{\lambda\alpha}^{\alpha}-\tilde{\Gamma}_{\mu\lambda}^{\alpha}\tilde{\Gamma}_{\nu\alpha}^{\lambda}.$
(3)

In order to obtain the field equation related to the metric field it is
assumed that $\delta\tilde{R}_{\mu\nu}\left(\tilde{\Gamma}\right)=0$, that is,
the Palatini connection is constant, and the matter Lagrangian is independent
of $\partial_{\lambda}g_{\mu\nu}$. Then the variation of the action (1) with
respect to the metric gives

$G_{\mu\nu}+\tilde{R}_{\mu\nu}f_{R}-\frac{g_{\mu\nu}}{2}f=\kappa^{2}T_{\mu\nu}-\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)f_{T},$
(4)

where $f_{R}\equiv\frac{\partial f}{\partial R}$, $f_{T}\equiv\frac{\partial
f}{\partial T}$, $T_{\mu\nu}$ is the energy-momentum tensor that is defined as

$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\partial\left(\sqrt{-g}\mathcal{L}_{m}\right)}{\partial
g^{\mu\nu}}=-2\frac{\partial\mathcal{L}_{m}}{\partial
g^{\mu\nu}}+g_{\mu\nu}\mathcal{L}_{m},$ (5)

and the tensor $\Theta_{\mu\nu}$ describes the variation of the energy-
momentum tensor with respect to the metric

$\Theta_{\mu\nu}\equiv\frac{\delta T_{\alpha\beta}}{\delta
g^{\mu\nu}}g^{\alpha\beta}.$ (6)

Using eq.(5) this tensor is written as

$\Theta_{\mu\nu}=-2T_{\mu\nu}+g_{\mu\nu}\mathcal{L}_{m}-2g^{\alpha\beta}\frac{\partial^{2}\mathcal{L}_{m}}{\partial
g^{\mu\nu}\partial g^{\alpha\beta}}.$ (7)

Contracting eq.(4) with the metric, the Ricci-Palatini scalar is given as

$R\left(g,\tilde{\Gamma}\right)f_{R}=\kappa^{2}T-\left(T+\Theta\right)f_{T}+2f+R,$
(8)

where $\Theta=\Theta^{\mu}\,_{\mu}$. Using the Einstein-Palatini tensor

$G_{\mu\nu}\left(g,\tilde{\Gamma}\right)=\tilde{R}_{\mu\nu}\left(\tilde{\Gamma}\right)-\frac{1}{2}g_{\mu\nu}\tilde{R}\left(g,\tilde{\Gamma}\right),$
(9)

and eqs.(4) and (8), the field equation from metric variation becomes

$\displaystyle G_{\mu\nu}\left(g,\tilde{\Gamma}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{f_{R}}\\{\kappa^{2}T_{\mu\nu}-f_{T}\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)-G_{\mu\nu}(g)-\frac{g_{\mu\nu}}{2}\left[f+\kappa^{2}T-f_{T}\left(T+\Theta\right)+R\right]\\}.$
(10)

As the action depends on both the metric and the connection, now the variation
of the action is performed in relation to the $\tilde{\Gamma}$, keeping the
metric constant. Then

$\displaystyle\delta f$ $\displaystyle=f_{R}\delta
R\left(\tilde{\Gamma}\right),$ (11)
$\displaystyle=f_{R}g^{\mu\nu}\delta\tilde{R}_{\mu\nu}\left(\tilde{\Gamma}\right),$
(12)
$\displaystyle=f_{R}g^{\mu\nu}\left[\tilde{\nabla}\left(\delta\tilde{\Gamma}_{\mu\nu}^{\lambda}\right)-\tilde{\nabla}\left(\delta\tilde{\Gamma}_{\mu\lambda}^{\lambda}\right)\right],$
(13)

where $\tilde{\nabla}$ is the covariant derivative associated with
$\tilde{\Gamma}$. Then the variation of the action becomes

