Some Static Fluid Spheres with Spin and Schwarzschild Solution

Some Static Fluid Spheres with Spin and Schwarzschild

Solution

Vinod Kumar1, Abhishek Kumar Singh2

1,2(PhD, M.Sc.)

1,2P.G. Department of Mathematics, Magadh University, Bodhgaya -824234, Bihar (INDIA)

[email protected],

[email protected]

Abstract : In the present paper we have found Some Static Fluid Spheres With Spin. In this

paper we have taken the problem of static fluid spheres in the frame work of Einstein-

Cartan theory. Adopting Tolman’s technique we have solved the field equations and thus

have metrics corresponding to well known Schwarzschild solution. We have obtained

pressure and density for the distribution. Also some physical and kinematical properties of

the models are discussed.

Keywords: Einstein-Cartan theory, Schwarzschild solution, Charged fluid sphere, Metric

potential, matter density, charged density.

1. INTRODUCTION

Static fluid spheres in Einstein-Cartan theory have attracted many research

workers in Relativity Theory. Infact the general theory of relativity which has been

considered as “most beautiful creation of single mind” has enjoyed success wherever a

test has been possible [23-25]. The general theory of relativity has also led under

general considerations to the existence of singularities in the universe. Since the

singularity is not a desirable feature for any physical theory, the question arises, is it

possible to keep this beautiful theory unmolested with regard to its success but at the

same time modify it so as to prevent singularities? The answer seems to be in

affirmative if one considers the most natural generalization of Einstein’s theory as

originally suggested by Elie Cartan [5,6] which is now known as Einstein-Cartan

theory (or E-C theory). In this theory the intrinsic spin of matter is incorporated as the

source of torsion of the space-time manifold. According to the relativistic quantum

mechanics mass and spin are two fundamental characters of an elementary particle

system. The energy momentum is source of curvature. By introducing torsion and

relating it to the density of intrinsic angular momentum the Einstein- Cartan theory

restores the analogy between mass and spin which extends to the principle of

By By

The International journal of analytical and experimental modal analysis

Volume XII, Issue I, January/2020

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mailto:[email protected]

mailto:[email protected]

equivalence at least in its weak form. According to this principle the world line of a

spin less test particle moving under the influence of gravitational fields only depends

on its initial position and velocity but not on its mass.

Since the predictions of E-C theory differ from those of general relativity only

for matter filled regions, therefore besides cosmology, an important application field

of E-C theory is relativistic astrophysics which deals with the theories of stellar

objects like neutron stars with some alignment of spins of the constituent particles.

Hence it is desirable to understand the full implication of the E-C theory for finite

distributions like fluid spheres with non-zero pressure. With this view many workers

have considered the problem of static fluid spheres in E-C theory (Prasanna [23],

Kerlick [13], Kuchowicz [17, 18, 19] Skinner & Webb [26] and Singh & Yadav [25],

Yadav.et.al. [35], Yadav, A.K. et.al [36-37], S.G. Ghosh et.al [38] and Maurya, S.K

et.al [39].

Following Trautman’s [31] reformulation of Einstein_ Cartan theory,

Kopczynski [14] was the first to obtain a solution, wherein he considered the problem

of studying the geometry of the space-time supporting the gravitational field produced

by a spherically symmetric distribution of incoherent matter composed of spinning

particles. Assuming a classical description of spin, i.e. the spin density-tensor i

jkS

i i k

jk jk jkS U S , U S 0

were in iU is the four-velocity vector and

jkS is the intrinsic angular momentum

tensor, he showed the existence of a two parameter family of world models of the

Friedman type without singularities. Following this Trautman [33] showed that by

starting from a Robertson-Walker type of line element, again for a classical

description of spin, the Friedman equation takes the form

2 2 2

2 2

R GM 3G S0

2 R 2C R

solving which he obtained a non-zero minimum radius at t 0 , as given by

12 3

2

3GSR

2MC

Stewart and Hajicek [28] commenting on this work showed that the

singularity in Trautman’s model was avoided mainly because of the perfect isotropy



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introduced. However as Hehl, Von der Heyde and Kerlick [10] have pointed out, one

can always rewrite Einstein-Cartan equations such that the torsion effects are include

in the energy momentum tensor of matter and in principle singularity might be

avoided by violating the positive energy condition of Penrose-Howking theorems.

