Mathematical modelling and analysis of gravitational collapse in curved geometry

Mathematical modeling and analysis of gravitate larvae colles in curved gemma

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Mathematical modeling and analysis of gravitate larvae colles in curved gemma S.Z. Abbas, H. Sun, H.H. Shah, W.A. Khan, S. Ahmad, M. Waqas PII: DOI: Reference:

S0169-2607(19)31928-5 https://doi.org/10.1016/j.cmpb.2019.105283 COMM 105283

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Computer Methods and Programs in Biomedicine

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28 October 2019 12 December 2019 16 December 2019

Please cite this article as: S.Z. Abbas, H. Sun, H.H. Shah, W.A. Khan, S. Ahmad, M. Waqas, Mathematical modeling and analysis of gravitate larvae colles in curved gemma, Computer Methods and Programs in Biomedicine (2019), doi: https://doi.org/10.1016/j.cmpb.2019.105283

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1 Highlights

• Modified gravity • Gravitational collapse • Singularities formation • Spatial Curvature

Mathematical modeling and analysis of gravitate larvae colles in curved gemma S. Z. Abbas,1, 2, ∗ H. Sun,1 H. H. Shah,3 W. A. Khan,1, † S. Ahmad,4 and M. Waqas5 1

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China 2 Department of Mathematics and Statistics,Hazara University Mansehra, Pakistan 3 Department of Mathematical Sciences, Baluchistan University of Information Technology, Engineering and Management Sciences, 87300 Quetta, Pakistan 4 Department of Mathematics Abbottabad University of Science and Technology, Abbottabad, KPK, Pakistan 5 NUTECH School of Applied Sciences and Humanities, National University of Technology, Islamabad, 4400, Pakistan Abstract: Background and objectives: In the visible universe, it is believed that mass and energy are interchangeable. However, the physical and chemical processes in the hidden world put the scientist into the deep thought of matter and energy contents that are responsible for these phenomena. These are regarded as dark matter and dark energy. In this article, we study the effects of spacetime curvature on the gravitational collapse of dark energy in modified gravity, considering the collapse of the spherically symmetric star, which is composed of perfect and homogeneous fluid. We studied the collapse for closed, flat and hyperbolic geometry. Method: As a result of mathematical modeling, we achieved highly non-linear differential equations. For the solution, we needed the assumption of physical significance. Specifically, we have taken the dark energy collapse. Then we achieved a simple system and solved for the analytic solutions of the field equations. Results: It is shown that the possible collapse is visibly influenced by spatial curvature. The collapse time is advanced for closed spacetime, delayed for the hyper-surface, and the flat space behaves in intermediately. We have taken here the equation of state in linear form to discuss the exhibition of fluid profile and a specific necessary criterion for the occurrence of spacetime singularity. Conclusion: In this paper, we study the mathematical model of gravitational collapse in modified gravity, which derives the field equations using the principle of least action. The significant outcomes are the influences of the spatial curvature on the collapsing process and the time of formation of spacetime singularity. The matching of boundary and the fundamental continuity of the 1-form and 2-form are discussed. Keywords: Modified gravity; Gravitational collapse;Singularity formation; Spatial Curvature.

I.

INTRODUCTION

In our Universe, the existence of Dark energy and Dark matter are the mysteries, which still require further investigation by the theoretic cosmologists. To model the physical problems of accelerating Universe, the thought of the existence of dark energy was initiated around the lat 900 s, former was the idea of dark matter which convinced the scientists for the same modeling. To carry out the modeling of dark matter, a variety of proposals in the theoretical literature were being studied. Remarkable results are achieved by the study of dark matter and dark energy in particle physics. The studies using modified gravity can be seen in [2, 19, 21, 23, 23, 25, 30, 31, 39, 40, 43–45, 47], which have several tempting attractive features for the study of dark matter that could exist in a geometric sense, named as mimetic gravity, see detailed

