some static fluid spheres with spin and schwarzschild solution
TRANSCRIPT
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)
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
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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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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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