Trautman [33] has proposed that spin and torsion may avert gravitational singularities

by considering a Friedman type of universe in the frame work of Einstein-Cartan

theory and obtaining a minimum radius R0 at t 0 . Isham, Salam, and Strathdee

[12] have shown that if one considers the Trautman model in the frame work of their

two-tensor theory then the minimum radius would increase substantially, giving a

reasonable density for the universe in the early stages. Applying the same arguments

for finite collapsing objects, Prasanna [22] has shown that it is possible to get a

minimum critical mass for black holes. Having seen that the new idea regarding

prevention of catastrophic collapse could have an interesting role in astrophysical

situations. He has also discussed the implications of the Einstein-Cartan theory for

finite distributions like fluid spheres with non-zero pressure. Also, since spin is a very

important property of a particle, it is very relevant to consider its role in the study of

such configurations as one may find in the interior of a star.

Hehl, Heyde and Kerlick [11] have considered the field equations of general

relativity with spin and torsion U4 theory to describe correctly the gravitational

properties of matter on a macro physical level. They have shown how the singularities

theorems of Penrose [21] and Hawking [17] must be modified to apply in E-C theory.

Parsanna [23] has solved Einstein- Cartan field equations for prefect fluid distribution

and adopting Hehl’s [8,9] approach, and Tolman’s technique [30] obtatied a number

of solutions. Arkuszewski et al [3] described the junction conditions in Einstein-

Cartan theory. Raychaudhuri and Benerje [24] considered collapsing spheres in E-C

theory and showed that it bounces at a radius greater than the Schwarzschild radius.

Banerji [4] has pointed out that E-C sphere must bounce outside the Schwarzschild

radius if it bounces at all. Singh and Yadav [25] and Yadav. et. al [35] studied the

fluid spheres in E-C theory and obtained a solution in an analytic form by the method

of quadrature. Spatially homogenous cosmological models of Bianchi type VI & VII

based on Einstein-Cartan theory were considered by Tsoubelies [34]. Som and



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Bedram [27] got the class of solutions that represent a static incoherent spherical dust

distribution in equilibrium under the influence of spin. Mollah and et.al. [20] have also

given a physically meaningful solutions of the field equations for static spherical dust

distribution in E-C theory. Krori et.al [16] gave a singularity free solution for a static

sphere in Einstein- Cartan theory. Suh [29] considering the static spherically

symmetric interior solution in Einstein- Cartan theory closely compared with those in

the Einstein theory of gravitation.

In this chapter we have taken the problem of static fluid spheres in the frame

work of Einstein-Cartan theory. Adopting Tolman’s technique [30] we have solved

the field equations and thus have obtained in two more cases besides the case

corresponding to well known Schwarzschild solution. We have also obtained pressure

and density for the distribution.

2. THE FIELD EQUATIONS

We use Einstein- Cartan field equations given by

(2.1)

1R R kt

2

(2.2)

Q Q Q kS

Where

Q is torsion tensor,

t is the canonical asymmetric energy momentum

tensor,

S is the spin tensor. Here we consider a static spherically symmetric matter

distribution given by the metric.

(2.3) 2 2 2 2 2 2 2 2 2 2ds e dr r d r sin d e dt

Where & are functions of r alone. If represents an orthonormal conframe

we have then

(2.4) 1 2 3 4e dr, rd , rsin d , e dt

The metric (2.3) now become

2 1 2 2 2 3 2 4 2ds ( ) ( ) ( ) ( )

So that ijg diag{ 1, 1, 1,1}



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Assuming that the spins of the individual particles composing the fluid are all aligned

in the radical direction we get for the spin tensor

S the only independent non-zero

component to be S23 = K, (say). Since the fluid is supposed to be static, we have the

velocity four-vector 4u .

Thus the non-zero components of

S are

(2.5) 4 4

23 32S S K

Hence from the Cartan equation (2.2), we get for

Q the components

(2.6) 4 4

23 32Q Q kK

the others are zero.

