All Conformally Flat Einstein–Gauss–Bonnet static Metrics

Sudan Hansraj,1, ∗ Megandhren Govender,1, † Ayan Banerjee,1, ‡ and Njabulo Mkhize1, §

1Astrophysics and Cosmology Research Unit, School of Mathematics,Statistics and Computer Science, University of KwaZulu–Natal,

Private Bag X54001, Durban 4000, South Africa(Dated: February 12, 2021)

It is known that the standard Schwarzschild interior metric is conformally flat and generates aconstant density sphere in any spacetime dimension in Einstein and Einstein–Gauss–Bonnet gravity.This motivates the questions: In EGB does the conformal flatness criterion yield the Schwarzschildmetric? Does the assumption of constant density generate the Schwarzschild interior spacetime?The answer to both questions turn out in the negative in general. In the case of the constantdensity sphere, a generalised Schwarzschild metric emerges. When we invoke the conformal flatnesscondition the Schwarschild interior solution is obtained as one solution and another metric whichdoes not yield a constant density hypersphere in EGB theory is found. For the latter solution oneof the gravitational metrics is obtained explicitly while the other is determined up to quadraturesin 5 and 6 dimensions. The physical properties of these new solutions are studied with the use ofnumerical methods and a parameter space is located for which both models display pleasing physicalbehaviour.

I. INTRODUCTION

In the standard theory of gravity due to Einstein all conformally flat spacetimes are known. They are either theSchwarzcshild interior metric in the case of no expansion or the Stephani universes when expansion is permitted[1–3]. A similar result is not known in the more complicated Einstein-Gauss–Bonnet (EGB) theory so this is oneof the motivations behind this work. Conformal structures are important in gravitational field theory as conformalsymmetries generate constants of the motion or conserved quantities along null geodesics. Conformal flatness ischaracterised by the vanishing of the Weyl tensor and physically this means that a spacetime is conformal to theMinkowski metric at spatial infinity. This is a reasonable restriction when it comes to modelling relativistic compactobjects such as neutron stars, white dwarfs or cold planets. It is already known both in the four dimensional generaltheory of relativity as well as its extensions to higher dimensions, that that the interior Schwarzschild metric is botha necessary and sufficient condition for conformal flatness [4]. Whether this holds when higher curvature effects areat play will be discussed herein.

Recently it has been demonstrated by Dadhich et al [5] that the Schwarzschild interior metric is universal as aconstant density solution in Lovelock gravity of which the EGB theory is a special case. These authors claimednecessity and sufficiency. The assumption of the Schwarzschild interior spacetime does indeed generate a constantdensity fluid. However, the converse is not true in general. The assumption of constant density generates a solutionwhich generalises the Schwarzschild interior spacetime. That is, the prescription of constant density is only a necessarybut not sufficient condition for the Schwarzschild interior metric. The aforesaid authors argued that an integrationconstant must vanish in order to ensure regularity at the stellar centre and consequently the solution reduces tothe Schwarzschild metric. We shall show that this is not required in the five dimensional constant density casewhich possesses no central singularity however the six dimensional incompressible hypersphere metric does have apersistent singularity at the stellar centre. Nevertheless, in neither case is there a basis to remove integration constantsarbitrarily. These constants are necessary and may be determined through the boundary conditions. These aspectsare being mentioned in the context of conformal flatness because the Schwarzschild solution is conformally flat but itneeds to be checked whether all conformally flat static metrics are necessarily the Schwarzschild interior solution. Itmust be remarked that ordinarily it is not advisable to set integration constants to zero arbitrarily since these mustbe settled by matching of the interior and exterior spacetime across a common hypersurface. Effectively deciding thevalue of constants of integration rearly in modelling runs the risk of over-determining the model. This option may

∗hansrajs@ukzn.ac.za†megandhreng@dut.ac.za‡ayanbanerjeemath@gmail.com§mkhizen18@gmail.com

arX

iv:2

102.

0604

1v1

[gr

-qc]

10

Feb

2021

2

only succeed when there is no boundary and a universe model is in evidence. This is the case for the isothermalspherically symmetric universe of [6] which was clearly over-determined but successful. In this same spirit, Hansrajand Moodly [7] showed that the dual requirement of conformal flatness and spacetime being of embedding class one,gives a null-result.

