Shearfree condition and dynamical instability in f(R, T) gravity
Eur. Phys. J. C
Shearfree condition and dynamical instability in f ( R, T ) gravity
Ifra Noureen 2
M. Zubair 1
A.A. Bhatti 2
G. Abbas 0
0 Department of Mathematics, COMSATS Institute of Information Technology , Sahiwal 57000 , Pakistan
1 Department of Mathematics, COMSATS Institute of Information Technology , Lahore 54700 , Pakistan
2 Department of Mathematics, SST, University of Management and Technology , Lahore 54770 , Pakistan
The implications of the shearfree condition on the instability range of an anisotropic fluid in f (R, T ) are studied in this manuscript. A viable f (R, T ) model is chosen to arrive at stability criterion, where R is Ricci scalar and T is the trace of energymomentum tensor. The evolution of a spherical star is explored by employing a perturbation scheme on the modified field equations and contracted Bianchi identities in f (R, T ). The effect of the imposed shearfree condition on the collapse equation and adiabatic index is studied in the Newtonian and postNewtonian regimes.

In a recent work [1], we have studied the effect of an
anisotropic fluid on the dynamical instability of a spherically
symmetric collapsing star in f (R, T ) theory. Herein, we plan
to explore the instability range of anisotropic spherically
symmetric stars, considering shearfree condition. The role
of the shear tensor in the evolution of gravitating objects and
consequences of the shearfree condition have been studied
extensively. Collins and Wainwright [2] studied the impact of
shear on general relativistic cosmological and stellar models.
Herrera et al. [3,4] worked out the homology and shearfree
conditions for dissipative and radiative gravitational
evolution.
The features of the gravitational evolution and its final
outcome are of great importance in view of general relativity
(GR) as well as in modified theories of gravity. Shearfree
collapse accounting for heat flow is discussed in [5], where it is
established that shear plays a critical role in the gravitational
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evolution and may lead to the formation of naked
singularities [6]. It is mentioned in [6] that the occurrence of
shearing effects near collapsing stars avoids the apparent horizon
leading to the formation of a naked singularity. However,
vanishing shear gives rise to the formation of an apparent
horizon and so the evolving cloud ends in a black hole (BH).
Thus, the relevance of the shear tensor in structure formation
and its consequences on the dynamical instability range of a
selfgravitating body is a wellmotivated direction of study.
Stars shine by consuming their nuclear fuel; continuous
fuel consumption causes imbalance between inwardly
acting gravitational pull and outwardly drawn pressure, giving
rise to collapse [7]. The outcome of gravitational evolution is
dependent on the size as well as other physical aspects [8,9],
such as isotropy, anisotropy, shear, radiation, dissipation etc.
In comparison to the stars of mass around one solar masses,
massive stars tend to lose nuclear fuel more rapidly and so
they are more unstable. The pressure to density ratio, called
the adiabatic index, denoted by is utilized in the estimation
of the stability/instability range of the stars. Chandrasekhar
[10] explored the instability range of spherical stars in terms
of .
Herrera et al. [11–17] contributed substantially in
addressing the instability problem in GR, accompanying various
situations, i.e., isotropy, anisotropy, the shearfree
condition, radiation, dissipation, the expansionfree condition, and
shearing expansionfree fluids. In order to achieve a more
precise and generic description of the universe, the dark
energy components are incorporated by introducing
modified theories of gravity. Modified theories are significant in
the advancement toward accelerated expansion of the
universe and to present corrections to GR on large scales. The
modifications are introduced in the Einstein–Hilbert (EH)
action by inducing a minimal or nonminimal coupling of
matter and geometry [18–25].
The dynamical analysis of selfgravitating sources in
modified theories of gravity has been discussed extensively in
( fT + 1)Tu(vm) − ρguv fT +
+(∇u ∇v − guv ) f R .
Here Tu(vm) is the energymomentum tensor for the usual
matter taken to be locally anisotropic.
The three dimensional spherical boundary surface is
considered that constitutes two regions named ‘interior’ and
‘exterior’ spacetimes. The line element for the region inside
the boundary is
−C 2(t, r )(dθ 2 + sin2 θ dφ2).