$\delta
S=\frac{1}{\kappa^{2}}\int\sqrt{-g}f_{R}g^{\mu\nu}\left[\tilde{\nabla}\left(\delta\tilde{\Gamma}_{\mu\nu}^{\lambda}\right)-\tilde{\nabla}\left(\delta\tilde{\Gamma}_{\mu\lambda}^{\lambda}\right)\right]d^{4}x.$
(14)

Defining $A^{\mu\nu}\equiv f_{R}g^{\mu\nu}$ and integrating by parts, we get

$\displaystyle\kappa^{2}\delta S$ $\displaystyle=$
$\displaystyle\int\tilde{\nabla}_{\lambda}\left[\sqrt{-g}\left(A^{\mu\nu}\delta\tilde{\Gamma}_{\mu\nu}^{\lambda}-A^{\mu\lambda}\delta\tilde{\Gamma}_{\mu\alpha}^{\alpha}\right)\right]d^{4}x-\int\tilde{\nabla}_{\lambda}\left[\sqrt{-g}\left(A^{\mu\nu}\delta^{\lambda}_{\alpha}-A^{\mu\lambda}\delta^{\nu}_{\alpha}\right)\right]\delta\tilde{\Gamma}_{\mu\nu}^{\alpha}d^{4}x.$
(15)

Note that the first term is a total derivative that can be related to a
surface integral that becomes zero. This leads to

$\tilde{\nabla}_{\lambda}\left[\sqrt{-g}\left(A^{\mu\nu}\delta^{\lambda}_{\alpha}-A^{\mu\lambda}\delta^{\nu}_{\alpha}\right)\right]=0.$
(16)

The case $\lambda\neq\alpha$ implies that

$\tilde{\nabla}_{\lambda}\left[\sqrt{-g}f_{R}g^{\mu\nu}\right]=0$ (17)

while for case $\lambda=\alpha$, eq. (16) is identically zero.

The connection $\tilde{\Gamma}$ is compatible with the conformal metric
$\tilde{g}_{\mu\nu}$, which means that the geometry is not modified. Then the
Palatini connection is written as

$\tilde{\Gamma}_{\mu\nu}^{\lambda}=\frac{1}{2}\tilde{g}^{\lambda\rho}\left(\partial_{\nu}\tilde{g}_{\rho\mu}+\partial_{\mu}\tilde{g}_{\rho\nu}+\partial_{\rho}\tilde{g}_{\mu\nu}\right).$
(18)

In terms of the Levi-Civita connection $\Gamma$, the components of
$\tilde{\Gamma}$ are

$\tilde{\Gamma}_{\mu\nu}^{\lambda}=\Gamma_{\mu\nu}^{\lambda}+\frac{1}{2f_{R}}\left(\delta^{\lambda}_{\mu}\partial_{\nu}+\delta^{\lambda}_{\nu}\partial_{\mu}-g_{\mu\nu}\partial^{\lambda}\right)f_{R}.$
(19)

Using this conformal transformation and the properties of a conformal metric,
the Ricci tensor becomes

$\displaystyle\tilde{R}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}+\frac{1}{f_{R}}\Bigl{\\{}\frac{3}{2F_{r}}\nabla_{\mu}f_{R}\nabla_{\nu}f_{R}-\left(\nabla_{\mu}\nabla_{\nu}+\frac{g_{\mu\nu}}{2}\Box\right)f_{R}\Bigl{\\}}.$
(20)

The conformal Ricci scalar is given by
$\tilde{R}\equiv\tilde{g}^{\mu\nu}\tilde{R}_{\mu\nu}=f_{R}^{-1}R$ and
$\tilde{G}_{\mu\nu}\equiv\tilde{R}_{\mu\nu}-\tilde{g}_{\mu\nu}\tilde{R}/2=G_{\mu\nu}\left(g,\tilde{\Gamma}\right)$.
Remember that the quantities with tilde are related to the conformal metric.
Then the Ricci scalar is written as