∗ Electronic † Electronic

address: [email protected] address: [email protected]

study in [8], and this idea was initiated in [33]. Further, it was extended to cosmological studies of the universe, and various kinds of literature are available in [5–7, 9– 12, 12, 15, 16, 18, 20, 24, 26–28, 32, 34, 36–38, 42, 46]. In [13, 17, 29, 35, 41], similar studies for astrophysical purposes have been carried out later. Using the potential formulism and Lagrangian multiplier, the extension in f (R) for the theory of Einstein-Hilbert mimetic, is carried out in [37]. This theory has an interesting feature to be expressed in a unique manner, the inflation and the acceleration that detected in the universe, is investigated in [5, 6, 15]. The gravitation collapse in f (R) is studied by [14] then further rectified by [4] with the assumption of dark matter to participate in the collapsing process. In [3], the study for the effects of spacetime curvature on the gravitational collapse is carried out for non-attracting dark energy and dark matter. The purpose of this study is to investigate the effect of the curvature of spacetime geometry on the gravitational collapse in f(R) gravity formalisms. The study carries certain physical assumptions to be taken for the analytical solutions of the field equations. As the final result, we

3 found the black hole formation in the region bounded by the apparent horizon. We organized the paper as follows: Section II is the mathematical formulation for the underlying problem in modified gravity. Section III is the study of the gravitational collapse of dark energy by solving EFEs with the assumption ω = −1. Section IV is the analysis of the formation of the apparent horizon. Section V elaborates junction conditions. Finally, section V I discuss the over outcomes and the concluding remarks. II.

MATHEMATICAL FORMULATION

For the description of field equations in f(R) gravity, we used the principle of least action from the Lagrangian mechanics. The following two subsections are written to achieve it’s purpose. A.

Line element

where fR = df /dR. We can express the above equation as:  κ  (m) (D) Gαβ = Tαβ + Tαβ , (4) fR where (D)

Tαβ =

1 κ



 f − RfR gαβ + ∇α ∇β fR − gαβ ∇µ ∇µ fR .(5) 2

In the interior of the star, the energy-momentum tensor is given as: (m)

Tαβ = (µ + p) uα uβ − pgαβ ,

where (µ, p, uµ ) are the energy density, pressure, and the four-velocity vector, respectively. In the comoving system of coordinates for the spacetime (1), we have uα = δα0 . With these assumptions the field equations lead to the following differential equations: k + a˙ 2 3 a2




1 f − RfR 1 1 − kr2 00 µ+ fR − 2 + 2 fR 2 a a !   
 0 0 2 2 f˙R aa˙ + fR kr − 2 f˙R raa˙ + fR kr − 1 , (7) ra =

We are interested in the collapse of dark energy. Hence, we here take the spacetime interior to the collapsing body as expressed using FRW-metric   dr2 2 2 2 2 2 ds− = dt − a (t) + r dΩ , (1) 1 − kr2

1 k + a˙ 2 + 2a¨ a = a2 fR

−p+

where, a(t) is the scale function known as the universe scale factor and the scalar k = 0, 1, −1 corresponds to zero (flat-space), positive and negative curvature for the geometry of spacetime, respectively, and dΩ2 is the metric on 2-sphere which reads as dΩ2 ≡ dθ2 +sin2 θdϕ2 . The involved connection is assumed to be metric-compatible, symmetric and torsion-free, which is also called a LeviCivita connection.

k + a˙ 2 + 2a¨ a 1 = 2 a fR

−p+

B.

f − RfR 1 − kr2 00 − f¨ + fR 2 a2 ! 
 0 0 2 ˙ 1 ˙ 2 ,(8) − 2 fR aa˙ + fR kr − 2 fR raa˙ + fR kr − 1 a ra

with (f (R), Sm , GN ), a nonlinear expression in R(the scalar curvature), action of source(matter, which vanishes for vacuum) and gravitational constant(Newtonian) respectively. For simplicity, we identified the speed of light as unity, i.e., c ≡ 1. In these settings, the field equations are given as [1]: 1 fR Rαβ − f gαβ − ∇α ∇β fR + gαβ ∇µ ∇µ fR = 2 (m) −8πGTαβ ,

(3)

1  f − RfR − f¨ − 2 f˙R raa˙ 2 ra ! 
0 +fR kr2 − 1 ,(9)

0 f˙R a˙ = . fR a

Lagrangian action in GR and Field Equations

Here, we present the basic approach to derive EFEs by a modified form of gravity f (R). The action in f (R) gravity, reads as: Z √ 1 S= (2) dx4 −gf (R) + Sm , 16πGN

