# Natural Capital as a Stock Option

O. Bertolami<EMAIL_ADDRESS>Departamento de Física e Astronomia,
Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687,
4169-007, Porto, Portugal.  Centro de Física das Universidades do Minho e do
Porto, Rua do Campo Alegre s/n, 4169-007, Porto, Portugal.

###### Abstract

The unfolding climate crisis is a physical manifestation of the damage that
market economy, driven by the high intensity consumption of fossil fuels, has
inflicted on the Earth System and on the stability conditions that were
established by a complex conjugation of natural factors during the Holoecene.
The magnitude of the human activities and its predatory nature is such that it
is no longer possible to consider the Earth System and the services it
provides for the habitability of the planet, the so-called natural capital, as
an economical externality. Thus one is left with two main choices in what
concerns the sustaintability of the planet’s habitability: radical economic
degrowth or highly efficient solutions to internalise the maintenance and the
restoration of ecosystems and the services of the Earth System. It is proposed
that an interesting strategy for the latter is to consider the natural capital
as a stock option.

Earth System - Natural capital - Stock option

1\. The evidence of the systemic disruption of the Earth System (ES) is
overwhelming. It is manifested itself through the widespread and longer
periods of droughts, heat waves, flooding, wild fires and other deleterious
climate events that were once rare and are now the norm and tend to become
even more frequent.These are symptons of a serious misuse and/or overuse of
the resources provided by the multiple components of the ES and the
degradation of its main regulatory large scale ecosystems due to the ever
increasing expansion of the human activities. The most visible result of this
disruption is the change in the climate patterns due to the accumulation of
anthropogenic greenshouse gases in the atmosphere. In fact, multiple studies
and the accumulated data show that we are living a climate emergency with
unpreditacble implications for the future generations IPCC23 .

The international community is well aware of the seriousness of the problem
and on various international climate summits the necessity to decarbonise
economic activities has been acknowledged and some important agreements have
been reached in order to establish targets and engage in long term
commitments. Nevertheless, the implementation of these agreements has been
problematic to say the least. We face a civilizational challenge and at the
source of the problem are the basic working principles of the consumption
society driven by cheap fossil fuels and built upon the mistaken assumptions
that Earth’s resources are infinite and that it can absorb a limitless amount
of waste. These assumptions violate the most obvious principles of common
sense and thermodynamics, but are the working rules of the continuous
expansion of the human activity. The magnitude and the fundamental nature of
the problem at hand, suggest any set of solutions for a long term fixing of
the climate change crisis is not realistic as it involves no less than a
dramtic reduction of the emissions of greenhouse gases and encompassing socio-
economic changes. Furthermore, it is important to realise that the problem
must be addressed in little more than a decade or so and that we may be
running out of time to carry out the absolutely necessary long term
transformations. Given the current situation, adaptation and mitigation
strategies are in the category of the absolutely minimal necessary measures in
order to get us some extra time to fix the problem. These measures include
accelerating efforts to decarbonise human activities, extending and
generalising the use of renewable energies, setting up means to carbon capture
by afforestation, restoration of ecosystems and other chemical-mechanical
means, besides rational use of water and of vital natural resources. Given the
urgency of the situation, even radical solutions involving geoengineering
measures have been discussed, ranging from deflection of solar radiation by
aerosols to removal of $CO_{2}$ from the atmosphere using the space lift (see,
for instance, Refs. OB23 ; OB24  and references therein).

Of course, as there is consensus that a drastic decarbonisation of the human
activities is required, a sensible way to address the problem involves a
significant reduction of the consumption patterns of our society as well as a
coupled effort to change the brutal and disfunctional way the expansion of the
market economy destroys ecosystems. Thus a realistic path towards a
sustainable future asks for a conscious and well planned economic degrowth. Of
course, such a drastic change in the underlying ideology of the contemporary
civilisation is beyond reach under the prevailing socio-economic and political
conditions of our world nowadays.

Another alternative involves renewed and vigorous forms of internalising the
regulatory ecosystems into the main stream workings of the economic
activities. The discussion on how this transformation might take place has
been a matter of heated debate, but no effective workable solution has
emerged. The idea of a tax to cover a negative externality or a market
failure, a Pigovian tax, goes back to 1970s in what concerns $CO_{2}$
emissions. Early versions of an integrated model of climate and economy
appeared in late 1970s and in more detail in 1990s through the work of the
Nobel laureate economist William Nordhaus. More recent versions of this model,
DICE (Dynamic Integrated Climate-Economy mode) and RICE (Regional Integrated
Climate-Economy model), are a matter of intensive research and adaptation
Nordhaus1 ; Nordhaus2 . Back in 2006, the so-called Stern report proposed a
different approach and called attention to the need to tackle the climate
crisis, which was then still incipient, through a sizebale invertion of
capital, about 1 to 2$\%$ of the developed countries GNP Stern1 ; Stern2 .
This warning was dismissed and we are left today with the existing carbon
social tax, which has become just another instrument of the market economy
with little impact on the consumption of fossil fuels and in reducing the
existing $CO_{2}$ in the atmosphere.

The present state of affairs is such that after the failure of the major
international agreements and commitments, Kyoto, Paris, annual COPs,
adaptation and mitigation strategies are the only realistic alternatives to
motivate a deeply divided international community. Under the present
conditions is unlikely that any coherent and determined policy will emerge in
order to face the unfolding crisis. Resilience has emerged as a basic concept
and its multiple dimensions have been extensively discussed (see, for
instance, Refs. Cutter1 ; Cutter2 ). The need to incorporate the resilience
factor into DICE-type models have been discussed in Refs. OBG23a ; OBG23b  and
a resilience social tax upon consumption has been proposed in Refs. OB22 ;
OBG23a .

In the present work we argue that given the difficulties discussed above, one
should consider direct forms of restoring ecosystems that have themselves an
econonic and a financial dimension. This involves relating the improvement of
the ES performance so to ensure weather stabily and sustenability as a capital
asset whose value evolves according the rules of the market. This amounts to
create financial assets directly connected to the maintainance and improvement
of the services provided by the ES aligning a positive performance of these
financial assets with actions that promote a sustainable habitability of the
planet.

Notice that this goes beyond the tenets of the usual green or circular economy
as it directly relates the improvement of the ES services by assigning to them
a market value. This follows the logic of the proposal of evaluating the world
ecosystem services as a capital, the natural capital Costanza . Given the
direct impact of ecosystem’s restoring activities on Earth’s life-supporting
system and human welfare, it can have deeper implications than improvements on
the existing production processes, recicling and waste management, which are
of course greener than the tradicional methods, but are far away from being an
encompassing and effective solution for decarbonising the economy.

2\. In order to establish the restoration of ecosystems as an economic and
financial activity, one must endow it with a suitable market characterization.
As the restoration of ecosystems is a knowledge and a capital intensive
activity, it is fairly straightforward to regard it as a derivative, that is
an economic and engineering activity that can be furnished in a financial
instrument whose worth is supported by the value of an underlying asset. In
the stock market, the way to trade a given asset so to ensure stability and
equity for owners and shareholders in a well defined time framework is through
the concept of a stock option.

A stock option is a derivative that endows a trader with the right, although
not the obligation, to buy or sell shares of a stock at an agreed-upon price
and date. This means that an option settles an agreement contract between
parties so to have the option to sell or to buy the stock in the future at a
specified price. This specified price is referred to as the strike price or
the exercise price.

As an asset to be traded in the future, the value of an option, the so-called
pay-off, $V$, is a function of stock price, $S$, and time, $t$, that is,
$V=V(S,t)$, which is an inherently stochastic variable evaluatted through
suitable Itøcalculus manipulations and whose behaviour in modern stock trading
is described by the Black-Scholes-Merton partial diferential equation BS ;
Merton  :

${\partial V\over\partial t}+{1\over
2}\sigma^{2}S^{2}{\partial^{2}V\over\partial S^{2}}+rS{\partial V\over\partial
S}-rV=0,$ (1)

where $\sigma$ is the volatility of the stock, the square root of the log of
the stock prices, and $r$ the interest rate.

