Research Article A Relativistic Algorithm with Isotropic Coordinates · 2019. 7. 31. · University of KwaZulu-Natal, Private Bag X, Durban , South Africa Correspondence should be

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 905168 6 pageshttpdxdoiorg1011552013905168

Research ArticleA Relativistic Algorithm with Isotropic Coordinates

S A Ngubelanga and S D Maharaj

Astrophysics and Cosmology Research Unit School of Mathematics Statistics and Computer ScienceUniversity of KwaZulu-Natal Private Bag X54001 Durban 4000 South Africa

Correspondence should be addressed to S D Maharaj maharajukznacza

Received 9 June 2013 Accepted 18 November 2013

Academic Editor Stephen C Anco

Copyright copy 2013 S A Ngubelanga and S D Maharaj This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We study spherically symmetric spacetimes formatter distributions with isotropic pressuresWe generate new exact solutions to theEinstein field equations which also contain isotropic pressuresWe develop an algorithm that produces a new solution if a particularsolution is known The algorithm leads to a nonlinear Bernoulli equation which can be integrated in terms of arbitrary functionsWe use a conformally flat metric to show that the integrals may be expressed in terms of elementary functions It is important tonote that we utilise isotropic coordinates unlike other treatments

1 Introduction

We consider the interior of static perfect fluid spheres in gen-eral relativitywith isotropic pressuresThepredictions of gen-eral relativity have been shown to be consistent with obser-vational data in relativistic astrophysics and cosmology For adiscussion of the physical features of a gravitating model werequire an exact solution to the Einstein field equations Exactsolutions are crucial in the description of dense relativisticastrophysical problems Many solutions have been found inthe past For some comprehensive lists of known solutionsto the field equations refer to Delgaty and Lake [1] Finchand Skea [2] and Stephani et al [3] Many of these solutionsare not physically reasonable For physical reasonablenesswe require that the gravitational potentials and matter vari-ables are regular and well behaved causality of the spacetimemanifold is maintained and values for physical quantitiesfor example the mass of a dense star are consistent withobservation

Solutions have been found in the past bymaking assump-tions on the gravitational potentials matter distribution orimposing an equation of stateThese particular approaches doyield models which have interesting properties However inprinciple it would be desirable to have a general method thatproduces exact solutions in a systematic manner Some sys-tematic methods generated in the past are those of Rahman

and Visser [4] Lake [5] Martin and Visser [6] Boonsermet al [7] Herrera et al [8] Chaisi and Maharaj [9] andMaharaj and Chaisi [10] In general relativity we have thefreedomof using anywell-defined coordinate systemThe ref-erences mentioned above mainly use canonical coordinatesThe use of isotropic coordinates may provide new insightsand possibly lead to new solutions This is the approach thatwe follow in this paper We generate a new algorithm pro-ducing a new solution to Einstein field equations in isotropiccoordinates From a given solution we can find a new sol-ution with isotropic pressures

The objective of this paper is to find new classes of exactsolutions of the Einstein field equations with an unchargedisotropic matter distribution from a given seed metric InSection 2 we derive the Einstein field equations for neutralperfect fluids in static spherically symmetric spacetime Weintroduce new variables due to Kustaanheimo and Qvist [11]to rewrite the field equations and the condition of pressureisotropy in equivalent forms In Section 3 we introduce ouralgorithm and the master nonlinear second order differentialequation containing two arbitrary functions that has to besolved In Section 4 we present new classes of exact solutionsin terms of the arbitrary functions In Section 5 we givean example for a conformally flat metric showing that theintegrals generated in Section 4 may be explicitly evaluatedIn Section 6 we summarise the results obtained in this paper

2 Advances in Mathematical Physics

2 The Model

We are modelling the interior of a dense relativistic star instrong gravitational fields The line element of the interiorspacetime with isotropic coordinates has the following form

1198891199042= minus1198602(119903) 1198891199052+ 1198612(119903) [119889119903

2+ 1199032(1198891205792+ sin21205791198891206012)]

(1)

where 119860(119903) and 119861(119903) are arbitrary functions representingthe gravitational potentials Relativistic compact objects suchas neutron stars in astrophysics are described by this lineelementThe energy momentum tensor for the interior of thestar has the form of a perfect fluid

119879119886119887= (120588 + 119901) 119906

119886119906119887+ 119901119892119886119887 (2)

where 120588 is the energy density and 119901 is the isotropic pressureThese quantities are measured relative to a timelike unit four-velocity 119906119886 (119906119886119906

119886= minus1)

The Einstein field equations for (1) and (2) have the form

120588 = minus1

1198612[211986110158401015840

119861minus1198611015840

119861(1198611015840

119861minus4

119903)] (3a)

119901 = 21198601015840

119860(1198611015840

1198613+1

119903

1

1198612) +

1198611015840

1198613(1198611015840

119861+2

119903) (3b)