At the outset, we sketch motivations for considering higher dimensional spacetimes and the importance of theEGB theory in gravitational field theory. Interest in higher dimensional spacetimes originated with the seminal worksof Kaluza [8] and Klein [9]. A five dimensional manifold was introduced in the context of the Einstein–Maxwellequations and now there were 15 nontrivial components as opposed to 10 in the standard four dimensions. Four ofthe components of the metric tensor were associated with the electromagnetic field while ten were connected to theusual 4 dimensional space. The remaining component was given the interpretation of a scalar field termed a scalaronor dilatonic field. Subsequently many other ideas emerged that considered higher dimensions. In fact, in the quantumworld, it is now understood that the most promising idea superstring theory and its generalisation M -theory bothrequire dimensions of the order of 10 or 11 at least. Additionally, brane-world cosmology [10] requires spacetimesof dimension five. But what would be the explanation for our inability to access extra spatial dimensions? Theseare believed to be topologically curled into microscopic circles of very small size yet they exert an influence on thegravitational field [8, 9]. Probing gravitational waves for information on extra dimensions was considered in a recentpaper [11] that reported the non-detection of large extra dimensions. Earlier theories predicting large extra dimensionswere tested through the Large Hadron Collider (LHC) experiment but found to be inconclusive [12–15]. This however,does not eliminate the possibility of small extra dimensions [16] presumed to be of the planck length. It must beremarked that the extra dimensions are actually angular dimensions so they do not necessarily manifest overtly. Inlight of these, investigations into higher dimensional spaces are justifiable.

Why is the EGB gravity proposal of immense importance? Firstly it offers a natural generalization of Einstein’stheory to higher dimensions with the inclusion of higher curvature terms but without violating well establishedingredients of gravitational field theory namely the Bianchi identities, diffeomorphism invariance and second order(ghost–free) equations of motion. The energy conservation conditions also hold in the usual way adjusted for extradimensions. In fact EGB is superseded by its generalisation Lovelock [17, 18] theory which has the same aforementionedproperties. The Nth order Lovelock Lagrangian is constructed from invariants polynomial in the Riemann tensor,Ricci tensor and the Ricci scalar and to second order (N = 2) the EGB special case consists of quadratic invariants.The critical spacetime dimensions d in Lovelock theory are d = 2N + 1 and d = 2N + 2. In particular it is sufficientto consider only dimensions 5 and 6 in studying EGB spacetimes [19]. It is also well known that the Gauss-Bonnetterm features in heterotic string theory and the coupling constant carries the meaning of the string tension in thatarea. This is also a good reason to probe the behaviour of higher curvature invariants in gravitational field theoryin view of the long time project to merge quantum field theory, of which string theory is a leading candidate, andgravitational physics.

The exterior solution for EGB gravity was found by Boulware and Deser [20] in 1985 and a year later Wiltshirepromoted the solution to include the effects of the electrostatic field [21]. The constant density solution of Dadhichet al [5] was the first interior solution reported and happened to be the Schwarzschild solution independent of thedimension. As in Einstein theory the solution suffers physical pathologies such as an infinite speed of sound. Thesquare of the speed of sound is calculated with the formula dp

dρ where p and ρ are the pressure and density respectively.

A thorough discussion of the relativistic sound speed was considered by Ellis et al [22]. Clearly a constant density ρrenders the sound speed meaningless. Kang et al [23] proposed a solution however, the metric demanded a furtherintegration to fully reveal the nature of the spacetime manifold. The first reported explicit exact solutions thatsatisfied elementary conditions for physical plausibility were generated in [24–26]. It is seriously difficult to locateexact solutions for perfect fluid matter in EGB because the extra curvature terms make the governing differentialequations intractable. An additional solution for constant potentials in six dimensional EGB spacetimes was found in[27] and recently Hansraj and Mkhize generated a physically viable six dimensional model with variable potentials anddensity [28]. A greater number of the extra curvature terms survive in 6D as opposed to 5D making the differentialequations even more difficult to work with. However, the effects of extra curvature may be studied more efficientlycompared to the 5D case where such terms are suppressed.

