# Armed Conflict and Early Human Capital Accumulation: Evidence from
Cameroon’s
Anglophone Conflict

Hector Galindo-Silva
Pontificia Universidad Javeriana
Department of Economics, Pontificia Universidad Javeriana, E-mail:
<EMAIL_ADDRESS>Guy Tchuente
University of Kent School of Economics, University of Kent, E-mail:
<EMAIL_ADDRESS>
###### Abstract

This paper examines the impact of the Anglophone Conflict in Cameroon on human
capital accumulation. Using high-quality individual-level data on test scores
and information on conflict-related violent events, a difference-in-
differences design is employed to estimate the conflict’s causal effects. The
results show that an increase in violent events and conflict-related deaths
causes a significant decline in test scores in reading and mathematics. The
conflict also leads to higher rates of teacher absenteeism and reduced access
to electricity in schools. These findings highlight the adverse consequences
of conflict-related violence on human capital accumulation, particularly
within the Anglophone subsystem. The study emphasizes the disproportionate
burden faced by Anglophone pupils due to language-rooted tensions and
segregated educational systems.
Keywords: Anglophone Conflict, Cameroon, human capital accumulation,
educational outcomes, language-based conflicts.
JEL codes: I25, O15, D74, J24.

##  1 Introduction

Since 2016, Cameroon, a central African country, has been immersed in a
violent civil conflict between armed separatist groups and the government.
Referred to as the Anglophone Conflict or the Anglophone Crisis, this conflict
has been primarily concentrated in two regions where English is the
predominant language, encompassing approximately 20% of the country’s
landmass. Its consequences have been devastating, resulting in a significant
loss of life, with reported casualties exceeding 4,000, and causing the
displacement of more than 788,000 individuals, leading to widespread
population movements (Human Rights Watch, 2022).

The Anglophone Conflict in Cameroon is primarily driven by a linguistic
division rooted in the country’s colonial history. With English and French as
official languages,111It is worth noting that although Cameroon boasts over
200 local languages, French and English serve as the official languages,
aiming to unify its ethnically diverse population. the Francophone population
holds power in the government and elite circles, while some Anglophones have
long faced marginalization. The conflict, which began in 2016 as a peaceful
protest by English-speaking lawyers and teachers, has since escalated into a
violent clash between armed separatists and the Cameroonian
government.222Numerous reports indicate that English-speaking separatists,
advocating for the establishment of an independent English-speaking state
called Ambazonia, have terrorized civilians and engaged in attacks against
government forces (Human Rights Watch, 2021). Additionally, there are reports
of Cameroonian troops firing upon unarmed civilians and destroying their homes
(Human Rights Watch, 2018; Amnesty International, 2018).

A significant aspect of the violence associated with this conflict emerged in
2017 when Anglophone activists initiated ‘Operation Ghost Town.’ This protest
called for the closure of Anglophone schools in the north-west and south-west
regions to oppose the perceived assimilation of the Anglophone education
system into the French-speaking one.333As explained later in this paper, the
education system in Cameroon is divided into Anglophone and Francophone
subsystems. While Anglophone schools are primarily located in certain regions
of the country, there is geographic overlap with Francophone schools.
Nevertheless, despite this overlap, the call for school closures by Anglophone
activists specifically targeted Anglophone schools. The boycott has been
marked by violence, including student abductions and targeted killings of
school staff (Human Rights Watch, 2018, 2021; Amnesty International, 2018;
OCHA, 2021). These incidents have become critical flashpoints in the ongoing
conflict (The Guardian, 2018; The Washington Post, 2019).

This paper investigates the impact of violence associated with Cameroon’s
Anglophone Conflict on the acquisition of human capital by pupils. The unique
characteristics of this conflict, such as its localized geography and
linguistic grievances as the primary driver, present an exceptional
opportunity to analyze the consequences of armed conflicts associated with
language-related matters. Moreover, the coexistence of both Anglophone and
Francophone education systems in Cameroon, with overlapping geographic areas,
provides a natural laboratory to explore the implications for human capital
resulting from conflicts of this nature.

Our analysis utilizes two main data sources: pupil test scores in reading and
mathematics for Grades 2 and 6, obtained from the Programme d’Analyse des
Systemes Educatifs de la CONFEMEN (PASEC) for the years 2014 and 2019. This
data includes a representative sample of pupils from both the Francophone and
Anglophone subsystems, allowing for a comparison before and after the onset of
the Anglophone Conflict. Additionally, we rely on the Armed Conflict Location
& Event Data Project (ACLED) to gather information on violent events and
fatalities associated with the conflict. Spanning from 2000 to 2022, this
dataset provides comprehensive information including the date, location, and
actors involved in each event (such as Ambazonian rebels or the Cameroonian
army).

Our study focuses on pupils in the conflict-affected North-West and South-West
regions who are enrolled in the Anglophone subsystem. We compare this group,
referred to as the treatment group, with Anglophone subsystem pupils in
unaffected regions as the control group. Using a difference-in-differences
methodology, we analyze the causal impact of the Anglophone Conflict on human
capital accumulation. Our primary identification assumption is that, in the
absence of the conflict and given certain observable variables (included in
our analysis), the change in test scores between 2014 and 2019 would have been
the same for all pupils. We provide evidence consistent with this assumption
using data from pupils from the francophone subsystem during the same time
period.

Our main finding reveals a significant and negative causal relationship
between violence stemming from the conflict and the accumulation of human
capital among pupils in the North-West and South-West regions of the
Anglophone educational subsystem. Specifically, our estimates indicate that an
additional ten violent events involving Ambazonian rebels correspond to a 2.6%
decrease in reading test scores and a 2.1% decline in mathematics test scores
for students enrolled in the Anglophone educational subsystem. Furthermore,
for every ten deaths, there is a 1.9% decrease in language test scores and a
1.9% reduction in mathematics test scores for these pupils. These results
indicate that events resulting in loss of life have a detrimental impact on
the accumulation of human capital. Given the crucial role of early childhood
years in human capital development, these declines may have significant long-
term consequences (which we cannot directly study due to the temporal
limitation of our data).

Moreover, our analysis indicates that a rise in fatal events involving
Ambazonian rebels is linked to an increased rate of teacher absenteeism in the
Anglophone education subsystem. This finding implies that the presence of a
high-risk environment detrimentally impacts both the quality and quantity of
pupils’ learning experiences, ultimately leading to a diminished acquisition
of skills. Additionally, our research demonstrates a negative association
between conflict-related fatalities and schools’ access to electricity. This
observation implies that the presence of conflict exacerbates economic
hardships, potentially leading pupils to shift their focus from studying to
engaging in labor within either the legal or illegal economy. As a
consequence, this further widens the disparities in learning outcomes among
Anglophone pupils.

We also investigate the effect of the Anglophone conflict on educational
outcomes of pupils enrolled in the Francophone education subsystem (in the
Grand-Ouest and Grand-Nord regions). Due to the characteristics of the
Anglophone conflict, we anticipate that these students have been less directly
exposed to armed violence or have experienced it indirectly. Our analysis
reveals that fatal events involving Ambazonian rebels have considerably
smaller effects on their human capital accumulation compared to their
Anglophone counterparts. In fact, the observed effects are approximately 10
times smaller for these pupils.

Our research not only highlights the detrimental impact of violence related to
the Anglophone conflict on the accumulation of human capital but also sheds
light on a mechanism by which ethnic or linguistic disparities are exacerbated
within the context of armed conflict. Remarkably, this conflict either
originates from or is driven by the intent to reduce these differences. Our
study demonstrates that such exacerbation occurs not only in the short term,
where there is direct destruction of lives and infrastructure, particularly in
regions with a predominantly English-speaking population, but also in the long
term, as early-life experiences significantly shape individuals’ future
educational and economic trajectories (Heckman, Pinto, and Savelyev, 2013).
Specifically, we identify a discernible effect of the Anglophone conflict on
students within the Anglophone and Francophone educational subsystems.

The existing body of literature on the impact of armed conflict on human
capital is extensive. Previous studies, such as Bertoni, Di Maio, Molini, and
Nistico (2019) , have demonstrated the significant consequences of armed
conflicts on the accumulation of human capital. Furthermore, exposure to
violence during childhood has been linked to negative effects on various
aspects of individuals’ lives, including psychological well-being, health, and
education, as indicated by Jürges, Stella, Hallaq, and Schwarz (2022).

Numerous studies have provided evidence of the detrimental effects of
conflicts on school attainment, academic achievement, and educational
performance. These studies, including those by Akbulut-Yuksel (2014); Bertoni,
Di Maio, Molini, and Nistico (2019); Chamarbagwala and Morán (2011); Dabalen
and Paul (2014); Di Maio and Nandi (2013); Leon (2012); Pivovarova and Swee
(2015); Singh and Shemyakina (2016); Swee (2015); Verwimp and Van Bavel
(2014); Valente (2014); Brück, Di Maio, and Miaari (2019), have significantly
contributed to our understanding of the relationship between conflict and
human capital accumulation—the acquisition of skills, knowledge, and
experience through formal education that are valuable in the production
process.

However, the specific impacts of language-based violence remain poorly
understood. Existing research has only tangentially explored how ethnic
disparities in relate these effects, with limited attention paid to this
aspect in studies by Chamarbagwala and Morán (2011); Dabalen and Paul (2014);
Bertoni, Di Maio, Molini, and Nistico (2019). Importantly, none of these
studies have specifically focused on conflicts rooted in language identity,
which constitutes the primary focus of this paper.

Moreover, our study makes a significant contribution to this literature by
providing causal estimates of the effects of language-rooted conflicts. We
demonstrate that when educational systems are segregated,the minority group
disproportionately bears the burden of the conflict. To the best of our
knowledge, this paper is the first to propose a quasi-experimental design
using a difference-in-differences approach, complemented by high-quality
measures of individual skill levels, to examine the effects of the Anglophone
conflict in Cameroon. Consequently, our research contributes to the ongoing
interdisciplinary efforts aimed at assessing the consequences of the
Anglophone crisis and exploring potential peaceful resolutions (Willis,
Angove, and Mbinkar, 2023; Willis, McAulay, Ndeunyema, and Angove, 2019;
Nwati, 2020; Crawford, Kewir, Annan, and Beseng, 2021; Pelican, 2022;
Ousmanou, 2022; Pelican, Schumann, Plücken, and Drew, 2022).

The remainder of the paper proceeds as follows. Section 2 presents details
regarding the data sources and institutional background pertinent to early
human capital and the Anglophone Conflict. Section 3 describes our research
design. Section 4 examines the impact of armed conflict on early human capital
accumulation. Section 5 focuses on estimating the effects of the conflict on
the Francophone subsystem. Finally, Section 6 concludes the paper.