119901 =1

1198612(11986010158401015840

119860+1

119903

1198601015840

119860) +

1

1198612[11986110158401015840

119861minus1198611015840

119861(1198611015840

119861minus1

119903)] (3c)

in isotropic coordinates Primes denote differentiation withrespect to the radial coordinate 119903 On equating (3b) and (3c)we obtain the condition of pressure isotropy which has theform

11986010158401015840

119860+11986110158401015840

119861= (

1198601015840

119860+1198611015840

119861)(2

1198611015840

119861+1

119903) (4)

This is the master equation which has to be integrated toproduce an exact solution to the field equations

It is possible to write the system (3a)ndash(3c) in an equivalentform by introducing new variables We utilize a transforma-tion that has proven to be helpful in relativistic stellar physicsWe introduce the new variables

119909 equiv 1199032 119871 equiv 119861

minus1 119866 equiv 119871119860 (5)

The above transformation was first suggested by Kustaan-heimo and Qvist [11] On applying transformation (5) in thefield equations of (3a)ndash(3c) we obtain the equivalent system

120588 = 4 [2119909119871119871119909119909minus 3 (119909119871

119909minus 119871) 119871

119909] (6a)

119901 = 4119871 (119871 minus 2119909119871119909)119866119909

119866minus 4 (2119871 minus 3119909119871

119909) 119871119909 (6b)

119901 = 41199091198712119866119909119909

119866+ 4119871 (119871 minus 2119909119871

119909)119866119909

119866

minus 4 (2119871 minus 3119909119871119909) 119871119909minus 8119909119871119871

119909119909

(6c)

We note that (6a)ndash(6c) are highly nonlinear in both 119871 and 119866In this system there are three independent equations and fourunknowns 120588119901 119871 and119866 So we need to choose the functionalform for 119871 or119866 in order to integrate and obtain an exact solu-tionThe value of the transformation (5) is highlighted in thereduction of the condition of pressure isotropy On equating(6b) and (6c) we get

119871119866119909119909= 2119866119871

119909119909 (7)

which is the new condition of pressure isotropy which has asimpler compact form

3 The Algorithm

It is possible to find new solutions to the Einsteinrsquos equationsfrom a given seed metric Examples of this process are givenin the treatments of Chaisi and Maharaj [9] andMaharaj andChaisi [10] They found new models with anisotropic pres-sures from a given seed isotropic metric in Schwarzschildcoordinates Our intention is to find new models withisotropic pressures from a given solution in terms of theisotropic line element (1)

We can provide some new classes of exact solutions to theEinstein field equations by generating a new algorithm thatproduces a model from a given solutionWe assume a knownsolution of the form (119871 119866) so that

119871119866119909119909= 2119866119871

119909119909(8)

holds We seek a new solution (119871 119866) given by

119871 = 119871119890119892(119909)

119866 = 119866119890119891(119909)

(9)

where 119891(119909) and 119892(119909) are arbitrary functions On substituting(9) into (7) we obtain

(119871119866119909119909minus 2119866119871

119909119909) + 2 (119871119866

119909119891119909minus 2119866119871

119909119892119909)

+ 119871119866 (119891119909119909minus 2119892119909119909) + 119871119866 (119891

2

119909minus 21198922

119909) = 0

(10)

which is given in terms of two arbitrary functions 119891(119909) and119892(119909) Then realizing that (119871 119866) is a solution of (7) and using(8) we obtain the reduced result

(119891119909119909minus 2119892119909119909) + 2(

119866119909

119866119891119909minus 2

119871119909

119871119892119909) + (119891

2

119909minus 21198922

119909) = 0

(11)

We need to demonstrate the existence of functions 119891(119909) and119892(119909) that satisfy (11) In general it is difficult to integrate (11)since it is given in terms of two arbitrary functions which arenonlinear

4 New Solutions

We consider several cases of (11) for which we have been ableto complete the integration

Advances in Mathematical Physics 3

41 119892(119909) Is Specified We can integrate (11) if 119892(119909) is specifiedAs a simple example we take 119892(119909) = 1 Then (11) becomes

119891119909119909+ 2

119866119909

119866119891119909+ 1198912

119909= 0 (12)

which is nonlinear in 119891 This is a first order Bernoulli equa-tion in 119891

119909 We can rewrite (12) in the form

(1

119891119909

)

119909

minus 2(119866119909

119866)(

1

119891119909

) = 1 (13)

It is possible to integrate (13) since it is linear in 1119891119909to obtain

119891119909= 119866minus2

(int119866minus2

119889119909 + 1198881)

minus1

(14)

We can formally integrate (14) to obtain the function 119891(119909) as

119891 (119909) = int[

[

119866minus2

(int119866minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882 (15)

where 1198881and 1198882are arbitrary constants

Then the new solution to (7) has the form

119871 = 119871 (16a)