Employing an equation of state (EoS) to describe static compact objects in classical general relativity and modifiedgravity theories have proved not to be fruitful. The imposition of a barotropic equation of state of the form p = p(ρ)reduces the problem of finding exact solutions to the Einstein field or the EGB equations to a single-generatingfunction when pressure isotropy is dropped. Requiring the vanishing of the Weyl stresses at each interior point ofthe fluid sphere completes the gravitational description of the model. These models have been shown to describecompact objects such as pulsars, neutron stars and colour-flavoured locked-in quark stars [29–33]. In fact, somemodels predicting the existence of self-gravitating anisotropic fluid spheres have been investigated in the literature.For example, see [34–38] and the references therein.

The role of the Weyl tensor in dissipative collapse has been thoroughly investigated. Herrera and co-workers

3

demonstrated that departure from hydrostatic equilibrium (or quasi-static equilibrium) is sensitive to changes in theconditions surrounding the Weyl tensor. In particular, in the case of conformally flat self-gravitating bodies, departurefrom equilibrium depends on the deviation from conformal flatness. It is well known that density inhomogeneities andpressure anisotropy influence the outcome of dissipative collapse. It has been shown that the thermal behaviour ofradiating stars can be drastically altered in the presence of anisotropic stresses within the collapsing core. Furthermore,specific combinations of the Weyl tensor components, pressure anisotropy and dissipative fluxes (for example radialheat flux) give rise to density inhomogeneities within the collapsing fluid. The stability of the shear-free conditionhas come under scrutiny as it was shown that an initially shear-free configuration would evolve into a shearing-likeregime due to a combination of density inhomogeneity, pressure anisotropy and dissipation[39]. Several exact modelsdescribing conformally flat radiating stars have been obtained. These solutions have been shown to obey the physicalrequirements necessary for dissipative collapse. Gravitational collapse from an initial static configuration describedby the interior Schwarzschild solution has rewarded researchers richly in terms of insights into horizon formation,stability in the Newtonian and Post-Newtonian regimes and thermal evolution within the framework of extendedirreversible thermodynamics[40–42].

This paper is arranged as follows: After a brief sketch of the main ingredients of EGB theory in section II, we derivethe five dimensional field EGB field equations and consider the exterior metric in section III. The incompressiblehypersphere is examined in section IV and the new exact solution for conformal flatness together with its physicalproperties is presented in section V. In section VI we find he conformally flat metrics in six dimensions and study itsmain features. Finally in section VII we conclude by reiterating our main results.

II. EINSTEIN–GAUSS–BONNET GRAVITY

The Gauss–Bonnet action is written as

S =

∫ √−g[

1

2(R− 2Λ + αLGB)

]dNx+ S matter, (1)

where α is the Gauss–Bonnet coupling constant and N is the spacetime dimension. The advantage of the action LGB isthat despite the Lagrangian being quadratic in the Ricci tensor, Ricci scalar and the Riemann tensor, the gravitationalfield equations are second order quasilinear which is expected of a viable theory of gravity. The Gauss–Bonnet termexerts no influence for n ≤ 4 but becomes dynamic for n > 4.

The EGB field equations may be written as

Gab + αHab = Tab, (2)

with metric signature (−+ + + +) where Gab is the Einstein tensor. The Lanczos tensor is given by

Hab = 2(RRab − 2RacR

cb − 2RcdRacbd +RcdeaRbcde

)− 1

2gabLGB , (3)

where the Lovelock term has the form

LGB = R2 − 4RabRab +RabcdR

abcd. (4)

In what follows it will be desirable to divide the analysis into the 5 and 6 dimensional cases separately as theseembody different physical characteristics. The 5 dimensional case has the feature of eliminating some terms in thefield equations which are dynamic only in 6-d and consequently affect the physics of the distribution.

