2021

[1]A. C. Khunt

[1]Department of Physics, Sardar Patel University, Vallabh Vidynagar, 388 120,
Gujarat, India

2]Department of Mathematics, Faculty of Science, The Maharaja Sayajirao
University of Baroda, Vadodara, 390 002, Gujarat, India

3]Department of Physical Sciences, P. D. Patel Institute of Applied Sciences,
Charotar University of Science and Technology, Changa , 388 421, Gujarat,
India

# Relativistic stellar modeling with perfect fluid core and anisotropic
envelope fluid

<EMAIL_ADDRESS>V. O. Thomas<EMAIL_ADDRESS>P. C. Vinodkumar
<EMAIL_ADDRESS>*  [  [

###### Abstract

We investigate the effect of density perturbations and local anisotropy on the
stability of stellar matter structures in general relativity using the concept
of cracking. Adopting a core-envelope model of a super-dense star, we examine
the properties and stability conditions by introducing anisotropic pressure to
the envelope region. Furthermore, we propose self-bound compact stars with an
anisotropic envelope as a potential progenitor for starquakes. We show how the
difference between sound propagation in radial and tangential directions would
be used to identify potentially stable regions within a configuration. Due to
an increase in the anisotropic parameter, strain energy accumulates in the
envelope region and becomes a potential candidate for building-up quake like
situation. This stress-energy stored in the envelope region that would be
released during a starquake of a self-bound compact star is computed as a
function of the magnitude of anisotropy at the core-envelope boundary.
Numerical studies for spherically asymmetric compact stars indicate that the
stress energy can be as high as $10^{50}$ erg if the tangential pressure is
slightly more significant than the radial pressure. It is happened to be of
the same order as the energy associated with giant $\gamma$-ray bursts. Thus,
the present study will be useful for the correlation studies between
starquakes and GRBs.

###### keywords:

Compact stars, Stellar stability, Gamma-ray burst

##  1 Introduction

Properties of compact stars are one the most sought-after topics in the field
of astroparticle physics in recent times itoh1970hydrostatic ;
collins1975superdense ; de2011self ; horvath2021modeling . Stability studies,
starquakes, wobbling of stars and the processes such as giant gamma-ray burst
etc are few examples. They release thousands times more energy than
supernovae. Their distinctive $\gamma$-ray emission lasts from fraction of
seconds to few minutes. A typical burst releases as much energy as the Sun
would in its entire 10-billion-year lifespan graham2013metal ; gendre2013ultra
. Several satellite missions (e.g. HETE-2, FERMI, INTEGRAL, SWIFT, RXTE,
ULYSSES) have been launched since the first observation of GRB and recorded
flare of GRBs and provided useful data related to the origin of GRBs. The
genesis and astrophysical implications of all these processes are vehemently
debated. The energy released in such processes are estimated to be $\sim
10^{44-47}$ erg kumar2015physics ; nakar2007short ; berger2014short . While
the reliable nature of these sources is absolutely fascinating and not yet
entirely understood, more spectacular events definitely related with them
challenge the researchers’ inquisitiveness. Many researchers have suggested
models for $\gamma$-ray superflares, which are based on compact objects (e.g.
Neutron stars, Black holes binary mergers). The model based on anisotropic
fluid for the highly compact stars (i.e. solid quarks) studied by Xu et al.
xu2006superflares  have proposed an alternative model from the conventional
one (i.e. the magnetar model)111Magnetars are the most highly magnetized
neutron stars in the cosmos (with magnetic field $10^{13}$–$10^{15}$ G)., that
SGR superflares could be caused by substantial quakes in quark stars. Xu et
al. xu2006superflares  addressed the prospect of gamma-ray bursts resulting
from the phase transition of a neutron star to a quark star, with the energy
released on the scale of $10^{52}$ erg. The degradation of the superstrong
field ($B\geq 10^{14}$ G) in magnetars might fracture the crust and create
bursts of high gamma-ray emission thompson1995soft ; thompson1996soft .
Gravitational, magnetic, and superfluid forces could all exert stress on the
crust of a neutron star that is evolving. Fracture of the star’s crust as a
result of these forces might have an impact on the star’s spin dynamics and
result in high-energy bursts franco2000quaking .

The source of the released energy is one of the most essential and fundamental
problems (inferred isotropic equivalent values are given in Table 1, Column
3.). There are few options for free energy sources that are worth examining
for comparison and future research. One such attempt is related to the high-
density phase transition at the core-evelope interface that can cause
instabilities. Mechanical readjustment (crust-cracking) of the star crust can
occur, depending on the parameters and the dynamics in which matter undergoes
a phase transition which lead to potential release of significant amount of
energy marranghello2002phase . Based on such estimates, the amount of energy
released is limited to around $\Delta E=W(\Delta a/a)$, where $W$ signifies
the star’s original binding energy and $\Delta a/a$ signifies the radius’s
fractional change. It is estimated that energy of the order $10^{53}$ erg is
released in such processes. It has been recently improved to include an exotic
solid quark phase which constitutes the bulk of the interior of the neutron
star, which is believed to be a possible source of free energy xu2003solid ;
peng2008pulsar ; zhou2004quakes ; xu2006superflares . The burning of neutron
matter to strange matter may be a source of free energy. For example, if the
energy per baryon number of three-flavor quark matter is smaller than that of
squeezed hadronic matter, and if other astronomical conditions are satisfied
then it induces conversion of a neutron star to a strange star shortly after
its formation berezhiani2002gamma ; ouyed2002quark . However, it is
conceivable for a neutron star to evolve into a quark star, particularly a
strange quark star. Quark stars are hypothesised extreme compact objects made
of quark matter. They are widely considered as the actual ground state of
compact baryon stars due to their high density. The presence of strange quark
matter raises a significant condition for the selection of the equation-of-
state (EoS) of quark stars: the EoS must incorporate the component provided by
strange quark matter in order for the EoS to be considered credible. Strange
quark matter has a lower pressure and energy than conventional quark matter
with the same quark number density, permitting to be a little more stable.

Superflares from GRBs graham2009grb ; soderberg2006afterglow  and SGRs
horvath2005energetics  Object Date $E_{\gamma}$ (erg) GRB 070714b 2007 Jul 14
$1.2\times 10^{51}$ GRB 051221a 2015 Dec 21 $(1.2-1.9)\times 10^{49}$ SGR
0526-66 1979 Mar 5 $6\times 10^{44}$ SGR 1900+14 1998 Aug 27 $2\times 10^{44}$
SGR 1806-20 2004 Dec 27 $3.5\times 10^{46}$

In relativistic astrophysics, the stability of a compact star structure is a
major concern. Chandrasekhar chandrasekhar1964dynamical  investigated radial
perturbation for isotropic fluid spheres in general relativity and proposed
the pulsation equation to evaluate fluid sphere stability. Generalizing this
method for use with anisotropic matter distributions was accomplished by Dev
and Gleiser dev2003anisotropic . Cracking, also known as overturning, is a
notion that was first presented by Herrera herrera1992cracking  and Di Prisco
et al.di1994tidal , and it proposes a different approach ($-1\leq
v_{s\bot}^{2}-v_{sr}^{2}\leq 1$, where $v_{s\bot}$ and $v_{sr}$ are tangential
and radial speed of sound respectively) determining whether or not an
anisotropic matter distribution is stable in general relativity. The
fundamental assumption of this approach is that the fluid constituents on each
side of the cracking point experience a relative acceleration towards each
other. It was developed with the purpose of describing the behavior of a fluid
distribution away from its equilibrium state. Later on, Chan et al.
chan1993dynamical  showed that even very minor departures from local isotropy
might contribute significantly for its structural stability. In conjunction
with this, they found that changes in density by themselves do not throw the
system out of equilibrium when it is configured with the anisotropic matter.
Such kinds of deviations can only be induced by perturbations that affect both
the density and the local anisotropy di1994tidal ; di1997cracking .

Increasing theoretical studies suggest that fascinating physical events may
cause local anisotropy, such as uneven radial and tangential stresses
$p_{r}\neq p_{\bot}$ ( see herrera1997local ; mak2003anisotropic , and
references therein). In the framework of General Relativity, Lemaître
lemaitre1933univers  noted that local anisotropy can relax the upper limits on
the maximum surface gravitational potential. Since the early works of Bowers
and Liang bowers1974anisotropic  its significance in General Relativity has
been investigated. Hillebrandt and Steinmetz hillebrandt1976anisotropic
studied the stability of completely relativistic anisotropic neutron star
models and found stability criteria comparable to that of isotropic models.
Subsequently, Chan et al. chan1993dynamical  evaluated the influence of the
local anisotropy in the emergence of dynamical instabilities. They observed
that moderate anisotropies might have a significant impact on the system’s
evolution.

Many studies have been reported on core-envelope anisotropic relativistic
objects in recent times sagar_2022 ; pant_2020three . In which compact stars
are divided into two distinct regions in accordance with the structure of
various physical considerations thomas2005core ; tikekar2005relativistic ;
mafa2016anisotropic ; gedela2019relativistic ; mardan2021charged ; pant_2021 ;
gedela_2020 . Any model for an anisotropic compact object is ineffective if it
is unstable against variations of its physical variables. Distinct degrees of
stability or instability lead to different patterns of development in the
collapse of self-gravitating objects. In this paper, we shall investigate the
effect of fluctuations in density and local anisotropy have on the potential
cracking of local and non-local anisotropic matter configurations within the
general relativity framework. In particular, we concentrate on how these
disturbances cause phenomena that are conducive for starquakes. Here, we
incorporate Einstein’s equations that determine cracking, overturning,
expansion, or collapse. Certain occurrences might radically change the
system’s evolution. If a configuration does not really crack (or overturn), it
is indeed potentially stable (not definitely stable), since further
perturbations might trigger expansion or collapse. Abreu et al. abreu2007sound
, Gonzalez et al. gonzalez2015cracking , and Ratanpal ratanpal2020cracking
proved that the regions for which $-1\leq v_{s\bot}^{2}-v_{sr}^{2}\leq 0$ are
potentially stable and the regions for which $0\leq
v_{s\bot}^{2}-v_{sr}^{2}\leq 1$ are potentially unstable , where
$v_{s\bot}^{2}=\frac{\partial p_{\bot}}{\partial\rho}$ and
$v_{sr}^{2}=\frac{\partial p_{r}}{\partial\rho}$.

Thus, compact stars are astrophysical laboratories of many extreme physics.
The core and envelope of a highly compact stars are made up of distinct
physical materials, according to the current knowledge of the strong
interactions processes leading to a phase transition to quark matter, the
study becomes extremely difficult due to the non-perturbative aspects of
quantum chromodynamics (QCD)annala2020evidence . An alternative treatment to
such a self bound system is being studied recently khunt2021distinct  based on
geometrical approach wherein the core-envelope model has been employed. The
model with isotropic core and anistropic envelope naturally supports quake
formation. Following such studies xu2006superflares  and shu2017gamma , the
prospects of a progenitor for starquakes from a self bound compact stars are
being addressed in this study. The core-envelope models studied by Thomas,
Ratanpal and Vinodkumar (TRV model) have considered anisotropic pressure in
the envelope region and isotropic pressure in the core region thomas2005core .
Physically such a scenario is possible with a core containing pure quark phase
and the envelope accumulating quark-hadron mixed phase. The comprehensive
investigation of the TRV model is described in our previous work
thomas2005core ; khunt2021distinct , the structural properties like mass-
radius relation, gravitational red-shift, Keplerian frequency, and surface
gravity are studied using the TRV model. Their structural properties show that
this model is important for the study of highly compact self-bound stars. A
density perturbation at the core-envelope interface can lead to an event like
starquake. The rearrangement of the star’s mass distribution caused by a
starquake affects the star’s moment of inertia, resulting in precession and
polar drifting link1998starquake . As a result, unusual spin behaviour might
be an indicator of the occurrence of a stellar quake. Evidence of crust
cracking may already exist in few isolated pulsars, showing that the process
is underway.