119866 = 119866 exp(int[

[

119866minus2

(int119866minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882)

(16b)

Therefore we have shown that if a solution (119871 119866) to the fieldequations is known then a new solution (119871 119866) is given by(16a) and (16b)

42 119891(119909) Is Specified We can also integrate (11) if 119891(119909) isspecified As another simple example we take119891(119909) = 1Then(11) becomes

119892119909119909+ 2

119871119909

119871119892119909+ 1198922

119909= 0 (17)

which is nonlinear in119892This is a first order Bernoulli equationin 119892119909 The differential equation (17) has a form similar to (12)

in Section 41 Following the same procedure we obtain

119892 (119909) = int[

[

119871minus2

(int119871minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882 (18)


Then another new solution to (7) is given by

119866 = 119866 (19a)

119871 = 119871 exp(int[

[

119871minus2

(int119871minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882)

(19b)

Therefore we have determined that if a solution (119871 119866) to thefield equations is known then a new solution (119871 119866) is givenby (19a) and (19b) Note that the solution of (19a) and (19b) isdifferent from that of (16a) and (16b)

43 119892(119909) = 120572119891(119909) We can integrate (11) if a relationship bet-ween the functions 119891(119909) and 119892(119909) exists We illustrate thisfeature by assuming that

119892 (119909) = 120572119891 (119909) (20)

where 120572 is an arbitrary constant Then (11) becomes

119891119909119909+

2

1 minus 2120572(119866119909

119866minus 2120572

119871119909

119871)119891119909+ (

1 minus 21205722

1 minus 2120572)1198912

119909= 0

(21)

which is a first order Bernoulli equation in 119891119909 For conve-

nience we let

Θ = (1 minus 2120572

2

1 minus 2120572) 120578 =

2

1 minus 2120572 120572 =

1

2 (22)

so that we can write (21) as

(1

119891119909

)

119909

minus 120578(119866119909

119866minus 2120572

119871119909

119871)(

1

119891119909

) = Θ (23)

which is linear in 1119891119909 We integrate (23) to obtain

119891119909= (

1198712120572

119866)

120578

[Θint(1198712120572

119866)

120578

119889119909 + 1198881]

minus1

(24)

We now formally integrate (24) to obtain

119891 (119909) = int((1198712120572

119866)

120578

[Θint(1198712120572

119866)

120578

119889119909 + 1198881]

minus1

)119889119909 + 1198882

(25)

where 1198881and 1198882are constants

We now have a new solution of (7) given by

119871 = 119871 exp120572[[

[

int((1198712120572

119866)

120578

times [

[

Θint(1198712120572

119866)

120578

119889119909 + 1198881]

]

minus1

)119889119909 + 1198882

]]

]

(26a)

119866 = 119866 exp[[

[

int((1198712120572

119866)

120578

times [Θint(1198712120572

119866)

120578

119889119909 + 1198881]

minus1

)119889119909 + 1198882

]]

]

(26b)

where Θ and 120578 are given in (22) Therefore we have demon-strated that if a solution (119871 119866) to the field equations is


specified then a new solution (119871 119866) is provided by (26a) and(26b)

Some special cases related to (26a) and (26b) should bepointed out These relate to 120572 = 1 plusmn1radic2 12 We considereach in turn

Case i (120572 = 1) With 120572 = 1 we find that (26a) and (26b)become

119871 = 119871 exp(int[

[

1198662

1198714(int

1198662

1198714119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882) (27a)

119866 = 119866 exp(int[

[

1198662

1198714(int

1198662

1198714119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882) (27b)

which are a simple form

Case ii (120572 = plusmn1radic2) If we set 120572 = plusmn1radic2 then (26a) and(26b) become

119871 = 119871 exp[[

[

plusmn1

radic2(1198881int(

119871plusmnradic2

119866)

2(1minus(plusmnradic2))

119889119909 + 1198882)]]

]

(28a)

119866 = 119866 exp[[

[

1198881int(

119871plusmnradic2

119866)


119889119909 + 1198882

]]

]

(28b)

which are another simple case

Case iii (120572 = 12) If 120572 = 12 then (26a) and (26b) are notvalid For this case (11) becomes

119891119909[119891119909+ 4(

119866119909

119866minus119871119909

119871)] = 0 (29)

When 119891 is constant then 119892 is also constant by (20) then (7)does not produce a new solution because of (9) When 119891 isnot constant then we can integrate (29) to produce thesolution

119871 = 1198701198713

1198662 (30a)

119866 = 1198701198714

1198663 (30b)

where 119870 is a constant Thus 120572 = 12 generates another newsolution (119871 119866) to (11)

5 Example

We show by means of a specific example that the integralsgenerated in Section 4 may be evaluated to produce a newexact solution to the field equations in terms of elementaryfunctions In our example we choose