III. FIELD EQUATIONS IN 5-d

The generic 5–dimensional line element for static spherically symmetric spacetimes is customarily written as

ds2 = −e2νdt2 + e2λdr2 + r2(dθ2 + sin2 θdφ2 + sin2 θ sin2 φdψ2

), (5)

where ν(r) and λ(r) are the gravitational potentials. We utilise a comoving fluid velocity of the form ua = e−νδa0 andthe matter field is that of a perfect fluid with energy momentum tensor Tab = (ρ + p)uaub + pgab. Accordingly the

4

EGB field equations (2) assume the form

ρ = − 3

e4λr3(4αλ′ + re2λ − re4λ − r2e2λλ′ − 4αe2λλ′

), (6)

pr =3

e4λr3(−re4λ +

(r2ν′ + r + 4αν′

)e2λ − 3αν′

), (7)

pT =1

e4λr2

(−e4λ − 4αν′′ + 12αν′λ′ − 4α (ν′)

2)

+1

e2λr2

(1− r2ν′λ′ + 2rν′ − 2rλ′ + r2 (ν′)

2)

+1

e2λr2

(r2ν′′ − 4αν′λ′ + 4α (ν′)

2+ 4αν′′

). (8)

where pr and pT are the radial and tangential pressure respectively. Note that the system (6)–(8) comprises threefield equations in four unknowns for isotropic particle pressure pr = pT . This is similar to the standard Einstein casefor spherically symmetric perfect fluids. In order to close the system of equations, it is necessary to stipulate one morecondition. Traditionally, one of the metric potentials is prescribed in the hope of finding the remaining potential byintegrating the equation of pressure isotropy. Alternatively mathematical insights such as shown by Tolman [43] inrearranging the pressure isotropy condition in a convenient way and setting individual terms to vanish yielded eightclasses of exact solutions five of which were new. The attempts at implementing physically reasonable assumptionssuch as an equation of state has not worked in Einstein gravity and the works of Nilsson and Uggla [44, 45] in thisdirection had to be completed with numerical methods. Therefore it is not expected that an a priori equation ofstate p = p(ρ) will succeed in the more complicated situation of EGB at hand. Additionally, conditions such as beingable to embed a spacetime in a higher dimensional geometry such as through use of the Kamarkar condition on theRiemann tensor components may also be an interesting route to follow [38, 46–48]. Historically, Schlaefli [49] firstraised the question of embedding a Riemannian space into a higher dimensional Euclidean space and Kamarkar [50]established a condition on the components of the Riemann tensor. The Kamarkar condition was corrected in time torule out the vanishing of one of the components [51]. The existence of symmetries such as conformal Killing vectorsalso offer an avenue to pursue. In our case we have elected to impose the condition for conformal flatness. Thevacuum metric describing the gravitational field exterior to the 5–dimensional static perfect fluid may be describedby the Boulware–Deser [20] spacetime as

ds2 = −F (r)dt2 +dr2

F (r)+ r2

(dθ2 + sin2 θdφ+ sin2 θ sin2 φdψ

), (9)

where

F (r) = 1 +r2

4α

(1−

√1 +

8Mα

r4

), (10)

and where we have taken a negative sign before the square root since expansion in powers of α give the Schwarzschildsolution when α approaches zero. In the above M is associated with the gravitational mass of the hypersphere. Theexterior solution is unique up to branch cuts.

We invoke the transformation e2ν = y2(x), e−2λ = Z(x) and x = Cr2 (C being an arbitrary constant) which wasutilised successfully by Durgapal and Bannerji [52], Finch and Skea [53] and Hansraj and Maharaj [54] to generatenew exact solutions for neutral and charged isotropic spheres. The motivation for using this transformation lies inthe fact that the equation of pressure isotropy may be expressed as a second order linear differential equations thusincreasing our chances of locating exact solutions. The field equations (6)–(8) may be recast as

3Z +3(Z − 1)(1− 4αCZ)

x=

ρ

C, (11)

3(Z − 1)

x+

6Zy

y− 24αC(Z − 1)Zy

xy=

p

C, (12)

2xZ (4αC[Z − 1]− x) y −(x2Z + 4αC

[xZ − 2Z + 2Z2 − 3xZZ

])y

−(

1 + xZ − Z)y = 0, (13)

5

where the last equation is the equation of pressure isotropy. Equation (13) has been arranged as a second orderdifferential equation in y, which for some analyses in the 4–dimensional Einstein models, proves to be a useful form.Functional forms for Z(x) may be selected a priori so as to allow for the complete integration of the field equations.For the present work it should be noted that (13) may also be regarded as a first order ordinary differential equationin Z, and may be expressed in the form(

x2y + xy + 4αCxy − 12αCxyZ)Z + 8αC (y − xy)Z2 +

(2x2y + 8αCxy − 8αCy − y

)Z + y = 0. (14)

This is an Abel equation of the second kind for which few solutions are known. However note that choosing forms fory should in theory result in the expressions for Z by integration. Therefore we seek choices for the metric potential ywhich will allow for a complete resolution of the geometrical and dynamical variables.