##  2 Institutional Background and Data Sources

###  2.1  Cameroon’s Primary Schools Education System

Cameroon’s primary education sector is served by three main providers: the
government, responsible for public schools, private institutions, and private
confessional schools such as Catholic, Islamic, and Protestant establishments.
These providers deliver education in the country’s two official languages:
French and English, each with its own separate primary education subsystem.

The primary education system in Cameroon is characterized by the presence of
both French and English subsystems in all ten regions of the country. However,
the majority of regions (eight out of ten) are predominantly French-speaking,
resulting in a higher concentration of French schools. On the other hand,
English-speaking regions like the North-West and South-West have a greater
proportion of educational institutions that offer instruction in the English
language.

Data collected during the 2014-2015 academic year shows that 71.6% of pupils
were enrolled in the Francophone education system, while 28.4% were enrolled
in the Anglophone education system (Alemnge, 2019). It is worth noting that
75% of pupils attend public schools, and there is a higher proportion of
pupils in the Francophone system compared to the Anglophone system (Alemnge,
2019).

Figure I: Enrolment Rate 1994 to 2019 from the World Bank data.

Regarding the evolution of the educational system, Figure I illustrates a
consistent upward trend in primary school enrollment rates for both boys and
girls from 1996 to 2016. Notably, there were periods of accelerated growth in
enrollment rates in 2000 and 2010. However, in 2016, there was a decline in
enrollment rates for both genders. This decline may be attributed to the
ongoing conflict in the Anglophone regions since 2016. While this graph
indicates a reduction in the percentage of primary school enrollment around
the onset of the Anglophone crisis, the precise causal impact of the conflict
on human capital accumulation remains to be accurately determined. Moreover,
further analysis is necessary to gain a comprehensive understanding of the
effects of the conflict on primary school enrollment rates. This paper will
focus on its impact on the accumulation of human capital.

###  2.2 Cameroon’s Anglophone Conflict

Cameroon gained independence from French and English colonial rule in 1960 and
1961, respectively, for the Francophone and Anglophone regions, and has
generally enjoyed a peaceful environment without military coups. However,
recent years have seen the emergence of conflicts in Cameroon that pose
threats to its stability. To illustrate this, Figure 2(a) provides a visual
representation of conflict-related events and fatalities since 2001.

Among the conflicts experienced by Cameroon in recent years, the Anglophone
Conflict has emerged as one of the most intense.444In addition to the
Anglophone conflict, Cameroon has faced the insurgency of Boko Haram in the
Far North, as well as complex challenges related to refugees and transborder
issues in the East region. (see Pelican, 2022, for a detailed background and
literature review on the Anglophone conflict). It primarily stems from a
linguistic division, with its epicenter located in the North West and South
West regions. The main actors involved are the Cameroonian army and the
Ambazonian separatists. The origins of the Anglophone Conflict can be traced
back to the historical legacy of Cameroon’s two official languages, English
and French, inherited from the colonial era. The institutional framework
established by former colonial powers, resulted in the dominance of French
speakers in Cameroon’s government and elite circles, and contributed to the
marginalization of the Anglophone population. This has given rise to various
social issues commonly referred to as the ‘Anglophone problem’ (Konings and
Nyamnjoh, 1997).

In 2016, the ‘Anglophone problem’ transformed into a full-fledged violent
conflict. This escalation originated from peaceful protests organized by
English-speaking lawyers and teachers. They were driven by their frustrations
with the government’s practice of assigning French-speaking judges and
teachers to English-speaking courts and schools. The English-speaking
community argued that this forced assimilation into Francophone legal and
educational systems. While the government initially acknowledged the need for
some reforms, it simultaneously repressed activists by imprisoning moderate
leaders and employing violence against protesters. As moderate voices were
silenced, more extremist factions emerged, advocating for complete separation
from Cameroon and demanding independence (The Washington Post, 2019; Pelican,
2022). Subsequently, the conflict intensified, with separatist groups
increasing their attacks on the military, prompting retaliatory actions by
Cameroonian troops. This retaliation has included firing upon unarmed
civilians and demolishing their homes.

One notable series of events in this conflict is known as the ‘Operation Ghost
Town,’ which was initiated by Anglophone activists in 2017. This ‘operation’
aimed to achieve the closure of schools in the north-west and south-west
regions as a protest against the perceived assimilation of the Anglophone
education system into the French-speaking one. Notably, this ‘operation’
specifically targeted Anglophone schools in these regions, while excluding
Francophone schools operating in the same areas. Acts of violence associated
with this ‘operation’ have included forced closures of schools, the abduction
of students, and targeted killings of principals and staff members within
these Anglophone schools.

In addition to including targeted violence against Anglophone schools, the
Anglophone conflict is characterized by two central features. Firstly, it
unfolds within a relatively concise timeframe, with violence stemming from
this conflict first appearing in 2017 and rapidly escalating in 2018. This
pattern is evident in Figure 4(b), which illustrates the distribution of
conflict-related events and fatalities specifically attributed to Ambazonian
separatists from 2011 to 2022.555Figure 4(b) also highlights the escalation of
the conflict in 2020 and 2021, leading to a substantial rise in both
fatalities and events. Despite the Major National Dialogue held in October
2019, which engaged multiple stakeholders, the desired political
transformation and decrease in violence within the Anglophone regions were not
realized, as depicted in Figure II.

The second notable characteristic of the Anglophone conflict is its
geographical localization, confined exclusively to the North West and South
West regions of Cameroon. This remarkable geographic limitation is visually
depicted in Map A1 of the Appendix. Given the significance of this aspect to
our empirical analysis, we will provide a more comprehensive exploration in
Section 2.4.

Figure II: Conflict-Related Events in Cameroon

(a) Any group involved

(b) Ambazonian Separatists involved

###  2.3 Data sources

The objective of this study is to examine the effect of violence associated
with the Anglophone conflict on the academic achievements of pupils in
different regions of Cameroon. In order to evaluate this impact, we utilize
reading and mathematics test scores from the Programme d’Analyse des Systemes
Educatifs de la CONFEMEN (PASEC) dataset. This dataset, collected in 2014 and
2019, comprises a representative sample of pupils enrolled in schools across
French-speaking African nations and can be accessed publicly through
registration on the official PASEC website.666The PASEC dataset can be
accessed at https://pasec.confemen.org/en/

The PASEC data on Cameroon consists of information from approximately 180
schools and around 4,887 pupils. This dataset encompasses both the francophone
and anglophone education subsystems and includes data from all ten regions of
the country. The PASEC data is obtained from a sample of individuals organized
into ‘strates’, which represent the regions and educational subsystem. In
2014, there were six strates, but in 2019, this number increased to
twelve.777For a comprehensive list of these strates, please refer to Table A2
in Appendix A.2.1. To ensure comparability between the two years, as explained
in detail in Appendix A.2.1, we focus on the 2014 strates. Table I presents
these strates along with their corresponding regions and educational
subsystems.

Table I: Strates along with their corresponding regions and educational subsystems Strate | Region | Educational system
---|---|---
1 (Zone Anglophone) |  North-West, South-West | Anglophone (public)
2 (Zone Anglophone) |  North-West, South-West | Anglophone (private)
3 (Zone Francophone) |  Littoral, West, Centre, Sud, Est,
Adamaoua, North, Extrem-Nort | Anglophone
4 (Grand-Ouest) |  West, North-West, South-West, Littoral | Francophone
5 (Grand-Centre) |  Centre, Sud, Est | Francophone
6 (Grand-Nord) |  Adamaoua, North, Extrem-Nort | Francophone

As shown in Table I, strates 1 and 2 (referred to as Zone Anglophone)
encompass pupils from both public and private schools within the Anglophone
subsystem residing in the North-West and South-West regions of Cameroon (the
English-speaking regions). As previously mentioned, these regions are known to
be the theatre of the Anglophone conflict. Strate 3 (or Zone Francophone)
comprises pupils from the Anglophone subsystem attending public schools in all
regions except the North-West and South-West (i.e., the French-speaking
regions of Cameroon). Strate 4 (or Grand-Ouest), includes pupils from the
Francophone subsystem from the West, North-West, South-West, and Littoral
regions. It is worth noting that Strate 4 overlaps geographically with Strate
1. Additionally, Strate 5 (or Grand-Centre) encompasses pupils selected from
the Centre, Sud, and Est regions. Lastly, Strate 6 (or Grand-Nord) comprises
pupils from the Adamaoua, North, and Extrem-North regions.888It should be
noted that strates 1 and 4 overlap in terms of the geographical regions they
cover.

Our study focuses on the Cameroonian regions that have been directly impacted
by violence associated with the Anglophone conflict. Specifically, we aim to
analyze the effects of this violence on Anglophone public schools. Given this
objective, the information pertaining to strates 1 and 3 holds particular
relevance for our research. In the subsequent discussion, we will elaborate on
the definition of strate 1 as our control group, while strate 3 will serve as
our treatment group. Descriptive statistics for our primary academic outcome
variables are presented separately for these two strates, as well as by year,
in Panels A and C of Table 2.3.