119871 = 119887 + 119886119909 (31a)

119866 = 1 + 119888119909 (31b)

Then the corresponding line element is given by

1198891199042= minus(

1 + 1198881199032

119887 + 1198861199032)

2

1198891199052

+ (1

119887 + 1198861199032)

2

(1198891199032+ 1199032(1198891205792+ sin21205791198891206012))

(32)

which is conformally flat The energy density for the metric(32) is constant so that we have the Schwarzschild interiorsolution in isotropic coordinates

Conformally flat metrics are important in gravitationalphysics in a general relativistic setting They arise forinstance in the gravitational collapse of a radiating star asshown in the treatments of Herrera et al [12] Maharaj andGovender [13] Misthry et al [14] and Abebe et al [15] Forthe choice of (31a) and (31b) we find that (27a) and (27b)become

119871=(119887 + 119886119909) exp(int[

[

(1 + 119888119909)2

(119887 + 119886119909)4

times (int(1 + 119888119909)

2

(119887 + 119886119909)4119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882)

(33a)

119866=(1 + 119888119909) exp(int[

[

(1 + 119888119909)2

(119887 + 119886119909)4

times (int(1 + 119888119909)

2

(119887 + 119886119909)4119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882)

(33b)

The integrals in (33a) and (33b) can be evaluated and weobtain

119871 =1

(119887 + 119886119909)2119880 (119909) (34a)

119866 =(1 + 119888119909)

(119887 + 119886119909)3119880 (119909) (34b)

where 1198881= 0 and 119888

2= 1 and we have set

119880 (119909) = 11988721198882+ 119886119887119888 (1 + 3119888119909) + 119886

2(1 + 3119888119909 + 3119888

21199092)

(35)

Thus the known solution (119871 119866) in (31a) and (31b) produces anew solution (119871 119866) in (34a) and (34b) The line element forthe new solution has the form

1198891199042= minus(

1 + 1198881199032

119887 + 1198861199032)

2

1198891199052+ (

(119887 + 1198861199032)2

119880(119903))

2

times (1198891199032+ 1199032(1198891205792+ sin21205791198891206012))

(36)

where119880(119903) is given by (35)Thus our algorithm has produceda new (not conformally flat) solution to the Einsteinrsquos fieldequations This has been generated from a seed conformallyflat model


02 04 06 08 10r

05

10

15

20

120588

Figure 1 Energy density 120588

02 04 06 08 10r

20

30

40

50

60

p

Figure 2 Pressure 119901

00002 00004 00006 00008 00010r

0684365

0684365

0684366

0684366

dpd120588

Figure 3 Speed of sound 119889119901119889120588

6 Conclusion

We now comment on the physical properties of the exampleWe have generated plots for the energy density 120588 pressure119901 and the speed of sound in Figures 1 2 and 3 respectivelyThese graphical plots indicate that 120588 and 119901 are positive andwell behavedThe speed of sound is less than the speed of lightas required for causalityTherefore the algorithmpresented inthis paper produces new solutions which are physicallyreasonable

We have generated an algorithm to produce a newsolution to the Eistein field equations from a given seed met-ric We observe that the resulting model contains isotropicpressures unlike the approach of Chaisi and Maharaj [9] and

Maharaj and Chaisi [10] in their treatment the new modelhas anisotropic pressures Another advantage of our approachis the use of isotropic coordinates in the formulation ofthe condition of pressure isotropy This may leads to newinsights into the behaviour of gravity since previous treat-ments mainly utilised canonical coordinates The algorithmproduced a new solution in terms of integrals containingarbitrary functions We have shown with the help of aconformally flat metric that these integrals may be evaluatedin terms of elementary functions This example suggests thatour approach may be extended to other physically relevantmetrics

Acknowledgments

S A Ngubelanga thanks the National Research Foundationand the University of KwaZulu-Natal for financial supportS D Maharaj acknowledges that this work is based uponresearch supported by the South African Research Chair Ini-tiative of the Department of Science and Technology and theNational Research Foundation

References

[1] M S R Delgaty and K Lake ldquoPhysical acceptability of isolatedstatic spherically symmetric perfect fluid solutions of Einsteinrsquosequationsrdquo Computer Physics Communications vol 115 no 2-3pp 395ndash415 1998

[2] M R Finch and J E F Skea Unpublished notes 1998[3] H Stephani D Kramer M A H MacCallum C Hoenselaars

and E Herlt Exact Solutions of Einsteinrsquos Field equationsCambridge University Press Cambridge UK 2003

[4] S Rahman and M Visser ldquoSpacetime geometry of static fluidspheresrdquo Classical and Quantum Gravity vol 19 no 5 pp 935ndash952 2002