IV. INCOMPRESSIBLE 5-d HYPERSPHERE

Since it is true in standard Einstein gravity that conformal flatness, the interior Schwarzschild metric and constantenergy density are all equivalent [4], it is important to examine the incompressible sphere in the present context. Theform Z = 1 +x which is the Schwarzschild potential generates the constant density distribution in higher dimensionalEinstein gravity and also in higher curvature EGB gravity. That is the Schwarzschild solution is a sufficient conditionfor constant density. This corroborates the result of Dadhich et al [5] independent of spacetime dimension. Howeverthe question of necessaity arises. Does the assumption of constant density give the Schwarzschild metric? We shallanalyse this problem below. Moreover, it is known that the Schwarzschild interior metric is conformally flat. Thisis a purely geometric consequence irrespective of the gravity theory under consideration. In the Einstein theory it isalready well known that invoking the conformal flatness condition generates the Schwarzschild metric. It is thereforenatural to ask whether the prescription of conformal flatness to close the system of equations also generates theinterior Schwarzschild potential in EGB theory? Additionally does the general conformally flat EGB solution producea constant density fluid. We examine these questions separately for 5 and 6 dimensions.

We note that equation (11) can be integrated when the density is assumed to be constant ρ = ρ0 and gives

Z(x) = 1 +x

4αC

(1±

√(3− 4αρ0) +

c1α2C2

x2

), (15)

where c1 is an integration constant. Setting the constant density to vanish ρ0 = 0 regains the vacuum solution (9) ofBoulware and Deser [20]. In the event that c1 = 0 the solution of Dadhich et al is regained and it is then demandedthat the coupling constant satisfies α ≤ 3

4ρ0. In this case the form Z = 1 + x arises but this is not in general true for

EGB theory in general. Substituting (15) in the conformal flatness condition, to be displayed later, regretfully leadsto an intractable differential which cannot be solved for now. So the y potential remains unknown in this generalcase. Note that Dadhich et al claim the integration constant should vanish for regularity at the centre however thisis not a valid issue. If (15) is written as

Z = 1 +x

4αC± 1

4

√c1 +

x2(3− 4αρ0)

α2C2(16)

then it is clear that there is no singularity at the stellar centre specifically for the five dimensional case. In fact

Z(0) = 1 +√c14 . Generally it is not acceptable to set integration constants to vanish as their values must be settled

when matching across a common boundary interface of the interior and exterior spacetime. Note that switching offthe higher curvature effects by setting α = 0 is not possible. What is true from (15) is that expansion in powers ofα will give the interior Schwarzschild metric of standard Einstein theory as α → 0 and for arbitrary c1. It shall beseen later that the absence of a singularity in the metric at the centre of an incompressible sphere in EGB theory isonly peculiar to the 5-d case and not the 6-d constant density sphere. These results challenge the conclusion that theSchwarzschild metric is universal in EGB theory. Clearly the potential (15) generalises the Schwarzschild potential.

In what follows we shall analyse the impact of imposing conformal flatness on the field equations without anyfurther restrictions. It will be interesting to see whether the interior Schwarzschild metric is preserved and whethera constant density is an unavoidable consequence of conformal flatness in EGB theory.