We utilize the core-envelope model based on the geometrical approach of highly
compact self bound stars to compute the stress-energy stored due to the
pressure anisotropy at the core-envelope interface that would be released in
the event of a quake like scenario. In Sec.2, we present anisotropic
Tolman–Oppenheimer–Volkoff equations, and we describe briefly the TRV model
and express the pressure anisotropy parameter $(S)$ at the interface of the
core and envelope. In Sec. 3, we discuss general relativity inspired EoS and
stellar stability. We then investigate the impact of possible anistropy in the
stability and cracking in Sec.4. In Sec.5, we estimate the amount of radius,
gravitational energy and moment of inertia due to change of anisotropy.
Finally, we summarize our main findings in Sec. 6 and conclude in Sec. 7.

##  2 Anisotropic matter configuration in general relativity

The following spherically symmetric line element in the Schwarzchild
coordinates $(x^{i})$=$(t,r,\theta,\phi)$ describes the interior of an
anisotropic fluid sphere. We start therefore with a metric of the form

$ds^{2}=e^{\nu(r)}dt^{2}-e^{\lambda(r)}dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta
d\phi^{2},$ (1)

where $\nu$ and $\lambda$ are functions of the radial coordinate $(r)$ only,
and a fluid stress-energy tensor $T^{\mu}_{\nu}=$ diag
$[\rho,-p_{r},-p_{\bot},-p_{\bot}]$, where $p_{r}$ and $p_{\bot}$ are the
radial and tangential pressures, respectively. The metric’s proper boundary
condition has to be matched with the Schwarchild exterior metric on the star’s
surface. It’s done as follows

$ds^{2}=\bigg{(}1-\frac{2GM}{ac^{2}}\bigg{)}dt^{2}-\bigg{(}1-\frac{2GM}{ac^{2}}\bigg{)}^{-1}dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta
d\phi^{2}$ (2)

$\nu(r=a)=\ln\bigg{(}1-\frac{2GM}{ac^{2}}\bigg{)}$ (3)

$\lambda(r=a)=-\ln\bigg{(}1-\frac{2GM}{ac^{2}}\bigg{)}.$ (4)

The radius and mass of the star are represented by $a$ and $M$, respectively.

The Einstein field equation is given by weinberg1972gravitation

$\mathcal{R}_{\mu\nu}-\frac{1}{2}\mathcal{R}g_{\mu\nu}=-\frac{8\pi
G}{c^{4}}T_{\mu\nu}$ (5)

for an energy–momentum tensor relevant for ideal fluid has been solved for the
metric provided by Eq. (1). The energy-momentum tensor for anisotropic fluid
distribution is taken as

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu}+\pi_{\mu\nu}$ (6)

where $\pi_{\mu\nu}$ denotes anistropic stress tensor given by thomas2005core

$\pi_{\mu\nu}=\sqrt{3}S\bigg{[}C_{\mu}C_{\nu}-\frac{1}{3}(u_{\mu}u_{\nu}-g_{\mu\nu})\bigg{]}$
(7)

where $S$=$S(r)$ is the magnitude of anisotropy stress tensor and
$C^{\mu}=(0,-e^{-\frac{\lambda}{2}},0,0)$, which is a radial vector.

The Einstein filed equations for this spacetime geometry and matter
distribution are

$\frac{(1-e^{-\lambda})}{r^{2}}+\frac{\lambda^{{}^{\prime}}e^{-\lambda}}{r}=8\pi\rho,$
(8)

$\frac{\nu^{{}^{\prime}}e^{-\lambda}}{r}-\frac{(1-e^{-\lambda})}{r^{2}}=8\pi
P_{r},$ (9)

$\frac{e^{-\lambda}}{4}\Big{(}2\nu^{{}^{\prime\prime}}+\nu^{{}^{\prime}2}-\nu^{{}^{\prime}}\lambda^{{}^{\prime}}+\frac{2\nu^{{}^{\prime}}}{r}-\frac{2\lambda^{{}^{\prime}}}{r}\Big{)}=8\pi
P_{\bot}.$ (10)

where prime denotes here differentiation with respect to the radial coordinate
$r$.

Using Eqs. (9) and (10), or correspondingly the conservation law
$T^{\mu}_{\nu\mathchar 24635\relax\;\mu}=0$, it is now convenient to transform
the above equations into a form where the hydrodynamical properties of the
system are more evident and that reduces to the TOV equations for systems with
isotropic pressure, i.e.,

$e^{-\lambda}=1-\frac{2m(r)}{r},$ (11)

$\nu^{{}^{\prime}}=\frac{2m(r)+8\pi r^{3}P_{r}}{r(r-2m(r))},$ (12)

$P^{{}^{\prime}}_{r}=-(\rho+P_{r})\frac{\nu^{{}^{\prime}}}{2}+\frac{2}{r}(P_{\bot}-P_{r}),$
(13)

where the mass function $m(r)$ is defined by

$m(r)=4\pi\int_{0}^{r}\rho~{}\bar{r}^{2}d\bar{r}.$ (14)

Moreover, it corresponds to the mass inside a sphere of radius $r$ as
perceived by a distant observer.

Combining (12) and (13), we finally obtain

$P^{{}^{\prime}}_{r}=-(\rho+P_{r})\frac{m(r)+4\pi
r^{3}P_{r}}{r(r-2m(r))}+\frac{2{\Delta}}{r}P_{r}.$ (15)

which is the generalized Tolmann-Oppenheimer-Volkoff (TOV) equation. It can be
seen that Eq. (15) readily reduces to the standard TOV when $P_{r}=P_{\bot}$.
Where $P_{\bot}=(1+\Delta)P_{r}$ is introduced here to account for the
anisotropy in the envelope. Interestingly, Eq. (15) which recalls the
Newtonian hydrostatic-equilibrium equation and where the final term is
obviously zero in the case of isotropic pressures, is evocative of the
Newtonian hydrostatic equilibrium equation, i.e., $P_{\bot}=P_{r}$.

###  2.1 The TRV core-envelope model

The model in the core-envelope family, as discussed by Thomas et al.
thomas2005core , is our primary focus in this paper. To solve Einstein’s
equations, they used ansatz for a pseudospheroidal geometry of spacetime. The
potential for this spacetime geometry is expressed as thomas2005core .

$e^{\lambda(r)}=\frac{1+K\frac{r^{2}}{R^{2}}}{1+\frac{r^{2}}{R^{2}}}$ (16)

K and R are geometric variables, respectively. With anisotropic stress tensor
$\pi_{\mu\nu}$, the energy–momentum tensor components (6) have nonvanishing
components.

$T^{0}_{0}=\rho,\,\,\ T^{1}_{1}=-\bigg{(}p+\frac{2S}{\sqrt{3}}\bigg{)},\,\,\
T^{2}_{2}=T^{3}_{3}=-\bigg{(}p-\frac{p}{\sqrt{3}}\bigg{)}.$ (17)

The magnitude of anisotropic stress is calculated using the following equation

$S=\frac{\mid p_{r}-p_{\perp}\mid}{\sqrt{3}}$ (18)

The core and envelope region’s boundary conditions are as follows

$\begin{split}S(r)=0\,\,\mbox{for}\,\,\,\,0\leq r\leq R_{C},\\\ S(r)\neq
0\,\,\mbox{for}\,\,\,\,R_{C}<r\leq R_{E}\end{split}$ (19)

where $R_{C}$ represents the core boundary radius and $R_{E}$ represents the
envelope boundary radius which is same as the radius of the star $(a)$ under
consideration. Using the conditions (19) with Eqs. (1), (5), (16) and (17),
the equations for pressures are given by Einstein’s field equations.

The Einstein field equation(5) relating to the metric (1) employing ansatz
(16) is provided by a combination of three equations:

$8\pi\rho=\frac{K-1}{R^{2}}\bigg{(}3+K\frac{r^{2}}{R^{2}}\bigg{)}\bigg{(}3+K\frac{r^{2}}{R^{2}}\bigg{)}^{-2},$
(20)

$8\pi
p_{r}=\bigg{[}\bigg{(}1+\frac{r^{2}}{R^{2}}\bigg{)}\frac{\nu^{{}^{\prime}}}{r}-\frac{K-1}{R^{2}}\bigg{]}\bigg{(}1+K\frac{r^{2}}{R^{2}}^{-1}\bigg{)},$
(21)

$\begin{split}8\pi\sqrt{3}S&=-\bigg{(}\frac{\nu^{{}^{\prime\prime}}}{2}+\frac{\nu^{{}^{\prime}2}}{4}-\frac{\nu^{{}^{\prime}}}{2r}\bigg{)}\bigg{(}1+\frac{r^{2}}{R^{2}}\bigg{)}\bigg{(}1+K\frac{r^{2}}{R^{2}}\bigg{)}^{-1}+\\\
&\frac{(K-1)}{R^{2}}r\bigg{(}\frac{\nu^{\prime}}{2}+\frac{1}{r}\bigg{)}\bigg{(}1+K\frac{r^{2}}{R^{2}}\bigg{)}^{-2}+\\\
&\frac{K-1}{R^{2}}\bigg{(}1+K\frac{r^{2}}{R^{2}}\bigg{)}^{-1}.\end{split}$
(22)

Thomas et al. thomas2005core  considered a case with isotropic core and
anisotropic envelope with radial pressure $p_{r}$ and tangential pressure
$p_{\bot}$. The anisotropy begins forming from the core boundary has radius
$r=R_{C}$. The radial difference in pressure in the enveloping region and it
reduces to zero at the surface ($r=a$, where an is the radius of the star
under consideration). We present the core upto the radius $r=R_{C}$, during
which $S(r)=0$. The radius of the star is considered as a and we split it into
two parts as conditions present in Eq.(19).

####  2.1.1 The Core of the star

The isotropic distribution of matter defines the core of the compact star. The
radial pressure $p_{r}$ is thus equal to the tangential pressure $p_{\bot}$
across the core region $0\leq r\leq R_{C}$, and $S(r)=0$. Eq. (22) thus
becomes

$\begin{split}&\bigg{(}\frac{\nu^{{}^{\prime\prime}}}{2}+\frac{\nu^{{}^{\prime}2}}{4}-\frac{\nu^{{}^{\prime}}}{2r}\bigg{)}\bigg{(}1+\frac{r^{2}}{R^{2}}\bigg{)}\bigg{(}1+K\frac{r^{2}}{R^{2}}\bigg{)}^{-1}-\\\
&\frac{(K-1)}{R^{2}}r\bigg{(}\frac{\nu^{\prime}}{2}+\frac{1}{r}\bigg{)}\bigg{(}1+K\frac{r^{2}}{R^{2}}\bigg{)}^{-2}=0\end{split}$
(23)

Equation (23) is a non-linear differential equation, if we adopt new
independent variable $z$ and dependent variable $F$ described by:

$z=\sqrt{1+\frac{r^{2}}{R^{2}}},$ (24) $F=e^{\nu/2},$ (25)

Equation (23) has the linear form

$(1-K+Kz^{2})\frac{\mathrm{d}^{2}F}{\mathrm{d}z^{2}}-Kz\frac{\mathrm{d}F}{\mathrm{d}z}+K(K-1)F=0.$
(26)

We get the solution of the metric potential ($e^{\nu/2}$) by transforming this
differential equation and solving it, we get

$F=e^{\nu/2}=A\sqrt{1+\frac{r^{2}}{R^{2}}}+B\bigg{[}\sqrt{1+\frac{r^{2}}{R^{2}}}L(r)+\frac{1}{\sqrt{2}}\sqrt{1+2\frac{r^{2}}{R^{2}}}\bigg{]}$
(27)

As a result, the spacetime metric of the core region $0\leq r\leq R_{C}$ is
described as follows:

$\begin{split}\mathrm{d}s^{2}&=\bigg{(}A\sqrt{1+\frac{r^{2}}{R^{2}}}+B\bigg{[}\sqrt{1+\frac{r^{2}}{R^{2}}}L(r)+\frac{1}{\sqrt{2}}\sqrt{1+2\frac{r^{2}}{R^{2}}}\bigg{]}\bigg{)}^{2}dt^{2}+\\\
&\bigg{(}\frac{1+2\frac{r^{2}}{R^{2}}}{1+\frac{r^{2}}{R^{2}}}\bigg{)}dr^{2}-r^{2}d\theta^{2}-r^{2}\mathrm{sin}\theta
d\phi^{2}.\end{split}$ (28)

The pressure and metric coefficients must be continuous across the core
boundary, $r=R_{C}$, in order to determine the constants $A$ and $B$.
Accordingly, the density distribution (core and envelope region) is expressed
as

$\rho=\frac{1}{8\pi
R^{2}}\bigg{[}3+2\frac{r^{2}}{R^{2}}\bigg{]}\bigg{[}1+2\frac{r^{2}}{R^{2}}\bigg{]}^{-2}.$
(29)

where R is a geometrical parameter. Eq. (29) provides the density distribution
in core and envelope region by using boundary condition for $0\leq r\leq
R_{C}$ for core and $R_{C}\leq r\leq R_{E}$ for envelope region.