[5] K Lake ldquoAll static spherically symmetric perfect-fluid solutionsof Einsteinrsquos equationsrdquo Physical ReviewD vol 67 no 10 ArticleID 104015 2003

[6] D Martin and M Visser ldquoAlgorithmic construction of staticperfect fluid spheresrdquo Physical Review D vol 69 no 10 ArticleID 104028 2004

[7] P BoonsermMVisser and SWeinfurtner ldquoGenerating perfectfluid spheres in general relativityrdquo Physical Review D vol 71 no12 Article ID 124037 2005

[8] L Herrera A Di Prisco J Martin J Ospino N O Santosand O Troconis ldquoSpherically symmetric dissipative anisotropicfluids a general studyrdquo Physical Review D vol 69 no 8 ArticleID 084026 2004

[9] M Chaisi and S D Maharaj ldquoA new algorithm for anisotropicsolutionsrdquo Pramana vol 66 no 2 pp 313ndash324 2006

[10] S D Maharaj and M Chaisi ldquoNew anisotropic models fromisotropic solutionsrdquo Mathematical Methods in the Applied Sci-ences vol 29 no 1 pp 67ndash83 2006

[11] P Kustaanheimo and B Qvist ldquoA note on some general solu-tions of the Einstein field equations in a spherically symmetricworldrdquo Societas Scientiarum Fennica vol 13 no 16 p 12 1948

[12] L Herrera G Le Denmat N O Santos and A Wang ldquoShear-free radiating collapse and conformal flatnessrdquo InternationalJournal of Modern Physics D vol 13 no 4 pp 583ndash592 2004


[13] S D Maharaj and M Govender ldquoRadiating collapse with van-ishingWeyl stressesrdquo International Journal ofModern Physics Dvol 14 no 3-4 pp 667ndash676 2005

[14] S S Misthry S D Maharaj and P G L Leach ldquoNonlinearshear-free radiative collapserdquo Mathematical Methods in theApplied Sciences vol 31 no 3 pp 363ndash374 2008

[15] G Abebe K S Govinder and S D Maharaj ldquoLie symmetriesfor a conformally flat radiating starrdquo International Journal ofTheoretical Physics vol 52 pp 3244ndash3254 2013

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Stochastic AnalysisInternational Journal of


2 The Model

We are modelling the interior of a dense relativistic star instrong gravitational fields The line element of the interiorspacetime with isotropic coordinates has the following form

1198891199042= minus1198602(119903) 1198891199052+ 1198612(119903) [119889119903

2+ 1199032(1198891205792+ sin21205791198891206012)]

(1)

where 119860(119903) and 119861(119903) are arbitrary functions representingthe gravitational potentials Relativistic compact objects suchas neutron stars in astrophysics are described by this lineelementThe energy momentum tensor for the interior of thestar has the form of a perfect fluid

119879119886119887= (120588 + 119901) 119906

119886119906119887+ 119901119892119886119887 (2)

where 120588 is the energy density and 119901 is the isotropic pressureThese quantities are measured relative to a timelike unit four-velocity 119906119886 (119906119886119906

119886= minus1)

The Einstein field equations for (1) and (2) have the form

120588 = minus1

1198612[211986110158401015840

119861minus1198611015840

119861(1198611015840

119861minus4

119903)] (3a)

119901 = 21198601015840

119860(1198611015840

1198613+1

119903

1

1198612) +

1198611015840

1198613(1198611015840

119861+2

119903) (3b)

119901 =1

1198612(11986010158401015840

119860+1

119903

1198601015840

119860) +

1

1198612[11986110158401015840

119861minus1198611015840

119861(1198611015840

119861minus1

119903)] (3c)

in isotropic coordinates Primes denote differentiation withrespect to the radial coordinate 119903 On equating (3b) and (3c)we obtain the condition of pressure isotropy which has theform

11986010158401015840

119860+11986110158401015840

119861= (

1198601015840

119860+1198611015840

119861)(2

1198611015840

119861+1

119903) (4)

This is the master equation which has to be integrated toproduce an exact solution to the field equations

It is possible to write the system (3a)ndash(3c) in an equivalentform by introducing new variables We utilize a transforma-tion that has proven to be helpful in relativistic stellar physicsWe introduce the new variables

119909 equiv 1199032 119871 equiv 119861

minus1 119866 equiv 119871119860 (5)

The above transformation was first suggested by Kustaan-heimo and Qvist [11] On applying transformation (5) in thefield equations of (3a)ndash(3c) we obtain the equivalent system

120588 = 4 [2119909119871119871119909119909minus 3 (119909119871

119909minus 119871) 119871

119909] (6a)

119901 = 4119871 (119871 minus 2119909119871119909)119866119909

119866minus 4 (2119871 minus 3119909119871

119909) 119871119909 (6b)