6

V. CONFORMAL FLATNESS

Conformally flat spacetimes are distinguished by the vanishing of the Weyl tensor which in turn determines theconstraint equation

r2(ν′′ + 2ν′2 − 2ν′λ′)− r(ν′ − λ′) + (1− e2λ) = 0 (17)

for the line element (36). This may be converted to the form

4x2Zy + 2x2Zy − (Zx− Z + 1)y = 0, (18)

in our coordinates and must be satisfied for spherically symmetric spacetimes of any dimension. Equation (18) admitsthe separation of variables

y = A√x cosh

(1

2

∫dx

x√Z

+B

), (19)

where A and B are integration constants. Plugging the result (19) into the pressure isotropy condition (13) gives theneat factorisation (

xZ − Z + 1)(−2β√Z sinh f(x) + (β + 3x) cosh f(x)− 3βZ cosh f(x)

)= 0, (20)

where f(x) =

(B + 1

2

∫dx

x√Z(x)

). The vanishing of the first bracket gives

Zx− Z + 1 = 0, (21)

which is solved by Z = 1 + cx, c some constant, in general. It can easily be checked that this is the well knownSchwarzschild interior solution which yields a constant density or incompressible perfect fluid hypersphere. In standardEinstein gravity it is known that the Schwarzschild interior metric is conformally flat and conversely demanding thata spacetime be conformally flat yields the Schwarzschild solution. Note that on setting β = 0 in (20) it can easilybe checked that Schwarzschild remains the unique solution for 5D Einstein gravity and indeed for any dimensionthereafter. However the Schwarzschild solution is not the unique conformally flat solution in the context of Einstein-Gauss-Bonnet gravity. The second pair of brackets in (20) gives the relationship

tanh f(x) =3x+ β(1− 3Z)

2β√Z

(22)

on rearranging. To solve equation (22) for Z in its present form would be difficult so we endeavor to cast it as adifferential equation and profit from results on exact solutions of differential equations. Differentiating (22) permitsus to recast the equation as

β(β + 3βZ + 3x)Z +√Z(−(β + 3x)2 − 9β2Z2 − 6β

√Z + 2β(5β + 9x)Z

)= 0, (23)

where we used the result d(tanh f)dx = 1− tanh2 f . The general solution of equation (23) is given by

Z =(9x+ 5β)(e2x − k)2 + 8kβe2x ± 2(e2x + k)

√β√

(9x+ 4β)(e2x − k)2 + 4kβe2x

9β(e2x − k)2, (24)

where k is a constant of integration. It is easily checked that the metric potential remains finite at the stellar centrex = 0 however this is not the case for the dynamical quantities as shall be seen below. In order to establish thefunction y we use the form (19) however, in light of the complexity of the spatial potential Z, an explicit form of yfree of integrals could not be realised. Nevertheless, the problem is exacerbated by the presence of two square rootsin the integrand. It may still be possible to study the physical characteristics of the model generated by the metricwith the help of numerical methods.

Now it remains to find the physical dynamics that should be analyzed for their contributions to the 5 dimensionalmodel. To pursue this we plug (19) and (24) into the system (11)–(12) then find succinct forms for the energy density

7

and isotropic pressure as

ρ

C=

√β(e4x − k2

)K8 +

(e2x − k

) (9k2 − 2ke2x(2β + 18x+ 9) + e4x(20β + 36x+ 9)

)− 8√βe2xJ1

3β (e2x − k)3

+

√β(e2x + k

)K6J2 − 36e2x

(e2x − k

)2 ((k2 + e4x

)(5β + 9x) + 2ke2x(β + 9x)

)J1

27√βx (e2x − k)

5J1

, (25)

p

C=

K4 − 6β(e2x − k

) (K1 − 2

√β(k2 − 4ke2x + e4x

))tanh

(B + 3

2K3

)√K2 + 2

√βK1

27βx2 (e2x − k)4 , (26)

where for ease of reference we put

J1 =

√(k + e2x)

2(k2(4β + 9x)− 2ke2x(2β + 9x) + e4x(4β + 9x)),

J2 = 9k4 − 16k2e4x(β + 9x) + 2ke2x(k2(20β + 36x− 9) + 8

√βJ1

)+ 2ke6x(20β + 36x+ 9)− 9e8x,

K1 =√k4(4β + 9x) + 4βk3e2x − 18k2e4xx+ 4βke6x + e8x(4β + 9x),

K2 = e4x(5β + 9x)− 2ke2x(β + 9x) + k2(9x+ 5β),K4 = k4(9x− 2β)2 − 2β3/2k2K1 + 2e4x

(9k2

(4β2 + 27x2 + 4βx

)− β3/2K1

)− 4ke2x

(k2(26β2 + 81x2

)+ 7β3/2K1

)−

4ke6x(26β2 + 81x2

)+ e8x(9x− 2β)2,

K5K1 = 9k4 + 8βk3e2x − 18k2e4x − 72k2e4xx+ 24βke6x + 8e8x(4β + 9x) + 9e8x,K6 = 2

√βJ1 − 4βk2 + 9k2x− 2ke2x(9x− 8β) + e4x(9x− 4β),

K8J1 = 9k3 + (8β − 9)k2e2x − ke4x(8β + 72x+ 9) + e6x(32β + 72x+ 9),

K3 =∫ (

x√

K2+2√βJ1

β(e2x−k)2

)−1dx.