The line element for the region beyond
ds+2 =
dν2 + 2dr dν − r 2(dθ 2 + sin2 θ dφ2),
recent years. The null dust nonstatic exact solutions in f (R)
gravity are studied in [26], Cembranos et al. [27] studied the
evolution of gravitating sources in the presence of a dust
fluid. The instability range of spherically and axially
symmetric anisotropic stars has been established in the context
of f (R) gravity [28–30], leading us to conclude that
deviations from spherical symmetry complicate the subsequent
evolution.
Harko et al. [31] presented the f (R, T ) theory of gravity
as another alternative to GR and a generalization of f (R)
theory representing nonminimal matter to geometry coupling.
The action in f (R, T ) gravity includes an arbitrary function
of the Ricci scalar R and the trace of the energymomentum
tensor T to take into account the exotic matter. After the
introduction of f (R, T ) gravity, its cosmological and
thermodynamic implications were widely studied [32–40] including
the energy conditions. Recently, we have studied the
evolution of an anisotropic gravitating source with zero
expansion [41]. Herein, we are interested in the exploration of the
shearfree condition implications on a spherically symmetric
gravitating source in f (R, T ) gravity.
The modified action in f (R, T ) gravity is as follows [31]:
d x 4√−g
where L(m) denotes the matter Lagrangian, and g represents
the metric tensor. Various choices of L(m) can be taken into
account, each of which leads to a specific form of fluid. Many
people worked out this problem in GR and modified theories
of gravity, and the stability of general relativistic
dissipative axially symmetric and spherically symmetric systems
with a shearfree condition has been established in [42,43].
A dynamical analysis of the shearfree spherically symmetric
sources in f (R) gravity is presented in [44].
The organization of this article is as follows: Sect. 2
comprises the modified dynamical equations in f (R, T ) gravity.
Section 3 includes the model under consideration, the
perturbation scheme, and the corresponding collapse equation
along with the shearfree condition in the Newtonian and
postNewtonian eras. Section 4 contains concluding remarks
followed by an appendix.
2 Dynamical equations in f ( R, T )
In order to study the implications of the shearfree
condition on the evolution of spherically symmetric anisotropic
sources, modified field equations in f (R, T ) gravity are
formulated by varying the action (1.1) with the metric guv. Here,
we have taken L(m) = ρ [36], for this choice of L(m) the
modified field equations in f (R, T ) gravity take the
following form:
where ν is the retarded time and M denotes the total mass.
The expression for the anisotropic energymomentum
tensor Tu(vm) is given by
where ρ is the energy density, Vu describes the fourvelocity
of the fluid, χu is the radial four vector, and pr and p⊥
represent the radial and tangential pressure, respectively. These
physical quantities are linked by
1
σuv = V(u;v) − a(u Vv) − 3
au = V(u;v)V v,
= V;uu .
(guv − Vu Vv),
is expansion scalar, given
The components of the shear tensor are found by variation of
(2.7) and these are used to find the expression for the shear
scalar in the following form:
where a dot and a prime indicate time and radial derivatives,
respectively. From the shearfree condition we arrive at a
vanishing shear scalar, i.e., σ = 0, implying BB˙ = C˙ .
C
It is worth mentioning here that the expansion scalar and
a scalar function described in terms of the Weyl tensor and
the anisotropy of the pressure controls the departure from
the shearfree condition. Such a function is related to the
Tolman mass and appears in a natural way in the
orthogonal splitting of the Riemann tensor [45]. It is obvious that
pressure anisotropy and density inhomogeneities have
extensive implications on the stability of the shearfree
condition, but it is not intuitively clear that their specific
combination affects the stability [43]. Generically the shearfree
condition remains unstable against the presence of pressure
anisotropy. Alternatively, one can consider such a case that
pressure anisotropy and density inhomogeneity are present
in such a way that the scalar function appearing in an
orthogonal splitting of the Riemann tensor vanishes, implying
nonhomogeneous anisotropic stable shearfree flow. Since we
are dealing with a fluid evolving under the shearfree
condition, we shall make use of this condition while evaluating
the components of the field equations and also in the
conservation equations.