As a result, the pressure distribution (both core and envelope region) is
expressed as :

$\displaystyle 8\pi p_{\scaleto{C}{3pt}}=$

for the radial pressure in the core region.

Where

$L(r)=\ln\bigg{(}\sqrt{2}\sqrt{1+\frac{r^{2}}{R^{2}}}+\sqrt{1+2\frac{r^{2}}{R^{2}}}\bigg{)}.$
(30)

The constants $A$ and $B$ are to be determined by requiring that the pressure
and metric coefficient must be continuous across the core boundary $r=R_{C}$;
and this is done as follows : The anisotropy parameter vanishes at the core
boundary, when $r=R_{E}$. Therefore, $\frac{a^{2}}{R^{2}}>2$ is necessary due
to the tangential pressure’s positive sign. The consistency of metric
coefficients and pressure across the distribution at $r=R_{C}$ leads to

$\sqrt{3}A+B\left[\sqrt{3}L(R_{C})+\sqrt{2.5}\right]=5^{-3/4}\left[11\sqrt{3}C+D\right],$
(31)
$\sqrt{3}A+B\left[\sqrt{3}L(R_{C})+\sqrt{2.5}\right]=5^{1/4}\left[11\sqrt{3}C+D\right],$
(32)

where

$L(R_{C})=\text{ln}(\sqrt{5}+\sqrt{6}).$ (33)

$\displaystyle
A=\frac{[5\sqrt{5}-3\sqrt{2}(\sqrt{3}L(R_{c})-\sqrt{2}.5)]C+\frac{1}{\sqrt{3}}[5\sqrt{5}+2\sqrt{2}(\sqrt{3}L(R_{c})-\sqrt{2.5}]D}{5\frac{5}{4}},$
(34)

$B=\frac{\sqrt{2}}{5\frac{5}{4}}[3\sqrt{3}C-2D].$ (35)

The curvature parameter $R$ is given by

$\displaystyle R=\sqrt{\frac{3\lambda}{8\pi\rho(a)}}$
$\displaystyle\quad,\>\lambda=\frac{\rho(a)}{\rho(0)}=\frac{1+\frac{2a^{2}}{3R^{2}}}{(1+2\frac{a^{2}}{R^{2}})^{2}}.$
(36)

where $\lambda$ is density variation parameter.

####  2.1.2 The Envelope of the Star

The envelope of the star is defined by the anisotropic concentration of
matter. Thus all across envelop region $R_{C}\leq r\leq R_{E}$ the radial
pressure $p_{r}$ is different from the tangential pressure $p_{\bot}$, and
therefore $S(r)\neq 0$. To determine the solution of Eq. (22), in this
context, we introduce new variables $z$ and $\psi$ specified by:

$z=\sqrt{1+\frac{r^{2}}{R^{2}}}\;\;\;\;\;\;\;\;\psi=\frac{e^{\nu/2}}{(1-K+Kz^{2})^{1/4}}$
(37)

in terms of which Eq. (22) has the following form:

$\frac{d^{2}\psi}{dz^{2}}+\bigg{[}\frac{2K(2K-1)(1-K+Kz^{2})-5K^{2}z^{2}}{4(1-K+Kz^{2})^{2}}+\frac{8\sqrt{3}\pi
R^{2}S(1-K-Kz^{2})}{z^{2}-1}\bigg{]}\psi=0$ (38)

On prescribing

$8\pi\sqrt{3}S=-\frac{(z^{2}-1)[2K(2K-1)(1-K+Kz^{2})-5K^{2}z^{2}]}{4R^{2}(1-K+Kz^{2})^{3}}$
(39)

the second term in Eq. (38) eliminates and the resultant equation is

$\frac{d^{2}\psi}{dz^{2}}=0,$ (40)

It has solution of the form :

$\psi=Cz+D$ (41)

where $C$ and $D$ are constants of integration. From Eq. (37) we obtain

$e^{\nu/2}=\bigg{(}1+K\frac{r^{2}}{R^{2}}\bigg{)}^{1/4}\bigg{(}C\sqrt{1+\frac{r^{2}}{R^{2}}}+D\bigg{)}.$
(42)

Thereby the spacetime metric of the enveloping region $b\leq r\leq R_{E}$ is
described by:

$\begin{split}\mathrm{d}s^{2}&=\sqrt{1+K\frac{r^{2}}{R^{2}}}\bigg{(}C\sqrt{1+\frac{r^{2}}{R^{2}}}+D\bigg{)}^{2}dt^{2}+\bigg{(}\frac{1+K\frac{r^{2}}{R^{2}}}{1+\frac{r^{2}}{R^{2}}}\bigg{)}dr^{2}\\\
&-r^{2}d\theta^{2}-r^{2}\mathrm{sin}\theta d\phi^{2}.\end{split}$ (43)

We shall set $K=2$ such that the distribution follows the same 3-space
geometry.

The following are explicit formula for the radial pressure $p_{r}$, tangential
pressure $p_{\bot}$, and anisotropy parameter $S(r)$:

$8\pi
p_{\scaleto{E}{3pt}}=\frac{C\sqrt{1+\frac{r^{2}}{R^{2}}}\big{(}3+4\frac{r^{2}}{R^{2}}\big{)}+D}{R^{2}\big{(}1+2\frac{r^{2}}{R^{2}}\big{)}^{2}\big{(}C\sqrt{1+\frac{r^{2}}{R^{2}}}+D\big{)}},$
(44)

$8\pi p_{\scaleto{E}{3pt}\bot}=8\pi
p_{\scaleto{E}{3pt}}-\frac{\frac{r^{2}}{R^{2}}\big{(}2-\frac{r^{2}}{R^{2}}\big{)}}{R^{2}\big{(}1+2\frac{r^{2}}{R^{2}}\big{)}^{3}},$
(45)

and anisotropy $S$ is expressed as

$8\pi\sqrt{3}S=\frac{\frac{r^{2}}{R^{2}}\big{(}2-\frac{r^{2}}{R^{2}}\big{)}}{R^{2}\big{(}1+2\frac{r^{2}}{R^{2}}\big{)}^{3}}.$
(46)

The constants $C$ and $D$ are to be determined by matching the solution with
Schwarzchild exterior solution (2). The following relationships are implied by
the continuity of metric coefficients and pressure along radial direction
across $r=a$:

$e^{\nu(a)}=\frac{1+\frac{a^{2}}{R^{2}}}{1+2\frac{a^{2}}{R^{2}}}=1-\frac{2M}{a}$
(47)

$C\sqrt{1+\frac{a^{2}}{R^{2}}}\left(3+4\frac{a^{2}}{R^{2}}\right)+D=0.$ (48)

Eqs.(47) and (48) determine constants $C$ and $D$ as

$C=-\frac{1}{2}\bigg{(}1+2\frac{a^{2}}{R^{2}}\bigg{)}^{-\frac{7}{4}},$ (49)

$D=\frac{1}{2}\sqrt{1+\frac{a^{2}}{R^{2}}}\bigg{(}3+4\frac{a^{2}}{R^{2}}\bigg{)}\bigg{(}1+2\frac{a^{2}}{R^{2}}\bigg{)}^{-\frac{7}{4}}.$
(50)

According to Eq. (29), the matter density at the centre is explicitly related
to the curvature parameter $R$ as

$\lim_{r\to 0}\rho(r)=\rho(0)\mathchar
24635\relax\;\;\;\therefore\;\;8\pi\rho(0)=\frac{3}{R^{2}}$ (51)

Eq.(51) estimates $R$ in terms of $\rho(a)$ and $\lambda$.

Fig. 1 depicts the radial profile of the temporal and spatial metric
potentials, $e^{\nu(r)}$ and $e^{-\lambda(r)}$, respectively. Both the metric
potentials are clearly finite at their centres and maintain consistently at
all points. The redshift factor $e^{\nu(r)}$ steadily increases from the
center of the star towards the asymptotic region. The metric function
$e^{-\lambda(r)}$ is flat near the center and reaches a minimum near the
surface of the star, where it joins the redshift factor. The metric potentials
$e^{-\lambda(r)}$, $e^{\nu(r)}$ are continuous and found well behaved in the
core and the envelope regions.

Figure 1: (Color online) Metric function of static star as function of radius
in units of km. For a $\lambda=0.07$ . The redshift factor $e^{\nu(r)}$
steadily increases from the center of the star towards the asymptotic region.
The metric function $e^{-\lambda(r)}$ is flat near the center and reaches a
minimum near the surface of the star, where it joins the redshift factor.

The radial and tangential pressures differ due to local pressure anisotropy
($p_{r}\neq p_{\perp}$). In a conventional isotropic
Tolman–Oppenheimer–Volkoff equation, the difference between radial and
tangential pressure generates an additional force. The discussion on
anisotropic pressure and the physical conditions for anisotropic stars can be
found in Refs. sulaksono2015anisotropic ; setiawan2019anisotropic , and
further details can be seen in an anisotropic pressure review paper
herrera1997local . In the present work, we examine quantitatively, the changes
on the properties of the compact star due to the local anisotropic pressure.
In order to adopt the model suggested in thomas2005core , we have computed
anisotropic magnitude for the core-envelope regions using the conditions (19)
with Eqs. (2.1.1), (44), (45) and (46). For a density variation
$(\lambda=0.07)$, the distributed anisotropy for the core and the envelope
regions is shown Fig. 2. For the core region, the anisotropy $(S)$ is zero
according to the TRV model thomas2005core , while in the envelope region
anisotropy is non-zero. The anisotropy magnitude inside the envelope region
computed using the Eq. (46) and their result is represented as blue solid
curve in Fig.2. It can be noticed that the anisotropy is an increasing
function of $r$. For 10.617 km $\leq r$ $\leq$ 12.177 km, the anisotropy
varies in the range $[2.458\times 10^{-8},2.774\times 10^{-6}]$. It is
important to note its influence on the stability of the envelope region. The
main reason for choosing a thin crust is to study processes like cracking or
overturning in the medium of anisotropy. To avoid confusion, we provide the
anisotropy profile for various density variation parameter, $\lambda$ in
Appendix A (8).

Figure 2: (Color online) Variation of an anisotropy $S$ in km-2 with respect
to a radius of the star within a range of boundary condition as given in Eq.
(19). For a density variation $(\lambda=0.07)$.

###  2.2 Revisited physical plausibility

The interior solution should fulfill certain fundamental physical criteria.
Some of the physical acceptability conditions for anisotropic materials have
been described in herrera1997local ; thomas2005core ; mak2003anisotropic  as

  1. (i)

radial pressure $p_{r}$, tangential pressure $p_{\bot}$ and density $\rho$
should be positive everywhere within the configuration;

  2. (ii)

gradients for radial pressure and density should be negative,

$\frac{\partial p_{r}}{\partial r}\leq
0\;\;\;\text{and}\;\;\;\frac{\partial\rho}{\partial r}\leq 0$

  3. (iii)

Sound speed at the interface should be continuous and should satisfy the
causality condition at the core of a compact star model, as well as decreasing
monotonically outwards.

$\frac{\partial p_{r}}{\partial\rho}\leq 1\;\;\;\text{and}\;\;\;\frac{\partial
p_{\bot}}{\partial\rho}\leq 1$

  4. (iv)

The core and the envelope for the star should satisfy the energy conditions
besides being continuous at the interface.