$\tilde{R}=R-\frac{3}{f_{R}}\Box f_{R}+\frac{3}{2f_{R}^{2}}\left(\nabla
f_{R}\right)^{2},$ (21)

and the Einstein tensor is

$\displaystyle\tilde{G}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
G_{\mu\nu}+\frac{1}{f_{R}}\Bigl{\\{}\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R}+\frac{3}{2f_{R}}\left[\nabla_{\mu}f_{R}\nabla_{\nu}f_{R}-\frac{g_{\mu\nu}}{2}\left(\nabla
f_{R}\right)^{2}\right]\Bigl{\\}}.$ (22)

Finally, from eq. (10), the field equation for this theory becomes

$\displaystyle\left[1+f_{R}\right]G_{\mu\nu}$ $\displaystyle+$ $\displaystyle
J_{\mu\nu}=\kappa^{2}T_{\mu\nu}-f_{T}\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)-\frac{g_{\mu\nu}}{2}\left[f+\kappa^{2}T-f_{T}\left(T+\Theta\right)+R\right],$
(23)

where, for convenience, it is defined

$\displaystyle J_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R}+\frac{3}{2f_{R}}\left[\nabla_{\mu}f_{R}\nabla_{\nu}f_{R}-\frac{g_{\mu\nu}}{2}\left(\nabla
f_{R}\right)^{2}\right].$ (24)

In the next section, the field equations of the hybrid metric-Palatini
$f(\mathcal{R},T)$ gravity are solved for the Gödel-type metrics.

##  III Gödel-type Metric in Hybrid Metric-Palatini $f(\mathcal{R},T)$ gravity

In this section, the main features of a generalized class of metrics called
Gödel-type are briefly discussed. Then, the question of causality in the
hybrid metric-Palatini $f(\mathcal{R},T)$ gravity is investigated. The line
element that describes the Gödel-type metric in cylindrical coordinates takes
the following form

$ds^{2}=-dt^{2}-2H(r)dtd\phi+dr^{2}+G(r)d\phi^{2}+dz^{2},$ (25)

where $G(r)=D^{2}(r)-H^{2}(r)$ and the functions $D(r)$ and $H(r)$ satisfy the
relations

$\displaystyle\frac{H^{\prime}(r)}{D(r)}$ $\displaystyle=$ $\displaystyle
2\omega,$ $\displaystyle\frac{D^{\prime\prime}(r)}{D(r)}$ $\displaystyle=$
$\displaystyle m^{2},$ (26)

with the prime being the derivative with respect to the radius coordinate. It
is important to note that, the parameters $\omega$ and $m$ describe entirely
the conditions of homogeneity of space-time. These parameters assume the
values in the range $-\infty\leq m^{2}\leq+\infty$ and $\omega\neq 0$. As
shown in tipo_godel1 , Gödel-type metrics can be divided into three different
classes: (i) hyperbolic class, where $m^{2}>0$ and $\omega\neq 0$; (ii)
trigonometric class, with $m^{2}<0$ and $\omega\neq 0$ and (iii) linear class,
where $m^{2}=0$ and $\omega\neq 0$. The Gödel metric, which is obtained for
the case $m^{2}=2\omega^{2}$, belongs to the hyperbolic class.

The main characteristic of Gödel-type spaces is the possibility of CTCs that
are cicles defined by $t,r,z=\mathrm{const}$ and $\phi\in(0,2\pi)$ in a region
limited by the range $r_{1}<r<r_{2}$, with $G(r)$ being negative within this
range. For the hyperbolic class, the functions $H(r)$ and $D(r)$ are given as

$\displaystyle H(r)$ $\displaystyle=$
$\displaystyle\frac{4\omega}{m^{2}}sinh^{2}\left(\frac{mr}{2}\right),$ (27)
$\displaystyle D(r)$ $\displaystyle=$ $\displaystyle\frac{1}{m}sinh(mr),$ (28)

and CTCs arise within the region corresponding to $r>r_{c}$, where $r_{c}$ is
the critical radius separating the causal and non-causal regions. The explicit
form of the critical radius is given by