119901 = 41199091198712119866119909119909

119866+ 4119871 (119871 minus 2119909119871

119909)119866119909

119866

minus 4 (2119871 minus 3119909119871119909) 119871119909minus 8119909119871119871

119909119909

(6c)

We note that (6a)ndash(6c) are highly nonlinear in both 119871 and 119866In this system there are three independent equations and fourunknowns 120588119901 119871 and119866 So we need to choose the functionalform for 119871 or119866 in order to integrate and obtain an exact solu-tionThe value of the transformation (5) is highlighted in thereduction of the condition of pressure isotropy On equating(6b) and (6c) we get

119871119866119909119909= 2119866119871

119909119909 (7)

which is the new condition of pressure isotropy which has asimpler compact form

3 The Algorithm

It is possible to find new solutions to the Einsteinrsquos equationsfrom a given seed metric Examples of this process are givenin the treatments of Chaisi and Maharaj [9] andMaharaj andChaisi [10] They found new models with anisotropic pres-sures from a given seed isotropic metric in Schwarzschildcoordinates Our intention is to find new models withisotropic pressures from a given solution in terms of theisotropic line element (1)

We can provide some new classes of exact solutions to theEinstein field equations by generating a new algorithm thatproduces a model from a given solutionWe assume a knownsolution of the form (119871 119866) so that

119871119866119909119909= 2119866119871

119909119909(8)

holds We seek a new solution (119871 119866) given by

119871 = 119871119890119892(119909)

119866 = 119866119890119891(119909)

(9)

where 119891(119909) and 119892(119909) are arbitrary functions On substituting(9) into (7) we obtain

(119871119866119909119909minus 2119866119871

119909119909) + 2 (119871119866

119909119891119909minus 2119866119871

119909119892119909)

+ 119871119866 (119891119909119909minus 2119892119909119909) + 119871119866 (119891

2

119909minus 21198922

119909) = 0

(10)

which is given in terms of two arbitrary functions 119891(119909) and119892(119909) Then realizing that (119871 119866) is a solution of (7) and using(8) we obtain the reduced result

(119891119909119909minus 2119892119909119909) + 2(

119866119909

119866119891119909minus 2

119871119909

119871119892119909) + (119891

2

119909minus 21198922

119909) = 0

(11)

We need to demonstrate the existence of functions 119891(119909) and119892(119909) that satisfy (11) In general it is difficult to integrate (11)since it is given in terms of two arbitrary functions which arenonlinear

4 New Solutions

We consider several cases of (11) for which we have been ableto complete the integration



119891119909119909+ 2

119866119909

119866119891119909+ 1198912

119909= 0 (12)



(1

119891119909

)

119909

minus 2(119866119909

119866)(

1

119891119909

) = 1 (13)


119891119909= 119866minus2

(int119866minus2

119889119909 + 1198881)

minus1

(14)


119891 (119909) = int[

[

119866minus2

(int119866minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882 (15)



119871 = 119871 (16a)

119866 = 119866 exp(int[

[

119866minus2

(int119866minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882)

(16b)



119892119909119909+ 2

119871119909

119871119892119909+ 1198922

119909= 0 (17)



119892 (119909) = int[

[

119871minus2

(int119871minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882 (18)



119866 = 119866 (19a)

119871 = 119871 exp(int[

[

119871minus2

(int119871minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882)

(19b)



119892 (119909) = 120572119891 (119909) (20)


119891119909119909+

2

1 minus 2120572(119866119909

119866minus 2120572

119871119909

119871)119891119909+ (

1 minus 21205722

1 minus 2120572)1198912

119909= 0

(21)


nience we let

Θ = (1 minus 2120572

2

1 minus 2120572) 120578 =

2

1 minus 2120572 120572 =

1

2 (22)


(1

119891119909

)

119909

minus 120578(119866119909

119866minus 2120572

119871119909

119871)(

1

119891119909

) = Θ (23)


119891119909= (

1198712120572

119866)

120578

[Θint(1198712120572

119866)

120578

119889119909 + 1198881]

minus1

(24)


119891 (119909) = int((1198712120572

119866)

120578

[Θint(1198712120572

119866)

120578

119889119909 + 1198881]

minus1

)119889119909 + 1198882

(25)



119871 = 119871 exp120572[[

[

int((1198712120572

119866)

120578

times [

[

Θint(1198712120572

119866)

120578

119889119909 + 1198881]

]

minus1

)119889119909 + 1198882

]]

]

(26a)

119866 = 119866 exp[[

[

int((1198712120572

119866)

120578

times [Θint(1198712120572

119866)

120578

119889119909 + 1198881]

minus1

)119889119909 + 1198882

]]

]

(26b)






119871 = 119871 exp(int[

[

1198662

1198714(int

1198662

1198714119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882) (27a)

119866 = 119866 exp(int[

[

1198662

1198714(int

1198662

1198714119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882) (27b)