Suitable boundary conditions to solve the Black-Scholes-Merton equation can be
settled through conditions on the buying price of an (european) option,
$C(S,t)$, as follows: $C(0,t)=0$, for any $t$ and large $T$; $C(S,t)\to S(t)$,
for large S; $C(S,T)=max(S-K,0)$, where $K(T)$ is the exercise price for
$t=T$, the option time. Thus, through the change of variables, $\tau=T-t$,
$u(x,\tau)=C(S,t)e^{r\tau}$, and $x=ln(S/K)+(r-{\sigma^{2}\over 2})\tau$, the
Black-Scholes-Merton equation becomes the Fourier difusion equation:

${\partial u(x,\tau)\over\partial\tau}={\sigma^{2}\over
2}{\partial^{2}u(x,\tau)\over\partial x^{2}},$ (2)

which, with the initial condition $u(x,0)\equiv u_{0}(x)=Kmax(e^{x}-1,0)$,
admits the well known solution

$u(x,\tau)={1\over\sigma\sqrt{2\pi\tau}}\int^{\infty}_{-\infty}u_{0}(y)e^{-(x-y)^{2}/(2\sigma^{2}\tau)}dy~{}.$
(3)

It is now simple to show that a useful solution of Eq. (1), with the initial
condition $u(x,0)\equiv u_{0}(x)=Kmax(e^{x}-1,0)$, is the well-known Black-
Scholes formula:

$C(S,t)=S(t)N(d_{1})-K(T)e^{-r(T-t)}N(d_{2})~{},$ (4)

where

$N(x)={1\over\sqrt{2\pi}}\int^{x}_{-\infty}e^{-y^{2}/2}dy~{},$ (5)

is the normal distribution function,

$d_{1}={ln(S/K)+(r+\frac{\sigma^{2}}{2})(T-t)\over\sigma\sqrt{T-t}}~{},$ (6)

and

$d_{2}=d_{1}-\sigma\sqrt{T-t}~{}.$ (7)

On its hand, the selling price of an option, $P(S,t)$, is obtained from the
buying price of the option, $C(S,t)$, written above:

$P(S,t)=K(T)e^{-r(T-t)}N(-d_{2})-S(t)N(-d_{1})~{}.$ (8)

One can envisage several ways to internalise the operating conditions of the
ES within the wrapping of a stock option. In what follows, we shall present a
suggestion that is based on the idea that the demage on ES due to human
activities can be evaluated by the so-called planetary boundaries (PB)
Steffen:2015 , a set of at least nine parameters (rate of biosphere loss, land
system change, global fresh water use, biogeochemical flows (global Nitrogen
and Phosphorus cycles), ocean acidification, atmospheric aerosol loading,
stratospheric ozone depletion, climate change, chemical pollution) whose
values relative to their readings at the Holocene, the period during which the
ES was in a stable condition. This latter state is referred to as the safe
operating space (SOS) Rockstrom:2009 . In fact, it has been argued that a PB
quota-based system is the most direct way to measure the impact of the human
activities on the ES Meyer , but of course, different parametrisations can be
used and the proposal that is discussed below is just an example of the
approach that we suggest here.

3\. We have reached a key point of our discusssion. The stock is a financial
instrument meant to generate profit and that can be completly abstract with no
contact with the material world. Our suggestion is to attach a stock to the
improvement of the ES via the PB parametrisation. This parametrisation is an
evaluation element of a particular action on the environment and on the ES
with a direct impact on the natural capital. Thus, our suggestion can be
summarised in a simple principle: a stock can be created so that its price
yields a profit if it adds up to the natural capital and allows for the
improvement of the services the ES provides to the habitabiliy of the planet.

To proceed in the discussion we need to specify the features of the PB
parameters and how they can be used to set up the price of a stock. The Great
Acceleration of the human activities in 1950s Steffen:2014  has led the ES to
be predominantly driven by human activities. This dominance defines the
Anthropocene. If we denote the human drivers, collectively by $H$, it is
possible to consider, through a suitable model Bertolami:2018 ; Bertolami:2019
, the impact of the human activities on the ES through the value of the PB
parameters, $h_{i}$, relative to their values at the SOS. Thus, the most
general form of $H$, allowing for interaction between the various PB
parameters is given Bertolami:2019 :

$H=\sum_{i=1}^{9}h_{i}+\sum_{i,j=1}^{9}g_{ij}h_{i}h_{j}+\ldots,$ (9)

where the second and the following set of terms parametrise the interactions
among the various effects of the human action on the ES. Of course, higher
order interactions terms can be considered, but it us usual to restrict the
discussion up to second order. It is physically sensible and mathematically
convenient to assume that the $9\times 9$ matrix, $[g_{ij}]$ is symmetric,
$g_{ij}=g_{ji}$, and non-degenerate, $\det[g_{ij}]\neq 0$. It is natural to
assume that these interactions terms are sub-dominating, and their importance
has to be established through a suitable empirical analysis. As shown in Ref.
Bertolami:2019 , they can lead to new equilibria and suggest some mitigation
strategies depending on the sign and strength of the matrix entries, $g_{ij}$
Bertolami:2019 . In fact, this modelling was used to show that the interaction
term between the climate change variable ($CO_{2}$ concentration), $h_{1}$,
and the oceans acidity, $h_{2}$, is non-vanishing and about $10\%$ of the
value of the single contributions themselves Barbosa:2020 .

This parametrisation is crucial for assessing the state of the ES and, in
particular, for the physical model proposed in Refs. Bertolami:2018 ;
Bertolami:2019 , which assumes that the transition of the ES, for instance,
from the Holocene to other stable states was similar to a thermodynamic phase
transition and could be described by the Landau-Ginzburg Theory (LGT).
According LGT, the relevant thermodynamic variable required to specify the
state of the ES is the free energy, $F$, and phase transitions are expressed
through a relevant order parameter, $\psi$, in the case of the ES, the reduced
temperature relative to the Holocene average temperature, $\langle T_{\rm
H}\rangle$, $\psi=(T-\langle T_{\rm H}\rangle)/\langle T_{\rm H}\rangle$. This
framework allows for determining the state and equilibrium conditions of the
ES in terms of physical variables, $(\eta,H)$, where $\eta$ are associated
with the astronomical, geophysical and internal dynamical drivers, while $H$
corresponds to the human activities, introduced in the phase transition model
as an external field. The so-called Anthropocene equation corresponds to the
evolution equation of the ES once it is dominated by the human activities. The
transition from the Holocene to the Anthropocene took place through the so-
called great acceleration of the human activities, visible from the second
half of the 20th century onwards Steffen:2014 . The proposed approach allows
one to perform a phase space analysis of the temperature field, $\psi$, once
an assumption for the evolution oh $H$ is considered. If a linear evolution
hypothesis for $H$ is assumed then it is shown that the recently discussed
Hothouse Earth scenario Steffen:2018  corresponds to an attractor of the
trajectories of the ES dynamical system for regular trajectories111The
capability of the proposed model to describe these planetary transitions
suggests that it can provide also an interesting framework for the
classification of rocky exoplanets and their terraforming OBFFrancisco2022 ..
If instead, a logist map is chosen for the behaviour of $H$, then for a
sufficiently large rate of growth of the human activities, the ES can show
stability point bifurcations and even chaotic behaviour in the phase space
Bernardini:2022 .

Notice that the parametrisation of the state of ES in terms of the PB is
fairly general and Eq. (9) stands independently of the physical model
described above. It can therefore be used to set the price of a stock in terms
of its purpose and performance. We propose that the simplest way to set the
price of a stock whose purpose is to maintain and restore the services to
common welfare of a ecosystem is to relate it with the improvement of the
services of the ES in terms of the PB, that is:

$S(t)=S_{0}-\alpha\Delta H,$ (10)

where $S_{0}$ is an arbitrary initial stock price, $\alpha$ an arbitrary
positive constant, $\Delta H=H(T)-H(t)$, and the minus sign is introduced so
to make clear that the purpose of the proposed stock instrument is to lower
the impact of the human activities on the ES. Of course, one could break $H$
in its PB components and design a stock instrument based on a single or a set
of PB parameters. Obviously, considering the interaction terms is a matter of
choice, but given their non-negligible values and their physical significance,
it is fairly natural to consider them. Moreover, it is clear that the strike
price, $K(T)$, is set upon the performance with respect to the PB parameters.
If so, it would have a price for $t>T$ set by an equation like Eq. (10).

It is then evident that the improvement of natural capital consists in
diminishing the impact of the human activities, that is decreasing H or its
components, if interactions are such that the entries of $g_{ij}$ are
positive. This will increase the price, $S(t)$, and the buying prize,
$C(S,t)$, of the asset, and hence making its selling more atractive, which, on
its turn, further boosts actions to improve the workings of the ES. Of course,
more involved financial strategies could be considered and devised. To further
illustrate this point let ius consider two simple examples:

i) Direct capture of $CO_{2}$ in the atmosphere, which lowers the climate
change variable, say $h_{1}(t)$, so that $\Delta H<0$ and through Eq. (10)
this action boosts the price of the option to higher values;

ii) The somewhat more involved case of the interaction between climate change
variable, $h_{1}(t)$ and the oceans acidity, $h_{2}(t)$ studied in Ref.
Barbosa:2020 . Oceans absorb atmospheric $CO_{2}$ which is turned in carbonic
acid, thus raising their acidity. To maximise the price $C(S,t)$ the
followiing conditions must be satisfied at the extremun
$(h_{1}^{0},h_{2}^{0})$ 222The general case requires studying the eigenvalues
and the determinant of the Hessian matrix,
$(H_{S})_{i,j}=\left({\partial^{2}S\over\partial h_{i}\partial
h_{j}}\right)$.:

${\partial^{2}S(h_{1}^{0},h_{2}^{0})\over\partial
h_{1}^{2}}{\partial^{2}S(h_{1}^{0},h_{2}^{0})\over\partial
h_{2}^{2}}-\left({\partial^{2}S(h_{1}^{0},h_{2}^{0})\over\partial
h_{1}\partial h_{2}}\right)^{2}>0,$ (11)

and

${\partial^{2}S(h_{1}^{0},h_{2}^{0})\over\partial h_{1}^{2}}<0~{},$ (12)

meaning that restoring the capacity of the oceans to absorb atmospheric
$CO_{2}$ by lowering their acidity is a profitable action. Needless to say
that a task of this sort would have on a global scale a huge positive impact
on marine ecosystems and on the overall state of the ES.