Table II: Descriptive Statistics | Strates 1 and 3 | Strate 1 | Strate 3
---|---|---|---
| obs. | mean | st.dev. | obs. | mean | st.dev. | obs. | mean | st.dev.
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
| 2014
_Panel A : Academic Outcomes (PASEC)_ |  |  |  |  |  |  |  |  |
Reading score | 2088 | 53.1 | 9.2 | 1578 | 51.3 | 8.8 | 510 | 58.6 | 8.4
Reading score (females) | 1058 | 53.6 | 9.2 | 802 | 51.8 | 8.5 | 256 | 59.4 | 8.9
Reading score (males) | 1030 | 52.6 | 9.2 | 776 | 50.9 | 9.0 | 254 | 57.7 | 7.7
Math score | 2088 | 50.4 | 8.7 | 1578 | 48.5 | 8.0 | 510 | 56.2 | 8.0
Math score (females) | 1058 | 50.6 | 8.7 | 802 | 48.6 | 7.8 | 256 | 56.9 | 8.5
Math score (males) | 1030 | 50.1 | 8.6 | 776 | 48.4 | 8.2 | 254 | 55.5 | 7.4

_Panel B : Armed Conflict (ACLED) _ Events involving any group 2088 34.4 44.7
1578 9.0 0.0 510 113.0 0.0 Events involving Amazonian rebels 2088 0.0 0.0 1578
0.0 0.0 510 0.0 0.0 Events involving Cameroonian army 2088 4.6 8.2 1578 0.0
0.0 510 19.0 0.0 Fatalities by any group 2088 344.4 569.0 1578 21.0 0.0 510
1345.0 0.0 Fatalities by Amazonian rebels 2088 0.0 0.0 1578 0.0 0.0 510 0.0
0.0 Fatalities by Cameroonian army 2088 43.2 76.1 1578 0.0 0.0 510 177.0 0.0
_Panel C : Academic Outcomes (PASEC)_ 2019 Reading score 2497 55.9 9.5 727
51.9 9.4 1770 57.6 9.1 Reading score (females) 1279 56.4 9.6 364 52.3 9.3 915
58.0 9.2 Reading score (males) 1218 55.5 9.5 363 51.5 9.5 855 57.2 9.0 Math
score 2497 52.4 8.3 727 48.9 7.5 1770 53.8 8.1 Math score (females) 1279 52.2
8.3 364 48.4 7.2 915 53.8 8.2 Math score (males) 1218 52.5 8.2 363 49.5 7.8
855 53.8 8.1  _Panel D : Armed Conflict (ACLED) _ Events involving any group
2497 435.9 12.3 727 455.0 0.0 1770 428.0 0.0 Events involving Amazonian rebels
2497 47.8 66.8 727 152.0 0.0 1770 5.0 0.0 Events involving Cameroonian army
2497 104.4 89.5 727 244.0 0.0 1770 47.0 0.0 Fatalities by any group 2497 591.1
131.3 727 796.0 0.0 1770 507.0 0.0 Fatalities by Amazonian rebels 2497 45.1
70.4 727 155.0 0.0 1770 0.0 0.0 Fatalities by Cameroonian army 2497 226.0
248.1 727 613.0 0.0 1770 67.0 0.0  Notes: The sample in all columns is
restricted to the period 2014 to 2019 and to pupils in the Anglophone
subsystem, from pubic schools. The sample in columns (1)-(3) includes strates
1 and 3 (see Table I for the definition of these strates). The sample in
columns (4)-(6) is limited to the strate 1 (Zone Anglophone, public schools)
and sample in columns (7)-(9) is limited to the strate 2 (Zone Francophone).
The data on education outcomes comes from PASEC. The data on conflict comes
from the ACLED.

To estimate the impact on pupil learning of Cameroon’s Anglophone conflict, we
integrate the PASEC data with data on conflict from the Armed Conflict
Location & Event Data Project (ACLED). The ACLED is a publicly available
database that offers extensive information on armed conflicts, political
violence, and protest events worldwide. It is a collaborative project between
researchers, analysts, and organizations working in the fields of political
violence, conflict resolution, and human rights.999The ACLED data can be
accessed publicly at https://acleddata.com

ACLED gathers and examines real-time information regarding the location,
participants, casualties, and other pertinent details of conflict incidents.
Their research encompasses various types of political violence, including
state-based conflict, non-state conflict, and one-sided violence. The data
undergoes regular updates, enabling the analysis of conflict trends and
patterns over time.

In our study, we utilize ACLED’s data specifically pertaining to Cameroon,
with a particular emphasis on the Anglophone conflict. Consequently, our scope
is limited to violence associated with the actors involved in this conflict
during the period from 2014 to 2019. The actors we identify as being
associated with this conflict include the Ambazonian Separatists, the
Ambazonian Defense Forces (ADF), and the armed and police forces of Cameroon.

As we describe in detail in Appendix A.2.2, in order to combine the ACLED data
with the PASEC data, we construct indicators of conflict-related violence at
the regional (or strate) level, however distinguishing between the armed
groups involved. Descriptive statistics for our primary conflict violence
variables are presented in Panels B and D of Table 2.3, organized by year.

###  2.4 Distribution of Conflict Related Events

Utilizing data extracted from the ACLED, Figure 3(a) provides an overview of
the temporal distribution of all conflict-related incidents across the
entirety of Cameroon’s territory over the past two decades. As mentioned
earlier, the majority of these incidents occurred after 2014. Complementing
this, Figure 3(b) specifically focuses on the number of fatalities resulting
from these events, revealing significant regional variation between 2000 and
2022. Notably, Strates 1 and 4, identified as the epicenter of the Anglophone
conflict, experienced the highest number of fatalities between 2014 and 2022.

Shifting our attention to events involving the Ambazonian Separatists, Figure
IV emphasizes the occurrence and intensity of such events exclusively within
Strates 1 and 4 (North-West and South-West regions). Importantly, these events
began escalating from 2016 onwards, and resulted in a significant number of
fatalities between 2014 and 2022.

Figure IV, when compared with Figure 3(a), highlights a crucial fact: prior to
2016: while other conflict-related events occurred before 2016, they took
place outside the strata of interest and involved armed actors unrelated to
the Anglophone conflict (e.g., Boko Haram in the Far North region). This
important finding, i.e. the localized nature of the Ambazonian separatists’
actions and the timing of their occurrence, will play a pivotal role in our
identification strategy.

Figure III: Conflict-Related Events and Fatalities Involving any Armed Group

(a) Events

(b) Fatalities

Figure IV: Conflict-Related Events and Fatalities Involving Ambazonian
Separatists

(a) Events

(b) Fatalities

##  3 Identification and Estimation Strategy

To assess the impact of violent conflict events during the Anglophone crisis
on pupil learning in Cameroon, we propose a quasi-experimental approach that
takes into account the level of exposure to conflict-related violence. Our
analysis will specifically examine fatalities in the North-West and South-West
regions, primarily involving actors such as the Ambazonian Separatists.
Initially, our main focus will be on students enrolled in the Anglophone
education subsystem. The presence of pupils in the anglophone subsystem both
in the rest of the country and in the North-West and South-West regions allows
for the design of a quasi-experiment. Subsequently, we will expand our
analysis to include the Francophone education subsystem, which will provide
supplementary results and will play a crucial role in evaluating our main
identification assumption.

Specifically, for our main model, we define the following two groups:

  * (i)

Pupils who resided in regions where there was violence related to the
Anglophone conflict and who were enrolled in the Anglophone education system.
These pupils belong to strate 1.

  * (ii)

Pupils who resided in regions unaffected by violence related to the Anglophone
conflict and who also were enrolled in the Anglophone education system. These
pupils belong to strate 3.

By comparing outcomes between two distinct groups - one exposed to conflict-
related violence and the other unexposed - we can estimate the effect of
violence on pupil learning using a quasi-experimental difference-in-
differences approach.

More formally, let $Y_{ist}$ denote an outcome of interest, representing math
or reading test scores for individual $i$ residing in region/stratum $r$ at
time $t$. Here, $t$ can take values of 2014 or 2019, while $r$ can be either 1
or 3, corresponding to the two treatment groups. The index $i$ ranges from 1
to $N$. To estimate the impact of the Anglophone conflict, we utilize the
following model:

$Y_{irt}=a_{0}+\alpha AMB\\_CE_{rt}+cMIL\\_CE_{rt}+bX_{irt}+a_{1}dT+u_{irt}$
(1)

where $AMB\\_CE_{rt}$ measures the intensity of conflict-related violent
events involving Ambazonian separatists, such as the number of fatalities, in
the region where individual $i$ resided at time $t$. The dummy variable $dT$
indicates whether the data was collected in 2014 or 2019. $MIL\\_CE_{rt}$
captures the intensity of conflict-related violent events involving the
Cameroon army.101010Fatalities involving the military are recorded in all
regions and time periods. The variable $X_{irt}$ represents a set of exogenous
characteristics, including factors like sex, age, and grade, and $u_{irt}$ is
the error term.

To clearly establish the main identification assumptions of the model in
equation (1), it is important to note that $AMB\\_CE_{rt}$ is only observed in
the North-West and South-West regions and only in 2019. Hence, we can rewrite
equation (1) as follows:

$Y_{irt}=a_{0}+a_{1}dT+\alpha dT\times
AMB\\_CE_{r}+cMIL\\_CE_{rt}+bX_{irt}+dAMB\\_CE_{r}+u_{irt}$ (2)

In equation (2), the variable $AMB\\_CE_{r}$ represents the intensity of
conflict-related violent events in region $r$. It is important to note that
for each individual $i$, region $r$, and time $t$, the expression $dT\times
AMB\\_CE_{r}$ in (2) is equivalent to $AMB\\_CE_{rt}$ in (1). This equivalence
implies that the model in (1) can be seen as a Difference-in-Difference style
representation, which is essentially the same as the model (2).

Assuming that military-induced violence, including violent events and
fatalities, has a consistent impact on the outcome variable $Y_{irt}$ across
all regions, the parameter $\alpha$ captures the effect of the Anglophone
conflict on $Y_{irt}$. Specifically, it measures the average effect of
conflict violence, whether in terms of events or fatalities, on pupils
enrolled in the Anglophone subsystem and residing in the North-West and South-
West regions. This parameter represents the average treatment on the treated
(ATT), providing an understanding of the average differential impact
experienced by these pupils.

To evaluate this impact, we rely on the parallel trend assumption, which
posits that in the absence of conflict-related events involving Ambazonian
separatists, the trajectory of human capital accumulation would have been the
same for pupils in the Anglophone system, regardless of their region of
residence, between 2014 and 2019. This assumption is based on the lack of
regional reforms in the educational system in Cameroon during that time
period.

Moreover, we conduct estimations to demonstrate that the violence associated
with the conflict has negligible or no effect on pupils in the Francophone
subsystem. These estimations serve as additional evidence, highlighting the
specific influence of the Anglophone conflict on pupil outcomes and
illustrating the divergent experiences between the Anglophone and Francophone
subsystems.

In this study, we adopt a similar model to examine the impact of the conflict
on human capital accumulation, explore potential transmission mechanisms, and
conduct identifying assumption tests. Depending on the research question at
hand, such as investigating transmission mechanisms, analyzing the effect of
the conflict on human capital accumulation, or conducting identifying
assumption tests, we adapt the outcome variable and regions while maintaining
the core explanatory variables.

##  4 Estimation of the Armed-Conflict Effects

###  4.1 Effects on Human Capital Accumulation

In this section, we present the key findings regarding the impact of the
Anglophone Armed conflict on early human capital accumulation. To capture the
varying intensity of violence, we examine the estimated effects using two
variables: the level of violence and the rate per 100,000 inhabitants.