(6)

(10)

Integration of the last equation by t reads: 0

fR = a(t)W0 (r),

(11)

where, Wo is due to integration, a function of r. Again performing the integration on Eq. (11) by r we have: R

fR = a(t)W1 (r),

(12)

here, W0 dr = W1 . By combining Eqs. (7), (8), and (9) we get the following:
k + a˙ 2 a ¨ 1 2 −2 = µ + p + f¨ (13) 2 a a fR !
 0 1 ˙ 2 − 2 fR raa˙ + fR kr − 1 . ra

4 Substituting Eq. (12) in (13) we get:
0 a W1 kr2 − 1 2k + 3a˙ − 3¨ aa − (µ + p) = − (. 14) W1 W1 r 2

III.

GRAVITATIONAL COLLAPSE OF DARK ENERGY

Here, we assume the linear equation of state p = ωµ with ω = −1 to discuss the collapse in this scenario Eq. (11) reduced to
0 W kr2 − 1 2k + 3a˙ 2 − 3¨ . (15) aa = − 1 W1 r

From Eq. (16) it is evident that for t = t0 , a(t) tends to zero, thus there appears a singularity of zero proper 3 2 √ sin(θ) volume which is measured as −g = a√r1−kr here g 2 presents the determinant of the metric in Eq. (1). The Ricci scalar R takes the following form: R = −24η −

(16)


0 W1 kr2 − 1 = ξ. W1 r

(17)

−

(24)

Now, the Kretschmann scalar defined as K = Rαβγδ Rαβγδ in our case it has the form: K = 48η 2 −

Choosing the separation constant ξ we get the following equations 2k + 3a˙ 2 − 3¨ aa = ξ,

2 (k + ξ) . a2

4 a8



2 1 ξ K + 2a2 η + 3 3  
1 2 2 kr2 − 1 + 4 r .

(25)

From Eq. (24) and (25) we observed that as a(t) approaches 0 both the curvature tends to infinity, which is regarded as the true curvature singularity (rather than the coordinates singularity). The evolution of all these quantities can be seen in Figs. (a, b, c).

Solving Eq. (13) for the first derivative we get a˙ 2 = 2a2 η −

1 (ξ − 2k) , 3

IV.

where η is the constant of integration. As we are interested in gravitational collapse, so a(t) should be the decreasing function of time, i.e., a˙ < 0 and with this criterion we get the analytic solution as follows:  p  1 p a(t) = √ ξ − 2k sinh 2η (t0 − t) , (19) 6η for the existence of real phenomena, ξ − 2k must be nonnegative. Now solving Eq. (17) for W1 we get: W1 =

ξ/2k

,

(26)

where R(p) is the proper radius. For the line element in Eq. (1) it takes the form: r2 a˙ 2 + kr2 − 1 = 0,

a˙ 2 + k =

The mass of the collapsing body bounded by radius r at the moment t as a function r and t, is reads as:
1 R 1 + R,α R,β g αβ . 2

(22)

For line element in Eq. (1) it reads as: m(r, t) =

(p)

(p) g αβ R,α R,β = 0,

(20)

where c is the constant of integration. Form these solutions we get:

√ √ √1 ξ − 2k sinh 2η (t0 − t) 6η fR = c . (21) ξ/2k (kr2 − 1)

m(r, t) =

The existence of Apparent Horizon is the consequence of the formation of the trapped surface, where the outward normal is null. Mathematically it reads as:

(27)

it can be written as:

c (kr2 − 1)

APPARENT HORIZON

(18)


ar3 2 a˙ + k . 2

(23)

1 = σ2 , r2

(28)

where σ 2 is the separation constant. The time of forming apparent horizon can be determined by Eq. (18) and (28) as: q p σ 2 − k + ξ ± σ 2 − 13 k + 2ξ 1 tapp = t0 − √ ln( ).(29) 2η 2η It is obvious by the above relation that the apparent horizon will appear for the non-negative value of σ 2 − 13 k − 3ξ . Using Eq. (19) the time of appearance of singularity a = 0 read as: 1 −2/3k + ξ/2 ts = t0 − √ ln| |. 2η 2 2η