We stress once again that our proposal concerning the connection between the
maintainance and improvement of the natural capital and the services it
provides and the stock price as an option can be achieved through any other
parametrisation system to evaluate the former. In fact, for this purpose, one
could very well consider any ecological footprint signature of the human
activity on a given component of the ES.

4\. In this work we have argued that an interesting strategy to internalise
the natural capital is through the creation of financial instruments with
direct impact on the ES through the restoration of ecosystems. Interprises to
restore ecosystems already exist, but to our knowledge their activities are
not profit oriented and are not funded, as we suggest, by financial
instruments that materialise in the stock market as options. This endows the
somewhat abstract concept of natural capital as a palpable reality in the
financial world allowing for a direct internalisation of the services of the
ES to the habitality conditions of the planet. Given that the global and
international nature of the market is an unchallenged reality and that its
expansion is much more effective than its regulation through international
climate and environment agreemets, the internalisation of the ES as we propose
might hopefully have a relevant impact. Thus, if the restoration of the ES can
be successfully internalised via a profitable financial asset according to the
very rules of the market, its impact can be complementary to the one of the
international climate agreements. If an economic and financial instrument can
be evaluated through parameters of the ES as suggested by Eq. (10), then its
financial success will be aligned with the maintainance and improvements of
the life-supporting services of the ES, in opposition to the usual market
activities which are profitable through the destruction or the dumping of
waste on the existing ecosystems. We believe that this can be a relevant
change in paradigm. Any attempt to make the restoration of ecosystems a
profitable business is helpful to fix the climate change crisis, which
according to some is the biggest market blunder ever. Another potential
benefit of the proposal discussed here is that it may be an additional
incentive to ecosystem restoring business at community level, a fundamental
cornerstone of the mitigation and adaptation activities OBG21 , meaning that
these activities can be advantgeously coupled together. Furthermore, putting
forward a financial asset based on a real world action of restoring ecosystems
requires the conjugation of a wide range of skills from scientific to
adminstrative, from engineering to financial and planing ones, which, on its
own, helps in creating the ground for the multi-dimensional approach that is
required to design solutions for coping with the climate change crisis.

For sure, none of the exposed hinders the need for stewardship measures to
rationaly manage the ES. The discussion about the need for stewardship
measures goes back to the very discussion about the onset of the Anthropocene
itself as a new geological epoch whose habitability conditions of the planet
are conditioned by the extent of the human activities Steffen:2011 ;
Steffen:2018 . In fact, it is widely believed that some form of stewardship is
necessary given the fragmented nature of the interenational juridical order
and the urgency of the climate crisis. In the opposite direction, it has been
argued that stewardship measures would be an obstacle to the expansion of the
free market hindering its benefits and its underlying liberal values. Given
that objective historic developments do not support these claims and that
international economic and financial crises arise quite regularly, it is no
longer possible to hide the need for a new international economic order to
prevent an irreversible planetary climate catastrophy. Furthermore, since
market economy cannot avoid widespread poverty and deep social inequalities,
it cannot, on its own, avoid the disfuncionalities it creates. These
disfuncionalities link together social inequalities and climate crisis OB22 ,
Thus, stewardship measures may function as regulatory boundaries which can be
exerted through the form of a new ES justice Gupta  or via descentralised
means based, for instance, on digital contracts OB:2020 . In any case, it is
clear that concerted and focussed action is required to face the ongoing
climate crisis and the social challeges it ensues. Internalising the life-
supporting services provided by the ES does not address the main flaws of the
market economy, but it might render an alternative to the more logical,
although unfeasible economic degrowth. Therefore, devicing ways to introduce
the natural capital into the financial market in order to fund the specialised
task of restoring ecosystems appears to be a fairly sensible approach,
wrapping these restoring activities as a derivative like a stock option seems
just natural.

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# Calculation of the Structure Properties of a Strange Quark Star in the
Presence of Strong Magnetic Field Using a Density Dependent Bag Constant

Gholam Hossein Bordbar1,2 111Corresponding author. E-mail:
<EMAIL_ADDRESS>Hajar Bahri1 and Fatemeh Kayanikhoo1 1Department
of Physics, Shiraz University, Shiraz 71454, Iran
and
2Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box
55134-441, Maragha, Iran

###### Abstract

In this article we have calculated the structure properties of a strange quark
star in static model in the presence of a strong magnetic field using MIT bag
model with a density dependent bag constant. To parameterize the density
dependence of bag constant, we have used our results for the lowest order
constrained variational calculation of the asymmetric nuclear matter. By
calculating the equation of state of strange quark matter, we have shown that
the pressure of this system increases by increasing both density and magnetic
field. Finally, we have investigated the effect of density dependence of bag
constant on the structure properties of strange quark star.

##  I Introduction

The core of a neutron star has been formed from nuclear matter, composed of
the neutrons, protons, electrons (for assurance negation of electric charge)
and other particles like pions, mesons and etc (Lattimer & Prakash rk1
(2004)). It is known that nuclear matter is meta-stable, that after releasing
a lot of energy converts into strange quark matter (SQM) to achieve stability.
This quark matter is the most stable state of matter that has been known until
now. Thus, there is a new class of compact stars that come from the collapse
of neutron stars, and are more stable compared to the neutron stars (Farhi &
Jaffe rk2 (1984)). The best candidates for this conversion are the neutron
stars with masses of $1.5-1.8\ M_{\odot}$ and quick spin (Drake et al. rk3
(2002); Li et al. rk4 (1999); Weber rk5 (2005)).

The collapse of a neutron star may lead to a strange quark star (SQS) or a
hybrid star. Also under special conditions, an SQS may be directly born from
the core collapse of a type II supernova. An SQS, from its center to surface
is made from SQM, and on its surface may exist a nuclear layer (Glendenning &
Weber rk6 (1992)). Hybrid stars are the ones with cores composed of SQM
(Bhattacharyya et al. rk7 (2006)). Here, we just consider the structure
properties of SQS.

The mass and density of an SQS is between the mass and density of a neutron
star and that of a black hole. The mass-radius relation for an SQS is as
$M\propto R^{3}$ which is different from that of a neutron star. This star
does not have the minimum mass. For an SQS with $1M_{\odot}\leq M\leq
2M_{\odot}$, the radius is about $10\ km$ (Farhi & Olinto rk8 (1986); Shapiro
& Tenkolsky rk9 (1983)).

Recent observations indicate that the object SWIFT J1749.4-2807 may be an SQS
(Yu & Xu rk17 (2010)). The given results by Chandra observations also show
that the objects RX J185635-3754 and 3C58 may be bare strange stars (Prakash
et al. rk55 (2003)). It is known that the compact objects such as neutron
star, pulsars, magnetars and strange quark stars, are under the influence of
strong magnetic fields which are typically about $10^{15}-10^{19}\ G$
(Kouveliotou et al. rk10 (1998), rk11 (1999); Haensel et al. rk12 (2007);
Glendenning rk13 (2000); Weber rk14 (2007); Camenzind rk15 (2007)). Therefore,
in astrophysics, it is of special interest to study the effect of a strong
magnetic field on the properties of SQM. We note that in the presence of a
magnetic field, the conversion of neutron stars to bare quark stars cannot
take place unless the value of the magnetic field exceeds $10^{20}\ G$ (Ghosh
& Chakrabarty rk16 (2001)).

In recent years, we have calculated the maximum gravitational mass and other
structure properties of a neutron star with a quark core at zero (Bordbar et
al. rk53 (2006)) and finite temperatures (Yazdizadeh & Bordbar rk54 (2011)).
We have also computed the structure properties of SQS at zero temperature
(Bordbar et al. rk18 (2009)) and finite temperature (Bordbar et al. rk19
(2011)). We have also calculated the structure of a magnetized SQS using MIT
bag model with a fixed bag constant ($90\ \frac{MeV}{fm^{3}}$) (Bordbar &
Peyvand rk20 (2011)). In the present work, we investigate the effect of
density dependence of bag constant on the structure of an SQS in the presence
of strong magnetic field.

##  II Computation of strange quark matter equation of state in the presence
of magnetic field

The equation of state (EOS) of strange quark matter (SQM) plays an important
role for determining the structure of stars at high densities. To obtain EOS
of SQM, there are different models based on Quantum Chromodynamics (QCD). At
present, it isn’t possible to achieve an exact EOS of SQM by primary
principles of QCD. Therefore, scientists have tried to find approximate
methods by combining the basic features of QCD, for example, MIT bag model
(Chodos et al. rk21 (1974); Weber rk14 (2007); Peshier et al. rk45 (2000);
Alford et al. rk46 (2005)), NJL model (Rehberg et al. rk47 (1996); Hanauske et
al. rk48 (2001); R¨uster & Rischke rk49 (2004); Menezes et al. rk50 (2006)),
and perturbative QCD model (Baluni rk51 (1978); Fraga et al. rk52 (2001);
Farhi & Jaffe rk2 (1984)).