Using (2.6) in (2.3) we can obtain the torsion two-form (H) to be

(2.7) 1 2 3 4 2 3(H) 0, (H) 0, (H) 0, (H) kK

Once we have the torsion form we can use it in (2.3) along with (2.4) and solve the

components of

which in the present case turn out to be

1 4 4 2 1 2

4 1 1 2

ee ' ,

r

(2.8)

2 4 3 3 1 3

4 2 1 3

1 eKk ,

2 r

3 4 2 3 2 4 3

4 3 2 3

1 1 cotKk , Kk

2 2 r

Using (2.8) in (2.4) we get the curvature form

to be

(2.9) 1 2 2 1 4

4 [e ( '' ' ' ')]( )

2 3kKe ( )

r

2 1 3

4

1 Kke k' ( )

2 r

22 2 2 4e 1

' k K ( )r 4



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3 1 2

4

1 Kke k' ( )

2 r

22 2 3 4e 1

' k K ( )r 4

21 1 2 4 3

2

e 1 1'( ) kKe ( ' )( )

r 2 r

21 1 3 4 2

3

e 1 1'( ) kKe ( ' )( )

r 2 r

22 2 2 2 3

3 2

1 e 1k K

4r

1 41ke K' K '

2

Equation (2.4) and (2.9) together give

(2.10) 1 2 2

44R e '' ' ' '

22 3 2 2

424 434

1 e 'R R k K

4 r

21 1

212 313

e 'R R

r

22 2 2

323 2

1 e 1R k K

4r

1

423

kKR e

r

2 3

413 412

1 KR R ke K'

2 r

1 1

243 342

1 1R R kKe '

2 r

2

314

1R e K' K '

2

The Ricci tensor

R and scalar of curvature R are therefore given by



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2 2

11

2 'R e '' ' ' '

r

(2.11)

2

22 33 2 2

e 1R R 1 r ' '

r r

2 2 2 2

44

2 ' 1R e '' ' ' ' k K

r 2

(2.12)

2

2

1R 2 e

r

2

2

1 2'' ' ' ' ' '

rr 21

kK2

With R 0, . Hence the Einstein tensor

1G R Rg

2 is found to

have the components

2 2 2

11 2 2

1 2 ' 1 1G e k K

r 4r r

(2.13)

2 2 2 2

22 33

1 1G G e '' ' ' ' ' ' k K

r 4

2 2 2

44 2 2

1 2 ' 1 1G e k K

r 4r r

Since we are considering a perfect fluid distribution with isotropic pressure p and

matter density we have for fore

t

(2.14)

k m

k m k kt R p u u u S u p

Using (2.6) we get then the non-zero components

(2.15) 1 2 3 4

1 2 3 4t t t p, t

Hence the field equation (2.1) may be written using (2.13) and (2.15) as

(2.16)

2 2 2

2 2

1 2 ' 1 1e k K kp

r 4r r

(2.17)

2 2 2 21 1e '' ' ' ' ' ' k K k

r 4



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(2.18)

2 2 2

2 2

1 2 ' 1 1e k K k

r 4r r

The conservation laws give us the relations

(2.19)

p u 0 ( matter conservation)

(2.20) Ku 0 ( spin conservation), &

(2.21) dp 1

p ' kK K' K ' 0dr 2

If we assume the equation of hydrostatic equilibrium to hold as in general relativity,

namely

(2.22) dp

p ' 0dr

We get the additional equation.

(2.23) K' K ' 0

Solving for K we get

(2.24) K He

Where H is a constant of integrations to be determined. Setting

2

8 Gk

c with G = l, c = l

We can write the field equation as

(2.25)

2 2 2

2 2

1 2 ' 18 p 16 K e

rr r

(2.26)

2 2 2

2 2

1 2 ' 18 16 K e

rr r

(2.27)

2

2 3 2

'' ' 1 2 ' 1 ' 'e ' '

r r rr r r

3

10

r

In principle we now have a completely determined system if an equation of state is

specified. However, it is well known that in practice this set of equations is formidable

to solve using a pre-assigned equation of state, except perhaps for the case p ,



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which may not be physically meaningful. Secondly, we have the question of boundary

conditions to be chosen for fitting the solutions in the interior and exterior of the state.

A very interesting aspect of the Einstein-Cartan theory is that outside the fluid

distribution the equations reduce to Einstein’s equations for empty space, viz.,

ijR 0 , since there is no spin density.