##  3 The geometrical EoS of extreme dense matter and stellar stability

In this work, we use geometrically deduced EoSs which are based on our
previous work khunt2021distinct  to study the canonical properties of compact
stars. In which, the pressure and density profile are computed using the Eqs.
(29), (2.1.1) and (44) for different choices density parameter ($\lambda=0.01$
and $\lambda=0.07$). The best fit for the pressure-density curve is found to
be in the quadratic form

$p=\gamma+\alpha\rho+\beta\rho^{2}$ (52)

where $\gamma$, $\alpha$ and $\beta$ are the fitted parameters. For two
different density variation parameters and their fitted value of EoSs are
represented in Table 3. Based on such an equation of state, the mass-radius
relation has been reported in our earlier study khunt2021distinct . The TRV
model based upon pseudo-spheroidal geometry, is also valid for different
choice of $\lambda$. Thomas et al. thomas2005core  shows that $\lambda$ can
varies from $0.01$ to $0.09$ for this particular model of superdense stars.
Looking for a thin crust situation, we present our results for the choice of
$\lambda=0.07$. The results of core radius and envelope size ($R_{E}=a-R_{c}$)
for two different values of $\lambda$ can be viewed from Table 3.

[h] Values of constants that generate EoSs, core and envelope radii ($R_{C}$,
$R_{E}$) for two different values of density variation parameters $\lambda$
$\gamma$ $\alpha$ $\beta$ $R_{C}$ (km) $R_{E}$ (km) 0.01 $-9.30\times 10^{-4}$
1.69 406 4.00 7.72 0.07 $-2.09\times 10^{-4}$ $0.77$ $1291.49$ 10.60 1.57

Figure 3: (Color online) Mass–central density relationship (panel (a)) and
Mass–radius relationship (panel (b)) for the two geometrical EoSs models
considered in these plots for comparison. The hatched (Yellow color) region in
panel (a) and (b) represents the range of precisely measured masses of binary
radio pulsar lattimer2012nuclear . The orange regions are the mass–radius
constraints from the GW170817 event abbott2018gw170817 . The blue region
represent the pulsar, is the NICER estimations of PSR J0030+0451 miller2019psr
.

In Fig.3, we show the Mass-Radius diagram with the TRV EoS considered in this
work. The density profile at a stellar structure’s centre, $r=0$, determines
how stable 222Not all branches of sequence $M=M(\rho_{c})$ are stable. This
can be unstable by means of radial oscillations. Degenerate stars with
$dM/d\rho_{c}<0$ are found to be unstable and will finally collapse towards
Neutron stars, or Black holes. it is against its own gravitational attraction.
At an equilibrium radius, the confined mass will be larger as the centre
density is higher. The mass of a self - gravitating compact star increases
with central density, in the stable region, in accordance with the Harrison-
Zel’dovich-Novikov criterion zeldovich_1971 ; harrison1_965 ; saklany_2023 .
The static stability condition necessitates that $dM/d\rho_{c}>0$ for all
points within the confining region, where $r<a$. As can be seen from Fig.3
(panel (a)), the mass of the model star $M({\rho_{c}})$ increases
monotonically with $dM/d\rho_{c}>0$, resulting to a stable stellar structure
as the central density $\rho_{c}$ increases. As a consequence, we find by
solving the TOV equation (Eq.(15) with $\Delta=0$) for a given EoS a maximum
possible mass corresponding to the central maximal density. Mass-radius
decreases with decreasing density variation parameter $\lambda$. The maximum
mass at maximum central density is about $1.68M_{\odot}$ and the corresponding
stellar radius 8.76 km for $\lambda=0.01$, while for $\lambda=0.07$ the
stellar mass is about $1.42M_{\odot}$ and radius 8.33 km. The TRV model of
superdense stars is very useful for classification of different class of
compact stars. For stars with
$M\geq\text{\begin{minipage}[t]{433.62pt}\rm\ignorespaces
\trivlist\par\trivlist@item@[]\hbox{\hbox{\hbox to0.0pt{\indent\hskip
0.0pt\hss}}\hskip 0.0pt\hbox{\hbox{}\hbox{\ignorespaces$\ignorespaces
M_{\odot}$}}}\par\endtrivlist\ignorespaces\end{minipage}}$, the radius changes
very little with $M$, $a\simeq 8-10$ km, comparable to neutron stars with a
moderately stiff EoS. However, at smaller $M$ the radius of bare strange star
behaves in an entirely different manner. It falls monotonically as $M$
decreases, with $a\propto M^{1/3}$ for $M\leq
0.5\text{\begin{minipage}[t]{433.62pt}\rm\ignorespaces
\trivlist\par\trivlist@item@[]\hbox{\hbox{\hbox to0.0pt{\indent\hskip
0.0pt\hss}}\hskip 0.0pt\hbox{\hbox{}\hbox{\ignorespaces$\ignorespaces
M_{\odot}$}}}\par\endtrivlist\ignorespaces\end{minipage}}$. as expected from
the fact that low-mass strange stars can be described by the Newtonian theory,
which gives $M\simeq\frac{4\pi}{3}\rho a^{3}$ . The decrease of $a$ with
decreasing $M$ is unique feature of strange star and quark stars or compact
star built of abnormal matter. It’s important to note that the mass-radius
constraint from the gravitational wave event GW170817 represented by the
orange clouded region corresponding to the heavier and lighter neutron star,
respectively abbott2018gw170817  (see insert in the Fig. 3. In this figure,
the blue clouded region show the NICER (Neutron Star Interior Composition
Explorer) mass-radius measurements on PSR J0030+0451 miller2019psr ;
riley2019nicer . Besides, the yellow horizontal band show the range of
precisely measured masses of binary radio pulsar lattimer2012nuclear . Because
we have the results of various EoSs in GR, we use the geometrically deduced
EoS to compute the mass-radius relation for different density variation
parameters, $\lambda$ that matches with the observational data. Strictly
speaking we investigated the mass-radius relation of neutron star in a general
relativistic framework for the class of highly compact self-bound stars (e.g,.
quark stars).

After calculating an equilibrium stellar model we analyze its stability as
stable equilibrium are of astrophysical relevance. Hereafter we briefly
address the stability with respect to density perturbations. For simplicity,
we restrict ourselves to non-rotating spherically symmetric equilibrium
models.

##  4 Cracking Stability

The sound speed is a microscopic quantity of immense importance, particularly
the potential upper bounds that might be obtained, because it impacts the EoS.
One approach to quantify dense materials is by the velocity of sound, which is
given by $v_{s}^{2}=dp/d\rho$, where $p$ is the pressure and the $\rho$ matter
density. Causality demands an absolute restriction on $v_{s}\leq 1$, and
thermodynamic stability ensures that $v_{s}^{2}\geq 0$ lattimer2001neutron .
The fundamental assumption concerning the upper bound of the velocity of sound
is that it satisfies causality ($v_{s}^{2}=dp/d\rho\leq c^{2}$), i.e. the
speed of sound cannot reach the speed of light, which is the significant
barrier imposed on the speed of sound by general principles according to
Zel’dovich et al. YBZ . Moreover, Bedaque and Steiner bedaque2015sound  showed
that the possibility of neutron stars with masses of around two solar masses,
together with knowledge of the EoS of hadronic matter at low densities, is
incompatible with the constraint $c/\sqrt{3}$. The speed of sound and its
impact on tidal deformability have also been analyzed by many authors
reed2020large ; moustakidis2017bounds ; van2017upper ; ma2019sound .

When examining arbitrary and independent density and anisotropy perturbations
(as in all earlier cracking investigations herrera1992cracking ; di1994tidal ;
di1997cracking ; abreu2007sound ; ratanpal2020cracking ), there are few
physical criteria to determine the magnitude (absolute and/or relative) of the
perturbation, i.e., relatively small (or huge) the perturbations should be a
cracking-like scenario could be produced by perturbations of varying orders of
magnitude (and relative size $\delta\Delta/\delta\rho$). In addition, these
earlier works have only considered continuous perturbations in their studies.
It is possible that variable perturbations have the potential to be more
effective in inducing cracking within a specific matter configuration.
However, we are looking for physical characteristics that can be tested to see
whether cracks are forming.

$\frac{\delta\Delta}{\delta\rho}\sim\frac{\delta(p_{\bot}-p_{r})}{\delta\rho}\sim\frac{\delta
p_{\bot}}{\delta\rho}-\frac{\delta p_{r}}{\delta\rho}\sim
v_{s\bot}^{2}-v_{sr}^{2}$ (60)

$v_{s\bot}^{2}$ and $v_{sr}^{2}$ stand for the radial and tangential sound
speeds, respectively abreu2007sound .

This fundamental idea employed in reviewing Herrera’s technique to identify
possibly unstable anisotropic matter structures based on the concept of
cracking. Now, by addressing the sound speeds and calculating Eq. (60), we may
not only have a more precise notion of the relative order of magnitude of the
perturbations ($\delta\Delta$ and $\delta\rho$) but also what regions are more
likely to be potentially unstable inside a matter configuration. There are two
possible stable and two possible unstable regions: those where the gradient of
anisotropy with respect to radial variable $r$ is larger than or equal to
zero, and those where it is negative. It is clear that because $0\leq
v_{sr}^{2}\leq 1$ and $0\leq v_{s\bot}^{2}\leq 1$, we have $\mid
v_{s\bot}^{2}-v_{sr}^{2}\mid\leq 1$ abreu2007sound . Hence,

$\displaystyle-1\leq v_{s\bot}^{2}-v_{sr}^{2}\leq
1\Rightarrow\begin{cases}-1\leq v_{s\bot}^{2}-v_{sr}^{2}\leq
0&\text{potentially stable}\\\ 0<v_{s\bot}^{2}-v_{sr}^{2}\leq
1&\text{potentially unstable}\end{cases}$ (61)

Accordingly, we may now examine possible stable/unstable regions inside
anisotropic models based on the difference of the propagation of sound within
the matter configuration. Those regions where $v_{sr}^{2}\geq v_{s\bot}^{2}$
will be unstable. On the other hand, if $v_{sr}^{2}\leq v_{s\bot}^{2}$
everywhere within a matter distribution, no cracking will occur. For
physically plausible models, the size of perturbations in the anisotropy
should always be lower than those in density, i.e. $\mid
v_{s\bot}^{2}-v_{sr}^{2}\mid$ $\leq$ $1$ $\Rightarrow$ $\mid\delta\Delta\mid$
$\leq$ $\mid\delta\rho\mid$. When $\delta\Delta/\delta\rho>0$, such
perturbations lead to the possibility of unstable configuration. Profile for
the $\delta\Delta/\delta\rho$ is displayed in Fig.4 for the anisotropic
envelope region. For the TRV core-envelope model under consideration , the
perturbation relation $\delta\Delta/\delta\rho\equiv v_{s\bot}^{2}-v_{sr}^{2}$
satisfies the physical limit as given by Eq.(61).

In Fig. 5, the radial ($v_{sr}$) and tangential sound velocity ($v_{s\perp}$)
as function of radii and density are shown. From the computed results, we
found that $v_{sr}$ as well as $v_{s\perp}$ is monotonically decreasing
outward with the continuity at the interface for the $v_{sr}$ and
discontinuity at interface for the $v_{s\perp}$. This discontinuity occurs at
the core-envelope interface actually comes from anisotropic pressure in the
envelope region, and thus interferes with sound velocity.

Figure 4: (Color online) Variation of the difference of radial and tangential
sound speeds for the anisotropic envelope configuration.

An important parameter which characterizes the stiffness of the EoS with
respect to density perturbations is the adiabatic index $\Gamma$. The physical
consistency of a relativistic anisotropic sphere is determined by $\Gamma$,
which is defined as the ratio of two specific heats bondi1964contraction . The
relativistic adiabatic indices also are important parameters that affect the
stability of any stellar system. These are defined as

$\Gamma_{r}(r)=\frac{p_{r}(r)+\rho(r)}{p(r)}\frac{\partial
p_{r}(r)}{\partial\rho(r)}$
$\Gamma_{\bot}(r)=\frac{p_{\bot}(r)+\rho(r)}{p_{\bot}(r)}\frac{\partial
p_{\bot}(r)}{\partial\rho(r)}$ (62)

The profile of $\Gamma$ of the core and envelope of both pressures are plotted
in Fig.6. Bondi bondi1964contraction  suggested that for a stable Newtonian
sphere, the $\Gamma$ should be greater than $\frac{4}{3}$. The $\Gamma$ is a
fundamental component of the instability criteria chandrasekhar1964dynamical .
In particular, the amount that contains all of the main attributes of the
equation of state on the instability formulas is described by Chandrasekhar
chandrasekhar1964dynamical , as a result, it serves as the link between the
relativist structure of a spherical static object and the equation of state of
the internal fluid. Specifically, $\Gamma$ varies from 2 to 4 in most
equations of state of neutron star matter. More specifically, in certain
circumstances, $\Gamma$ is a weak function of density, in other cases,
however, the density dependency is more complicated haensel2007neutron .