In MIT bag model, the quarks in the bag are considered as a free Fermi gas,
and the energy per volume for SQM is equal to the kinetic energy of the free
quarks plus a bag constant ($\mathcal{B}$) (Chodos et al. rk21 (1974)). The
bag constant $\mathcal{B}$ can be interpreted as the difference between the
energy densities of the noninteracting quarks and the interacting ones.
Dynamically, its role is as the pressure that keeps the quark gas in constant
density and potential. In the initial MIT bag model, different values such as
$55$ and $90\ \frac{MeV}{fm^{3}}$ are considered for the bag constant. As we
know, the density of SQM increases from surface to the core of SQS, therefore
using a density dependent bag constant instead of a fixed bag constant is more
suitable.

###  II.1 Density dependent bag constant

The analysis of the experimental data achieved at CERN shows that the quark-
hadron transition happens at a density about seven times the normal nuclear
matter energy density ($156\ MeVfm^{-3}$) (Heinz rk22 (2001); Heinz & Jacobs
rk44 (2000); Farhi & Jaffe rk2 (1984)). However theoretically, for no density-
independent value of bag constant the hadron to quark matter transition takes
place (Burgio et al. rk26 (2002)). Therefore, it is essential to use a density
dependent bag constant. Recently, a density dependent form has been also
considered for $\mathcal{B}$ (Adami & Brown rk23 (1993); Jin & Jenning rk24
(1997); Blaschke et al. rk25 (1999); Burgio et al. rk26 (2002)). The density
dependence of $\mathcal{B}$ is highly model dependent. According to the
hypothesis of a constant energy density along the transition line, Burgio et
al. tried to determine a range of possible values for B by exploiting the
experimental data obtained at the CERN SPS (Burgio et al. rk26 (2002)). By
assumption that the transition to quark-gluon plasma is determined by the
value of the energy density only, they estimated the value of bag constant and
its possible density dependence. They attempted to provide effective
parameterizations for this density dependence, trying to cover a wide range by
considering some extreme choices in such a way that at asymptotic densities,
the bag constant has some finite value. They employed a Gaussian form as
follows

$\mathcal{B}(\rho)=\mathcal{B}_{\infty}+(\mathcal{B}_{0}-\mathcal{B}_{\infty})e^{-\gamma(\rho/\rho_{0})^{2}}.$
(1)

The parameter $\mathcal{B}_{0}=\mathcal{B}(\rho=0)$ is constant and equal to
$\mathcal{B}_{0}=400\ \frac{MeV}{fm^{3}}$. In the above equation, $\gamma$ is
a numerical parameter which is usually equal to $\rho_{0}\approx 0.17\
fm^{-3}$, the normal nuclear matter density. $\mathcal{B}_{\infty}$ depends
only on the free parameter $\mathcal{B}_{0}$.

The value of the bag constant ($\mathcal{B}$) should be compatible with
experimental data. The experimental results at CERN-SPS confirms a proton
fraction $x_{pt}=0.4$ (Heinz rk22 (2001); Heinz & Jacobs rk44 (2000); Burgio
et al. rk26 (2002)). Therefore, we use the equation of state of asymmetric
nuclear matter to evaluate $\mathcal{B}_{\infty}$. We use the lowest order
constrained variational (LOCV) many-body method based on the cluster expansion
of the energy for calculating the equation of state of asymmetric nuclear
matter as follows (Bodbar & Modarres rk27 (1997), rk28 (1998); Modarres &
Bordbar rk29 (1998); Bordbar & Bigdeli 2007a , 2007b , 2008a , 2008b ; Bigdeli
et al. rk34 (2009); Bigdeli et al. rk35 (2010)).

The asymmetric nuclear matter is defined as a system consisting of $Z$ protons
($pt$) and $N$ neutrons ($nt$) with the total number density
$\rho=\rho_{pt}+\rho_{nt}$ and proton fraction
$x_{pt}=\frac{\rho_{pt}}{\rho}$, where $\rho_{pt}$ and $\rho_{nt}$ are the
number densities of protons and neutrons, respectively. For this system, we
consider a trial wave function of the form,

$\psi=F\phi,$ (2)

where $\phi$ is the slater determinant of the single-particle wave functions,
and $F$ is the A-body correlation operator ($A=Z+N$) which is given by

$F=\mathcal{S}\prod_{i>j}f(ij).$ (3)

In the above equation, $\mathcal{S}$ is a symmetrizing operator.

For the asymmetric nuclear matter, the energy per nucleon up to the two-body
term in the cluster expansion is as follows

$E([f])=\frac{1}{A}\frac{<\psi|H|\psi>}{<\psi|\psi>}=E_{1}+E_{2}.$ (4)

The one-body energy, $E_{1}$, is

$E_{1}=\sum_{i=1}^{2}\sum_{k_{i}}\frac{\hbar^{2}k_{i}^{2}}{2m},$ (5)

where labels 1 and 2 are used for proton and neutron respectively, and $k_{i}$
is the momentum of particle $i$. The two-body energy, $E_{2}$, is given by

$E_{2}=\frac{1}{2A}\sum_{ij}<ij|\mathcal{V}(12)|ij-ji>,$ (6)

where

$\mathcal{V}(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12).$
(7)

In Eq. (7), $f(12)$ and $V(12)$ are the two-body correlation and nucleon-
nucleon potential, respectively. In our calculations, we use $UV_{14}+TNI$
nucleon-nucleon potential (Lagaris & Pandharipande 1981a , 1981b ). We
minimize the two-body energy with respect to the variations in the correlation
functions subject to the normalization constraint. From the minimization of
the two-body energy, we get a set of differential equations. By numerically
solving these differential equations, we can calculate the correlation
functions. The two-body energy is obtained using these correlation functions
and then we can calculate the energy of asymmetric nuclear matter. The
procedure of these calculations has been fully discussed in reference (Bordbar
& Modarres rk28 (1998)).

The experimental results at CERN-SPS confirm a proton fraction $x_{pt}=0.4$
(Heinz rk22 (2001); Heinz & Jacobs rk44 (2000); Burgio et al. rk26 (2002)).
Therefore, to calculate $\mathcal{B}_{\infty}$, we use our results of the
above formalism for the asymmetric nuclear matter characterized by a proton
fraction $x_{pt}=0.4$. By assuming that the hadron-quark transition takes
place at the energy density equal to $1100\ MeVfm^{-3}$ (Heinz rk22 (2001);
Burgio et al. rk26 (2002)), we find that the baryonic density of nuclear
matter corresponding to this value of the energy density is $\rho_{B}=0.98\
fm^{-3}$ (transition density). At densities lower than this value, the energy
density of SQM is higher than that of the nuclear matter. With increasing the
baryonic density, these two energy densities become equal at the transition
density, and above this value, the nuclear matter energy density remains
always higher. Later, we determine $\mathcal{B}_{\infty}=8.99\
\frac{MeV}{fm^{3}}$ by putting the energy density of SQM and that of the
nuclear matter equal to each other.

###  II.2 Energy density calculation of strange quark matter in the presence
of magnetic field

We consider SQM composed of $u$, $d$ and $s$ quarks with spin up (+) and down
(-). We denote the number density of quark $i$ with spin up by
$\rho_{i}^{(+)}$ and spin down by $\rho_{i}^{(-)}$. We introduce the
polarization parameter $\xi_{i}$ by

$\xi_{i}=\frac{\rho_{i}^{(+)}-\rho_{i}^{(-)}}{\rho_{i}},$ (8)

where $0\leq\xi_{i}\leq 1$ and $\rho_{i}=\rho_{i}^{(+)}+\rho_{i}^{(-)}$. Under
the conditions of beta-equilibrium and charge neutrality, we get the following
relation for the number density,

$\rho=\rho_{u}=\rho_{d}=\rho_{s},$ (9)

where $\rho$ is the total baryonic density of the system.