The results presented in Table LABEL:tab:conflict_full specifically focus on
students enrolled in the Anglophone education subsystem. These findings
highlight that conflict-related violence in the Anglophone region
significantly impedes the accumulation of human capital among students. Our
analysis reveals that an additional ten violent events involving the
Ambazonian separatists resulted in a 2.5% decrease in language test scores and
a 2.1% decrease in mathematics test scores. Similarly, a ten-unit increase in
conflict-related deaths led to a 1.9% decline in language test scores and a
1.9% reduction in mathematics test scores. Notably, these negative effects
persist even when accounting for the region’s population by measuring event
occurrence and fatalities per 100,000 inhabitants.

Table LABEL:tab:conflit_hetero in the Appendix examines the heterogeneous
effects based on the grade and gender of the students. We expect a potentially
stronger effect on grade six pupils due to the conflict’s duration covering a
significant portion of their primary school education and the challenges they
faced in a war-affected learning environment. In contrast, second-grade pupils
experienced conflict throughout their entire primary schooling. However, the
estimated effects on grade six pupils show a slightly more negative impact,
although the difference is not statistically significant. Additionally, female
pupils also exhibit a slightly more negative effect of violence on their human
capital, although this difference is not statistically significant, at a 5%
level, either.

Dependent variable: math score Number Rate Number Rate Events involving
Ambazonian rebels -0.211∗∗∗ -38.420∗∗∗ (0.038) (7.284) Fatalities by
Ambazonian rebels -0.191∗∗∗ -34.753∗∗∗ (0.029) (5.643) R-sq 0.099 0.099 0.121
0.121 Observations 4585 4585 4585 4585  Notes: All columns in report the
estimates from Eq. (1). Rate is per 100 000 inhabitants. Robust standard
errors (in parentheses) are clustered by school. Do not include fixed effects
by school. * denotes statistically significant estimates at 10%, ** denotes
significant at 5% and *** denotes significant at 1%.

###  4.2 Discussion of Transmission Mechanisms

The findings presented in the previous section reveal a detrimental impact of
violent events stemming from the Anglophone conflict on the early accumulation
of human capital. In this section, we examine potential mechanisms through
which the acquisition of language and mathematics skills by pupils is
hindered, shedding light on their diminished learning outcomes. We explore and
discuss two distinct yet interconnected transmission channels.

Firstly, we analyze the effect of conflict-related fatalities on teachers’
absenteeism, as illustrated in Table LABEL:tab:mechanism (columns 1 and 2).
These estimates indicate that an increase in fatalities resulting from events
involving Ambazonian separatists leads to a higher probability of teacher
absenteeism. This decrease in teachers’ presence is observed in both grades 2
and 6. We contend that, from the perspective of educators and adults, the
heightened risk associated with the conflict detrimentally affects the quality
and quantity of pupils’ learning experiences, thereby impeding their skill
acquisition. For instance, the presence of conflict may trigger temporary
migration to safer areas among both pupils and teachers, resulting in learning
gaps.

Secondly, we posit that the presence of conflict contributes to economic
hardship, potentially diverting pupils from their studies to engage in work
within the legal or illegal economy, further exacerbating learning gaps. Table
LABEL:tab:mechanism (column 3) provides estimates that shed light on this
relationship. They reveal that an increase in conflict-related fatalities in
events involving Ambazonian separatists decreases the likelihood of a school
having access to electricity. This suggests that the violence associated with
the conflict may have led to the destruction of existing infrastructure or
created economic challenges that prioritize other urgent needs over
educational resources.

##  5 Identifying Assumptions Checks: Effects on the Francophone Subsystem

In this section, we analyze the impact of the Anglophone conflict on
Francophone pupils, focusing on two key objectives. Firstly, we assess the
implications arising from the existence of two separate educational systems in
the context of this conflict. By closely examining the experiences of
Francophone pupils, we aim to shed light on any disparities or divergences
that may emerge as a result of this unique socio-political situation.
Secondly, we use this analysis to validate the plausibility of our identifying
assumptions.

In our research design, outlined in Section 3, we primarily studied a sample
of pupils within the Anglophone subsystem. Now, our attention turns to
evaluating the influence of the Anglophone conflict on pupils belonging to the
Francophone subsystem. To achieve this, we designate the Grand-Ouest (4) and
Grand-Nord (6) strata as the treatment and control groups, respectively.

The estimation results, presented in Table LABEL:tab:placebo, yield
significant insights into the effect of conflict-related violence on human
capital accumulation within the Francophone subsystem. In particular, we find
that an increase in violent events or deaths does not have a significant
impact on the level of human capital accumulated in both language and
mathematics. Specifically, we observe a negligible decrease in mathematics
skill acquisition and a slight increase in language skill acquisition.
However, these effects are minimal compared to those observed in the sample of
Anglophone pupils residing in conflict-affected regions (North-West and South-
West). Thus, pupils in the Francophone educational subsystem were largely
unaffected. Indeed, the effects of conflicts are approximately 10 times
smaller than the effects observed in the Anglophone educational subsystem
sample.

Dependent variable: math score Number Rate Number Rate Events involving
Ambazonian rebels -0.019∗∗ -0.703∗∗ (0.008) (0.270) Fatalities by Ambazonian
rebels -0.020∗∗∗ -0.694∗∗∗ (0.008) (0.264) R-sq 0.033 0.035 0.035 0.035
Observations 4585 4585 4585 4585  Notes: All columns in report the estimates
from Eq. (1). Rate is per 100000 inhabitants. Robust standard errors (in
parentheses) are clustered by school. No model includes fixed effects by
school. * denotes statistically significant estimates at 10%, ** denotes
significant at 5% and *** denotes significant at 1%.

As mentioned earlier, the results in Table LABEL:tab:placebo not only provide
new and relevant evidence on the connection between the Anglophone conflict
and the educational outcomes of students in the Francophone education
subsystem, but also support the causal interpretation of our estimated effects
by aligning with our primary identification assumption: the parallel trend
assumption.

Specifically, Table LABEL:tab:placebo demonstrates that when comparing
educational outcomes between 2014 and 2019, within two groups that closely
resemble those examined in our baseline specification, but with significantly
lower exposure to the Anglophone conflict (as anticipated for students
residing in the same regions as those included in the baseline specification,
yet enrolled in the Francophone educational subsystem which was not directly
affected by the Anglophone conflict due to its concentrated impact on
Anglophone schools),111111It is worth noting that while some pupils from the
Grand-Ouest strata may originate from conflict-affected regions, such as the
North-West and South-West, the majority of them come from the West and
Littoral regions. there is no substantial effect of the Anglophone conflict on
the accumulation of human capital.

To summarize, this section’s results make two important contributions.
Firstly, they provide crucial evidence that the observed disparities found in
Section 4.1 can be attributed to the Anglophone conflict rather than other
confounding factors. Secondly, these findings shed light on how linguistic
disparities are further exacerbated within the context of armed conflict.
Specifically, they demonstrate that despite the Anglophone conflict
originating from the existence of such disparities and the aim to reduce them,
the conflict paradoxically leads to an increase in these disparities. This
increase is evident not only in the short term, where there was direct loss of
lives and destruction of infrastructure, particularly in English-speaking
regions, but also in the long term, affecting the academic outcomes of a
generation of Anglophone students, which significantly lag behind those of
their Francophone counterparts, and it is likely that these negative early-
life experiences play a crucial role in shaping individuals’ future
educational and economic trajectories.

##  6 Conclusion

This paper aims to quantify the consequences of conflict-related violence in
the Anglophone crisis on human capital accumulation, with a specific focus on
the educational outcomes of Cameroonian Anglophone pupils. By utilizing high-
quality individual-level data on pupils in Cameroon collected by the PASEC
from 2014 to 2019, combined with precise information on conflict-related
violent events from the ACLED, we present a causal estimate of the Anglophone
conflict’s effects.

Employing a difference-in-difference design, our findings reveal that an
additional hundred violent events linked to the Ambazonian rebels result in a
2.5% reduction in language test scores and a 2.1% decrease in mathematics test
scores. Likewise, a hundred more conflict-related deaths lead to a 1.9%
decline in language test scores and a 1.9% reduction in mathematics test
scores. The burden of the conflict is disproportionately borne by pupils
within the Anglophone educational subsystem. Empirical evidence further
suggests that conflict violence increases absenteeism among both teachers and
pupils, thereby diminishing the quality and quantity of learning interactions
and exacerbating the learning gap. Additionally, we provide evidence of
heightened economic hardship stemming from conflict-related violence.

The short-term impact of the Anglophone conflict on human capital accumulation
is substantial, and the consequences could extend beyond the immediate
learning gap, given the critical role of early human capital in adults’
health, economic, and social well-being. The evidence presented in this paper
offers valuable insights for post-conflict reconstruction efforts in the realm
of human capital capacity building in the Anglophone conflict in Cameroon.

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##  Appendix A Appendix

###  A.1 Additional Figures and Tables

Figure A1: Fatalities by Ambazonian Rebels between 2014 and 2019

Dependent variable: math score Number Rate Number Rate Fatalities by
Ambazonian rebels -0.178∗∗∗ -29.731∗∗∗ -0.178∗∗∗ -32.825∗∗∗ (0.044) (8.496)
(0.029) (5.662) Fatalities by Ambazonian rebels $\times$ grade 6 -0.015 -6.351
(0.044) (8.468) Fatalities by Ambazonian rebels $\times$ female -0.029∗ -3.834
(0.016) (3.037) R-sq 0.125 0.125 0.106 0.121 Observations 4585 4585 4585 4585
Notes: All columns in report the estimates from Eq. (1). Rate is per 100000
inhabitants. Robust standard errors (in parentheses) are clustered by school.
Do not include fixed effects by school. * denotes statistically significant
estimates at 10%, ** denotes significant at 5% and *** denotes significant at
1%.

###  A.2 Education and Conflict Data

####  A.2.1 Data on educational outcomes

The data used in this paper to examine educational outcomes is obtained from
the Programme d’analyse des Systèmes éducatifs de la Confemen (PASEC). This
dataset encompasses information for two distinct time periods, specifically
2014 and 2019, and concentrates on two specific grade levels, namely grade 2
and grade 6. It encompasses data concerning the regional geographical location
of students and their enrollment in either the francophone or anglophone
educational subsystem. In this study, we use the term ‘strate’ to refer to the
combination of a student’s region and their educational subsystem.

Importantly, the information concerning these "strates" differs between the
two available years of data. In 2019, PASEC classifies students into 12
distinct strates, as depicted in column 1 of Table A2. Conversely, for the
year 2014, the classification comprises only 6 strates, as shown in column 2
of Table A2.