The results of $v_{s}^{2}$ and $\Gamma$ for the compact star matter are
presented in Fig.5 and Fig.6, respectively. The results show that there are
possible stable regions inside the anisotropic envelope of the star. Because
$\delta\Delta/\delta\rho<0$, the sound speed stability criterion, Eq.(61)
suggests that in the TRV anisotropic model, no cracking will occur. Fig.6
clearly indicate that the radial adiabatic indexes are continual at the
interface and satisfies the Bondi condition bondi1964contraction , but in the
tangential case it does not obey the Bondi condition. It will be the main
reasons for the deformation and cracks for the Newtonian sphere.

Figure 5: (Color online) Left: Speed of sound $v_{s}^{2}$ as a function radial
distance for the TRV EoS. Right: Speed of sound $v_{s}^{2}$ versus density.
The vertical dotted line shows the core-envelope interface and the horizontal
dotted line represents for the conformal limit ($v_{s}^{2}<1/3$). The red
solid line represents radial velocity in the core and the envelope regions,
respectively,. The solid blue line described tangential velocity in the
envelope region, as well as grey upper band shows causality region.

Figure 6: (Color online) The adiabatic index $\Gamma$ versus radii and
density(in geometrical unit) in a compact star core and envelope. The red-
solid line is for radial $\Gamma$ in the core and evelope regions,
respectively, the blue-solid line corresponds to the case in which the
tangential velocity the envelope region. The horizontal orange band is
$\Gamma=4/3$ characteristic of a free ultra-relativistic Fermi gas.

##  5 Estimates of stellar parameters $:$ $\Delta a$, $\Delta E_{g}$ and
$\Delta I$

To account for the small anisotropy parameter, S of the order of $10^{-8}$ to
$10^{-6}$, high numerical precision is required to arrive at the solution of
the anisotropic TOV equation. Different anisotropic scales are presented here
in the form of $\Delta$, and their values are chosen by the domain of
anisotropy of Eq.(46) (for $\lambda=0.07$). In the case of isotropic pressure,
$S=0$, Eq. (15) turn out to be the TOV equation. We have incorporated $\Delta$
with Eq.(15) for solving the anisotropic TOV equations, we employ the TRV
equation of state deduced from our earlier study of a core-envelope model of
superdense stars khunt2021distinct ; thomas2005core . To begin with, the
star’s core is considered to be homogeneous, with the density,
$\rho=\rho_{0}$. Here, we have fixed the central density for the TRV EoS at
$\rho_{0}=1.34\times 10^{15}$ g cm-3, Eqs. (14) and (15) are integrated
numerically to determine the global structure of (e.g. mass and radius) of a
compact star.

The total gravitational energy ($E_{g}$) as well as the stellar moment of
inertia ($I$) are computed as follows :

(i) Gravitational Energy:

A refined formula to compute the $E_{g}$ containing the  compactness parameter
$x_{GR}=r_{g}/a$, as proposed by lattimer2001neutron  is written as

$E_{g}\simeq 1.6\times
10^{53}\Big{[}\frac{M(\Delta)}{M_{\odot}}\frac{x_{GR}(\Delta)}{0.3}\Big{]}\frac{1}{1-0.25~{}x_{GR}(\Delta)}\,\,\text{erg}.$
(63)

Here, $x_{GR}=2GM/ac^{2}=r_{g}/a$, where $r_{g}$ is the Schwarzschild radius
which depends implicitly on the anisotropic parameters, $\Delta$ through $M$
and $a$. The importance of relativistic effects for a neutron star mass $M$
and radius $a$ is characterized by the compactness parameter $r_{g}/a$.

(ii) Moment of inertia versus $M$ and $a$ :

To quantify moment of inertia, we have used approximate formula relating $I$
to stellar mass and radius ravenhall1994neutron . They showed that the ratio
$I/Ma^{2}$ depends mostly on the compactness parameter ($x_{GR}$) and is
expressed as thorne1977relativistic

$\displaystyle I\simeq 0.21M(\Delta)a^{2}_{\infty}\mathchar
24635\relax\;\;\;\;\;a_{\infty}=\frac{a}{\sqrt{1-x_{GR}(\Delta)}}.$ (64)

where $a_{\infty}$ is an apparent radius related to the circumferential radius

In Fig.7 we show the behaviour of the moment of inertia with respect to the
TOV mass for $\lambda=0.07$ with $\Delta=0$. The moment of inertia at maximum
stable mass, $M_{max}$ is about $8.24\times 10^{44}$ g cm2.

Figure 7: (Color online) Total moment of inertia versus total mass for
$\lambda=0.07$ without anisotropy ($\Delta=0$).

In the computation, it is vital to enhance precision in order to accomplish
the differences that we had in $E_{g}(\Delta\neq 0)-E_{g}(\Delta=0)$ and
$I(\Delta\neq 0)-I(\Delta=0)$. To optimize integration precision, we have
divided the domain into smaller and smaller sections at the boundary, until
the computed values are apparent.

Figure 8: (Color online) The distinction $a(\Delta\neq 0)-a(\Delta=0)$ in
stellar radius as a function of stellar mass. Where black-dash line for
$\Delta=10^{-6}$, red-dot line for $\Delta=10^{-7}$ and for $\Delta=10^{-8}$
is shown in blue dash-dot line.

Figure 9: (Color online) As in Fig. 8, but for difference in gravitational
energy.

Figure 10: (Color online) As in Fig. 8, but for the ratio difference in moment
of inertia.

The computed values of $a(M)$, $E_{g}(M)$ and $I(M)$ are shown in Fig. 8, 9
and 10, respectively. In the mass-radius calculation, we considered $\Delta$
from $10^{-6}$ to $10^{-8}$. From Fig.8, it is seen that as $\Delta$ varies
from $10^{-8}$ to $10^{-6}$, the radius of the star in this range varies by
$10^{-5}$ to $10^{-1}$ km.

Starquakes can cause a drastic change in the evelope region, releasing both
gravitational and tangential strain energy franco2000quaking ;
horvath2005energetics ; xu2003solid . In general, the changes in stellar
radius, $E_{g}$, and $I$ have direct correlation to stellar mass and
anisotropic parameter $\Delta$. This indicates that in a highly compact self-
bound star with a greater mass, a quake-like event should be more significant
for larger change in $\Delta$. Moreover, the disruptive instability of the
radius may lead to more chaotic events at the core-envelope interface and can
cause huge energy release. Fig.9 shows the computed gravitational energy
difference corresponds to the anisotropy parameter $\Delta\sim 10^{-6}$ and
$M_{max}\sim 1.43M_{\odot}$. It is observed that the released energy is as
high as $10^{50}$ erg. The energy released decreases with a decreasing
magnitude of anisotropy (for $\Delta$ varies from $10^{-6}$ to $10^{-8}$).
This scale of energy released is comparable with the energy observed in SGRs
and GRBs (see in Table 1). From our calculation as shown in Fig.9, we find
that a giant starquake with $\Delta\leq 10^{-6}$ could produce such an
energetic flare. Based on different choices of the anisotropy parameters,
xu2006superflares  have shown for quarks stars that the energy released on
$\Delta=10^{-5}$ is of the order of $10^{49}$ erg. A change in $\Delta$ might
cause a change in its spin frequency, $\Delta\Omega/\Omega$ $=-\Delta I/I$
leading to glitches. We observed from Fig.10 that such glitches with
$\Delta\Omega\sim 10^{-5}$ to ${10^{-1}}$ could appear for mass variability of
$M=0.2-1.43M_{\odot})$ and $\Delta=10^{-8}$ to $10^{-6}$ and a giant energetic
flare is considered to be associated with a high-amplitude glitch.

##  6  Discussion

To summarize, we have systematically investigated the compact stars having a
thin crust particularly its gravitational energy, the moment of inertia as an
implicit function of the anisotropy magnitude ($\Delta$) predicted by the TRV
core-envelope model of compact star with anisotropy at the envelope region. We
modeled the compact star as a self-bound core of incompressible fluid with a
fragile envelope. The anisotropy variation as shown in Fig.2 indicates that
the anisotropy starts at the core boundary increases up to a radial distance
of 10 to 12 km . It further decreases towards the envelope boundary of the
star. In Fig.8, we have shown the mass-radius relation for different values of
the anisotropy parameter in the range $10^{-8}\leq\Delta\leq 10^{-6}$ and
computed the difference in the radius of the star. Further the gravitational
energy differences and moment of inertia difference due to the radial
fluctuations caused by the anisotropy parameter are also estimated (see
Fig.9). It is found that the magnitude of this energy is comparable to the
energy produced by SGRs and GRBs.

Cracking is a concept that was first suggested for self-gravitating
anisotropic matter configurations by L Herrera and collaborators
herrera1992cracking ; di1994tidal ; di1997cracking . This concept has been
reintroduced in our analysis. In this study, we followed their approach and
studied the cracking and stability of self-bound anisotropic stars. It has
been revealed that, for certain dependent perturbation in particular, the
ratio for fluctuations in anistropy to energy density,
$\delta\Delta/\Delta\rho$, could be well understood in terms of the difference
between the velocities of sound (i.e. $\delta\Delta/\delta\rho\equiv
v_{s\bot}^{2}-v_{sr}^{2}$). Cracking points in a configuration that satisfies
$\mid\delta\Delta/\delta\rho\mid>1$, would not lead to physically unstable
models. From the present study, we are able to clearly determine, based on
equation (61), the region within the matter distribution are more likely to be
potentially stable/unstable. From the Fig.4, we see that $-1\leq
v_{s\bot}^{2}-v_{sr}^{2}\leq 0$ condition is satisfied in the anisotropic
configuration. Our results indicate that anisotropic structure is potentially
stable (as shown in Fig.4). Moreover the work done by abreu2007sound  argued
that when the system is entirely anisotropic ($p_{\bot}\neq 0$ and $p_{r}=0$)
the system may be considered possibly stable and it can be essential for
cracking like event. The faults generated due to the cracking events in the
evelope region can lead to quakes. The superflares of $\gamma$-ray emission
(e.g., GRBs, SGRs) could be caused by quakes in such cases.

##  7 Conclusion

In conclusion, we are able to propose a viable model towards our understanding
of the production of GRBs to the conventional framework. According to our
theoretical calculation and numerical results : if the tangential pressure is
slightly larger [only $(1+10^{-6})$ times] than the radial pressure in a self
bound star with mass $1.43M_{\odot}$, the released gravitational energy during
a quake could be as high as $10^{50}$ erg, for spherically symmetric compact
stars. We have examined the evolution of anisotropy in the envelope of highly
compact star. The envelope develops cracks under the pressure stress caused by
density fluctuations induced by the anisotropy. The amount of energy emitted
from this event is very remarkable. It is possible for stresses to build in
the envelope of a high compact star resulting the crust being fractured (i.e.,
a starquake), which may have an impact on the star’s spin development and
generate glitches.

A star-quake is a possible activation source for stellar instabilities, e.g.,
it might be linked to a pulsar glitch or a magnetar flare. The idea is that
the star’s evolution, such as magnetic field loss, produces strain in the
elastic crust, and the stored energy is abruptly released when the system
reaches a critical threshold. The usual energy released in this process would
consequently be of the order of the maximum mechanical energy that can be
stored in the crust, which is estimated to be $10^{-9}-10^{-7}M_{\odot}c^{2}$
blaes1989neutron ; mock1998limits . This is significant considering magnetars
appear to be associated with rather regular flare outbursts
watts2016colloquium . Internal phase transitions are another possibility
linked with the star’s evolution. From the point of view of the present study
, it is important to note that the anisotropy largely at the envelope region
that causes deformation at the envelope (star crust) leading to matter being
thrown out and falling back.

## Declarations

  * •

Funding : Not applicable

  * •

Conflict of interest/Competing interests: Not applicable

  * •

Availability of data and materials: No data has been used in this paper and
the materials used have been cited in appropriate places

  * •

Code availability: Not applicable

  * •

Authors’ contributions: (i) Khunt: Conceptualization, methodology, Analytical
and Numerical calculation, manuscript preparation ; (ii) Thomas :
investigation, writing-review, ; (iii) Vinodkumar: writing-review,
calculation-analysis

##  8 Appendix A : Anisotropy profile

In this appendix, we show anisotropy plot for values of $\lambda=$ 0.1, 0.3,
0.5, and 0.9 from the TRV model presented in Subsection 2.1.2.

