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International Journal of Theoretical Physics manuscript No. (will be inserted by the editor) Applications of Lie Symmetries to Higher Dimensional Gravitating Fluids A. M. Msomi · K. S Govinder · S. D. Maharaj 3 1 0 2 Received: date/Accepted: date n a J Abstract We consider a radiating shear-free spherically symmetric metric in 8 higher dimensions. Several new solutions to the Einstein’s equations are found systematically using the method of Lie analysis of differential equations. Using ] the five Lie point symmetries of the fundamental field equation, we obtain either c an implicit solution or we can reduce the governing equations to a Riccati equa- q - tion.WeshowthatknownsolutionsoftheEinsteinequationscanproduceinfinite r familiesofnewsolutions.Earlierresultsinfourdimensionsareshowntobespecial g [ cases of our generalised results. 1 Keywords Gravitatingfluids Symmetries Higher dimensionalphysics · · v 9 7 1 Introduction 4 1 The spherically symmetric radiating spacetimes with vanishing shear are impor- . 1 tantfor applicationsin relativisticastrophysics,radiatingstarsand cosmology.In 0 the literature, there exists a large number of studies of various models involving 3 gravitationalcollapse with radiativeprocesses. Studies modeling relativisticstars 1 show that a necessary requirement for these models is that the interior radiat- : v ing spacetime has to be matched at the boundary, with the radial pressure being i X nonzero,totheexteriorVaidyaradiatingspacetime.Krasinski[1]pointedoutthe significanceof relativisticheat conducting fluids in modeling inhomogeneous pro- r a cesses. Some exact solutions in the presence of heat flow have been developed by Bergmann [2],Maiti [3] and Modak [4]. In consideringspherical gravitationalcollapse,theappearanceofsingularitiesand theformationofhorizons,Banerjeeand Chatterjee [5] and Banerjee et al. [6] have investigated heat conducting fluids in A.M.Msomi·K.S.Govinder·S.D.Maharaj Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, Private Bag X54001,UniversityofKwaZulu-Natal,Durban4000, SouthAfrica E-mail:[email protected] A.M.Msomi Department of Mathematical Sciences, Mangosuthu University of Technology, P. O. Box 12363, Jacobs4026, SouthAfrica 2 A.M.Msomietal. higherdimensionalcosmologicalmodels.DavidsonandGurwich[7]andMaartens and Koyama [8] highlighted the role of heat flow in gravitational dynamics and perturbationsin the frameworkof brane world cosmologicalmodels. A proper and completemodel of a radiatingrelativisticstarrequires thepres- ence of heat flux. The result given by Santos et al. [9] indicates that the interior spacetime must contain a nonzero heat flux so that the matching of the interior at the boundary to the exteriorVaidya spacetimeis possible. Models of heat flow in astrophysics have been used in gravitational collapse, black hole physics, for- mation of singularities and particle production at the stellar surface in four and higher dimensions. Herrera et al. [10], Maharaj and Govender [11] and Misthry et al. [12] showed that heat conducting relativistic radiating stars are also useful intheinvestigationofthecosmiccensorshiphypothesisandin describingcollapse withvanishingtidalforces.SolutionstotheEinsteinfieldequationsforashear-free spherically symmetric spacetime with a homothetic vector, together with radial heatflux,havebeenpresentedbyWaghetal.[13].Analyticalsolutionstothefield equationsforradiatingcollapsingspheresinthediffusionapproximationhavebeen found by Herrera et al. [14]. Recent examples of radiating stars, with generalised energymomentumtensors,aregivenbyHerreraetal.[15]andPinheiroandChan [16]. Shear-free fluids, in the presence of heat flux, are also important in modeling inhomogeneous cosmological processes. The need for radiating models in the formationofstructure,evolutionofvoids,thestudyofsingularities,andinvestiga- tions of the cosmic censorship hypothesis, has been pointed out by Krasinski [1]. Banerjee et al.