Table A2: Strates in 2014 and 2019 (1) | (2)
---|---
Strates in 2019 | Strates in 2014
1 Ouest francophone | 1 Zone anglophone publique
2 Littoral francophone | 2 Zone anglophone privée
3 Centre francophone | 3 Zone francophone
4 Est francophone | 4 Grand-Ouest
5 Sud francophone | 5 Grand-Centre
6 Adamaoua francophone | 6 Grand-Nord
7 Extreme-Nord francophone |
8 Nord francophone |
9 Ouest anglophone |
10 Centre anglophone |
11 Littoral anglophone |
12 Reste anglophone |

In our identification strategy we exploit regional, education subsystem and
temporal variations. Therefore, understanding the process of matching
‘strates’ between different years is crucial. It is important to note that the
matching between ‘strates’ across years is feasible because the ‘strates’
defined in one year are always subsets of the ‘strates’ defined in another
year. This strict overlapping allows us to successfully match strata from one
year to another.

Table A3 provides a comprehensive breakdown of the methodology employed for
the matching process. Specifically, the table delineates the mapping between
different elements. The initial eight lines pertain to the francophone system,
illustrating the precise geographic correspondence between each of the first
eight strates in column (1) of Table A2 and a single strate in column (2).
Similarly, lines 9 to 11 delineate the mapping for the anglophone system.

Table A3: Strate matching between strates in 2014 and 2019 (1) |  | (2)
---|---|---
Strates in 2019 dataset |  | Strates in 2014 dataset
1 Ouest francophone | $\rightarrow$ | 4 Grand-Ouest
2 Littoral francophone | $\rightarrow$ | 4 Grand-Ouest
3 Centre francophone | $\rightarrow$ | 5 Grand-Centre
4 Est francophone | $\rightarrow$ | 5 Grand-Centre
5 Sud francophone | $\rightarrow$ | 5 Grand-Centre
6 Adamaoua francophone | $\rightarrow$ | 6 Grand-Nord
7 Extreme-Nord francophone | $\rightarrow$ | 6 Grand-Nord
8 Nord francophone | $\rightarrow$ | 6 Grand-Nord
9 Ouest anglophone | $\rightarrow$ | 3 Zone francophone
10 Centre anglophone | $\rightarrow$ | 3 Zone francophone
11 Littoral anglophone | $\rightarrow$ | 3 Zone francophone
12 Reste anglophone | $\rightarrow$ | 1 Zone anglophone publique
|  | 2 Zone anglophone privée
Figure A2: Map of Cameroon with representation of the different zones
(strates)

(a) Francophone subsystem

(b) Anglophone subsystem

The geographic correspondences, specific to an educational subsystem, are
illustrated in Figure A2. The left map (map 2(a)) displays the matching for
the Francophone subsystem, while the right map (map 2(b)) represents the
matching for the anglophone subsystem. Additionally, referring to line 12 in
Table A3, which represents two anglophone subregions in 2019, these subregions
were classified as the ’Zone anglophone’ (both public and private) in 2014. As
the 2019 data does not differentiate between these two subregions, we
associate them with the ’reste anglophone’ strate in 2019.

####  A.2.2 Conflict data

The paper relies on conflict data obtained from the Armed Conflict Location
and Event Data Project (ACLED). ACLED offers comprehensive information on a
wide range of violent and non-violent events involving political agents. Our
analysis focuses on the period between 2014 and 2019. The conflict-related
event data includes various aspects such as event type (e.g., battles,
violence against civilians, explosions), date (day, month, year), location
(geographic coordinates and municipality), and the armed actors involved
(e.g., Ambazonian Separatists, Military Forces of Cameroon).

As the educational outcome data (from PASEC) only provides information at the
‘strate’ and year levels (as discussed in Sub-section A.2.1), we initially
aggregated the conflict-related events indicators at these levels. For each
year and strate, we tallied the number of events and fatalities by armed group
and type of violence. This allowed us to create conflict-related indicators
that could be matched with the educational outcomes data.

Next, we focused on identifying the specific armed actors directly linked to
Cameroon’s anglophone conflict. Our analysis distinguished between two types
of armed groups: rebel groups and groups associated with state forces. The
rebel groups identified were the Ambazonian Separatists and the Ambazonia
Defense Forces (ADF). The groups associated with state forces were the
Military Forces of Cameroon and the Police Forces of Cameroon. We then
included the number of conflict events and fatalities within these groups,
encompassing all types of violence. Importantly, the violence perpetrated by
these two types of armed groups accounted for 100% of the conflict-related
events that took place within the regions (or strates) concentrated in our
study, as stated in the main text.


# What makes a shear-free spherical perfect fluid be inhomogeneous with tidal
effects?

Jonathan Hakata<EMAIL_ADDRESS>Astrophysics Research Centre, School
of Mathematics, Statistics and Computer Science, University of KwaZulu–Natal,
Private Bag X54001, Durban 4000, South Africa  Rituparno Goswami
<EMAIL_ADDRESS>Astrophysics Research Centre, School of Mathematics,
Statistics and Computer Science, University of KwaZulu–Natal, Private Bag
X54001, Durban 4000, South Africa  Chevarra Hansraj<EMAIL_ADDRESS>Astrophysics Research Centre, School of Mathematics, Statistics and Computer
Science, University of KwaZulu–Natal, Private Bag X54001, Durban 4000, South
Africa  Sunil D. Maharaj<EMAIL_ADDRESS>Astrophysics Research Centre,
School of Mathematics, Statistics and Computer Science, University of
KwaZulu–Natal, Private Bag X54001, Durban 4000, South Africa

###### Abstract

This is an important and natural question as the spacetime shear,
inhomogeneity and tidal effects are all intertwined via the Einstein field
equations. However, as we show in this paper, such scenarios are possible for
limited classes of equations of state that are solutions to a highly non-
linear and fourth order differential equation. To show this, we use a
covariant semitetrad spacetime decomposition and present a novel geometrical
classification of shear-free Locally Rotationally Symmetric (LRS-II) perfect
fluid self-gravitating systems, in terms of the covariantly defined fluid
acceleration and the fluid expansion. Noteworthily, we deduce the governing
differential equation that gives the possible limited equations of state of
matter.

Keywords:  spacetime decomposition, shear-free fluids, equation of state

##  I Introduction

Spherical shear-free self-gravitating systems of perfect fluids are perhaps
the most well studied systems in the historical quest for exact solutions of
the Einstein field equations, and as rightly pointed out by exact , is rich in
rediscoveries. All the known solutions are presented in McVittie ; Kustaan ;
Wyman ; wafo2 , while in all later works these solutions are always contained
as special cases. Generally, there are three different approaches to find
exact solutions of the field equations for shear-free and spherical perfect
fluid distributions. The first approach is via an adhoc ansatz for one of the
metric functions, while the second one is to look for symmetries including Lie
point symmetries, contact symmetries and Noether symmetries in the field
equations. The third approach is more general and rigorous, where the
solutions have Painlevé properties Wyman . As a matter of fact, all the known
solutions belong to this latter class.

In this paper, we investigate this well studied system through a more
geometrical perspective. In fact, we generalize the spacetime geometry to
Locally Rotationally Symmetric (LRS-II) spacetimes vanElstEllis , of which
spherical symmetry is a subclass. By covariantly decomposing the spacetime
using the fluid flow congruence and a preferred spherical congruence which is
guaranteed by LRS-II symmetry, we classify all the possible solutions of the
field equations in terms of the covariantly defined fluid acceleration and the
fluid expansion.

Shear-free matter distributions have been used to model both static and
radiating stars, justifying its assumption. Spherical symmetry simplifies the
relativistic equations even further. Spherically symmetric shear-free
solutions have been used to model many physical applications leading to many
classes of solutions stephani ; srivastava ; deng ; sussman1 ; sussman2 ;
maharaj1 ; havas ; maharaj2  where it is demonstrated that zero shear
restricts the possible choices of equations of state sussman1 ; sussman2 .
Recent applications of this model relate to gravitational collapse sharif ;
pinheiro ; naidu ; charan ; symmetry analysis wafo ; gumede  and methods msomi
; moopanar ; and particular solutions wagh ; bhatti ; shah .

Interestingly, our analysis sheds new light on the class of solutions where
fluid acceleration and expansion are strictly non-vanishing. It can easily be
seen that the Weyl curvature is also non-vanishing for this class. This class
is interesting as it is dynamical and necessarily inhomogeneous but still
shear-free. It is well known that spacetime shear and the electric part of the
Weyl form a feedback loop with each other. The electric part of the Weyl is a
source term for the shear evolution equation while the shear is the source
term for the evolution of the electric part of the Weyl.

_Therefore, constraining one of the positive feedback loop quantities to
vanish identically, while the other is strictly non-vanishing, would imply a
stringent constraint on other geometrical and thermodynamical quantities in
the spacetime._

From our analysis, we transparently show that indeed such a stringent
constraint exists on the possible equations of state that give rise to this
class. We find the governing ODE, which is highly nonlinear, the solution to
which gives the possible classes of equations of state that may give rise to
this class of solutions. We present a numerical solution for this ODE and
clearly show that the usual equations of state that are linear, or a finite
combination of power law solutions, will not generate the given class.

The paper is organized as follows: In the next two sections we discuss the
covariant semitetrad decomposition of the spacetime manifold, and the field
equations written in terms of geometrical and thermodynamical variables that
emerge due to the decomposition. In the subsequent two sections we discuss the
special case of shear-free spacetimes, and show how all possible solutions can
be classified in terms of the fluid acceleration and the fluid expansion. The
next section is dedicated to the interesting class of inhomogeneous and shear-
free spacetimes, and the differential equation that governs the equation of
state of the perfect fluid. Finally, we conclude our results in the last
section.

##  II 1+1+2 formalism

The 1+1+2 covariant approach, first introduced by Greenberg in greenberg ,
further reviewed by van Elst and Ellis vanElstEllis  and subsequently expanded
by Clarkson and Barrett clarksonbarrett ; clarkson2007  is an extension of the
1+3 formalism ehlers ; ellis1+3 . The latter is one of the most widely used
tetrad approaches which formulates the equations of general relativity, using
a timelike congruence, as first order differential equations unlike the
coordinate approach involving second order partial derivatives of the metric
functions.