Figure 11: (Color online) Variation of an anistropy $S$ in km-2 with respect
to a envelope radius of the star. For a density variation of
$\;\;\lambda=0.1,0.3,0.5\;\;\text{and},0.9$.

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# Schreier’s type formulae and two scales for growth of Lie algebras and
groups

Victor Petrogradsky  Department of Mathematics, University of Brasilia,
70910-900 Brasilia DF, Brazil<EMAIL_ADDRESS>Dedicated to 70th
anniversary of Vesselin Drensky

###### Abstract.

Let $G$ be a free group of rank $n$ and $H\subset G$ its subgroup of finite
index. Then $H$ is also a free group and the rank $m$ of $H$ is determined by
Schreier’s formula $m-1=(n-1)\cdot|G:H|.$

Any subalgebra of a free Lie algebra is also free. But a straightforward
analogue of Schreier’s formula for free Lie algebras does not exist, because
any subalgebra of finite codimension has an infinite number of generators.

But the appropriate Schreier’s formula for free Lie algebras exists in terms
of formal power series. There exists also a version in terms of exponential
generating functions. This is a survey on how these formulas are applied to
study 1) growth of finitely generated Lie algebras and groups and 2) the
codimension growth of varieties of Lie algebras. First, these formulae allow
to specify explicit formulas for generating functions of respective types for
free solvable (or more generally, polynilpotent) Lie algebras. Second, these
explicit formulas for generating functions are used to derive asymptotic for
these two types of the growth. These results can be viewed as analogues of the
Witt formula for free Lie algebras and groups. In case of Lie algebras, we
obtain two scales for respective types of growth. We also shortly mention the
situation on growth for other types of linear algebras.

###### Key words and phrases:

Identical relations, growth, generating functions, codimension sequence,
solvable Lie algebras, polynilpotent Lie algebras

###### 2000 Mathematics Subject Classification:

16R10, 16P90, 17B01, 17B65

##  1\. Analogue of Schreier’s formula for free Lie algebras

Denote the ground field by $K$. Let $X$ be an at most countable set supplied
with a weight function $\mathop{\mathrm{wt}}\nolimits:X\to\mathbb{N}$, namely
we assume that

$\displaystyle X=\mathop{\cup}_{i=1}^{\infty}X_{i};\quad X_{i}=\\{x\in X\
|\mathop{\mathrm{wt}}\nolimits x=i\\},\quad|X_{i}|<\infty,\ i\in\mathbb{N}.$

We say that $X$ is finitely graded. Assume that an algebra $A$ is generated by
$X$, then we naturally define the weight of a monomial $a\in A$. Let $Y$ be a
set of monomials in $X$, then one defines the Hilbert-Poincaré series of $Y$
(with respect to $X$), see e.g. [4]

${\mathcal{H}}_{X}(Y)={\mathcal{H}}_{X}(Y,t):=\mathop{\sum}_{i=1}^{\infty}|Y_{i}|t^{i};\qquad
Y_{i}=\\{y\in Y\ |\mathop{\mathrm{wt}}\nolimits y=i\\},\ i\in\mathbb{N}.$

Consider a subspace $V\subset A$, then we define ${\mathcal{H}}_{X}(V)$ using
a homogeneous basis of $\mathop{\rm gr}\nolimits V$; where $\mathop{\rm
gr}\nolimits V$ being the associated graded space.

Next, we introduce the operator ${\mathcal{E}}$ on power series
$\phi(t)=\sum_{n=1}^{\infty}b_{n}t^{n}$, $b_{n}\in\\{0,1,2,\dots\\}$ (see [25,
27]):

${\mathcal{E}}\,:\,\phi(t)=\sum_{n=1}^{\infty}b_{n}t^{n}\ \longrightarrow\
{\mathcal{E}}(\phi(t)):=\sum_{n=0}^{\infty}a_{n}t^{n}=\prod_{n=1}^{\infty}\frac{1}{(1-t^{n})^{b_{n}}}.$

Assume that $L$ is a Lie algebra generated by $X$ and $U(L)$ its universal
enveloping algebra. Let

${\mathcal{H}}_{X}(L,t)=\sum_{n=1}^{\infty}b_{n}t^{n},\quad{\mathcal{H}}_{X}(U(L),t)=\sum_{n=0}^{\infty}a_{n}t^{n}.$

One has a well-known formula that explains importance of the operator above
${\mathcal{H}}_{X}(U(L))={\mathcal{E}}({\mathcal{H}}_{X}(L))$ [36]. The
following is the natural analogue of Schreier’s formula, introduced by the
author. For basic facts on free Lie (super)algebras see [1, 2].

######  Theorem 1 ([27]).

Assume that $L$ is a free Lie algebra generated by a finitely graded set $X$.
Let $H$ be a subalgebra and $Y$ is a set of its free generators. Then

${\mathcal{H}}(Y,t)-1=({\mathcal{H}}(X,t)-1)\cdot{\mathcal{E}}({\mathcal{H}}(L/H),t).$

Let $L$ be a Lie algebra. One defines the lower central series as $L^{1}=L$,
$L^{i+1}=[L,L^{i}]$ for $i=1,2,\dots$. Now, $L$ is nilpotent of class $s$ iff
$L^{s+1}=\\{0\\}$ while $L^{s}\not=\\{0\\}$. All Lie algebras nilpotent of
class at most $s$ form the variety denoted by $\mathbf{N}_{s}$. A Lie algebra
$L$ is called polynilpotent with a tuple of integers
$(s_{q},\dots,s_{2},s_{1})$ iff there exists a chain of ideals

$\\{0\\}=L_{q+1}\subset L_{q}\subset\dots\subset L_{2}\subset L_{1}=L,\qquad
L_{n}/L_{n+1}\in\mathbf{N}_{s_{n}},\ n=1,\ldots,q.$

All polynilpotent Lie algebras with a fixed tuple form a variety denoted by
$\mathbf{N}_{s_{q}}\dots\mathbf{N}_{s_{2}}\mathbf{N}_{s_{1}}$. In the
particular case $s_{q}=\cdots=s_{1}=1$, one obtains the variety
${\mathbf{A}}^{q}$, consisting of solvable Lie algebras of length at most $q$.
On the other hand, a polynilpotent Lie algebra is solvable. Moreover, the free
polynilpotent Lie algebras, a tuple being fixed, provide interesting examples
of solvable Lie algebras. The definitions in case of group theory are similar.
Let $G$ be a group, denote by $\\{\gamma_{n}(G)|n=1,2,\ldots\\}$, terms of the
lower central series (warning: below $\gamma$ has also a different meaning!).

Suppose that $L$ is the free Lie algebra of rank $k$,
$L=\oplus_{n=1}^{\infty}L_{n}$ its natural grading, and $G$ the free group of
rank $k$. Then the lower central series factors
$\gamma_{n}(G)/\gamma_{n+1}(G)$ are free abelian groups and their ranks are
given by the classical Witt formula [1]:

$\psi_{k}(n):=\operatorname{rank}_{\mathbb{Z}}\gamma_{n}(G)/\gamma_{n+1}(G)=\dim_{K}L_{n}=\frac{1}{n}\sum_{a|n}k^{a}\mu\left(\frac{n}{a}\right)\approx\frac{k^{n}}{n},\quad
n\in\mathbb{N},$ (1)

where $\mu(*)$ is the Möbius function. Theorem 1 allows to derive the
following explicit formulas.

######  Theorem 2 ([27, 29]).

Consider the free polynilpotent Lie algebra
$L=F({\bf{N}}_{s_{q}}\cdots{\bf{N}}_{s_{1}},k)$ of finite rank $k\geq 2$,
where $q\geq 1$. Set $\beta_{0}(z):=0$, $\alpha_{0}(z):=kz$, and define the
following functions recursively

$\displaystyle\begin{split}\beta_{i}(z)&:=\beta_{i-1}(z)+\sum_{m=1}^{s_{i}}\frac{1}{m}\sum_{a|m}\mu\left(\frac{m}{a}\right)\left(\alpha_{i-1}(z^{m/a})\right)^{a}\\!\\!,\\\
\alpha_{i}(z)&:=1+(kz-1)\cdot{\mathcal{E}}(\beta_{i}(z)),\end{split}\qquad
1\leq i\leq q.$

Then ${\mathcal{H}}(L,z)=\beta_{q}(z).$

For example, we have a particular case.

######  Corollary 3 ([27, 29]).

Let $L:=F({\bf{A}}{\bf{N}}_{d},k)$. Then

${\mathcal{H}}(L,z)=\psi_{k}(1)z+\dots+\psi_{k}(d)z^{d}+1+\frac{kz-1}{(1-z)^{\psi_{k}(1)}\cdots(1-z^{d})^{\psi_{k}(d)}}.$

##  2\. Scale 1 and Growth of free solvable (polynilpotent) finitely generated
Lie algebras and groups

Now we describe applications of the analogue of Schreier’s formula for free
Lie algebras (Theorem 1 and its application Theorem 2) to specify the growth
of free solvable (more generally, polynilpotent) Lie algebras and groups of
finite rank.

Assume that $L$ is a relatively free algebra of some multihomogeneous variety
of (associative, or Lie) algebras, generated by $X=\\{x_{1},\dots,x_{k}\\}$.
Then we have a natural grading $L=\mathop{\oplus}_{n=1}^{\infty}L_{n}$ by
degree in $X$. One defines the growth function with respect to $X$ as
$\gamma_{L}(X,n):=\sum_{s=1}^{n}\dim_{K}L_{s}$.

######  Theorem 4 (Berele, [3]).

The growth function of a finitely generated associative PI-algebra is bounded
by a polynomial function.

For more details on proofs of this important result see [4, 16]. But the
growth of finitely generated Lie PI-algebras is more complicated. In this
case, the author introduced scale (2) of functions of intermediate growth and
suggested that it is complete in the sense of Conjecture 1 below. Define
functions

$\displaystyle\begin{split}\ln^{(0)}x:=x,\qquad&\ln^{(s+1)}x:=\ln(\ln^{(s)}x),\\\
\exp^{(0)}x:=x,\qquad&\exp^{(s+1)}x:=\exp(\exp^{(s)}x),\end{split}\qquad\qquad
s=0,1,2,\dots$

Recall standard notations $f(x)\approx g(x)$, $x\to\infty$, denotes that
$\mathop{{\rm lim}\,}_{x\to\infty}f(x)/g(x)=1$; $f(x)=o(g(x))$ means that
beginning with some number $f(x)=\alpha(x)g(x)$ and $\mathop{{\rm
lim}\,}_{x\to\infty}\alpha(x)=0$. Also, $\zeta(x)$ is the zeta function.

Consider the scale 1 consisting of a a series of functions
$\Phi^{q}_{\alpha}(n)$, $q=1,2,3,\dots$ of a natural argument with a parameter
$\alpha\in\mathbb{R}^{+}$:

$\displaystyle\text{{scale
1:}}\qquad\begin{split}\Phi^{1}_{\alpha}(n)&:=\alpha,\\\
\Phi^{2}_{\alpha}(n)&:=n^{\alpha},\\\
\Phi^{3}_{\alpha}(n)&:=\exp(n^{\alpha/(\alpha+1)}),\\\
\Phi^{q}_{\alpha}(n)&:=\exp\bigg{(}{n\over(\ln^{(q-3)}n)^{1/\alpha}}\bigg{)},\quad\quad
q=4,5,\dots.\end{split}$ (2)

Now, we specify the growth of free solvable (more generally, polynilpotent)
Lie algebras of finite rank with the respect to scale (2) by giving the
following asymptotic.