$\displaystyle
sinh^{2}\left(\frac{mr_{c}}{2}\right)=\left(\frac{4\omega^{2}}{m^{2}}-1\right)^{-1}.$
(29)

It is interesting to note two special cases: (i) for $m^{2}=2\omega^{2}$ the
standard Gödel solution is recovered and its critical radius becomes

$r_{c}=\frac{2}{m}sinh^{-1}(1);$ (30)

and (ii) for $m^{2}=4\omega^{2}$ there are no CTCs, since the critical radius
$r_{c}=\infty$, that is, the violation of causality is avoided. In addition,
similarly, linear and trigonometric classes also display CTCs.

Therefore, in this theory, there is the possibility of a causal rather than a
non-causal solution. These regions are determined from the free metric
parameters, i.e., $m$ and $\omega$, and limited by a critical radius $r_{c}$.
Then, the Gödel-type solutions bring more details to the problem of causality.

In order to make calculations simpler, let’s define a local set of tetrad
basis $\theta^{A}=e^{A}\,_{\mu}dx^{\mu}$, so that the line element takes the
form

$ds^{2}=\eta_{AB}\theta^{A}\theta^{B}=(\theta^{0})^{2}-(\theta^{1})^{2}-(\theta^{2})^{2}-(\theta^{3})^{2},$
(31)

where the capital Latin letters denote the transformed space, $\eta_{AB}$ is
the Minkowski metric and

$\displaystyle\theta^{(0)}$ $\displaystyle=dt+H(r)d\phi,$ (32)
$\displaystyle\theta^{(1)}$ $\displaystyle=dr,$ (33)
$\displaystyle\theta^{(2)}$ $\displaystyle=D(r)d\phi,$ (34)
$\displaystyle\theta^{(3)}$ $\displaystyle=dz.$ (35)

The non-zero components of the Einstein tensor in the flat (local) space-time
take the form,

$\displaystyle G_{(0)(0)}$ $\displaystyle=3\omega^{2}-m^{2},$ (36)
$\displaystyle G_{(1)(1)}$ $\displaystyle=\omega^{2},$ (37) $\displaystyle
G_{(2)(2)}$ $\displaystyle=\omega^{2},$ (38) $\displaystyle G_{(3)(3)}$
$\displaystyle=m^{2}-\omega^{2},$ (39)

where $G_{AB}=e^{\mu}_{A}e^{\nu}_{B}G_{\mu\nu}$ has been used. Here,
$e^{\mu}\,_{B}$ is the inverse of $e^{A}\,_{\mu}$ and they satisfy the
relation $e^{A}\,_{\mu}e^{\mu}\,_{B}=\delta^{A}_{B}$. The non-zero components
of the tetrads are

$e^{(0)}\ _{0}=e^{(1)}\ _{1}=e^{(3)}\ _{3}=1,e^{(0)}\ _{2}=H(r),e^{(2)}\
_{2}=D(r),$ (40)

and the non-zero components of the inverse of the tetrad are

$\displaystyle e^{0}\ _{(0)}$ $\displaystyle=$ $\displaystyle e^{1}\
_{(1)}=e^{3}\ _{(3)}=1,$ $\displaystyle e^{0}\ _{(2)}$ $\displaystyle=$
$\displaystyle-\frac{H(r)}{D(r)},e^{2}\ _{(2)}=D^{-1}(r).$ (41)

Thus, the field equation (23) in the tetrad basis becomes

$\displaystyle\left[1+f_{R}\right]G_{AB}$ $\displaystyle+$ $\displaystyle
J_{AB}=\kappa^{2}T_{AB}-f_{T}\left(T_{AB}+\Theta_{AB}\right)-\frac{g_{AB}}{2}\left[f+\kappa^{2}T-f_{T}\left(T+\Theta\right)+R\right].$
(42)

To complete the field equations, it is necessary to fix the matter content. In
tipo_godel1  well-motivated matter sources such as a perfect fluid, a scalar
field, and an electromagnetic field are investigated. Here, a perfect fluid, a
scalar field, and a combination of both are considered.