The components of the modified Einstein tensor are
pr + (ρ + pr ) fT −
p⊥ + (ρ + p⊥) fT −
f¨R f R f˙R
+ A2 − B2 − A2
The dynamical equations extracted from the conservation
laws are vital in the study of stellar evolution. The
conservation of the full field equations is considered to incorporate the
nonvanishing divergence terms; the Bianchi identities are
Guvv Vu = 0, G;uvvχu = 0,
;
and on simplification of (2.14), we have dynamical equations
as follows:
A
(ρ + pr ) fT + (1 + fT ) pr + ρ A
where Z1(r, t ) and Z2(r, t ) are provided in the appendix as
(5.1) and (5.2), respectively. Deviations from equilibrium in
the conservation equations with the time transition leads to
the stellar evolution, and a perturbation approach is devised
to estimate the instability range.
3 Perturbation scheme and shearfree condition
We consider a particular f (R, T ) model of the form
f (R, T ) = R + α R2 + λT ,
where α and λ can be any positive constants. The
perturbation approach is utilized to estimate the instability range of
a spherical star with the shearfree condition. This scheme
is utilized in the determination of more generic analytical
constraints on the collapse equation, or rather to establish a
dynamical analysis of special cases numerically. Also, the
field equations are highly nonlinear differential equations; in
such a scenario the application of a perturbation is beneficial
to gaining insight.
It is assumed that initially all quantities are independent of
time and with the passage of time the perturbed form depends
on both time and radial coordinates. Taking 0 < ε 1, the
physical quantities and their perturbed form can be arranged
as
A(t, r ) = A0(r ) + ε D(t )a(r ),
B(t, r ) = B0(r ) + ε D(t )b(r ),
pr (t, r ) = pr0(r ) + ε p¯r (t, r ),
p⊥(t, r ) = p⊥0(r ) + ε p¯⊥(t, r ),
R(t, r ) = R0(r ) + ε D1(t )e1(r ),
T (t, r ) = T0(r ) + ε D2(t )e2(r ),
Considering the Schwarzschild coordinate C0 = r and
implementing the perturbation scheme on the vanishing shear
scalar implies
Bb0 = rc¯ .
c
+ λ1 ¯ (2ρ0 + pr0 + 4 p⊥0) + Y Z1 p D˙ = 0,
r
] + Y Z2 p = 0,
where Z1 p and Z2 p are given in the appendix. For the sake
of simplicity we put Y in place of 1 + 2α R0 and λ1 = λ + 1,
assuming that D1 = D2 = D and e1 = e2 = e. The
above mentioned perturbed dynamical equations and
perturbed field equations shall be used to arrive at perturbed
physical quantities such as ρ¯, p¯r , and p¯⊥.
The expression for ρ¯ can be found from (3.31), as follows:
c
+ r¯ (3ρ0 + pr0 + 4 p⊥0) + Y Z1 p D.
The Harrison–Wheeler type equation of state relates ρ¯ and
p¯r ; it is given by
p¯r =
p¯r = −
+ pr0 + 4 p⊥0) + Y Z1 p D.
1 Y c¯
+ 4 p⊥0) + Y Z1 p
Z3 + Z2 p = 0.
Matching conditions at the boundary surface together with
the perturbed form of (2.13) can be written in the simplified
form as follows:
Using (3.18)–(3.29) and (3.30) in the dynamical equations
i.e., (2.15) and (2.16), leads to the following expressions:
Substitution of ρ¯, p¯r , and p¯⊥ from (3.33), (3.35), and
(3.36) into (3.32) leads to a collapse equation,
The perturbed form of the field equation (2.13) yields an
expression for p¯⊥ that turns out to be
Z3 is the effective part of the field equation given in the
appendix as (5.5).
D¨ (t ) − Z4(r )D(t ) = 0,
provided that
+4 p⊥0) + Y Z1 p + λZ13 .
The valid solution of (3.38) turns out to be
D(t ) = −e√Z4t .
The terms of Z4 must be constrained in such a way that
all terms maintain positivity. The impact of the shearfree
condition on the dynamical instability of N and pN regimes
is covered in the following subsections.
3.1 Newtonian regime
In order to establish the instability range in the Newtonian
era, we set ρ0 pr0, ρ0 p⊥0, and A0 = 1, B0 = 1.
Insertion of these assumptions and (3.40) into (3.37) leads to
the instability condition, relating the usual matter and dark
source contribution,
X3 = −2α2b R0 + Y b,
Z4 = λ1 ρ0a + 2( pr0 + p⊥0)b
The quantities Z1 p(N ) and Z2 p(N ) are terms of Z1 p and
Z2 p belonging to the Newtonian era. The gravitating source
remains stable in the Newtonian approximation until the
inequality for is satisfied, for which the following
constraints must be met:
The case when α → 0 and λ → 0 leads to GR corrections
and results for f (R) can be retrieved by setting λ → 0.