(30)

5 aHtL

RHtL

14 12

-20

10

0.0

-30

8

-40

k=-1,0,1

6

-50

4 2 0.5

1.0

1.5

t -0.5

2.0

2.5

3.0

t

-2000 KHt,rL -4000

k=1,0,-1

-60 t

-1.0 0

0.5

1.0

1.5

(a)

2.0

2.5

3.0

t

50 r

100

(b)

0

-6000

(c)

FIG. 1: Developments universe scale factor a(t), Ricci scalar curvature R(t), and Kretchman curvature K(t, r) of collapsing body in different geometries (closed, flat and hyperbolic).

From Eq. (29) and (30) the visibility of central singularity is calculated as:

ts − tapp

1 = √ ln| 2η

p

q σ 2 − k + ξ ± σ 2 − 13 k + 2ξ p |.(31) −2/3k + ξ/2

By the above relation, we noticed that the singularity is dependent on the curvature of spacetime and it is free of radius r of collapsing body Which proved the nonoccurrence of a naked singularity, according to [22].

V.

JUNCTION CONDITIONS

Here, we will present the junction conditions between the interior and exterior of the collapsing star. For the interior of the star, we will take the FRW-metric, while, for the exterior, we will choose flat spacetime, and as a particular case, we will suppose the Schwarzschild spacetime. Junction conditions proposed by Israel-Darmois[29] will be utilized for the said purpose. Let the interior region, exterior region and boundary of the star, are denoted by N − , N + , and σ, respectively. For the exterior, we take the metric as: ds2+ = Y 2 (R, T )dT 2 − z 2 (R, T )(dR2

+R2 (dθ2 + sin2 θdφ2 )).

(32)

From Eq. (4) and (25) the non-vanishing components of ± extrinsic curvature Kαβ read as − Kθθ =

1 2 − r Kφφ = (1 − kr2 ). sinθ a

(33)

The continuity of the second fundamental form reads as + − Kαβ − Kαβ = 0 gives + Kθθ =

r (1 − kr2 )σ . a

(34)

The rest of the results are similar as given in [4], and we will not discuss them here in detail.

VI.

CONCLUSION

In current work, the process of gravitational collapse is studied, considering the perfect fluid, which is homogeneous, using f (R) theory of gravity in the presence of the spatial curvature. Our study derived the field equations by the principle of Least action and then we studied them by FRW-metric. The black hole formation or the occurrence of naked singularity i.e., the singularity without apparent horizon. The result varied by the chosen values of the equation of the State’s parameter. The role of the curvature of spacetime is studied for all possible physical parameters, which are the active participants in the process of gravitational collapse. Notably, for spacetime singularity, the scale function a(t) of the universe, the Ricci scalar R(t), and Kretschmann scalar K(t, r), are deeply observed to be influenced by the curvature of spacetime. The study revealed that a(t) approaches zero at the highest rate in case of positive curvature(closed geometry) corresponding to k = 1 followed by the zero curvature(flat geometry) and then negative curvature(hyperbolic geometry). Consequently, the time to approach the spacetime singularity is favored for positive curvature then the zero curvature, the negative curvature provide much delay in the formation of the singularity. The evolution can be checked graphically in Fig. 1(a). The Ricci scalar has the visible effect by the curvature of spacetime, the divergence of Ricci scalar to infinity is earlier for the positively curved spacetime as compare to the flat case and negatively curved space; it is showing that the true singularity is the result of gravitational collapse for all values of curvature parameter k, and it is clearly visible in Fig. 1(b). Since the Ricci scalar only dependant on time while the Kretschmann scalar K(t, r) is the function of time as well as the radius of the collapsing body, and the effects by the curvature parameter are still apparent for both the time and radius. Much more delay is shown by the negative curvature and the flat spacetime behaves in between them, for the graphical illustration see Fig. 1(c). In future studies, we will try to discuss the cases corresponds to ω 6= −1 for the equation of the state’s parameter by using some numeric technique.

6 Declaration of Competing Interest

AND PROGRAMS IN BIOMEDICINE.

The authors declared no conflict of interest regarding this manuscript submitted to COMPUTER METHODS

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