###  III.1 Perfect Fluid

Let’s start by assuming that the content of matter is a perfect fluid whose
energy-momentum tensor in the local frame is given as

$T_{AB}=(\rho+p)u_{A}u_{B}-p\eta_{AB},$ (43)

where $u_{A}=(1,0,0,0)$ is the four-velocity of the fluid and $\rho$ and $p$
are the fluid density and pressure, respectively. It is important to note
that, the Lagrangian of matter that describes a perfect fluid in gravitational
models with non-minimal coupling between matter and geometry has been
intensely explored Faraoni ; Bertolami ; Bertolami1 . These theories carry an
arbitrariness, that is, the Lagrangian of matter is not uniquely defined.
However, there are different studies in which it is shown that such
arbitrariness can be avoided Harko10 ; Moraes .

For this matter source, the field equations are given as

$\displaystyle 2\left[1+f_{R}\right]f_{R}\left(3\omega^{2}-m^{2}\right)+f$
$\displaystyle=8\pi\left(\rho+3p\right)+\left(\rho+p\right)f_{T}+R,$ (44)
$\displaystyle 2\left[1+f_{R}\right]f_{R}\omega^{2}-f$
$\displaystyle=8\pi\left(\rho-p\right)+\left(\rho+p\right)f_{T}-R,$ (45)
$\displaystyle 2\left[1+f_{R}\right]f_{R}\left(m^{2}-\omega^{2}\right)-f$
$\displaystyle=8\pi\left(\rho-p\right)+\left(\rho+p\right)f_{T}-R,$ (46)

where the Ricci scalar for the Gödel-type metrics takes a constant value
$R=2(m^{2}-\omega^{2})$. From eq.(45) and eq.(46) is found that

$2\left(1+f_{R}\right)(2\omega^{2}-m^{2})f_{R}=0.$ (47)

Note that, for $f_{R}>0$, eq. (47) leads to $m^{2}=2\omega^{2}$ which defines
the Gödel metric and the remaining field equations provide

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{-f+m^{2}}{16\pi}+\frac{(1+f_{R})f_{R}}{8\pi+f_{T}}m^{2},$
(48) $\displaystyle p$ $\displaystyle=$ $\displaystyle\frac{f-m^{2}}{16\pi}.$
(49)

Analyzing these solutions, it is observed that the Gödel solution is allowed.
Then, inevitably closed timelike curves are exhibited, i.e. there are non-
causal Gödel’s circles which are limited by the critical radius $r_{c}$. In
the framework of hybrid metric-Palatini $f(\mathcal{R},T)$ gravity, $r_{c}$ is
given as

$\displaystyle
r_{c}=\frac{2}{m}\sinh^{-1}(1)=2\sinh^{-1}(1)\sqrt{\frac{(1+f_{R})f_{R}}{(\rho+p)(f_{T}+8\pi)}}.$
(50)

This result can be divided as follows. First, the Gödel-type metric is a
solution of the hybrid metric-Palatini $f(\mathcal{R},T)$ gravity. Second, the
critical radius depends on both gravity theory and matter content. Therefore,
the main conclusion is that the violation of causality is allowed in this
gravitational theory. Now let’s investigate whether other sources of matter
can generate causal solutions.