Within the MIT bag model, the total energy of SQM in the presence of magnetic
field ($B$) can be written as

$E_{tot}=E_{K}+\mathcal{B}+E_{M},$ (10)

where $E_{M}$ is the contribution of magnetic energy, $\mathcal{B}$ is the bag
constant (in this article, we use a density dependent bag constant (Eq. (1)),
and $E_{K}$ is the total kinetic energy of SQM. The total kinetic energy of
SQM is as follows,

$E_{K}=\sum_{i=u,d,s}E_{i},$ (11)

where $E_{i}$ is the kinetic energy of quark $i$,

$E_{i}=\sum_{p=\pm}\sum_{k^{(p)}}\sqrt{m_{i}^{2}c^{4}+\hbar^{2}k^{(p)^{2}}c^{2}}$
(12)

we ignore the masses of quark $u$ and $d$, while we assume $m_{s}=150\ MeV$
for quark $s$. After performing some algebra, supposing that
$\xi_{s}=\xi_{u}=\xi_{d}=\xi$, we obtain the following relation for the total
kinetic energy density ($\varepsilon_{K}=\frac{E_{K}}{V}$) of SQM,

$\displaystyle\varepsilon_{K}$ $\displaystyle=$
$\displaystyle\frac{3}{16\pi^{2}\hbar^{3}}\sum_{p=\pm}[\frac{\hbar}{c^{2}}k_{F}^{(p)}E_{F}^{(p)}(2\hbar^{2}k_{F}^{(p)^{2}}c^{2}+m_{s}^{2}c^{4})-m_{s}^{4}c^{5}ln(\frac{\hbar
k_{F}^{(p)}+E_{F}^{(p)}/c}{m_{s}c})]$ (13) $\displaystyle+$
$\displaystyle\frac{3\hbar
c\pi^{2/3}}{4}\rho^{4/3}[(1+\xi)^{4/3}+(1-\xi)^{4/3}]$

where

$k_{F}^{(\pm)}=(\pi^{2}\rho)^{1/3}(1\pm\xi)^{1/3},$ (14)

and

$E_{F}^{(\pm)}=(\hbar^{2}k_{F}^{(\pm)^{2}}c^{2}+m_{s}^{2}c^{4})^{1/2}.$ (15)

For SQM, the contribution of magnetic energy is as $E_{M}=-M.B$. If we assume
the magnetic field is along the $z$ direction, the contribution of the
magnetic energy of SQM is given by

$E_{M}=-\sum_{i=u,d,s}M_{z}^{(i)}B,$ (16)

where $M_{z}^{(i)}$ is the magnetization of the system corresponding to
particle $i$ which is given by

$M_{z}^{(i)}=N_{i}\mu_{i}\xi_{i}.$ (17)

In the above equation, $N_{i}$ and $\mu_{i}$ are the number and magnetic
moment of particle $i$, respectively. By some simplification, the contribution
of the magnetic energy density ($\varepsilon_{M}=\frac{E_{M}}{V}$) of SQM can
be obtained as follows

$\varepsilon_{M}=-\sum_{i=u,d,s}\rho_{i}\mu_{i}\xi_{i}B.$ (18)

Using the above equation and $\rho=\rho_{u}=\rho_{d}=\rho_{s}$ and with the
assumption that $\xi=\xi_{u}=\xi_{d}=\xi_{s}$, we have

$\varepsilon_{M}=-(\rho B\xi\mu_{s}+\rho B\xi\mu_{u}+\rho B\xi\mu_{d}).$ (19)

Now, we take advantage of numerical values of the magnetic moment for quarks (
Wong Samuel rk40 (1990)) :
$\mu_{s}=-0.581\ \mu_{N},\mu_{u}=1.852\ \mu_{N},\mu_{d}=-0.972\ \mu_{N}$.
Using Eq. (19) and above values, we conclude that

$\varepsilon_{M}=-0.299\rho\xi\mu_{N}B,$ (20)

where $\mu_{N}=5.05\times 10^{-27}\ (J/T)$ is nuclear magneton.

###  II.3 The results of energy of strange quark matter in the presence of
magnetic field

We have calculated the properties of strange quark matter in the presence of
magnetic field with the density dependent bag constant (Eq. (1)). Our results
are as follows.

Our results for the total energy density
($\varepsilon_{tot}=\frac{E_{tot}}{V}$) of strange quark matter (SQM) in the
presence of magnetic field have been plotted versus the polarization parameter
in Fig. 1 for various densities at $B=5\times 10^{18}\ G$. We see that there
is a minimum point in the energy curve for each density which shows a meta-
stable state for this system. As the density increases, the minimum point of
energy shifts to the lower values of the polarization, and finally it
disappears at high densities in which the system becomes nearly unpolarized.

In Fig. 2, we have plotted the polarization parameter corresponding to the
minimum point of energy versus density for two magnetic fields $B=5\times
10^{18}\ G$ and $B=5\times 10^{19}\ G$. We can see that the value of $\xi$
decreases by increasing the density, and it becomes nearly zero at high
densities. We have also drown the polarization parameter as a function of the
magnetic field at different densities in Fig. 3. As this figure shows, the
polarization parameter increases by increasing the magnetic field for all
densities.

For the strange quark matter, our results for the total energy density at
$B=5\times 10^{18}\ G$, which calculated with the density dependent bag
constant, have been shown as a function of density in Fig. 4. The results for
$\mathcal{B}=90\ \frac{MeV}{fm^{3}}$ at $B=5\times 10^{18}\ G$ (Bordbar &
Peyvand rk20 (2011)) are also given for comparison. It can be observed that
the total energy density has an increasing rate by increasing the density.
Also, it can be found that for $\rho$ greater (lower) than about $0.6\
fm^{-3}$, the energy of SQM with the density dependent bag constant is lower
(greater) than that with the fixed bag constant. From Fig. 4, it is seen that
for $\rho<0.6\ fm^{-3}$, the increasing of energy has a slow slope, whereas
for $\rho>0.6\ fm^{-3}$ this increasing is accomplished with a more quick
slope.

Fig. 5 shows the phase diagram for the strange quark matter. We can see that
as the density increases, the magnetic field grows monotonically. It
explicitly means that at higher densities, the ferromagnetic phase transition
occurs at higher values of the magnetic field.

###  II.4 The equation of state for strange quark matter in the presence of
magnetic field

In this section, we calculate the equation of state (EoS) of strange quark
matter (SQM) in the presence of magnetic field with density dependent bag
constant. Generally, we can calculate EoS using the following relation,

$P(\rho)=\rho\frac{\partial\varepsilon_{tot}}{\partial\rho}-\varepsilon_{tot},$
(21)

where $P$ is the pressure and $\varepsilon_{tot}$ is the energy density, which
in the presence of magnetic field, is obtained from Eq. (10). In Fig. 6, we
have compared our results for EoS of SQM at different magnetic fields. This
shows that for all magnetic fields, by increasing the density, pressure has an
increasing rate. Also, we can see that with increasing magnetic field, the
pressure increases. In Fig. 7, we have drawn EOS of SQM by density dependent
bag constant at $B=5\times 10^{18}\ G$. The results for $\mathcal{B}=90\
\frac{MeV}{fm^{3}}$ at $B=5\times 10^{18}\ G$ (Bordbar & Peyvand rk20 (2011))
are also given for comparison. This figure indicates that for $\rho$ greater
than about $0.52\ fm^{-3}$, when the bag constant is density dependent, the
pressure of SQM is greater than that of the density independent case.

##  III Structure of strange quark star

Quark stars are relativistic objects, therefore we used the general relativity
for calculation of their structures. Since most of the massive general
relativistic objects have some forms of rotation (very rapid in the case of
pulsars). In this calculations, we are interested in the investigation of the
strong magnetic field effects on the structure of a static strange quark star.
Using the the equation of state (EoS) of strange quark matter (SQM), we can
obtain the structure of these stars by numerically integrating the general
relativistic equations of hydrostatic equilibrium, the Tolman-Oppenheimer-
Volkoff (TOV) equation (Shapiro & Teukolski rk9 (1983)),

$\displaystyle\frac{dP}{dr}$ $\displaystyle=$
$\displaystyle-\frac{G(\frac{\varepsilon(r)m(r)}{r^{2}})(1+\frac{P(r)}{c^{2}\varepsilon(r)})(1+\frac{4\pi
r^{3}P(r)}{m(r)c^{2}})}{(1-\frac{2Gm(r)}{c^{2}})}$
$\displaystyle\frac{dm}{dr}$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}\varepsilon(r).$ (22)

In the above equations, $P$ is pressure and $\varepsilon(r)$ is the energy
density, $G$ is the gravitational constant and $m(r)$ is the mass inside
radius $r$ which is calculated as follows

$\displaystyle m(r)=\int_{0}^{r}4\pi r^{\prime
2}\varepsilon(r^{\prime})dr^{\prime}.$ (23)

Now, by selecting a central energy density $\varepsilon_{c}$, under the
boundary conditions $P(0)=P_{c}$ and $m(0)=0$, we integrate TOV equations
outwards to a radius $r=R$ at which $P$ vanishes (Shapiro & Teukolski rk9
(1983)).

In this section, we calculate the structure of the SQS with density dependent
bag constant in the presence of a magnetic field. We should note that a strong
magnetic field changes the spherical symmetry of the system and for magnetic
fields less than $10^{19}\ G$, this effect is ignorable (Felipe & Martinez
rk42 (2009); Gonzalez Felipe et al. rk43 (2011)). Considering the anisotropy
of the strange quark matter pressures in the presence of magnetic field, it
has been shown that for vanishing AMM (anomalous magnetic moments), the
perpendicular component of the pressure $P_{\perp}$ goes to zero at about
$2\times 10^{19}G$ (Gonzalez Felipe et al. rk56 (2008)). Thus in the case of
SQM, for $B<10^{19}G$ the anisotropy in the pressures is relatively small,
i.e, $P_{\perp}=P_{\parallel}$.