119871 = 119871 exp[[

[

plusmn1

radic2(1198881int(

119871plusmnradic2

119866)


119889119909 + 1198882)]]

]

(28a)

119866 = 119866 exp[[

[

1198881int(

119871plusmnradic2

119866)


119889119909 + 1198882

]]

]

(28b)



119891119909[119891119909+ 4(

119866119909

119866minus119871119909

119871)] = 0 (29)


119871 = 1198701198713

1198662 (30a)

119866 = 1198701198714

1198663 (30b)


5 Example


119871 = 119887 + 119886119909 (31a)

119866 = 1 + 119888119909 (31b)


1198891199042= minus(

1 + 1198881199032

119887 + 1198861199032)

2

1198891199052

+ (1

119887 + 1198861199032)

2

(1198891199032+ 1199032(1198891205792+ sin21205791198891206012))

(32)



119871=(119887 + 119886119909) exp(int[

[

(1 + 119888119909)2

(119887 + 119886119909)4

times (int(1 + 119888119909)

2

(119887 + 119886119909)4119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882)

(33a)

119866=(1 + 119888119909) exp(int[

[

(1 + 119888119909)2

(119887 + 119886119909)4

times (int(1 + 119888119909)

2

(119887 + 119886119909)4119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882)

(33b)


119871 =1

(119887 + 119886119909)2119880 (119909) (34a)

119866 =(1 + 119888119909)

(119887 + 119886119909)3119880 (119909) (34b)

where 1198881= 0 and 119888


119880 (119909) = 11988721198882+ 119886119887119888 (1 + 3119888119909) + 119886

2(1 + 3119888119909 + 3119888

21199092)

(35)


1198891199042= minus(

1 + 1198881199032

119887 + 1198861199032)

2

1198891199052+ (

(119887 + 1198861199032)2

119880(119903))

2

times (1198891199032+ 1199032(1198891205792+ sin21205791198891206012))

(36)



02 04 06 08 10r

05

10

15

20

120588


02 04 06 08 10r

20

30

40

50

60

p


00002 00004 00006 00008 00010r

0684365

0684365

0684366

0684366

dpd120588


6 Conclusion




Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891119909119909+ 2

119866119909

119866119891119909+ 1198912

119909= 0 (12)



(1

119891119909

)

119909

minus 2(119866119909

119866)(

1

119891119909

) = 1 (13)


119891119909= 119866minus2

(int119866minus2

119889119909 + 1198881)

minus1

(14)


119891 (119909) = int[

[

119866minus2

(int119866minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882 (15)



119871 = 119871 (16a)

119866 = 119866 exp(int[

[

119866minus2

(int119866minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882)

(16b)



119892119909119909+ 2

119871119909

119871119892119909+ 1198922

119909= 0 (17)



119892 (119909) = int[

[

119871minus2

(int119871minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882 (18)



119866 = 119866 (19a)

119871 = 119871 exp(int[

[

119871minus2

(int119871minus2

119889119909 + 1198881)

minus1

]

]

119889119909 + 1198882)

(19b)



119892 (119909) = 120572119891 (119909) (20)


119891119909119909+

2

1 minus 2120572(119866119909

119866minus 2120572

119871119909

119871)119891119909+ (

1 minus 21205722

1 minus 2120572)1198912

119909= 0

(21)


nience we let

Θ = (1 minus 2120572

2

1 minus 2120572) 120578 =

2

1 minus 2120572 120572 =

1

2 (22)


(1

119891119909

)

119909

minus 120578(119866119909

119866minus 2120572

119871119909

119871)(

1

119891119909

) = Θ (23)


119891119909= (

1198712120572

119866)

120578

[Θint(1198712120572

119866)

120578

119889119909 + 1198881]

minus1

(24)


119891 (119909) = int((1198712120572

119866)

120578

[Θint(1198712120572

119866)

120578

119889119909 + 1198881]

minus1

)119889119909 + 1198882

(25)



119871 = 119871 exp120572[[

[

int((1198712120572

119866)

120578

times [

[

Θint(1198712120572

119866)

120578

119889119909 + 1198881]

]

minus1

)119889119909 + 1198882

]]

]

(26a)

119866 = 119866 exp[[

[

int((1198712120572

119866)

120578

times [Θint(1198712120572

119866)

120578

119889119909 + 1198881]

minus1

)119889119909 + 1198882

]]

]

(26b)






119871 = 119871 exp(int[

[

1198662

1198714(int

1198662

1198714119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882) (27a)

119866 = 119866 exp(int[

[

1198662

1198714(int

1198662

1198714119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882) (27b)



119871 = 119871 exp[[

[

plusmn1

radic2(1198881int(

119871plusmnradic2

119866)


119889119909 + 1198882)]]

]

(28a)