######  Theorem 5 ([25]).

Consider the free polynilpotent Lie algebra
$L:=F({\bf{N}}_{s_{q}}\cdots{\bf{N}}_{s_{1}},k)$ of finite rank $k\geq 2$,
where $q\geq 2$, generated by $X=\\{x_{1},\dots,x_{k}\\}$. Then

$\displaystyle\gamma_{L}(X,n)=\left\\{\begin{array}[]{ll}\displaystyle\frac{A+o(1)}{N!}\,n^{N},&q=2,\\\\[10.0pt]
\displaystyle\exp\Big{(}(C{+}o(1))\,n^{{N}/(N+1)}\Big{)},&q=3,\\\\[10.0pt]
\displaystyle\exp\Biggl{(}(B^{1/N}{+}o(1))\,{\frac{n}{(\ln^{(q-3)}n)^{1/N}}}\Biggr{)},&q\geq
4,\end{array}\right.\qquad n\to\infty;$

where the constants are:

$\displaystyle N:=s_{2}\dim_{K}F({\bf{N}}_{s_{1}},k),\quad
A:=\frac{1}{s_{2}}\left(\frac{k-1}{\prod_{q=2}^{s_{1}}q^{\psi_{k}(q)}}\right)^{s_{2}},$
$\displaystyle B:=s_{3}A\zeta(N{+}1),\qquad\qquad
C:=(1+1/N)(BN)^{\frac{1}{1+N}}.$

This result has the following application to group theory. Let $G$ be a group.
Due to Lazard, one constructs the related Lie algebra [19]:

$\displaystyle
L_{K}(G):=\mathop{\oplus}\limits_{n=1}^{\infty}(\gamma_{n}(G)/\gamma_{n+1}(G))\otimes_{\mathbb{Z}}K.$

If $G$ is a free polynilpotent group, then $L_{K}(G)$ is the free
polynilpotent Lie algebra of the same rank and with the same tuple, moreover,
the lower central series factors are free abelian groups (A. Shmelkin [35]).

######  Corollary 6 ([25, 29]).

Let $G=G(\mathbf{N}_{s_{q}}\dots\mathbf{N}_{s_{1}},k)$, $q\geq 2$, be the free
polynilpotent group of rank $k$. Consider ranks of the lower central series
factors, namely, denote
$b_{n}:=\operatorname{rank}_{\mathbb{Z}}\gamma_{n}(G)/\gamma_{n+1}(G)$, $n\geq
1$. Then we get the asymptotic:

$b_{n}=\left\\{\begin{array}[]{ll}\displaystyle\frac{A+o(1)}{(N-1)!}\,n^{N-1},&q=2,\\\\[10.0pt]
\displaystyle\exp\Big{(}(C{+}o(1))\,n^{{N}/(N+1)}\Big{)},&q=3,\\\\[10.0pt]
\displaystyle\exp\Biggl{(}(B^{1/N}{+}o(1))\,{n\over(\ln^{(q-3)}n)^{1/N}}\Biggr{)},&q\geq
4,\end{array}\right.\qquad n\to\infty;$

where $N,A,B,C$ are the same as in Theorem 5.

Observe that just by setting $s_{q}=\cdots=s_{1}=1$, Theorem 5 and its
Corollary 6 are turned into results on the free solvable Lie algebra and group
of rank $k$ and length $q$. A similar observation is valid concerning the
results (e.g. Theorem 20) on the codimension growth below.

M.I.Kargapolov raised problem 2.18 in [17] to specify the lower central series
ranks for free polynilpotent finitely generated groups. Exact recursive
formulae were given by Egorychev [6]. We suggest another answer to this
problem by specifying the asymptotic behaviour of that ranks. We consider to
view the asymptotic of Theorem 5 and its Corollary 6 as an analogue of the
Witt formula (1), now for the free solvable (more generally, polynilpotent)
Lie algebras and groups.

Since the growth of finitely generated solvable Lie algebras is intermediate
[20], the respective generating function converges in the unit circle. It is
important to study an asymptotic of the generating function when we approach
the unit circumference from inside. Since the coefficients of the series are
real nonnegative numbers, it is sufficient to study this behaviour while
$z=t\to 1-0$. The crucial step to prove Theorem 5 is the following asymptotic
of the generating function ${\mathcal{H}}_{X}(L,t)$.

######  Theorem 7 ([25]).

Let $L:=F({\bf{N}}_{s_{q}}\cdots{\bf{N}}_{s_{1}},k)$ be the free polynilpotent
Lie algebra of finite rank $k\geq 2$, where $q\geq 2$, generated by
$X=\\{x_{1},\dots,x_{k}\\}$. Then

$\begin{array}[]{ll}\displaystyle\lim_{t\to
1{-}0\,}(1-t)^{N}{\mathcal{H}}_{X}(L,t)=A,&q=2,\\\\[8.0pt]
\displaystyle\lim_{t\to
1{-}0\,}(1-t)^{N}\ln^{(q-2)}({\mathcal{H}}_{X}(L,t))=s_{3}\,\zeta(N{+}1)\,A,&q\geq
3;\end{array}$

where $N,A$ are the same as in Theorem 5.

Now, we describe an important idea to prove Theorem 7. The explicit formula of
the generating function (Theorem 2) shows that ${\mathcal{H}}_{X}(L,t)$ is
”roughly speaking” a $(q-1)$-iteration of ${\mathcal{E}}$ applied to
$\beta_{1}(z)$. We can easily describe the first application:

$\displaystyle\beta_{1}(z)$
$\displaystyle=\psi_{k}(1)z+\psi_{k}(2)z^{2}+\dots+\psi_{k}(s_{1})z^{s_{1}};$
$\displaystyle{\mathcal{E}}(\beta_{1}(z))$
$\displaystyle=\frac{1}{(1-z)^{\psi_{k}(1)}\cdots(1-z^{s_{1}})^{\psi_{k}(s_{1})}}\approx\frac{\mu}{(1-t)^{M}},\quad\text{as}\quad
t\to 1{-}0;$ $\displaystyle\text{where}\quad M$
$\displaystyle=\psi_{k}(1)+\dots+\psi_{k}(s_{1})=\dim_{K}F({\bf{N}}_{s_{1}},k),\qquad\mu=\prod_{q=2}^{s_{1}}q^{-\psi_{k}(q)}.$

Next, we use the fact that ${\mathcal{E}}$ is ”approximately” the exponent,
thus ${\mathcal{H}}_{X}(L,t)$ behaves like

$\exp^{(q-2)}\left(\frac{\bar{\mu}+o(1)}{(1-t)^{N}}\right),\qquad t\to
1{-}0\,.$

Another idea in the proof of Theorem 5 is to specify a connection between the
growth of a function analytic in the unit circle with asymptotic of its
coefficients [25].

A similar version of Schreier’s formula for free Lie superalgebras was
established as well [27, 29]. Also, the asymptotic of Theorem 5 was extended
to the case of free solvable (polynilpotent) Lie superalgebras of finite rank
[13].

Below we see that the scale (4) for the superexponential codimension growth
for varieties of Lie algebras is rather complete (Theorem 12). So, we
conjecture that the scale (2) for the intermediate growth of finitely
generated Lie PI-algebras is complete as well.

######  Conjecture 1 ([22]).

Let $L$ be a finitely generated Lie PI-algebra. Then there exist numbers
$q,N_{0}$ such that

$\gamma_{L}(n)\leq\exp\bigg{(}\frac{n}{\ln^{(q)}n}\bigg{)},\quad n\geq N_{0}.$

This bound was confirmed for almost solvable (more generally, almost
polynilpotent) Lie algebras [14, 15].

##  3\. Exponential analogue of Schreier’s formula for free Lie algebras

Let us consider complexity functions, refereed to also as exponential
generating functions in combinatorics. Assume that we are given a set $A$ of
monomials in $X=\\{x_{i}\,|\,i\in\mathbb{N}\\}$. Consider a set of distinct
elements $\widetilde{X}=\\{x_{i_{1}},\dots,x_{i_{n}}\\}\subset X$, denote by
$P_{n}(A,\widetilde{X})$ the set of all multilinear elements of degree $n$ on
$\widetilde{X}$ belonging to $A$. Suppose that the number of these elements
$c_{n}(A,\widetilde{X})$ does not depend on the choice of $\widetilde{X}$, but
depends only on $n$. In this case, we denote
$c_{n}(A):=c_{n}(A,\widetilde{X})$ and say that $A$ is $X$-uniform and define
the complexity function with respect to $X$:

${\mathcal{C}}_{X}(A,z):=\sum_{n=1}^{\infty}\frac{c_{n}(A)}{n!}z^{n},\quad
z\in\mathbb{C}.$ (3)

(the sum is taken from $n=0,\ c_{0}=1$ for associative algebras and groupoids
with unity). Remark that $A$ need not consist of multilinear elements. We also
omit the variable $z$ and (or) the set $X$ and write
${\mathcal{C}}_{X}(A,z)={\mathcal{C}}(A)$. These definitions are naturally
extended to algebras, subspaces, and their dimensions with respect to their
generating sets. We illustrate this notion by examples. Let $X$ be a countable
set and $A=A(X)$, $L=L(X)$ the free associative and Lie algebras,
respectively. Then

$\displaystyle{\mathcal{C}}_{X}(A,z)$
$\displaystyle=\sum_{n=0}^{\infty}z^{n}=\frac{1}{1-z},$
$\displaystyle{\mathcal{C}}_{X}(L,z)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{z^{n}}{n}=-\ln(1-z),$
$\displaystyle{\mathcal{C}}_{X}({\bf{N}}_{s},z)$
$\displaystyle=\sum_{n=1}^{s}\frac{z^{n}}{n}.$

The author established an exponential analogue of Schreier’s formula for free
Lie algebras as follows.

######  Theorem 8 ([26]).

Assume that $L$ is the free Lie algebra generated by a countable generating
set $X$. Assume that $H$ is an $X$-uniform subalgebra. Then $H$ has an
$X$-uniform set of free generators $Y$ and

${\mathcal{C}}_{X}(Y,z)-1=(z-1)\cdot\exp({\mathcal{C}}_{X}(L/H,z)).$

##  4\. Scale 2 for the codimension growth of Lie PI-algebras

For the theory of varieties of associative and Lie algebras see [1, 4, 10].
Let ${\bf{V}}$ be a variety of Lie algebras, and $F({\bf{V}},X)$ its free
algebra generated by $X=\\{x_{i}|i\in\mathbb{N}\\}$. Let
$P_{n}({\bf{V}})\subset F({\bf{V}},X)$ be the subspace of all multilinear
elements in $\\{x_{1},\dots,x_{n}\\}$ and consider the codimension growth
sequence $c_{n}({\bf{V}})=c_{n}(F({\bf{V}},X),X):=\dim_{K}P_{n}({\bf{V}})$,
$n=1,2,\dots$.

In case of associative algebras the fundamental fact is as follows:

######  Theorem 9 (Regev, [34]; Latyshev [18]).

Let an associative algebra $A$ satisfies a nontrivial identical relation of
degree $d$. Then $c_{n}(A)\leq C^{n}$, $n\geq 1$; where $C:=(d-1)^{2}.$

Another crucial fact on the codimension growth of associative algebras in
characteristic zero is that the exponent, defined as
$\operatorname{Exp}(A):=\mathop{\lim}\limits_{n\to\infty}\sqrt[n]{c_{n}(A)}$
always exists and is integral [8].

Now we start discussing the codimension growth for Lie algebras. The
integrality of the exponent of the codimension growth for finite dimensional
Lie algebras over a field of characteristic zero was proved by Zaitsev [38].
In general, the exponents for Lie algebras are not always integral [39, 9].
Moreover, the codimension growth in case of Lie algebras is more versatile.
Unlike the associative case, the codimension growth of a rather small variety
$\mathbf{A}\mathbf{N}_{2}$ is overexponential (Volichenko [37]). On the other
hand, the following upper bound was found.

######  Theorem 10 (Grishkov [12]).

Let $L$ be a Lie algebra satisfying a nontrivial identity. Then for any $r>1$
there exists $N_{0}$ such that

$c_{n}(L)\leq\frac{n!}{r^{n}},\quad n\geq N_{0}.$

Razmyslov introduced the complexity functions (3) and reformulated the upper
bound as follows.

######  Theorem 11 (Razmyslov [33]).

Let ${\bf{V}}$ be a nontrivial variety of Lie algebras. Then the complexity
function ${\mathcal{C}}({\bf{V}},z)$ is an entire function of complex
variable.

The author established a better and optimal general bound for the series that
allowed to prove an upper bound for the codimension growth sequence as well.
The estimate of the first item was recently obtained in [31].