In Fig. 8, we have drawn the gravitational mass versus the central density
($\varepsilon_{c}$) for an SQS in the magnetic fields $B=0$ and $5\times
10^{18}\ G$. We see that as the central density increases, the gravitational
mass of an SQS increases, and finally it reaches a limiting value which is
called maximum gravitational mass. Fig. 8 shows that by presenting the
magnetic field, the gravitational mass decreases. The results for
$\mathcal{B}=90\ \frac{MeV}{fm^{3}}$ at $B=5\times 10^{18}\ G$ (Bordbar &
Peyvand rk20 (2011)) are also given in Fig. 8 for the sake of comparison. This
indicates that the density dependence of bag constant leads to substantially
higher values for the gravitational mass of SQS. With the density dependent
bag constant, we have found that the maximum gravitational mass of SQS is
about $1.62\ M_{\odot}$, while with the fixed bag constant, it is about $1.33\
M_{\odot}$.

We have plotted the gravitational mass of SQS as a function of the radius
(mass-radius relation) for the magnetic fields $B=0$ and $5\times 10^{18}\ G$
in Fig. 9. It is seen that for all cases of SQS, the gravitational mass
increases by increasing the radius. In Fig. 9, we have also compared our
results for the density dependent case of bag constant with those of density
independent case. We can see that for the case of fixed bag constant, the
increasing rate of gravitational mass versus radius is higher than that of
density dependent case. However, it will be more constructive to consider the
effects of rotation on the properties of the star which is beyond our present
investigation. Some authors have shown that considering the rotation of the
star leads to the larger maximum mass for strange quark stars (Shen et al.
rk57 (2005)).

##  IV Summary and conclusions

We have investigated a cold static strange quark star in the presence of a
strong magnetic field. For this purpose, some of the bulk properties of the
strange quark matter such as the energy density and equation of state have
been computed using the MIT bag model with the density dependent bag constant.
Calculations of the energy for different magnetic polarization in the presence
of a magnetic field demonstrated that as the density increases, the minimum
point of energy shifts to the lower values of the polarization. We have shown
that the value of the polarization parameter decreases by increasing the
density, and it also increases by increasing the magnetic field. Our results
at $B=5\times 10^{18}\ G$ show that for both cases of the density dependent
bag constant and fixed bag constant, the total energy density have an
increasing rate by increasing the density. We have shown that there is a
ferromagnetic phase transition at high magnetic fields. Our computations
indicate that the pressure increases by increasing the density and magnetic
field. In this work, we have also studied the structure properties of the
strange quark stars. Our results show that the gravitational mass of the
strange quark star increases by increasing the central energy density. It was
shown that this gravitational mass reaches a limiting value at higher values
of the central energy density. We have shown that the maximum mass of the
strange quark star reduces by presenting the magnetic field. Finally, a
comparison has been also made between the results of density dependent bag
constant and those of fixed bag constant. Our calculation with the density
dependent bag constant shows a higher maximum mass with respect to that of
fixed bag constant.

One of the possible astrophysical implications of our results is calculation
of the surface redshift $(z_{s})$ of SQS. This parameter is of special
interest in astrophysics and can be obtained from the mass and radius of the
star using the following relation (Camenzind rk15 (2007)),

$z_{s}=(1-\frac{2GM}{Rc^{2}})^{-\frac{1}{2}}-1.$ (24)

Our results corresponding to the maximum mass and radius of SQS calculated by
density dependent bag constant lead to $z_{s}=0.534\ ms^{-1}$ for the magnetic
field $B=5\times 10^{18}G$.

## Acknowledgements

This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha. We wish to thank Shiraz University Research Council.

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Figure 1: Total energy density ($\varepsilon_{tot}$) as a function of the
polarization parameter ($\xi$) for different densities ($\rho$).

Figure 2: Polarization parameter ($\xi$) versus density ($\rho$) for
$B=5\times 10^{18}$ and $5\times 10^{19}\ G$.

Figure 3: Polarization parameter ($\xi$) corresponding to the minimum points
of energy density versus the magnetic field ($B$) for different values of
density ($\rho$).

Figure 4: Total energy density versus density ($\rho$) calculated by density
dependent bag constant (full curve) at $B=5\times 10^{18}\ G$. The results for
$\mathcal{B}=90\ \frac{MeV}{fm^{3}}$ (dashed curve) at $B=5\times 10^{18}\ G$
have been also given for comparison.

Figure 5: Phase diagram for the strange quark matter in the presence of a
strong magnetic field.

Figure 6: The equation of state of SQM at $B=0$, $\ 5\times 10^{18}\ $ and
$5\times 10^{19}\ G$.

Figure 7: The equation of state of SQM in the case of density dependent bag
constant (full curve) at $B=5\times 10^{18}\ G$. The results for the case of
fixed bag constant ($\mathcal{B}=90\ \frac{MeV}{fm^{3}}$) (dashed curve) at
$B=5\times 10^{18}\ G$ have been also given for comparison.

Figure 8: Gravitational mass versus the central energy density
($\varepsilon_{c}$) at $B=0$ (full curve) and $B=5\times 10^{18}\ G$ (dashed
curve). The results for $\mathcal{B}=90\ \frac{MeV}{fm^{3}}$ (dashed dotted
curve) at $B=5\times 10^{18}\ G$ have been also given for comparison.

Figure 9: The gravitational mass versus radius at $B=0$ (full curve) and
$B=5\times 10^{18}\ G$ (dashed curve). The results for $\mathcal{B}=90\
\frac{MeV}{fm^{3}}$ (dashed dotted curve) at $B=5\times 10^{18}\ G$ have been
also given for comparison.


# Bounded conjugacy classes, commutators, and approximate subgroups

Pavel Shumyatsky  Pavel Shumyatsky: Department of Mathematics, University of
Brasilia, Brasilia-DF, 70910-900 Brazil<EMAIL_ADDRESS>
###### Abstract.

Given a group $G$, we write $g^{G}$ for the conjugacy class of $G$ containing
the element $g$. A famous theorem of B. H. Neumann states that if $G$ is a
group in which all conjugacy classes are finite with bounded size, then the
commutator subgroup $G^{\prime}$ is finite. We establish the following
results.

Let $K,n$ be positive integers and $G$ a group having a $K$-approximate
subgroup $A$.

If $|a^{G}|\leq n$ for each $a\in A$, then the commutator subgroup of $\langle
A^{G}\rangle$ has finite $(K,n)$-bounded order.

If $|[g,a]^{G}|\leq n$ for all $g\in G$ and $a\in A$, then the commutator
subgroup of $[G,A]$ has finite $(K,n)$-bounded order.

###### Key words and phrases:

Conjugacy classes, commutators, approximate subgroups

###### 2010 Mathematics Subject Classification:

20N99,20F12,20F24

This research was supported by the Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq), and Fundação de Apoio à Pesquisa do Distrito
Federal (FAPDF), Brazil.

##  1\. Introduction

Let $K$ be a positive integer. A symmetric subset $A$ of a group $G$ is a
$K$-approximate subgroup if there is a finite subset $E\subseteq G$ such that
$|E|\leq K$ and $AA\subseteq EA$. The formal definition of an approximate
subgroup was introduced by Tao in [13]. Since then many important results on
approximate subgroups have been established. In particular, Breuillard, Green
and Tao essentially described the structure of finite approximate subgroups
[4]. The reader is referred to the recent book [14], or the surveys [2, 3],
for detailed information on these developments.

An interesting principle was stated in [3]:

Group-theoretical arguments can often be successfully transferred to
approximate subgroups.

In the present article we check this principle against certain variations of
B. H. Neumann’s theorem that a BFC-group has finite commutator subgroup.

Given a group $G$ and an element $x\in G$, we write $x^{G}$ for the conjugacy
class containing $x$. More generally, if $X,Y\subseteq G$, we write $X^{Y}$
for the set of all $x^{y}$, where $x\in X$ and $y\in Y$. Of course, if the
number of elements in $x^{G}$ is finite, we have $|x^{G}|=[G:C_{G}(x)]$. A
group is called a BFC-group if its conjugacy classes are finite and have
bounded size. In 1954 B. H. Neumann discovered that the commutator subgroup
$G^{\prime}$ of a BFC-group $G$ is finite [9]. It follows that if $|x^{G}|\leq
n$ for each $x\in G$, then $G^{\prime}$ has finite $n$-bounded order.
Throughout the article we use the expression “$(a,b,\dots)$-bounded” to mean
that a quantity is finite and bounded by a certain number depending only on
the parameters $a,b,\dots$. A first explicit bound for the order of
$G^{\prime}$ was found by J. Wiegold [15], and the best known was obtained in
[8] (see also [10] and [12]). The article [6] deals with groups $G$ in which
conjugacy classes containing commutators are bounded. It is shown that if
$|x^{G}|\leq n$ for any commutator $x$, then the second commutator subgroup
$G^{\prime\prime}$ has finite $n$-bounded order. Later this was extended in
[5] to higher commutator subgroups. A related result for groups in which the
conjugacy classes containing squares have finite bounded sizes was obtained in
[7].

A stronger version of the Neumann theorem was recently established in [1]:

Let $n$ be a positive integer and $G$ a group having a subgroup $A$ such that
$|a^{G}|\leq n$ for each $a\in A$. Then the commutator subgroup of $\langle
A^{G}\rangle$ has finite $n$-bounded order.