###  III.2 Perfect fluid and scalar field

In this section, in order to find a causal Gödel-type solution, a combined
energy-momentum tensor is considered, that is,

$\displaystyle T_{AB}$ $\displaystyle=$ $\displaystyle T_{AB}^{m}+T_{AB}^{S},$
(51) $\displaystyle=$
$\displaystyle\left(\rho+p\right)u_{A}u_{B}-p\eta_{AB}+\nabla_{A}\phi\nabla_{B}\phi-\frac{1}{2}\eta_{AB}\eta^{CD}\nabla_{C}\phi\nabla_{D}\phi,$

where $T_{AB}^{m}$ and $T_{AB}^{S}$ are the energy-momentum tensor to a
perfect fluid and a scalar field, respectively. Here, $\nabla_{A}$ is the
covariant derivative relative to the local basis
$\theta^{A}=e^{A}_{\beta}dx^{\beta}$ and the scalar field is chosen as
$\phi(z)=\epsilon z+\epsilon$, with $\epsilon=\mathrm{const}$. The non-zero
components of the energy-momentum tensor associated with the scalar field are,

$T_{00}^{S}=-T_{11}^{S}=-T_{22}^{S}=T_{33}^{S}=\frac{\epsilon^{2}}{2},$ (52)

and its trace is given by

$T^{S}=\rho-3p+\epsilon^{2}.$ (53)

Considering the contributions of the scalar field, the tensor $\Theta_{AB}$ is
rewritten as

$\Theta_{AB}=\Theta_{AB}^{m}+\Theta_{AB}^{S},$ (54)

where $\Theta_{AB}^{m}$ is associated with the perfect fluid and is given as

$\displaystyle\Theta_{AB}^{m}=-2T_{AB}-pg_{AB}.$ (55)

Here, eq. (7) is used and the matter Lagrangian is taken as ${\cal L}_{m}=-p.$
To calculate $\Theta_{AB}^{S}$, let’s consider the following Lagrangian,

$\mathcal{L}^{S}=\eta^{AB}\nabla_{A}\phi\nabla_{B}\phi,$ (56)

such that the tensor becomes,

$\Theta_{AB}^{S}=-T_{AB}^{S}+\frac{1}{2}T^{S}\eta_{AB}.$ (57)

Then the field equation (42) is written as

$\displaystyle\left[1+f_{R}\right]G_{AB}$ $\displaystyle=$
$\displaystyle\kappa^{2}\left[\left(\rho+p\right)u_{A}u_{B}-p\eta_{AB}+T_{AB}^{S}\right]-\frac{1}{2}\Bigl{\\{}\kappa^{2}\left(\rho-3p+\epsilon^{2}\right)+f$
(58) $\displaystyle+$ $\displaystyle
f_{T}\left(\rho+p-2\epsilon^{2}\right)+R\Bigl{\\}}\eta_{AB}+f_{T}\left[\left(\rho+p\right)u_{A}u_{B}-\frac{1}{2}\epsilon^{2}\eta_{AB}\right].$

Using the Gödel-type metric, a set of equations is obtained, i.e.,

$\displaystyle\kappa^{2}\epsilon^{2}$
$\displaystyle=\left(m^{2}-2\omega^{2}\right)\left[1+f_{R}\right]+R,$ (59)
$\displaystyle\kappa^{2}p+\frac{1}{2}\epsilon^{2}f_{T}$
$\displaystyle=\frac{1}{2}\left(2\omega^{2}-m^{2}\right)\left[1+f_{R}\right]+\frac{1}{2}f-R,$
(60)
$\displaystyle\kappa^{2}\rho+f_{T}\left(\rho+p-\frac{1}{2}\epsilon^{2}\right)$
$\displaystyle=\frac{1}{2}\left(6\omega^{2}-m^{2}\right)\left[1+f_{R}\right]-\frac{1}{2}f-R.$
(61)

Assuming that $f_{R}>0$ and $f_{T}>0$, these equations allow for a causal
Gödel-type class of solutions that is given as

$\displaystyle m^{2}$ $\displaystyle=$ $\displaystyle 4\omega^{2},$ (62)
$\displaystyle f_{R}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\epsilon^{2}-6\omega^{2}}{2\omega^{2}}-1,$ (63)
$\displaystyle f_{T}$ $\displaystyle=$
$\displaystyle-\frac{1}{\rho+p}[\kappa^{2}(\rho+p)-12\omega^{2}].$ (64)

Note that, eq. (62) leads to $r_{c}\rightarrow\infty$. Therefore, for matter
content, which is a combination of a perfect fluid and a scalar field,
causality is not violated for any hybrid metric-Palatini $f(\mathcal{R},T)$
gravity.