3.2 PostNewtonian regime
We assume A0 = 1 − mr0 and B0 = 1 + mr0 to evaluate the
stability condition in the pN regime. On substitution of these
assumptions in (3.37), we have the following inequality for
to be fulfilled for the stability range:
pr0 r(rm−0m0) + 2αYR0 + r2
r − m0
− 2( pr0 + p⊥0)b
ρ0m0
+e λ1( pr0 + r (r − m0) ) + pr0
2e r
X7 = Y + λ1b 2 + r + m0 ,
X8 =
Z1 p(P N ) and Z2 p(P N ) are terms of Z1 p and Z2 p that lie in
the postNewtonian era. The above inequality (3.42) holds for
positive definite terms and describes the stability range of the
subsequent evolution. The positivity of each term appearing
in (3.42) leads to the following restrictions:
r − m0
> 2( pr0 + p⊥0)b , ρ0 <
4 Concluding remarks In this manuscript, we carried out a study of the implications of the shearfree condition on the stability of spherically
symmetric anisotropic stars in f (R, T ). Our exploration
regarding the viability of the f (R, T ) model reveals that
the selection of f (R, T ) model for dynamical analysis is
constrained to the form f (R, T ) = f (R) + λT , where λ
is an arbitrary positive constant. The restriction on the form
of f (R, T ) originates from the complexities of nonlinear
terms of the trace in an analytical formulation of the field
equations. The model under consideration is of the form
f (R, T ) = R + α R2 + λT , representing a viable
substitute to dark source and the exotic matter, both satisfying the
viability criterion (positivity of radial derivatives up to
second order).
In f ( R, T ) gravity, the nonminimal matter–geometry
coupling includes the terms of the trace T in the action (1.1)
that is beneficent in the description of quantum effects or
socalled exotic matter. The components of the modified field
equations together with the implementation of the shearfree
condition are developed in Sect. 2. Further conservation laws
are considered in order to arrive at the dynamical equations
by means of the Bianchi identities. These equations are
utilized to estimate the variations in the gravitating system with
the passage of time.
The complexities of more generic analytical field
equations are dealt with by using a linear perturbation of the
physical quantities. The perturbation scheme induces a
significant ease in the description of the dynamical system, or
rather to present a stability analysis by means of numerical
simulations. The analytic approach we have employed here
is more general and substantially important in explorations
regarding structure formation. The perturbed shearfree
condition together with the dynamical and field equations leads
to the evolution equation, relating with the usual and dark
source terms. It is found that the induction of the trace of
the energymomentum tensor in the action (1.1) contributes
a positive addition to , which slows down the subsequent
evolution considerably.
The outcome of the gravitational evolution is size
dependent, and we have as well other physical aspects such as
isotropy, anisotropy, shear, radiation, dissipation, etc. The
instability range for N and pN approximations is considered,
which imposes some restrictions on the physical variables.
It is observed that the terms appearing in are less
constrained for both the regimes (N and pN) in comparison to
the anisotropic sources [1]. Thus, the shearfree condition
benefits in more stable anisotropic configurations.
Corrections to GR and f ( R) establishments can be made by setting
α → 0, λ → 0, and λ → 0, respectively. The local isotropy
of the model can be settled by assuming pr = p⊥ = p. The
extension of this work for a shearing expansion of the free
evolution of anisotropic spherical and cylindrical sources is
in process.
Acknowledgments The authors thank the anonymous referee for
fruitful comments and suggestions.
Open Access This article is distributed under the terms of the Creative
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Funded by SCOAP3.
Z1(r, t ) = f R A2
f˙R − AA f˙R − BB˙ f R
3 f¨R B˙ A˙
+ A2 B + A ( f − R f R ) −
2 A˙ B˙
3 A B 2C
3 A 2C B
Z2(r, t ) = f R B2
f˙R − AA f˙R − BB˙ f R
3 B C
Z2 p = B02Y
Z3 =
c
¯
+ r
e −
− R0
c
¯
+ 2 r
B0 − [λT0 − α R02]
e − e
− B0
+ e − e
− 2 R0
e − 2 R0
3 A0
e −
R0 − e
− 2
3 B0
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