119866 = 119866 exp[[

[

1198881int(

119871plusmnradic2

119866)


119889119909 + 1198882

]]

]

(28b)



119891119909[119891119909+ 4(

119866119909

119866minus119871119909

119871)] = 0 (29)


119871 = 1198701198713

1198662 (30a)

119866 = 1198701198714

1198663 (30b)


5 Example


119871 = 119887 + 119886119909 (31a)

119866 = 1 + 119888119909 (31b)


1198891199042= minus(

1 + 1198881199032

119887 + 1198861199032)

2

1198891199052

+ (1

119887 + 1198861199032)

2

(1198891199032+ 1199032(1198891205792+ sin21205791198891206012))

(32)



119871=(119887 + 119886119909) exp(int[

[

(1 + 119888119909)2

(119887 + 119886119909)4

times (int(1 + 119888119909)

2

(119887 + 119886119909)4119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882)

(33a)

119866=(1 + 119888119909) exp(int[

[

(1 + 119888119909)2

(119887 + 119886119909)4

times (int(1 + 119888119909)

2

(119887 + 119886119909)4119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882)

(33b)


119871 =1

(119887 + 119886119909)2119880 (119909) (34a)

119866 =(1 + 119888119909)

(119887 + 119886119909)3119880 (119909) (34b)

where 1198881= 0 and 119888


119880 (119909) = 11988721198882+ 119886119887119888 (1 + 3119888119909) + 119886

2(1 + 3119888119909 + 3119888

21199092)

(35)


1198891199042= minus(

1 + 1198881199032

119887 + 1198861199032)

2

1198891199052+ (

(119887 + 1198861199032)2

119880(119903))

2

times (1198891199032+ 1199032(1198891205792+ sin21205791198891206012))

(36)



02 04 06 08 10r

05

10

15

20

120588


02 04 06 08 10r

20

30

40

50

60

p


00002 00004 00006 00008 00010r

0684365

0684365

0684366

0684366

dpd120588


6 Conclusion




Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119871 = 119871 exp(int[

[

1198662

1198714(int

1198662

1198714119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882) (27a)

119866 = 119866 exp(int[

[

1198662

1198714(int

1198662

1198714119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882) (27b)



119871 = 119871 exp[[

[

plusmn1

radic2(1198881int(

119871plusmnradic2

119866)


119889119909 + 1198882)]]

]

(28a)

119866 = 119866 exp[[

[

1198881int(

119871plusmnradic2

119866)


119889119909 + 1198882

]]

]

(28b)



119891119909[119891119909+ 4(

119866119909

119866minus119871119909

119871)] = 0 (29)


119871 = 1198701198713

1198662 (30a)

119866 = 1198701198714

1198663 (30b)


5 Example


119871 = 119887 + 119886119909 (31a)

119866 = 1 + 119888119909 (31b)


1198891199042= minus(

1 + 1198881199032

119887 + 1198861199032)

2

1198891199052

+ (1

119887 + 1198861199032)

2

(1198891199032+ 1199032(1198891205792+ sin21205791198891206012))

(32)



119871=(119887 + 119886119909) exp(int[

[

(1 + 119888119909)2

(119887 + 119886119909)4

times (int(1 + 119888119909)

2

(119887 + 119886119909)4119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882)

(33a)

119866=(1 + 119888119909) exp(int[

[

(1 + 119888119909)2

(119887 + 119886119909)4

times (int(1 + 119888119909)

2

(119887 + 119886119909)4119889119909 + 119888

1)

minus1

]

]

119889119909 + 1198882)

(33b)


119871 =1

(119887 + 119886119909)2119880 (119909) (34a)

119866 =(1 + 119888119909)

(119887 + 119886119909)3119880 (119909) (34b)

where 1198881= 0 and 119888


119880 (119909) = 11988721198882+ 119886119887119888 (1 + 3119888119909) + 119886

2(1 + 3119888119909 + 3119888

21199092)

(35)


1198891199042= minus(

1 + 1198881199032

119887 + 1198861199032)

2

1198891199052+ (

(119887 + 1198861199032)2

119880(119903))

2

times (1198891199032+ 1199032(1198891205792+ sin21205791198891206012))

(36)



02 04 06 08 10r

05

10

15

20

120588


02 04 06 08 10r

20

30

40

50

60

p


00002 00004 00006 00008 00010r

0684365

0684365

0684366

0684366

dpd120588


6 Conclusion




Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 04 06 08 10r

05

10

15

20

120588


02 04 06 08 10r

20

30

40

50

60

p


00002 00004 00006 00008 00010r

0684365

0684365

0684366

0684366

dpd120588


6 Conclusion




Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Relativistic Algorithm with Isotropic Coordinates · 2019. 7. 31. · University of KwaZulu-Natal, Private Bag X, Durban , South Africa Correspondence should be