[17] generateda model of a heat conducting sphere which radiates energy during collapsewithoutthe appearanceof a horizonat any stage.This re- sultholdsinfourdimensionsbutmaybeextendedtomodelsinhigherdimensions. BanerjeeandChatterjee[5]studiedheatconductingfluidsincosmologicalmodels in higher dimensions, and determined that gravitational collapse is also possible without the appearance of an event horizon. The presence of heat flow in brane world models sometimes allows for more general behaviour than is the case in standard general relativity.Govender and Dadhich [18] proved that the analogue of the Oppenheimer-Snyder model of a collapsingdust permitsa radiatingbrane. In this paper we analysethe masterequationfor higher dimensionalradiating fluids studied by Banerjee and Chatterjee [5] applicable to a (n+2)-dimensional sphericallysymmetricmetric.In our analysis,we used the Lietheory of extended groups as a systematic approach to generalise known solutions and generate new solutionsofthesameequation.Thehigherdimensionalradiatingmodelisderived inSect.2.InSect.3,wegiveabriefoutlineoftheLietheory.InSect.4,wediscuss the new solutions of the master equation that can be found via Lie symmetries by taking one potential to be a function of the remaining potential. Also in this section, we systematically study other group invariant solutions admitted by the fundamental equation by taking specific ratios of the potentials. In Sect. 5, we extendknownsolutionstonewsolutionsofthefundamentalequationutilisingLie theory.Regardlessofthecomplexityofthegeneratingfunctionchosenitispossible to find new exact solutions; we demonstrate this in two cases. We conclude this paperwithsomebriefobservationsaboutthenatureofthenew exactsolutionsin Sect. 6. ApplicationsofLieSymmetriestoHigherDimensionalGravitatingFluids 3 2 Radiating Model We consider the shear-free, spherically symmetric line element with an exterior (n+2)-dimensionalmanifoldgiven by 1 ds2 = A2dt2+ dr2+r2dX2 (1) − F2 n h i where A=A(t,r) and F =F(t,r) and Xn2 =dθ12+sin2θ1dθ22+···+sin2θ1sin2θ2...sin2θn−1dθn2 (2) The energy momentumtensor for a nonviscous heat conducting fluid is given by T =(ρ+p)v v +pg +q v +q v (3) ij i j ij i j j i where ρ is the energy density of the fluid, p the isotropic fluid pressure, v is the i (n+2)-velocityandq istheheatflowvector.Usingequations(1)and(3)wefind i that the nontrivialEinstein field equations in comovingcoordinates are n(n+1)F2 n(n+1)F2 n2FF ρ= t r +nFF + r (4a) 2A2F2 − 2 rr r nA FF nA F2 n(n 1)F2 n(n 1)FF p= r r + r + − r − r − A rA 2 − r nF n(n+3)F2 nA F + tt t t t (4b) A2F − 2A2F2 − A3F F2A n(n 1)F2 (n 1)F2A p= Arr −(n−1)FFrr+ −2 r + −rA r (n 1)2FF (n 2)FF A nF − r − r r + tt − r − A A2F n(n+3)F2 nA F t t t (4c) − 2A2F2 − A3F nFF nF F nFF A q = tr + t r + t r (4d) − A A A2 The isotropyof pressure is given by equations (4b) and (4c) together in the form FA +2A F (n 1)AF =0 (5) xx x x xx − − with x=r2. Equation (5) is the master equation for the system of higher dimensional Einstein field equations with n 2. In this paper, we reduce the order of ≥ (5) via Lie analysis in order to find general solutions in higher dimensions. 3 Lie analysis The symmetry analysis for a system of ordinary differential equations in two dependent variablesrequires the determinationof theone– parameter(ε) Liegroup of transformations x¯ =f(x,F,A,ε) F¯ =g(x,F,A,ε) (6) A¯=h(x,F,A,ε) 4 A.M.Msomietal. thatleavesthesolutionsetofthesysteminvariant.Itisdifficulttocalculatethese transformationsdirectly, and as such, we must resort to approximationsvia x¯ =x+εξ(x,F,A)+O(ε2) F¯ =F +εη(x,F,A)+O(ε2) (7) A¯=A+εζ(x,F,A)+O(ε2) The transformations(7) can be obtained once we find the (symmetry)operator ∂ ∂ ∂ Z =ξ +η +ζ (8) ∂x ∂F ∂A whichisasetofvectorfields.Oncethesesymmetriesaredetermined,itispossible to regain the finite (global) form of the transformation, given by (6), on solving Lie’s equations dx¯ =ξ(x¯,F¯,A¯) dε dF¯ =η(x¯,F¯,A¯) (9) dε dA¯ =ζ(x¯,F¯,A¯) dε subject to initialconditions x¯ =x, F¯ =F, A¯ =A (10) ε=0 |ε=0 ε=0 Thefulldetailsonthe(cid:12) symmetryapproachtosolving(cid:12)differentialequationscanbe (cid:12) (cid:12) found in a number of excellent texts (Bluman and Kumei [19],Olver [20]). Thedeterminationofthegeneratorsisastraightforwardprocessandhasbeen automated by computer algebra packages (Dimas and Tsoubelis [21], Cheviakov [22]). In practice, we have found the package PROGRAM LIE (Head [23]) to be the mostuseful.Itisquiteaccomplishedgivenitsage- itoftenyieldsresultswhen its modern counterparts fail. Utilising PROGRAM LIE, we show that (5) admits the following Lie point symmetries/vectorfields: ∂ Z = (11a) 1 ∂x ∂ Z =x (11b) 2 ∂x ∂ Z =A (11c) 3 ∂A F ∂ Z = (11d) 4 n 1∂F − ∂ xF ∂ Z =x2 + (11e) 5 ∂x n 1∂F − where n > 2. It is normal practice to use the symmetries (11a)-(11e) to reduce the order of the equation in the hope of finding solutions of the master equation. We need to proceed with some caution due to the overdetermined nature of (5). Thereafter we indicate how known solutions can be extended using these symmetries. ApplicationsofLieSymmetriestoHigherDimensionalGravitatingFluids 5 4 New Solutions via Lie symmetries One of the main purposes of calculating symmetries is to use them for symmetry reductionsandhopefullyobtaingroupinvariantsolutions.Thegoalofthissection istoapplythesymmetriescalculatedinSect.3toobtainsymmetryreductionsand exact solutions where possible. The application of symmetries (11a)-(11e) to the master equation results in either an implicit solution of (5) or we can reduce the governing equations to complicated Riccati equations that are difficult to solve. However there are two cases in which we can find new solutions regardless of the complexityof the function chosen. 4.1 The choice A=A(F) An obvious case to consider in this subsection, is when one dependent variable in (5) is a function of the other. Usually such an approach results in a more complicatedequationtosolve.In spiteof this,wecan makesignificantprogress if weusetheLiesymmetryZ (whichgivesthesameresultasZ ).Forourpurposes 1 2 we use the partialset of invariants of ∂ Z = (12) 1 ∂x given by p =F q(p)=F (13) x r(p)=A This transformationreduces equation (5) to ′ ′ ′′ ′ q (p) (n 1)r(p) pr (p) =q(p) pr (p)+2r (p) (14) − − (cid:2) (cid:3) (cid:2) (cid:3) which can be integratedto give q =q0eR (n2−r′1+)rp−r′r′′pdp (15) Substituting for the metric functions via (13), we can integrateone more time to give the solution − 2AF+FAFF dF e R (n−1)A−FAF dF =q0x+x0 (16) Z (cid:20) (cid:21) whereq andx arearbitraryfunctionsoftime.Equation(16)suggeststhat,given 0 0 anyfunctionAdependingonarbitraryF,wecanworkoutF explicitlyfrom(16). Such an explicit relationship between F and A has not been found previously. Notethatsince(5)islinear,onceweobtainF via(16)wecanuseittoobtainthe general solution of (5) using standard techniques for solving linear equations. Weillustratethismethodwithsimpleexamples.UsingA=1,weevaluate(16) to obtain F =q (t)x+x (t) (17) 0 0 6 A.M.Msomietal. We can easily generatethe general solution to (5) as C A = q(x−+1qx) +C2 (18) 0 1+n 2 F = qC−(1+n)(x0+qx)1−n 1 1+n 1+n 1+n 1+nq(x0+qx)−1+nC1+(−C1+q(x0+qx)C2)−1+n C2 (19) (cid:20)− (cid:21) If we take A=F2, then equation (16) is reduced to 6 F(cid:16)3−n(cid:17)dF =q0x+x0 (20) Z and hence 3 n 9 n 9−n F = 3−n(q¯0x+x¯0) − (cid:20) − (cid:21) 6 2n 9 n 9−n A = 3−n(q¯0x+x¯0) − (21) (cid:20) − (cid:21) The functionalform of F can be easily extended to obtain F =C 93−−nn 39−−nn (9−10n+n2)(q0x+x0)39−−nn+−9−9+102nn+−nn22 1(cid:16) (cid:17) q ( 9+2n n2) 0 − − 3 n 9 n 9−n +C2 3−n(q0x+x0) − (22) (cid:20) − (cid:21) where C and C are arbitraryfunctions of time, which is the general solution to 1 2 (5) when 6 2n 9 n 9−n A= 3−n(q¯0x+x¯0) − (23) (cid:20) − (cid:21) 4.2 The choice W = F A1/(n−1) The combinationof symmetriesgiven by ∂ F ∂ Z +Z =A + (24) 3 4 ∂A n 1∂F − gives rise to the invariant F W = (25) A1/(n−1) Then equation (5) is transformedby (25) to the form nWA2 +(n 1)A2W =0 (26) − x − xx with solution (n 1)2W A=C1(t)exp − xxdx (27) Z ±p √nW ! ApplicationsofLieSymmetriestoHigherDimensionalGravitatingFluids 7 whichcomesasaresultoftreatingequation(26)asanonlinearfirstorderordinary differential equation in A. Given any function W we can integratethe right hand side of (27) and find a form for A. If we take W =a(t)x+b(t), then (27) gives A=C¯ (t) (28) 1 and F =C¯ (t)1/(n−1)(a(t)x+b(t)) (29) 1 which are new solutions of (5) for n>2. Alternatively,we could substitutethe inverseof (25),i.e. A1/(n−1) W = (30) F into (5) and obtain c nA 2W2 2(n 1)2A2W 2+(n 1)2A2WW =0 (31) x x xx − − − with solution c c cc 2(n 1)2W 2 (n 1)2WW x xx A=C (t)exp − − − dx (32) 2 ± q √nW  Z c cc   Again,givenanyfunctionW wecanintegratethecrighthandsideof(32)andfind a form for A. If we take W =a(t)x+b(t) as before we find that √2(n 1) A1 =C2(t)[a(t)x+b(t)] √n− (33) and 1 C2(t)(a(t)x+b(t))√2√(nn−1) n−1 F = (cid:20) (cid:21) (34) 1 a(t)x+b(t) which is essentially a new solution of the master equation in higher dimensional space. We can also have C¯ (t) A = 2 (35) 2 √2(n 1) (a(t)x+b(t)) √n− and 1/n−1 C¯2(t)(a(t)x+b(t))−√2√(nn−1) F = (cid:20) (cid:21) (36) 2 a(t)x+b(t) thus obtaining two different solutionsfrom the same seed function. Observe that (27) and (32) will contain all solutions of the master equation (5) for appropriately chosen seed functions W or W which are ratios of the metric functions. We are always able to reduce (5) to the quadratures (27) or (32) regardless of the complexityof the seed functions.c 8 A.M.Msomietal. 5 Extending known solutions Another use of Lie point symmetries is the extension of known solutions of differential equations. This is possible due to the fact that the symmetries generate transformations that leave the equations invariant. As a result, applying those transformationsto known solutions will (usually)result in new solutions. WeillustratetheapproachbyusingthesimpleinfinitesmalgeneratorZ ,where 1 we observe that ξ =1, η =0, ζ =0 (37) We solvethe Lie equations (9), subject to initialconditions (10), to obtain x¯=x+a 1 F¯ =F (38) A¯=A This means that using (38) we can map the equation (5) to the form F¯A¯ +2A¯ F¯ (n 1)A¯F¯ =0 (39) x¯x¯ x¯ x¯ x¯x¯ − − As a result of this mapping, any existing solution to equation (5) can be trans- formedtoasolutionof(39)by(38).Usually,a isanarbitraryconstant.However, 1 since F and A depend on x and t we take a to be an arbitraryfunction of time, 1 a =a (t). 1 1 If we now take each of the remaining symmetries successively, we obtain the general transformation x¯ = ea2(a1+x) 1 a5ea2(a1+x) − ea4F F¯ = (40) 1 a5ea2(a1+x) − A¯=ea3A where the a are all arbitraryfunctions of timeand n>2. i Thusanyknownsolutionofequation(5)canbetransformedtoanewsolution of equation (5) via (40).For example, if we start with the solution A=1, F =α(t)x+β(t) (41) the transformation(40) yields the new solution x¯ = ea2(a1+x) 1 a5ea2(a1+x) − ea4(α(t)x+β(t)) F¯ = (42) 1 a5ea2(a1+x) − A¯=ea3 All the new results that we derived in the previous section can be similarly extended via (40). ApplicationsofLieSymmetriestoHigherDimensionalGravitatingFluids 9 6 Conclusion In this work, we have provided symmetry reductions and exact solutions of the Einstein field equations governing shear-free heat conducting fluids in higher dimensions. Explicit relationships were provided between the gravitational potentials,obviatinga need tostartwith“simple”formsfor onetocalculatetheother. We were also able to provide a general transformation to extend our (and any other) known solution into new solutions. When n = 2 we regain the results of Deng [24] who developed a method to generate solutions when simple forms of A or F arechosen.Thecasen=2 alsocontainstheresultsofMsomi etal.[25]who adoptedamoregeometricand systematicapproachusingLietheorytogeneralise knownsolutionsandgeneratenew solutions.Thenewsolutionsofthis papermay be used to study the physics of radiating astrophysical and cosmological models in higher dimensions. 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