In the 1+3 covariant approach the essential ‘time’ coordinate is separated
from the 3-‘space’ so that the metric tensor takes the form

$g_{ab}=h_{ab}-u_{a}u_{b}.$ (1)

Two important derivatives are defined. The covariant time derivative along the
observers’ worldlines, denoted by ‘ ⋅ ’, is defined using the vector
${u^{a}}$, as

$\dot{Z}^{a...b}{}_{c...d}=u^{e}\nabla_{e}Z^{a...b}{}_{c...d},$ (2)

for any tensor ${Z^{a...b}{}_{c...d}}$. The fully orthogonally projected
covariant spatial derivative, denoted by ‘ ${D}$ ’, is defined using the
spatial projection tensor ${h_{ab}}$, as

$D_{e}Z^{a...b}{}_{c...d}=h^{r}{}_{e}h^{p}{}_{c}...h^{q}{}_{d}h^{a}{}_{f}...h^{b}{}_{g}\nabla_{r}Z^{f...g}{}_{p...q},$
(3)

with total projection on all the free indices. In the context of Locally
Rotationally Symmetric class II (LRS-II) spacetimes vanElstEllis , the
geometrical quantities defined for the timelike congruence are the expansion
scalar $(\Theta=D_{a}u^{a})$, the acceleration 3-vector
$(\dot{u}^{a}=u^{b}\nabla_{b}u^{a})$ and the shear 3-tensor
$[\sigma_{ab}=\left(h^{c}{}_{(a}h^{d}{}_{b)}-\frac{1}{3}h_{ab}h^{cd}\right)D_{c}u_{d}]$.
The timelike congruence uniquely defines the electric part of the Weyl tensor
$(E_{ab}=C_{abcd}u^{c}u^{d}=E_{<ab>})$ and the magnetic part vanishes
identically. The energy momentum tensor is also decomposed according to the
timelike congruence to produce the energy density $(\mu=T_{ab}u^{a}u^{b})$,
the isotropic pressure $(p=\frac{1}{3}h_{ab}T^{ab})$, the heat flux 3-vector
$(q_{a}=q_{<a>}=-h^{c}{}_{a}T_{cd}u^{d})$ and the anisotropic stress 3-tensor
$(\pi_{ab}=T_{cd}h^{c}{}_{<a}h^{d}{}_{b>})$. Angle brackets denote orthogonal
projections of covariant time derivatives along ${u^{a}}$ as well as represent
the projected, symmetric and trace-free part of tensors as follows

$\displaystyle v_{<a>}$ $\displaystyle=$ $\displaystyle
h^{b}{}_{a}\dot{v}_{b},$ $\displaystyle Z_{<ab>}$ $\displaystyle=$
$\displaystyle\left(h^{c}{}_{(a}h^{d}{}_{b)}-\frac{1}{3}h_{ab}h^{cd}\right)Z_{cd}.$
(4)

All these quantities have a direct geometrical and physical meaning and are
described by tensorial quantities that remain valid in all coordinate systems
which is the principal advantage of using spacetime decomposition.

A further decomposition of the 1+3 quantities gives rise to the useful 1+1+2
set of variables featured in clarksonbarrett . The splitting of the LRS-II
spacetime is performed with respect to the timelike unit vector $u^{a}$
($u^{a}u_{a}=-1$) as well as the isolation of a preferred spatial direction
$e^{a}(e^{a}e_{a}=1)$ chosen orthogonal to $u^{a}(u^{a}e_{a}=0)$. The metric
tensor (1) is decomposed further into

$g_{ab}=-u_{a}u_{b}+e_{a}e_{b}+N_{ab},$ (5)

where $N_{ab}$ is the 2-dimensional metric on the spherical 2-shell. We
introduce two new derivatives for any tensor ${\Psi_{a...b}{}^{c...d}}$:

$\displaystyle\hat{\Psi}_{a...b}{}^{c...d}$ $\displaystyle\equiv$
$\displaystyle e^{f}D_{f}\Psi_{a...b}{}^{c...d},$ (6)
$\displaystyle\delta_{f}\Psi_{a...b}{}^{c...d}$ $\displaystyle\equiv$
$\displaystyle
N_{f}{}^{j}N_{a}{}^{l}...N_{b}{}^{g}N_{h}{}^{c}...N_{i}{}^{d}D_{j}\Psi_{l...g}{}^{h...i},$
(7)

defined by the spatial congruence ${e^{a}}$. The hat-derivative (6) is the
spatial derivative along the ${e^{a}}$ vector field in the surfaces orthogonal
to ${u^{a}}$, and the delta-derivative (7) is the projected spatial derivative
onto the 2-sheet, with projection on every free index. In a similar way, there
are geometrical quantities generated for the preferred spatial congruence with
the only non-vanishing quantity for LRS-II spacetimes being the 2-sheet volume
expansion $(\phi=\delta_{a}e^{a})$. Using the preferred spatial congruence we
can then extract a set of covariant scalars that completely govern the
dynamics of the system in the following manner

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\dot{u}^{a}e_{a},\quad\Sigma=\sigma_{ab}e^{a}e^{b},\mathcal{E}=E_{ab}e^{a}e^{b},$
$\displaystyle Q$ $\displaystyle=$ $\displaystyle
q^{a}e_{a},\quad\Pi=\pi_{ab}e^{a}e^{b}.$ (8)

Evidently the 1+1+2 decomposition method is well suited to spacetimes that
have a preferred spatial direction such as LRS-II spacetimes where $u^{a}$ and
$e^{a}$ are hypersurface orthogonal. These spacetimes have the inherent
property that there exists a unique preferred spacial direction at each point
that creates a local axis of symmetry. Hence all the physics and geometry of
the spacetime are described by well defined kinematic and dynamic scalar
variables that generate the field equations. The 1+1+2 covariant method has
generated new results in studies involving LRS spacetimes sayuri , the Kerr
spacetime kerr , the Vaidya spacetime vaidya  and a general spacetime
admitting conformal symmetry CKV .

##  III Field equations for LRS-II spacetimes

As pointed out by Goswami and Ellis birkhoff , the only variables in the 1+1+2
formalism which characterize the kinematics are

$\mathcal{D}=\left\\{\mathcal{A},\Theta,\phi,\Sigma,\mathcal{E},\mu,p,\Pi,Q\right\\}.$
(9)

For these variables the propagation, evolution and constraint equations can be
derived as

$\displaystyle\hat{\phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\phi^{2}+\left(\frac{1}{3}\Theta+\Sigma\right)\left(\frac{2}{3}\Theta-\Sigma\right)$
(10) $\displaystyle-\frac{2}{3}\mu-\frac{1}{2}\Pi-\mathcal{E},$
$\displaystyle\hat{\Sigma}-\frac{2}{3}\hat{\Theta}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\phi\Sigma-Q,$ (11)
$\displaystyle\hat{\mathcal{E}}-\frac{1}{3}\hat{\mu}+\frac{1}{2}\hat{\Pi}$
$\displaystyle=$
$\displaystyle-\frac{3}{2}\phi\left(\mathcal{E}+\frac{1}{2}\Pi\right)$ (12)
$\displaystyle+\left(\frac{1}{2}\Sigma-\frac{1}{3}\Theta\right)Q,$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-\left(\Sigma-\frac{2}{3}\Theta\right)\left(\mathcal{A}-\frac{1}{2}\phi\right)+Q,$
(13) $\displaystyle\dot{\Sigma}-\frac{2}{3}\dot{\Theta}$ $\displaystyle=$
$\displaystyle-\mathcal{A}\phi+2\left(\frac{1}{3}\Theta-\frac{1}{2}\Sigma\right)^{2}-\mathcal{E}$
(14) $\displaystyle+\frac{1}{3}(\mu+3p)+\frac{1}{2}\Pi,$
$\displaystyle\dot{\mathcal{E}}-\frac{1}{3}\dot{\mu}+\frac{1}{2}\dot{\Pi}$
$\displaystyle=$
$\displaystyle\left(\frac{3}{2}\Sigma-\Theta\right)\mathcal{E}+\frac{1}{4}\left(\Sigma-\frac{2}{3}\Theta\right)\Pi$
(15) $\displaystyle+\frac{1}{2}\phi
Q-\frac{1}{2}(\mu+p)\left(\Sigma-\frac{2}{3}\Theta\right),$
$\displaystyle\dot{\mu}+\hat{Q}$ $\displaystyle=$
$\displaystyle-\Theta(\mu+p)-\frac{3}{2}\Sigma\Pi$ (16)
$\displaystyle-(\phi+2\mathcal{A})Q,$ $\displaystyle\dot{Q}+\hat{p}+\hat{\Pi}$
$\displaystyle=$
$\displaystyle-\left(\frac{3}{2}\phi+\mathcal{A}\right)\Pi-(\mu+p)\mathcal{A}$
(17) $\displaystyle-\left(\frac{4}{3}\Theta+\Sigma\right)Q,$
$\displaystyle\hat{\mathcal{A}}-\dot{\Theta}$ $\displaystyle=$
$\displaystyle-(\mathcal{A}+\phi)\mathcal{A}+\frac{1}{3}\Theta^{2}$ (18)
$\displaystyle+\frac{3}{2}\Sigma^{2}+\frac{1}{2}\left(\mu+3p\right),$

according to clarkson2007 . The field equations are not a closed set of
equations because there are no explicit equations for $\dot{\mathcal{A}}$
(evolution equation for $\mathcal{A}$) and $\hat{\Theta}$ (propagation
equation for $\Theta$). Hence an equation of state, governing the
thermodynamical quantities, is needed of the general form $F(\mu,p,\Pi,Q)=0$.
This form becomes simplified for a shear-free perfect fluid model where there
are no shear stresses or heat flux $(\Pi=Q=0$). Hence our equation of state
takes the form

$p=p(\mu),$ (19)

where the isotropic pressure $p$ is a function of $\mu$, the effective energy
density. At this stage, we need to point out that the derivative operators
‘$\delta^{a}$’, ‘⋅’ and ‘ $\hat{}$ ’ do not commute and give rise to an
interesting result later. Instead, according to clarkson2007 , the commutation
relation when acting on some scalar $\beta$ for our imposed LRS-II spacetime
conditions is given by

$\displaystyle\hat{\dot{\beta}}-\dot{\hat{\beta}}=-\mathcal{A}\dot{\beta}+\left(\frac{1}{3}\Theta+\Sigma\right)\hat{\beta},$
(20)

which will be used in our calculations in the subsequent sections.