###  III.3 Scalar field

If only the scalar field $\phi(z)$ is considered, the field equations become

$\displaystyle\left[1+f_{R}\right]\left(3\omega^{2}-m^{2}\right)+\frac{f}{2}$
$\displaystyle=\frac{1}{2}\epsilon^{2}f_{T}+R,$ (65)
$\displaystyle\left[1+f_{R}\right]\omega^{2}-\frac{f}{2}$
$\displaystyle=-\frac{1}{2}\epsilon^{2}f_{T}-R,$ (66)
$\displaystyle\left[1+f_{R}\right]\left(m^{2}-\omega^{2}\right)-\frac{f}{2}$
$\displaystyle=8\pi\epsilon^{2}-\frac{1}{2}f_{T}\epsilon^{2}-R.$ (67)

These equations allow the solution

$\displaystyle m^{2}$ $\displaystyle=$ $\displaystyle 4\omega^{2}.$ (68)

Therefore, when only the scalar field is considered, this theory defines a
class of solutions without causality violation, since the critical radius goes
to infinity. Furthermore, this result holds for an arbitrary
$f(\mathcal{R},T)$ function with $f_{R}\neq 0$ and $f_{T}\neq 0$.

##  IV Discussions and Final Remarks

In this paper, the hybrid metric-Palatini $f(\mathcal{R},T)$ gravity is
considered. This modified theory of gravity consists of adding to the
Einstein–Hilbert Lagrangian a function $f(\mathcal{R},T)$ that is constructed
in the Palatini formalism. Thus, this theory combines the metric and Palatini
approaches to gravity and is an extension of $f(R,T)$ theories, which define a
non-minimal coupling between matter and geometry. Then the modified Einstein
field equations are obtained. It is important to note that if this theory
describes gravity, there are several issues that must be investigated in this
context. A question that must be re-examined is whether this theory of gravity
allows for Gödel-type solutions, which lead to the violation of causality.
Although locally the general relativity has the same causal structure as the
flat space-time of special relativity, on a non-local scale it permits
solutions to Einstein field equations that show causal anomalies such as
CTC’s. Here, the question arises: is the causality violation allowed in the
hybrid metric-Palatini $f(\mathcal{R},T)$ gravity? In order to answer this
question, a combination of physical well-motivated sources of matter is
considered. For the perfect fluid as the content of matter is shown that the
violation of causality is an unavoidable characteristic of any hybrid metric-
Palatini $f(\mathcal{R},T)$ gravity. Then, an expression for the critical
radius has been derived. It depends on both the gravity theory and the content
of matter. To find a causal Gödel-type solution, two different types of matter
have been introduced: (i) a combination of perfect fluid and scalar field, and
(ii) just a scalar field. In both cases, causal solutions are permitted, that
is, there is a condition between the metric parameters $m$ and $\omega$ such
that the critical radius becomes infinity. Therefore, the existence of causal
and non-causal Gödel-type solutions makes it clear that hybrid metric-Palatini
$f(\mathcal{R},T)$ gravity does not prevent causal anomalies in the form of
CTCs that are allowed in Einstein theory of gravity. It is important to note
that in Azizi  the Gödel-type solution and the causality violation is studied
in Palatini $f(R)$ gravity with a non-minimal curvature-matter coupling.
However, this model is a type of $f(R,{\cal L}_{m})$ gravity. Here, our model
is different, it is an extension of $f(R,T)$ in the hybrid formalism, while in
this reference only the Palatini formalism is considered. Therefore the
theories are different and the conditions required for a causal and non-causal
solution are different.

## Acknowledgments

This work by A. F. S. is supported by CNPq projects 430194/2018-8 and
313400/2020-2; J. S. Gonçalves thanks CAPES for financial support.

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