Applications of Lie Symmetries to Higher Dimensional Gravitating Fluid

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by  A. M. Msomi

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 gravitationalcol-
lapse,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 model-
ing 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 dimen-
sional 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 de-
pendent 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 sym-
metries/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 equa-
tion. 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 met-
ric 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 dif-
ferential 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 ex-
tended 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 di-
mensions. Explicit relationships were provided between the gravitational poten-
tials,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.
It is interesting to observe an important feature of the solutions admitted by
equation (5). When F =0 we find that (5) becomes
xx
FA +2A F =0
xx x x
This equation has the remarkablefeature that it is independent of the dimension
n. Thus any solution with F = 0 presented by Deng [24] and Msomi et al. [25]
xx
in four dimensions willalso be applicablein higher dimensions.This direct appli-
cation of four dimensional solutions into higher dimensional spacetimes is rather
unusual in general relativity.
Acknowledgements AMMandKSGthanktheNationalResearchFoundationandtheUni-
versity of KwaZulu–Natal for ongoing support. AMM also thanks Mangosuthu University of
Technology forsupport. SDM acknowledges that this workis basedupon research supported
bytheSouthAfricanResearchChairInitiativeoftheDepartment ofScienceandTechnology
andtheNationalResearchFoundation.
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