8

Invoking (25) and (26) renders the ratio

S =dp

dρ=

[8e4x

(9k2

(27x2 + 4βx+ 4β2

)−K1β

3/2)− 24e6xk

(81x2 + 26β2

)− 8e2xk

(7K1β

3/2 + k2(81x2 + 26β2

))27 (e2x − k)

4x2β

+2(9k2(54x+ 4β)− k2K5β

3/2 − 12β

3/2K5

)e4x + 8e8x(9x− 2β)2 − 4e2xk

(72K5β

3/2 + 162k2x)− 648e6xkx

27 (e2x − k)4x2β

+8e2xK9 tanh

(B + 3

2K3

)−√

2√βK1 +K2

(K5 − 4

(4e4x − 8e2xk

)√β)

tanh(B + 3

2K3

)9 (e2x − k)

3x2

− 8e2xK4

27 (e2x − k)5x2β

−K9β

3/2sech2(B + 3

2K3

)3 (e2x − k)x3

√2√βJ1 +K2

− 2K4 − 18e8x(9x− 2β)− 18k4(9x− 2β)

27 (e2x − k)4x3β

+4K9 tanh

(B + 3

2K3

)9 (e2x − k)

2x3

−K9β

3/2K10 tanh(B + 3

2K3

)9x2

(2√βK1 +K2

) ]÷[

2J32

(k + e2x

) (K11 +K7

√β)− J2

(k + e2x

)24e2xK6

((9x+ 4β)k2 − 2e2x(9x+ 2β)k + e4x(9x+ 4β)

)54 (e2x − k)

5x√βJ3

1

+2e2x

(9k2 − 2e2x(18x+ 2β + 9)k + e4x(36x+ 20β + 9)

)− 8e2x

((k2 + e4x

)(9x+ 5β)− 2e2xk(9x+ β)

)3 (e2x − k)

3β

+2e2x(9k2 − 2e2x(18x+ 2β + 9)k + e4x(36x+ 20β + 9)

)+ 2e2x

(e2x − k

)K8

√β + 2e2x

(k + e2x

)K8

√β

3 (e2x − k)3β

+(e2x − k

) (36e4x − 36e2xk − 4e2x(18x+ 2β + 9)k + 4e4x(36x+ 20β + 9)

)3 (e2x − k)

3β

−J2(k + e2x

)3K6K11

54 (e2x − k)5x√βJ3

1

+

(k + e2x

)K6

(72e6xk − 144e4xk2 − 64e4x(9x+ β)k2 + 2e2x

(36k2 + 4K7

√β)k)− J2

(k + e2x

)K6

27J1 (e2x − k)5x√β

+

(k + e2x

)K6

(12e6x(36x+ 20β + 9)k + 4e2x

((36x+ 20β − 9)k2 + 8J1

√β)k − 72e8x

)+ 2e2xJ2K6

27J1 (e2x − k)5x√β

+

(e4x − k2

) (2e2x(8β − 9)k2 − 72e4xk − 4e4x(72x+ 8β + 9)k + 72e6x + 6e6x(72x+ 32β + 9)

)3 (e2x − k)

3√βJ1

+2e2x

(8e2x√βJ1 + 4e2x

((k2 + e4x

)(9x+ 5β) + 2e2xk(9x+ β)

))(e2x − k)

4β

−10e2xJ2

(k + e2x

)K6

27J1 (e2x − k)6x√β

+4e4x(9x+ 5β)− 16e2x

√βJ1 − 4e2x

√βK7 − 4e2x

(9(k2 + e4x

)− 18e2xk − 4e2x(9x+ β)k

)3 (e2x − k)