Here, as usual, $\langle X\rangle$ denotes the subgroup generated by the set
$X$ and so $\langle A^{G}\rangle$ denotes the minimal normal subgroup
containing $A$.

In the present paper we extend the above result as follows.

######  Theorem 1.1.

Let $K,n$ be positive integers and $G$ a group having a $K$-approximate
subgroup $A$ such that $|a^{G}|\leq n$ for each $a\in A$. Then the commutator
subgroup of $\langle A^{G}\rangle$ has finite $(K,n)$-bounded order.

For a subset $X$ of a group $G$ we write $[G,X]$ to denote the subgroup
generated by all commutators $[g,x]$, where $g\in G$ and $x\in X$. It is well-
known that $[G,X]$ is a normal subgroup of $G$. Moreover, $[G,X]=[G,\langle
X\rangle]$. We examine approximate subgroups $A\subseteq G$ such that the
conjugacy classes of commutators $[g,a]$ have bounded sizes whenever $g\in G$
and $a\in A$.

######  Theorem 1.2.

Let $K,n$ be positive integers and $G$ a group having a $K$-approximate
subgroup $A$ such that $|[g,a]^{G}|\leq n$ for all $g\in G$ and $a\in A$. Then
the commutator subgroup of $[G,A]$ has finite $(K,n)$-bounded order.

It is worthwhile to mention that Theorem 1.2 was unknown even in the case
where $A$ is a subgroup of $G$. It can be regarded as an extension of the
aforementioned result in [6] that if $|x^{G}|\leq n$ for any commutator $x$,
then the second commutator subgroup $G^{\prime\prime}$ has finite $n$-bounded
order.

##  2\. Preliminaries

Let $G$ be a group generated by a set $X$ such that $X=X^{-1}$. Given an
element $g\in G$, we write $l_{X}(g)$ for the minimal number $l$ with the
property that $g$ can be written as a product of $l$ elements of $X$. A proof
of the following result can be found in [6, Lemma 2.1].

######  Lemma 2.1.

Let $G$ be a group generated by a set $X=X^{-1}$ and let $L$ be a subgroup of
finite index $m$ in $G$. Then each coset $Lb$ contains an element $g$ such
that $l_{X}(g)\leq m-1$.

The next lemma is almost obvious.

######  Lemma 2.2.

Let $k,n,s\geq 1$, and let $G$ be a group containing a set $X=X^{-1}$ such
that $|x^{G}|\leq n$ for any $x\in X$. Let $g_{1},\dots,g_{s}\in\langle
X\rangle$ and assume that $l_{X}(g_{i})\leq k$. Then
$C_{G}(g_{1},\dots,g_{s})$ has finite $(k,n,s)$-bounded index in $G$.

###### Proof.

Since $l_{X}(g_{i})\leq k$, we can write $g_{i}=x_{i1}\ldots x_{ik}$, where
$x_{ij}\in X$ and $i=1,\ldots,s$. By the hypothesis the index of
$C_{G}(x_{ij})$ in $G$ is at most $n$ for any such element $x_{ij}$. Set
$U=\cap_{i,j}C_{G}(x_{ij})$. We have $[G:U]\leq n^{ks}$. Since $U\leq
C_{G}(g_{1},\dots,g_{s})$, the lemma follows. ∎

The next observation will play a crucial role in the proof of Theorems 1.1 and
1.2.

######  Lemma 2.3.

Let $X$ be a normal subset of a group $G$ such that $|x^{G}|\leq n$ for any
$x\in X$, and let $H=\langle X\rangle$. Then the subgroup
$\langle[H,x]^{G}\rangle$ has finite $n$-bounded order.

###### Proof.

Without loss of generality we can assume that $X=X^{-1}$. Let $m$ be the
maximum of indices of $C_{H}(x)$ in $H$ for $x\in X$. Of course, $m\leq n$.
Take $x\in X$. Since the index of $C_{H}(x)$ in $H$ is at most $m$, by Lemma
2.1 we can choose elements $y_{1},\ldots,y_{m}$ in $H$ such that
$l_{X}(y_{i})\leq m-1$ and the subgroup $[H,x]$ is generated by the
commutators $[y_{i},x]$, for $i=1,\ldots,m$. For any such $i$ write
$y_{i}=y_{i1}\ldots y_{i(m-1)}$, with $y_{ij}\in X$. By using standard
commutator identities we can rewrite $[y_{i},x]$ as a product of conjugates in
$H$ of the commutators $[y_{ij},x]$. Let $\\{h_{1},\ldots,h_{s}\\}$ be the
conjugates in $H$ of all elements from the set $\\{x,y_{ij}\mid 1\leq i\leq
m,\ 1\leq j\leq m-1\\}.$ Note that the number $s$ here is $m$-bounded. This
follows form the fact that $C_{H}(x)$ has index at most $m$ in $H$ for each
$x\in X$. Put $T=\langle h_{1},\ldots,h_{s}\rangle$. Observe that the centre
$Z(T)$ has index at most $m^{s}$ in $T$, since the index of $C_{H}(h_{i})$ in
$H$ is at most $m$ for any $i=1,\ldots,s$. Thus, by Schur’s theorem [11,
10.1.4], we conclude that the commutator subgroup $T^{\prime}$ has finite
$m$-bounded order. Since $[H,x]$ is contained in $T^{\prime}$, deduce that the
order of $[H,x]$ is $m$-bounded. Further, the subgroup $[H,x]$ is normal in
$H$ and there are at most $n$ conjugates of $[H,x]$ in $G$. Therefore
$\langle[H,x]^{G}\rangle$ is a product of at most $n$ normal subgroups, each
of which has $n$-bounded order. Hence, the result. ∎

######  Lemma 2.4.

Let $G$ be a group and $x,y\in G$. Assume that $|x^{G}|=m$ and $|(yx)^{G}|\leq
m$. Suppose that there are $b_{1},\dots,b_{m}\in G$ such that
$x^{G}=\\{x^{b_{1}},\dots,x^{b_{m}}\\}$ and $y\in C_{G}(b_{1},\dots,b_{m})$.
Then $[G,y]\leq[G,x]$.

###### Proof.

First we note that $(yx)^{G}=\\{yx^{b_{1}},\dots,yx^{b_{m}}\\}$. This is
because all elements $yx^{b_{1}},\dots,yx^{b_{m}}$ are different and since by
the hypothesis $(yx)^{G}$ contains at most $m$ elements, the class $(yx)^{G}$
must coinside with $\\{yx^{b_{1}},\dots,yx^{b_{m}}\\}$. Therefore for any
$g\in G$ there is $b_{i}\in\\{b_{1},\dots,b_{m}\\}$ such that
$(yx)^{g}=yx^{b_{i}}$. So $y^{g}x^{g}=yx^{b_{i}}$ and
$[y,g]=x^{b_{i}}x^{-g}\in[G,x]$. The lemma follows. ∎

##  3\. Proof of Theorem 1.1

Recall that $A$ is a $K$-approximate subgroup of a group $G$ such that
$|a^{G}|\leq n$ for any $a\in A$. We wish to show that $H=\langle
A^{G}\rangle$ has finite commutator subgroup of $n$-bounded order.

Let $E=\\{e_{1},\dots,e_{K}\\}$ be a set of size $K$ such that $AA\subseteq
EA$. It will be assumed that $K$ is chosen as small as possible and so for
each $i=1,\dots,K$ there are $x_{i1},x_{i2},x_{i3}\in A$ such that
$x_{i1}x_{i2}=e_{i}x_{i3}$. By Lemma 2.3 each of the subgroups
$\langle[H,x_{ij}]^{G}\rangle$ has $n$-bounded order. Hence, also their
product has finite $(K,n)$-bounded order. Now we can pass to the quotient over
the product of all $\langle[H,x_{ij}]^{G}\rangle$ and without loss of
generality assume that the set $E$ is contained in the centre of $H$.

Denote by $X$ the set $A^{G}$. Let $m$ be the maximum of indices of $C_{H}(x)$
in $H$ for $x\in X$. Of course, $m\leq n$. Select $a\in A$ such that
$|a^{H}|=m$. Choose $b_{1},\ldots,b_{m}$ in $H$ such that $l_{X}(b_{i})\leq
m-1$ and $a^{H}=\\{a^{b_{i}};i=1,\ldots,m\\}$. The existence of the elements
$b_{i}$ is guaranteed by Lemma 2.1. Set $U=C_{G}(\langle
b_{1},\ldots,b_{m}\rangle)$. In view of Lemma 2.2 note that the index of $U$
in $G$ is $n$-bounded.

Let $r$ be the minimal number for which there are elements
$d_{1},\dots,d_{r}\in A$ such that $A$ is contained in the union of the left
cosets $d_{i}U$. We fix the elements $d_{i}$ and denote by $S$ the product of
the subgroups $\langle[H,d_{i}]^{G}\rangle$ for $i=1,\dots,r$. By Lemma 2.3
$S$ is a product of at most $r$ normal subgroups of finite $n$-bounded order.
Taking into account that $r$ is $n$-bounded conclude that $S$ has finite
$n$-bounded order.