Following Hehl’s approach, if we define

(2.28) 2 2p p 2 K , 2 K

We find that the equations take the usual general relativistic form for a static fluid

sphere as given by

(2.29)

2

2 2

1 2 ' 18 e

rr r

(2.30)

2

2 2

1 2 ' 18 e

rr r

With (2.27) remaining the same. The equation of continuity (2.21) now becomes

(2.31) dp

p ' 0dr

It is clear from these equations that it is the p and not the p which is continuous

across the boundary r = a of the fluid sphere. The continuity of p across the boundary

ensures that of ' exp.2 . Further with p and replacing p and

respectively we are assured that the metric coefficients are continuous across the

boundary. Hence we shall apply the usual boundary conditions to the solution of

equations (2.27), (2.29) and (2.30). We use the boundary conditions

(2.32)

2 2

r r

2me e 1

(2.33) p 0 at r = a

Where is the radius of the fluid sphere and m is the mass of the fluid sphere. The

total mass, as observed by an external observer, inside the fluid sphere of radius is

given by



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(2.34)

2 2 2 2 2

0 0 0

m r dr 4 r dr 8 K r r dr

Thus the total mass of the fluid sphere is modified by the correction

a

2 2 2

0

8 K r r dr

Equations (2.29), (2.30), and (2.31) are the same as obtained by Tolman [30], so we

can use the same solutions for our discussion. Assuming that the sphere has a finite

radius r = a for r > a. Since the equations are Rij = 0. We have by Birkhoff’s theorem

the space-time metric represented by the Schwarzschild solution

(2.35)

1

2 2 2 22mds 1 dr r d

r

2 2 2 22mr sin d 1 dt

r

Where m is a constant associated with the mass of the sphere.

3. SOLUTION OF THE FIELD EQUATIONS

Here we consider the case corresponding to the well- known Schwarzschild solution

(3.1)

12

2 2 2 2

2

rds 1 dr r d

R

21

2 22 2 2

2

rr sin d A B 1 dt

R

Where

(3.2)

12 22

2 2

3 1 2mA 1 , B

2 2R R

The pressure and the density are given by

(3.3)

1 12 22 2

2 2

2 2 2

21

2 2

2

1 r r16 H 3B 1 A A B 1

R R R

8 p

rA B 1

R



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(3.4)

21

2 22 2

2 2

21

2 2

2

3 r16 H A B 1

R R

8 p

rA B 1

R

Unlike in the case of general relativity, the fluid sphere is now no longer of uniform

density. The constant H can be found in terms of the central density 0 to be

(3.5)

112 22

0 2 2

1 3H 8 3 1 1

8 R R

Again as on the case of Einstein’s theory we find that a singularity are r = 0 occurs

only for the case A = B i.e. m/a = 4/9 From (3.4) we can calculate r in terms of and

substituting the value so obtained in (3.3) we get the equation of state.

(3.6)

1

2

2 2 2

A 3 68 8 8 p

2 HR R R

Here for different value of R we get different equation of state.

4. DISCUSSION

In general the dependence of the spin on the radial distance r is not determined in the

absence of a magnetic field. This dependence can therefore be chosen arbitrarily.

Prasanna [23] introduced an assumption the equation (2.22) to determine the radial

dependence of spin. In the present chapter also we have used the same assumption.

Further we observe that the continuity of p (not of p) across the surface 𝑟 = 𝑎

ensures the continuity of 'as required by equation (2.31), whereas ' is

discontinuous. The discontinuity of ' is due to the curvature coordinates employed

and hence the same as in general relativity. However, since the spin density is

discontinuous the pressure p is discontinuous across 𝑟 = 𝑎. Thus we find that the

usual general- relativistic boundary conditions, namely that (I) the metric potentials

are cI and the (2) the hydrostatic pressure is continuous, are not satisfied. This , in our

opinion should not be surprising, as in this theory spin does not influence the

geometry outside the distribution. As could be seen the presence of spin density

induces non uniformity in density in a Schwarzschild sphere, and consequently the

equation of state is charged. The other three cases considered by Tolman [30]



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(i) 𝑒2𝑝 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, (ii) 𝑒−2𝛼−2𝛽 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, (iii) 𝑒2𝛼 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 do not give us

any interesting distributions. Here our solution represents the static Einstein universe.