##  IV Shear-free perfect fluid LRS-II equations

Thus far and later in this paper, we consider a shear-free perfect fluid LRS-
II spacetime with an equation of state $\eqref{eos}$. Perfect fluids were
studied in the context of the 1+3 decomposition method vanElstEllis  and for
the 1+1+2 decomposition method clarkson2007 . Such a fluid appears to be a
good description of the observed universe on a large scale. The absence of
shear stresses, viscosity and heat conduction is a great advantage as the
relativistic equations become simpler. For this shear-free perfect fluid LRS-
II model we have

$\Sigma=Q=\Pi=0.$ (21)

Hence the above general LRS-II equations (10)-(18) simplify to

$\displaystyle\hat{\phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\phi^{2}+\frac{2}{9}\Theta^{2}-\frac{2}{3}\mu-\mathcal{E},$
(22) $\displaystyle\hat{\Theta}$ $\displaystyle=$ $\displaystyle 0,$ (23)
$\displaystyle\hat{\mathcal{E}}-\frac{1}{3}\hat{\mu}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}\phi\mathcal{E},$ (24) $\displaystyle\dot{\phi}$
$\displaystyle=$
$\displaystyle\frac{2}{3}\Theta\mathcal{A}-\frac{1}{3}\Theta\phi,$ (25)
$\displaystyle-\frac{2}{3}\dot{\Theta}$ $\displaystyle=$
$\displaystyle-\mathcal{A}\phi+\frac{2}{9}\Theta^{2}+\frac{1}{3}(\mu+3p)-\mathcal{E},$
(26) $\displaystyle\dot{\mathcal{E}}-\frac{1}{3}\dot{\mu}$ $\displaystyle=$
$\displaystyle-\Theta\mathcal{E}+\frac{1}{3}\Theta(\mu+p),$ (27)
$\displaystyle\dot{\mu}$ $\displaystyle=$ $\displaystyle-\Theta(\mu+p),$ (28)
$\displaystyle\hat{p}$ $\displaystyle=$ $\displaystyle-\mathcal{A}(\mu+p),$
(29) $\displaystyle\hat{\mathcal{A}}-\dot{\Theta}$ $\displaystyle=$
$\displaystyle-(\mathcal{A}+\phi)\mathcal{A}+\frac{1}{3}\Theta^{2}+\frac{1}{2}(\mu+3p).$
(30)

We immediately note that due to our imposed conditions (21) we obtain an
explicit expression for $\hat{\Theta}$ in (23); however, we still do not have
an explicit expression for $\dot{\mathcal{A}}$. At this point, we make
$\mathcal{A}$ the central focus of our study and use it to classify shear-free
perfect fluid LRS-II solutions in the next section.

##  V Classes of shear-free perfect fluid LRS-II solutions

For the equations outlined in the preceding section, we obtain various classes
of shear-free perfect fluid LRS-II solutions characterized by the acceleration
scalar $\mathcal{A}$. First, it is pertinent to consider the simplest case
where acceleration is absent.

###  V.1 LRS-II spacetime with no acceleration: $\mathcal{A}=0$

In this case, we immediately see from equation (29) that the pressure must be
homogeneous with $\hat{p}=0$. Now comparing equations (26) and (30), implies
$\mathcal{E}=0$. Equation (24) then implies $\hat{\mu}=0$, and hence this
scenario describes a dynamic and homogeneous matter distribution. Since we
have $\mathcal{A}=\mathcal{E}=0$ and the matter is homogeneous, the spacetime
has to strictly be Friedmann-Lemaître-Robertson-Walker (FLRW). It is also
important to note that from the general LRS-II field equations (10)-(18), we
cannot have a perfect fluid solution where $\Theta=\mathcal{A}=0$, so we do
not consider that solution branch.

###  V.2 LRS-II spacetime with acceleration: $\mathcal{A}\neq 0$

From the equations (22)-(30), we noted earlier that we do not have an explicit
equation for $\dot{\mathcal{A}}$. However with the imposed equation of state
(19), we can obtain this crucial evolution equation so that we have a closed
set of equations, and we now describe this process in detail.

Considering the definition of the isentropic speed of sound
$c_{s}^{2}=\left(\partial p/\partial\mu\right)_{s=\text{constant}}$, we note
that we can write

$\displaystyle\hat{p}$ $\displaystyle=$ $\displaystyle
c_{s}^{2}\hat{\mu}=\frac{\partial p}{\partial\mu}\,\hat{\mu},$ (31)
$\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle
c_{s}^{2}\dot{\mu}=\frac{\partial p}{\partial\mu}\,\dot{\mu}.$ (32)

Therefore, from equation (29),

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle-\frac{\hat{p}}{(\mu+p)},$
$\displaystyle\implies\dot{\mathcal{A}}$ $\displaystyle=$
$\displaystyle-\frac{\dot{\hat{p}}}{(\mu+p)}+\mathcal{A}\Theta\left(1+\frac{\partial
p}{\partial\mu}\right),$ (33)

using (28) and (32) for simplification. Obtaining an expression for
$\dot{\hat{p}}$ from the commutation relation (20) and substituting into
equation (33) we get

$\dot{\mathcal{A}}=\mathcal{A}\Theta\left(\frac{\partial
p}{\partial\mu}-\frac{1}{3}-\frac{(\mu+p)}{c_{s}^{2}}\frac{\partial^{2}p}{\partial\mu^{2}}\right)=\mathcal{A}\Theta\mathcal{F},$
(34)

after simplification using (32) and where we have set

$\mathcal{F}\equiv\mathcal{F}(\mu)=\left(\frac{\partial
p}{\partial\mu}-\frac{1}{3}-\frac{(\mu+p)}{c_{s}^{2}}\frac{\partial^{2}p}{\partial\mu^{2}}\right).$
(35)

We now investigate how the obtained acceleration evolution equation (34) is
compatible with the system from the integrability condition; to achieve this
we make use of the commutation relation (20). According to the left hand side
of equation (20), we need to obtain $\hat{\dot{\mathcal{A}}}$ and
$\dot{\hat{\mathcal{A}}}$. Using (34), we have

$\displaystyle\hat{\dot{\mathcal{A}}}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{A}}\Theta\mathcal{F}+\mathcal{A}\hat{\Theta}\mathcal{F}+\mathcal{A}\Theta\hat{\mathcal{F}}$
(36) $\displaystyle=$
$\displaystyle\Theta\mathcal{F}\left(-\mathcal{A}^{2}+\frac{1}{2}\mathcal{A}\phi+\frac{3}{2}\mathcal{E}\right)$
$\displaystyle-\mathcal{A}^{2}\Theta\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime},$

where $\mathcal{F}^{\prime}=\frac{\partial\mathcal{F}(\mu)}{\partial\mu}$.
Using (26) in the Raychaudhuri equation (30) we get

$\displaystyle\hat{\mathcal{A}}$ $\displaystyle=$
$\displaystyle-\mathcal{A}^{2}+\frac{1}{2}\mathcal{A}\phi+\frac{3}{2}\mathcal{E},$
$\displaystyle\implies\dot{\hat{\mathcal{A}}}$ $\displaystyle=$
$\displaystyle-2\mathcal{A}^{2}\Theta\mathcal{F}+\frac{1}{3}\mathcal{A}^{2}\Theta-\frac{1}{6}\mathcal{A}\Theta\phi$
(37)
$\displaystyle+\frac{1}{2}\mathcal{A}\Theta\phi\mathcal{F}-\frac{3}{2}\Theta\mathcal{E}.$

Then subtracting (37) from (36), as required by the commutation relation (20),
yields

$\displaystyle\hat{\dot{\mathcal{A}}}-\dot{\hat{\mathcal{A}}}$
$\displaystyle=$
$\displaystyle\mathcal{A}^{2}\Theta\mathcal{F}+\frac{3}{2}\Theta\mathcal{E}\mathcal{F}-\mathcal{A}^{2}\Theta\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime}$
(38)
$\displaystyle-\frac{1}{3}\mathcal{A}^{2}\Theta+\frac{1}{6}\mathcal{A}\Theta\phi+\frac{3}{2}\Theta\mathcal{E}.$

However according to the commutation relation (20), we expected to obtain

$\displaystyle\hat{\dot{\mathcal{A}}}-\dot{\hat{\mathcal{A}}}$
$\displaystyle=$
$\displaystyle-\mathcal{A}\dot{\mathcal{A}}+\frac{1}{3}\Theta\hat{\mathcal{A}}$
(39) $\displaystyle=$
$\displaystyle-\mathcal{A}^{2}\Theta\mathcal{F}-\frac{1}{3}\mathcal{A}^{2}\Theta+\frac{1}{6}\mathcal{A}\Theta\phi+\frac{1}{2}\Theta\mathcal{E}.$

We immediately notice the interesting discrepancy between (38) and (39). They
do not match and this needs to be resolved for the existence of solutions. The
condition for $\dot{\mathcal{A}}$ to be compatible with the system is obtained
by setting the difference between (38) and (39) to zero as follows

$2\mathcal{A}^{2}\Theta\mathcal{F}+\frac{3}{2}\Theta\mathcal{E}\mathcal{F}-\mathcal{A}^{2}\Theta\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime}+\Theta\mathcal{E}=0,$
(40)

which simplifies to

$\displaystyle\Theta\left[\mathcal{A}^{2}\left(2\mathcal{F}-\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime}\right)+\mathcal{E}\left(\frac{3}{2}\mathcal{F}+1\right)\right]=0.$
(41)

Equation (41) implies that either $\Theta=0$ or the square-bracketed term is
zero.

####  V.2.1 Case 1: Static fluid $\left(\Theta=0\right)$

Since we have $\Sigma=Q=\Pi=\Theta=0$ then the spacetime is necessarily
static. This means that any reasonable equation of state of the form
$p=p(\mu)$ will solve the system.

####  V.2.2 Case 2: Nonstatic fluid $\left(\Theta\neq 0\right)$

In this case $\Theta$ is strictly not equal to zero. Hence the square-
bracketed term in (41) has to be zero, that is

$\left[\mathcal{A}^{2}\left(2\mathcal{F}-\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime}\right)+\mathcal{E}\left(\frac{3}{2}\mathcal{F}+1\right)\right]=0.$
(42)

Two possible subcases arise.

  * •

Subcase 1: If both $\mathcal{A}=\mathcal{E}=0$ then the spacetime is
necessarily Friedmann-Lemaître-Robertson-Walker (FLRW). This type reduces to
the case contained in section V.1.