Choose any $u\in A\cap U$. Since $AA\subseteq EA$ write $ua=ex$ for suitable
$e\in E$ and $x\in A$. Since $e\in Z(H)$ and $|x^{H}|\leq m$, it follows that
$|(ua)^{H}|\leq m$. Recall that $U=C_{G}(\langle b_{1},\ldots,b_{m}\rangle)$.
Lemma 2.4 implies that $[H,u]\leq[H,a]$. This happens for every choice of
$u\in A\cap U$ and so $[H,(A\cap U)]\leq[H,a]$.

Let $T=\langle[H,a]^{G}\rangle$ and observe that by virtue of Lemma 2.3 $T$
has finite $n$-bounded order. Choose an arbitrary element $d\in A$. There is
an index $j$ such that $d\in d_{j}U$ and ${d_{j}}^{-1}d\in U$. Since
${d_{j}}^{-1}d\in AA$, write ${d_{j}}^{-1}d=ey$ for suitable $e\in E$ and
$y\in A$. Observe that $e\in U$, whence $y\in U$. Taking into account that
$[H,d_{j}]\leq S$, $[H,e]=1$, and $[H,y]\leq T$ deduce

$[H,d]=[H,d_{j}ey]\leq[H,d_{j}][H,e][H,y]\leq ST.$

Since $d$ was chosen in $A$ arbitrarily, conclude that $[H,A]\leq ST$. Recall
that $H=\langle A^{G}\rangle$. We therefore conclude that $H^{\prime}\leq ST$.
Now the theorem follows from the fact that the order of $ST$ is $n$-bounded.
This completes the proof of the theorem.

##  4\. Proof of Theorem 1.2

Throughout this section $A\subseteq G$ is a $K$-approximate subgroup of a
group $G$ such that $|[g,a]^{G}|\leq n$ for each $g\in G$ and $a\in A$. We
need to prove that $[G,A]$ has finite commutator subgroup of $(K,n)$-bounded
order.

Let $X$ be the set of all conjugates of commutators $[g,a]$, where $g\in G$
and $a\in A$. Note that the set $X$ is symmetric. Put $H=\langle X\rangle$. By
Lemma 2.3 the subgroup $\langle[H,x]^{G}\rangle$ has finite $n$-bounded order
whenever $x\in X$.

######  Lemma 4.1.

For any $a\in A$ the subgroup $[H,[G,a]]$ has finite $n$-bounded order.

###### Proof.

Choose $a\in A$. Let $m_{0}$ be the maximum of indices of $C_{H}(x)$ in $H$,
where $x$ ranges through the set of commutators $[g,a]$ with $g\in G$. Select
$g_{0}\in G$ such that $|[g_{0},a]^{H}|=m_{0}$. Choose
$b_{1},\ldots,b_{m_{0}}$ in $H$ such that $l_{X}(b_{i})\leq m_{0}-1$ and
$[g_{0},a]^{H}=\\{[g_{0},a]^{b_{i}};i=1,\ldots,m_{0}\\}$. (The existence of
the elements $b_{i}$ is guaranteed by Lemma 2.1.) Set $U=C_{G}(\langle
b_{1},\ldots,b_{m_{0}}\rangle)$. Note that by Lemma 2.2 the index of $U$ in
$G$ is $n$-bounded. Let $U_{0}=\cap_{g\in G}U^{g}$ be the maximal normal
subgroup of $G$ contained in $U$. Obviously, the index of $U_{0}$ in $G$ is
$n$-bounded as well. For any $g\in G$ observe that
$[gg_{0},a]=[g,a]^{g_{0}}[g_{0},a]$. Choose $g\in U_{0}$ and set
$[g,a]^{g_{0}}=u$. Lemma 2.4 shows that $[H,u]\leq[H,[g_{0},a]]$.

Let $c_{1},\dots,c_{k}$ be a transversal of $U_{0}$ in $G$. For $i=1,\dots,k$
let $T_{i}$ denote the subgroup $\langle[H,[c_{i},a]]^{G}\rangle$. In view of
Lemma 2.3 each subgroup $T_{i}$ has finite $n$-bounded order. Further, let
$T_{0}$ denote the subgroup $\langle[H,[g_{0},a]]^{G}\rangle$. Likewise,
$T_{0}$ has finite $n$-bounded order. Let $N$ be the product of all $T_{i}$
for $i=0,1,\dots,k$.

Any element $g\in G$ can be written as a product $g=xc_{j}$ for suitable $x\in
U_{0}$ and $j\leq k$. Then we have $[g,a]=[xc_{j},a]=[x,a]^{c_{j}}[c_{j},a]$.
We now know that the images in $G/N$ of both $[x,a]$ and $[c_{j},a]$ are
central in $H/N$. It follows that also the image $[G,a]$ is central in $H/N$.
Since $N$ has finite $n$-bounded order, the lemma follows. ∎

Let $E=\\{e_{1},\dots,e_{K}\\}$ be a set of size $K$ such that $AA\subseteq
EA$. It will be assumed that $K$ is chosen as small as possible and so for
each $i=1,\dots,K$ there are $x_{i1},x_{i2},x_{i3}\in A$ such that
$x_{i1}x_{i2}=e_{i}x_{i3}$. By Lemma 4.1 for each $x_{ij}$ the subgroup
$N_{ij}=[H,[G,x_{ij}]]$ has finite $n$-bounded order. Let $N$ be the product
of all these subgroups $N_{ij}$ and observe that $N$ has finite
$(K,n)$-bounded order. Pass to the quotient $G/N$ and assume that
$[H,[G,x_{ij}]]=1$ for all $i,j$. Then of course $[H,[G,e_{i}]]=1$ for all
$i=1,\dots,K$. Therefore in what follows, without loss of generality, we will
assume that

$[G,E]\leq Z(H).$

Let $m$ be the maximum of indices of $C_{H}(x)$ in $H$, where $x$ ranges
through the set $X$.

######  Lemma 4.2.

For any $b\in\langle A\rangle$ and $g\in G$ we have $|[g,b]^{H}|\leq m$.

###### Proof.

Indeed, since $b\in\langle A\rangle$, we can write $b=ea$ for suitable $a\in
A$ and $e\in\langle E\rangle$. Then $[g,b]=[g,ea]\in[G,E]X$. Now use that
$[G,E]\leq Z(H)$ and $|x^{H}|\leq m$ for any $x\in X$ and deduce the lemma. ∎

Now fix $h_{0}\in G$ and $a_{0}\in A$ such that $|[h_{0},a_{0}]^{H}|=m$.
Choose $h_{1},\ldots,h_{m}$ in $H$ such that $l_{X}(h_{i})\leq m-1$ and

$[h_{0},a_{0}]^{H}=\\{[h_{0},a_{0}]^{h_{i}};i=1,\ldots,m\\}.$

The existence of the elements $h_{i}$ follows from Lemma 2.1. Set
$V=C_{G}(\langle h_{1},\ldots,h_{m}\rangle)$. Note that by Lemma 2.2 the index
of $V$ in $G$ is $n$-bounded. Let $V_{0}=\cap_{g\in G}V^{g}$ be the maximal
normal subgroup of $G$ contained in $V$ and note that the index of $V_{0}$ in
$G$ is $n$-bounded as well. By Lemma 4.1 the subgroup $S=[[H,[G,a_{0}]]$ has
finite $n$-bounded order.

######  Lemma 4.3.

$[H,[G,V_{0}\cap\langle A\rangle]]\leq S$.

###### Proof.

Let $b\in V_{0}\cap\langle A\rangle$. Lemma 4.2 tells us that
$|[g,a_{0}b]^{H}|\leq m$ for any $g\in G$. Moreover, observe that
$[g,a_{0}b]=[g,b][g,a_{0}]^{b}$ while $[g,b]\in V_{0}$. Lemma 2.4 shows that
$[H,[g,b]]\leq[H,[g,a_{0}]]\leq S$. This happens for every $g\in G$ so the
lemma follows. ∎

Let $r$ be the minimal number for which there are elements
$d_{1},\dots,d_{r}\in A$ such that $A$ is contained in the union of the left
cosets $d_{1}V_{0},\dots,d_{r}V_{0}$. We fix the elements $d_{i}$ and for each
$i=1,\dots,r$ put $M_{i}=[H,[G,d_{i}]]$. By Lemma 4.1 the product of all
$M_{i}$ has finite $n$-bounded order. Pass to the quotient $G/\prod_{i}M_{i}$
and, without loss of generality, assume that $[G,d_{i}]\leq Z(H)$ for each
$i$.

For an arbitrary element $a\in A$ there is $i\leq r$ such that $a\in
d_{i}V_{0}$. We have $[G,a]\leq[G,d_{i}][G,{d_{i}}^{-1}a]$. By assumptions,
$[H,[G,d_{i}]]=1$. Taking into account that ${d_{i}}^{-1}a\in V_{0}\cap\langle
A\rangle$ and using Lemma 4.3 deduce that $[H,[G,{d_{i}}^{-1}a]]\leq S$. Thus,
$[H,[G,a]]\leq S$ whenever $a\in A$. Since $H=\prod_{a\in A}[G,a]$, it follows
that $H^{\prime}\leq S$. This completes the proof of the theorem.

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