5. REFERENCES

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2. Bowers, R. L. and Liang, E. P.(1934) : Astrophys. J. 188, 657

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4. Benerji. S.(1978) : G. R. G. 9, 783

5. Cartan, E.(1922) : Coptes Rendus(Paris), 174, 593

6. Cartan, E.(1923) : Ann. Ec. Norm. Sup.(3), 40, 325

7. Hawking, S. W.(1966) : Proc. Roy. Soc. Lond. A295, 490

8. Hehl, F. W.(1973) : G. R. G. 4, 333

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11. Hehl, F. W. Vounder Heyde, P. and Kerlick, G. D. and Nester, J. N.(1976) : Rev. Mod.

Phys., 48, 393

12. Isham, C. J., Salam A. and Strathdee, J.(1973) : Nature(Lond.) Phys. Sci., 244, 82

13. Kerlick, G. D.(1975) : Spin and torsion in general relativity. Foundations and implications

for astrophysics and cosmology, Ph. D Thesis, Princeton University.

14. Kopczynski, W.(1972) : Phys. Letters A39, 219

15. Kopczynski, W.(1973) : Phys. Letters 43A, 63.

16. Krori, K. D., Sheikh, A. R. and Mahanta, L.(1981) : Can. J. Phys., 59, 425

17. Kuchowicz, B.(1975 a) : Acta Cosmologyica, 3,109

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http://www.scirp.org/journal/ijaa.

21. Penrose, R.(1965) : Phys Lett., 14, 57

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34. Tsoblis, D.(1979) : Phys. Rev., D 20, 3004.

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AUTHORS PROFILE

Dr. Vinod Kumar is an

Indian researcher in the

Field of Mathematics.

He is Young Research

Scholar who completed

Ph.D Programme from

University Deptt. of

Mathematics, Magadh

University, Bodhgaya,

Bihar (INDIA). He has completed M.Sc. in

Mathematics from Magadh University, Bodhgaya.

He contributes in the field of applied mathematics

and specially Relativity. He became a very good

academic index. His Ph.D topic is “SOME

ASPECTS AND PROBLEMS IN EINSTEIN-

CARTAN THEORY”. He is a life member of

Indian Science Congress since 2016 and of

Mathematical Society of B.H.U since last year. He

has more than 06 years of Research and Teaching

experience in the field of Mathematics and has

published more than 18 Research papers in

reputed international and national journals, it’s

also available online. He has presented more than

15 research article in international and national

seminars and got BEST PAPER and other Award.

His main research work focuses on Einstein-

Cartan Theory, Theory of relativity And

Einstein’s Field Equations.

AUTHORS PROFILE

Dr. Abhishek Kumar

Singh is a bona fide

Indian researcher and

author in the Field of

Mathematics. He is

Young and dynamic

Research Scholar who

currently pursuing

D.Sc. Programme on

the topic of “Cosmological Aspects of the

Universe in Higher Dimensions” from University

Deptt. of Mathematics, Magadh University,

Bodhgaya, Bihar (INDIA). He has completed

Ph.D and M.Sc. in Mathematics from Magadh

University, Bodhgaya. He became a Gold-

Medalist in M.Sc. (Mathematics) programme. He

is highly interested in applied Mathematics and

especially in Relativity & Cosmology. He obtained

his Ph.D. degree in due line due to his hard

labour and quickly picking up the relevant

knowledge. He achieves “BHARAT RATNA Dr.

ABDUL KALAM GOLD MEDAL AWARD”,

BEST PAPER AWARD by INDIAN SCIENCE

CONGRESS ASSOCIATION & IPA

CHANDIGARH, YOUNG SCIENTIST AWARD

and more. He is a life member of Indian Science

Congress since 2013 and of Mathematical Society

of B.H.U since last year. He has more than 07

years of Research and Teaching experience in the

field of Mathematics and has published more than

25 Research papers in reputed international and

national journals, it’s also available online. He

has presented more than 20 research article in

international and national seminars and got 02

times BEST PAPER AWARD. His main research

work focuses on Theory of relativity,

Cosmological models, String Cosmological

Models and Universal Extra Dimensions (UED)

Models.



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Some Static Fluid Spheres with Spin and Schwarzschild Solution

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