######  Theorem 12 ([21, 23, 24, 31]).

Let $L$ be a Lie algebra satisfying a nontrivial identity of degree $m\geq 4$.
Then

  1. (1)

The following coefficientwise bound for the series holds:

$\mathcal{C}(L,z)\prec
z\underbrace{\exp(z\exp(\ldots(z\exp(z\exp}\limits_{m-2\text{\rm{ times
}}\exp}(z)))\ldots)).$

  2. (2)

there exists an infinitesimal such that

$c_{n}(L)\leq\frac{n!}{(\ln^{(m-3)}n)^{n}}(1+o(1))^{n},\quad n\to\infty.$

Thus, we have a vast area of overexponential growths for Lie algebras, lying
between the exponent and the factorial functions. To describe such a growth we
introduce the scale 2 consisting of a series of functions
$\Psi^{q}_{\alpha}(n)$, $q=2,3,\dots$, with a real parameter $\alpha$ [21]:

$\text{{scale
2:}}\qquad\Psi^{q}_{\alpha}(n):=\left\\{\begin{array}[]{lll}\displaystyle\big{(}n!\big{)}^{\frac{\scriptstyle\alpha-1}{\scriptstyle\alpha}},&\alpha>1,&q=2;\\\
\displaystyle\frac{n!}{(\ln^{(q-2)}n)^{n/\alpha}},&\alpha>0,&q=3,4\dots\end{array}\right.$
(4)

The upper bounds of Theorem 12 are ”adequate” and the scale (4) for the
codimension growth of Lie PI-algebras is complete, because the free solvable
Lie algebras do have such an asymptotic behaviour, see Theorem 20 below.
Actually we obtain a more fine scale formed by a series of functions with two
real parameters $\alpha,\beta$:

$\text{{scale
2${}^{\prime}$:}}\qquad\Psi^{q}_{\alpha,\beta}(n):=\left\\{\begin{array}[]{lll}\displaystyle\big{(}n!\big{)}^{\frac{\scriptstyle\alpha-1}{\scriptstyle\alpha}}\beta^{n/\alpha},&\alpha\geq
1,\ \beta>0;&q=2;\\\\[7.0pt]
\displaystyle\frac{n!\cdot(\beta/\alpha)^{n/\alpha}}{(\ln^{(q-2)}n)^{n/\alpha}},&\alpha>0,\
\beta>0;&q=3,4\dots\end{array}\right.$ (5)

Observe that in terms of scale (5) the exponential growth is a subcase of
level $q=2$ when $\alpha=1$.

###  4.1. The codimension growth for another classes of linear algebras

The codimension growth of arbitrary linear algebras can be weird [7]. The
varieties of absolutely free (commutative, or anticommutative) algebras have
Schreier’s type formulae in terms of generating function of both types, the
regular generating functions and exponential generating functions (i.e.
complexity functions), i.e. we have natural analogues of both, Theorem 1 and
Theorem 8), see [30]. We describe both, the generating functions and the
growth functions, for two kinds of growth, for different versions of
nilpotency and solvability for three types of linear algebras above [30]. But
we do not get an analogue of Theorem 13 and these results do not lead us to
something like scale 1 and scale 2 for the respective types of growth [30].

It was recently shown that the same scale (4) stratifies the ordinary
codimension growth of Poisson PI-algebras, but here it is essential to assume
that a Poisson algebra satisfies a nontrivial Lie identical relation [31]. If
a Poisson algebra is satisfying so called mixed identities only, then the
ordinary codimension growth has a factorial behaviour [32, 31].

The codimension growth for Jordan PI-algebras can be overexponential [5, 11].
We conjecture that something like scale 2 (see (4)) should appear in case of
Jordan PI-algebras as well.

##  5\. Explicit formulae for complexity functions and asymptotic for Lie PI-
algebras

Let ${\bf{M}},{\bf{V}}$ be varieties of Lie algebras. Their product
${\bf{M}}\cdot{\bf{V}}$ is the class of all Lie algebras $L$ such that there
exists an ideal $H\subset L$ satisfying $H\in{\bf{M}}$ and $L/H\in{\bf{V}}$,
see [1]. Using the exponential Schreier’s formula (Theorem 8) the following
explicit formula was proved.

######  Theorem 13 ([26]).

Let ${\bf{M}}\cdot{\bf{V}}$ be the product of varieties of Lie algebras, where
${\bf{M}}$ is multihomogeneous. Then

${\mathcal{C}}({\bf{M}}\cdot{\bf{V}},z)={\mathcal{C}}({\bf{V}},z)+{\mathcal{C}}({\bf{M}},1+(z-1)\exp({\mathcal{C}}({\bf{V}},z))).$

Roughly speaking, the formula says that
${\mathcal{C}}({\bf{M}}\cdot{\bf{V}},z)$ is ”almost” a composition of three
functions ${\mathcal{C}}({\bf{M}})\circ\exp\circ\,{\mathcal{C}}({\bf{V}})$.
The variety ${\bf{V}}={\bf{N}}_{s_{q}}\cdots{\bf{N}}_{s_{1}}$ can be viewed as
the product
${\bf{V}}={\bf{N}}_{s_{q}}\cdots{\bf{N}}_{s_{2}}\cdot{\bf{N}}_{s_{1}}$. As
application, the following explicit formula was derived.

######  Theorem 14 ([26, 29]).

Consider the variety of polynilpotent Lie algebras
${\bf{V}}:={\bf{N}}_{s_{q}}\cdots{\bf{N}}_{s_{1}}$, $q\geq 1$. Define
functions

$\displaystyle\begin{split}\beta_{1}(z)&:=\displaystyle\sum_{m=1}^{s_{1}}\frac{z^{m}}{m},\\\
\beta_{i}(z)&:=\displaystyle\beta_{i-1}(z)+\sum_{m=1}^{s_{i}}\frac{(1+(z-1)\exp(\beta_{i-1}(z)))^{m}}{m},\qquad
2\leq i\leq q.\end{split}$

Then ${\mathcal{C}}({\bf{V}},z)=\beta_{q}(z).$

Consider particular cases.

######  Corollary 15.

Fix $d\in\mathbb{N}$. Then

${\mathcal{C}}({\bf{A}}{\bf{N}}_{d},z)=z+\frac{z^{2}}{2}+\dots+\frac{z^{d}}{d}+1+(z-1)\exp\left(z+\frac{z^{2}}{2}+\dots+\frac{z^{d}}{d}\right).$

######  Corollary 16 ([23]).

Fix $q\in\mathbb{N}$. Consider the variety of solvable Lie algebras of length
$q$, denoted as ${\bf{A}}^{q}$. Set $\beta_{1}(z)=z$, and
$\beta_{i+1}(z)=\beta_{i}(z)+1+(z-1)\exp(\beta_{i}(z))$ for $i=1,2,\dots,q-1$.
Then

${\mathcal{C}}({\bf{A}}^{q},z)=\beta_{q}(z).$

######  Corollary 17 ([23]).

Fix $c\in\mathbb{N}$. Then

${\mathcal{C}}({\bf{N}}_{c}{\bf{A}},z)=z+\sum^{c}_{m=1}\frac{1}{m}\Big{(}1+(z-1)\exp(z)\Big{)}^{m}.$

Complexity functions are useful for computation of the codimension growth. For
example, the last result yields an asymptotic.

######  Corollary 18 ([23]).

$c_{n}({\bf{N}}_{c}{\bf{A}})\approx c^{n-c-1}n^{c}$, as $n\to\infty$.

Let $f(z)$ be an entire function of complex variable, denote ${\rm
M}_{f}(r):=\max_{|z|=r}|f(z)|$. Observe that in case of complexity functions,
we have ${\rm M}_{f}(r)=f(r),r\in{\mathbb{R}}^{+}$, since all coefficients are
nonnegative.

By Theorem (14), ${\mathcal{C}}({\bf{N}}_{s_{q}}\cdots{\bf{N}}_{s_{1}},z)$ is
”almost” $q-1$ iterations of $\exp(*)$ applied to
$\beta_{1}(z)=\sum_{m=1}^{s_{1}}{z^{m}}/m$, thus one has something like
$\exp^{(q-1)}(z^{s_{1}}/s_{1})$. More, precisely, we derive the following
asymptotic.

######  Theorem 19 ([23]).

Consider the variety of polynilpotent Lie algebras
${\bf{V}}:={\bf{N}}_{s_{q}}\cdots{\bf{N}}_{s_{1}}$, $q\geq 2$, and its
complexity function $f(z):={\mathcal{C}}({\bf{V}},z)$. Then

$\lim_{r\to\infty}\frac{\ln^{(q-1)}{\mathrm{M}}_{f}(r)}{r^{s_{1}}}=\frac{s_{2}}{s_{1}}.$

Next, we establish a relation between the growth of fast growing entire
functions and an asymptotic of their coefficients. This fact helped us to
match the upper bounds in the next result. But a connection between the lower
bounds require more direct estimates.

######  Theorem 20 ([23]).

Consider the variety of polynilpotent Lie algebras
${\bf{V}}:={\bf{N}}_{s_{q}}\cdots{\bf{N}}_{s_{1}}$, $q\geq 2$. Then there
exists an infinitesimal such that

$\displaystyle
c_{n}({\bf{V}})=\left\\{\begin{array}[]{ll}\displaystyle(n!)^{\frac{\scriptstyle
s_{1}-1}{\scriptstyle
s_{1}}}\big{(}s_{2}+o(1)\big{)}^{n/s_{1}}\\!,&q=2;\\\\[10.0pt]
\displaystyle\frac{n!}{(\ln^{(q-2)}n)^{n/s_{1}}}\Bigl{(}\frac{s_{2}+o(1)}{s_{1}}\Bigr{)}^{n/s_{1}}\\!,&q=3,4,\dots;\end{array}\right.\quad
n\to\infty.$

In the following two cases we obtain somewhat more precise asymptotic, but
they look rather complicated.

######  Theorem 21 ([26]).

Fix $d\in\mathbb{N}$. Then

$\displaystyle c_{n}({\bf{A}}{\bf{N}}_{d})$
$\displaystyle\approx\mu\left(n!\right)^{1-1/d}\exp\bigg{(}\sum_{k=1}^{d-1}\lambda_{k}n^{1-k/d}\bigg{)}n{\rule{0.0pt}{10.00002pt}}^{\frac{3-d}{2d}}\\!,\quad
n\to\infty,\quad\text{where}$ $\displaystyle\lambda_{k}$
$\displaystyle:=\frac{\left(\frac{k}{d}+1\right)\cdots\left(\frac{k}{d}+k-1\right)}{k!(d-k)},\qquad
k=1,\dots,d-1;$ $\displaystyle\mu$
$\displaystyle:=\exp\bigg{(}-\frac{1}{d}\sum_{k=2}^{d}\frac{1}{k}\bigg{)}(2\pi){\rule{0.0pt}{10.00002pt}}^{\frac{1-d}{2d}}d^{-1/2}.$

It is well known that $c_{n}({\bf{A}}^{2})=n-1\approx n$, this coincides with
our asymptotic. The particular cases are.

$\displaystyle\begin{split}\displaystyle
c_{n}({\bf{A}}{\bf{N}}_{2})&\approx\sqrt{n!}\,\frac{\exp\left(\sqrt{n}\right)\sqrt[4]{n}}{\sqrt[4]{8\pi
e}},\\\ \displaystyle
c_{n}({\bf{A}}{\bf{N}}_{3})&\approx\left(n!\right)^{2/3}\,\frac{\exp\left({\frac{1}{2}}n^{2/3}+{\frac{5}{6}}n^{1/3}\right)}{\sqrt{3}\sqrt[3]{2\pi}e^{5/18}},\end{split}\qquad
n\to\infty.$

######  Theorem 22 ([26]).

Consider the variety ${\bf{A}}^{3}$ of solvable Lie algebras of length 3 and
its codimension growth sequence $c_{n}:=c_{n}({\bf{A}}^{3})$. Then

$c_{n}=\frac{n!}{(\ln n)^{n}}\exp\left(\frac{n}{\ln n}\left(2\ln\ln
n+1+\frac{2(\ln\ln n)^{2}-2\ln\ln n-1}{\ln n}+o\left(\frac{1}{\ln
n}\right)\right)\right),\quad n\to\infty.$

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