  * •

Subcase 2: If $\mathcal{A}$ and $\mathcal{E}$ are well defined and both
nonzero then from (41) we have

$\displaystyle\mathcal{A}^{2}\left(2\mathcal{F}-\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime}\right)+\mathcal{E}\left(\frac{3}{2}\mathcal{F}+1\right)=0,$
$\displaystyle\implies\frac{\mathcal{E}}{\mathcal{A}^{2}}=\frac{\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime}-2\mathcal{F}}{\frac{3}{2}\mathcal{F}+1}.$
(43)

Subcase 2 gives us a new constraint equation (43) that needs to be
consistently satisfied by the equation of state. Now for this constraint
equation to time-evolve consistently, its time derivative must also be zero,
and evolving (41) once yields

$\displaystyle\Theta\left[\mathcal{A}^{2}\left(2\mathcal{F}\mathcal{G}-(\mu+p)\mathcal{G}^{\prime}\right)\right.$
$\displaystyle\left.-\mathcal{E}\left(\frac{3}{2}\mathcal{F}+1+\frac{3}{2}(\mu+p)\mathcal{F}^{\prime}\right)\right]=0,$
(44)

where we have set

$\mathcal{G}\equiv\mathcal{G}(\mu)=2\mathcal{F}-\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime},$
(45)

and $\mathcal{G}^{\prime}=\frac{\partial\mathcal{G}(\mu)}{\partial\mu}$. As we
shall see in the next section, any further time evolution of the above
constraint is not required as the constraints (43) and (44) give us the
governing differential equation for the equation of state of the matter that
gives rise to this given subcase. We summarize the findings in this section in
a diagram depicting the various classes of spherically symmetric shear-free
perfect fluid solutions characterized by the acceleration scalar $\mathcal{A}$
in Figure 1.

Figure 1: Diagram illustrating the various configurations of the property of
acceleration and the spacetime classification.

##  VI Dynamic and inhomogeneous fluid

In the event that $\Theta\neq 0$ and $\mathcal{A}\neq 0$, setting the square-
bracketed term in equation (44) to zero yields

$\frac{\mathcal{E}}{\mathcal{A}^{2}}=\frac{2\mathcal{F}\mathcal{G}-(\mu+p)\mathcal{G}^{\prime}}{\frac{3}{2}\mathcal{F}+1+\frac{3}{2}(\mu+p)\mathcal{F}^{\prime}}.$
(46)

For consistency, equations (43) and (46) need to be equivalent and this
condition produces a master differential equation for the equation of state
given by

$\displaystyle\frac{\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime}-2\mathcal{F}}{\frac{3}{2}\mathcal{F}+1}=\frac{2\mathcal{F}\mathcal{G}-(\mu+p)\mathcal{G}^{\prime}}{\frac{3}{2}\mathcal{F}+1+\frac{3}{2}(\mu+p)\mathcal{F}^{\prime}},$

which simplifies to

$\displaystyle\left(\frac{(\mu+p)}{c_{s}^{2}}\mathcal{F}^{\prime}-2\mathcal{F}\right)\left(\frac{3}{2}\mathcal{F}+\frac{3}{2}(\mu+p)\mathcal{F}^{\prime}+1\right)-\left(\frac{3}{2}\mathcal{F}+1\right)\left(2\mathcal{F}\mathcal{G}-(\mu+p)\mathcal{G}^{\prime}\right)=0.$
(47)

Equation (47) imposes a constraint on the equation of state. Finding an
equation of state which is a solution to (47) will then necessarily make the
two constraints (43) and (44) identically compatible. Since these two
constraints are identically satisfied by the equation of state (which is the
solution of the above equation), any further time evolution of these
constraints will also be identically satisfied and the system of equations is
well posed. From equation (46), both $\Theta$ and $\mathcal{A}$ are nonzero
hence $\hat{\mu}$ and $\hat{p}$ are also nonzero and so these solutions are
necessarily dynamic and inhomogeneous but shear-free.

The resulting differential equation (47) is not autonomous as it stands.
However, it can be transformed into that type via a change of variables.
Letting $X=(\mu+p)$ yields the autonomous ordinary differential equation given
by

$\displaystyle-\frac{3}{2}\left(X^{\prime}-1\right)\left(XX^{\prime\prime}-\left(X^{\prime}\right)^{2}+\frac{5}{3}X^{\prime}-\frac{2}{3}\right)X^{3}X^{\prime\prime\prime\prime}+\frac{3}{2}XX^{\prime\prime\prime}\left\\{-\left(X-2X^{\prime}+2\right)\left(XX^{\prime\prime}\right)^{2}\right.$
$\displaystyle+\left(X^{\prime}-1\right)\left[-4\left(X^{\prime}\right)^{2}+\left(4X+\frac{16}{3}\right)X^{\prime}+X^{2}-\frac{11}{3}X-\frac{4}{3}\right]XX^{\prime\prime}$
$\displaystyle\left.-4\left(X^{\prime}-1\right)^{2}\left(X^{\prime}-\frac{2}{3}\right)\left(-\frac{1}{2}\left(X^{\prime}\right)^{2}+\left(X+\frac{1}{2}\right)X^{\prime}-\frac{7}{6}X\right)\right\\}-\frac{3}{2}\left(-13X^{\prime}+X+9\right)\left(XX^{\prime\prime}\right)^{3}$
$\displaystyle-6\frac{\left(XX^{\prime\prime}\right)^{4}}{X^{\prime}-1}-3\left(X^{\prime}-1\right)\left(8\left(X^{\prime}\right)^{2}+\left(X-12\right)X^{\prime}-\frac{3}{2}X+\frac{35}{9}\right)\left(XX^{\prime\prime}\right)^{2}$
$\displaystyle+3\left(X^{\prime}-1\right)^{2}\left(6\left(X^{\prime}\right)^{3}+\left(X-\frac{95}{6}\right)\left(X^{\prime}\right)^{2}+\left(-\frac{7}{3}X+\frac{233}{18}\right)X^{\prime}+\frac{4}{3}X-\frac{29}{9}\right)XX^{\prime\prime}$
$\displaystyle-6\left(X^{\prime}-1\right)^{4}\left(X^{\prime}-\frac{2}{3}\right)\left(X^{\prime}-\frac{4}{3}\right)\left(X^{\prime}-\frac{5}{6}\right)=0.$
(48)

This equation is not satisfied by either linear or power laws of the form
$p=\kappa\mu$ or $p=\kappa\mu^{\gamma},(\gamma>2)$ respectively. This infers
that it has a complicated solution and so carrying out a numerical analysis on
the mathematical software Maple maple  yields the solution given in Figure 2.

Figure 2: Numerical solution for the autonomous system obtained from the
mathematical software Maple.

Now in order to conduct a dynamical analysis, we investigate the stationary
points of the differential equation (VI). We first create a system of four
differential equations by letting

$\displaystyle X^{\prime}$ $\displaystyle=$ $\displaystyle Y,$ $\displaystyle
Y^{\prime}$ $\displaystyle=$ $\displaystyle Z,$ $\displaystyle Z^{\prime}$
$\displaystyle=$ $\displaystyle W,$ $\displaystyle W^{\prime}$
$\displaystyle=$ $\displaystyle f(X,Y,Z,W).$ (49)

Further using (VI), $X^{\prime\prime\prime}\equiv W^{\prime}$, the function
$f$ in (49) can be written explicitly giving

$\displaystyle W^{\prime}$ $\displaystyle=$
$\displaystyle\left\\{-\frac{2}{3}\left(-\frac{3}{2}XW\left(-\left(X^{2}Z-2XYZ+2\right)^{2}+\left(Y\left(-4Y^{2}+\left(4X+\frac{16}{3}\right)Y+X^{2}-\frac{11}{3}X-\frac{4}{3}\right)-1\right)XZ\right.\right.\right.$
(50)
$\displaystyle-4\left(Y\left(Y-\frac{2}{3}\right)\left(-\frac{1}{2}Y^{2}+\frac{1}{6}\left(6X+3\right)Y\right.-\frac{7}{6}X-1\right)^{2}+\frac{3}{2}\left(-13XYZ+X^{2}Z+9\right)^{3}+6\frac{\left(XZ\right)^{4}}{Y-1}$
$\displaystyle+3\left(Y\left(8Y^{2}+\left(X-12\right)Y-\frac{3}{2}X+{\frac{35}{9}}\right)XZ-1\right)^{2}-3\left(Y\left(\frac{1}{18}\left(18Y^{2}-42Y+24\right)X+6Y^{3}\right.\right.$
$\displaystyle\left.\left.\left.\left.\left.-{\frac{95}{6}}Y^{2}+\frac{233}{18}Y-\frac{29}{9}\right)-1\right)^{2}XZ+6\left(Y\left(Y-\frac{2}{3}\right)\left(Y-\frac{4}{3}\right)\left(Y-\frac{5}{6}\right)-1\right)^{4}\right)\right)\right\\}\Bigg{/}$
$\displaystyle\left\\{\left(Y\left(XZ-Y^{2}+\frac{5}{3}Y-\frac{2}{3}\right)-1\right)X^{3}\right\\}.$

At the stationary points $X^{\prime}=Y^{\prime}=Z^{\prime}=0$ which imply that
$Y=Z=W=0$. Substituting $Y=Z=W=0$ into (50) and simplifying yields

$\displaystyle W^{\prime}$ $\displaystyle=\frac{40}{9X^{3}}.$ (51)

Clearly $W^{\prime}=0$ only if $X=(\mu+p)\to\infty$. This means that the
stationary points for the autonomous system exist at infinity indicating the
presence of a singularity there.

##  VII Discussion

In this paper, we investigated the behaviour of the self-gravitating system of
a shear-free perfect fluid with LRS-II symmetry through a geometrical
perspective. We have provided a comprehensive answer to the question posed in
the title of our paper, in the most generic covariant way, to extract the
highly nonlinear differential equation that governs the equation of state. The
novel results that emerged from our analysis can be summarized as follows:

  * •

We can completely characterize all spherically symmetric shear-free perfect
fluid spacetimes under a single classification that depends on the covariantly
defined fluid acceleration and expansion.

  * •

One of the most important classes of shear-free spherical symmetry is the
inhomogeneous dynamical class. We pinpoint the exact constraint, given by a
differential equation (VI), governing the equation of state. It is interesting
to note that the autonomous ODE (VI) obtained places a restriction solely on
the equation of state. It does not place a restriction on any of the
kinematical variables.

  * •

Since the governing ODE is highly nonlinear in nature, we easily showed that
usual equations of state used in astrophysical settings (for example, a linear
equation of state or a finite combination of power law equations of state)
will not satisfy this equation. Hence all the solutions belonging to this
class are extremely special in terms of the matter content. We also presented
a numerical solution for this ODE.

###### Acknowledgements.

CH is supported by the Oppenheimer Memorial Trust (OMT) and the University of
KwaZulu-Natal. JH, RG and SDM are supported by the National Research
Foundation (NRF), South Africa, and the University of KwaZulu-Natal.

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