3β

−2e2x

(e2x − k

) (9k2 − 2e2x(18x+ 2β + 9)k + e4x(36x+ 20β + 9)

)+(e4x − k2

)K8

√β

(e2x − k)4β

−(e4x − k2

) (9k3 + e2x(8β − 9)k2 − e4x(72x+ 8β + 9)k + e6x(72x+ 32β + 9)

)K7

6 (e2x − k)3√

βJ21

], (27)

9

which represents the square of the speed of sound and where we make the additional substitutions

K7J1 =√β((k + e2x

)2 (9k2 − 4ke2x(2β + 9x)− 18ke2x + 4e4x(4β + 9x) + 9e4x

)+4e2x

(k + e2x

) (k2(4β + 9x)− 2ke2x(2β + 9x) + e4x(4β + 9x)

)),

K8J1 = 9k3 + (8β − 9)k2e2x − ke4x(8β + 72x+ 9) + e6x(32β + 72x+ 9),

K9 =(K1 − 2

(k2 − 4e2xk + e4x

)√β)√2

√βK1 +K2

(e2x − k)2β,

K10 =9k2 − 18e2xk − 4e2x(9x+ β)k + 9e4x + 4e4x(9x+ 5β) +K5

√β

(e2x − k)2β

−4e2x

(2√βK1 +K2

)(e2x − k)

3β

,

K11 = 9k2 − 18e2xk − 4e2x(9x+ 2β)k + 9e4x + 4e4x(9x+ 4β),

to alleviate the complexity of the expression. Using our expressions (25) and (26) we are now able to find expressionsfor the weak, strong and dominant energy conditions in the forms

ρ− pC

=2√βK13

(K1 − 2

(k2 − 4e2xk + e4x

)√β)

tanh(B + 3

2K3

)9 (e2x − k)

2x2

− K12

27 (e2x − k)4x2β

+J2K6

(e2x + k

)27 (e2x − k)

5xJ1√β

+

(e4x − k2

)√βK8 +

(e2x − k

) (9k2 − 2e2x(18x+ 2β + 9)k + e4x(36x+ 20β + 9)

)− 8e2x

√βJ1 − 4K2e

2x

3 (e2x − k)3β

,

(28)

ρ+ p

C=

2√βK13

(2(k2 − 4e2xk + e4x

)√β −K1

)tanh

(B + 3

2K3

)9 (e2x − k)

2x2

−4e2x

((k2 + e4x

)(9x+ 5β) + 2e2xk(9x+ β)

)3 (e2x − k)

3β

+

(e4x − k2

)√βK8 +

(e2x − k

) (9k2 − 2e2x(18x+ 2β + 9)k + e4x(36x+ 20β + 9)

)− 8e2x

√βJ1

3 (e2x − k)3β

+K12

27β (e2x − k)4x2

+J2K6

(e2x + k

)27 (e2x − k)

5xJ1√β, (29)

ρ+ 4p

C=

K13

(2(k2 − 4e2xk + e4x

)√β −K1

)tanh

(B + 3

2K3

)9 (e2x − k)

2x2

−4e2x

((k2 + e4x

)(9x+ 5β) + 2e2xk(9x+ β)

)3 (e2x − k)

3β

+

(e4x − k2

)√βK8 +

(e2x − k

) (9k2 − 2e2x(18x+ 2β + 9)k + e4x(36x+ 20β + 9)

)− 8e2x

√βJ1

3 (e2x − k)3β

+4K12

27β (e2x − k)4x2

+J2K6

(e2x + k

)27 (e2x − k)

5xJ1√β, (30)

respectively. We have additionally setK12 = 2e4x

(9k2

(27x2 + 4βx+ 4β2

)− β3/2J1

)− 4e2xk

(7J1β

3/2 + k2(81x2 + 26β2

))+ e8x(9x− 2β)2 − 2k2J1β

3/2 +

k4(9x− 2β)2 − 4e6xk(81x2 + 26β2

), K13 =

√K2+2

√βJ1

(e2x−k)2β and K14 = K4

27(e2x−k)4x2β.

10

The Chandrasekhar stability index is given by

Γ =(e2x − k

) ((e2x + k

)K15

)3/2( βK16

(9x+ 4β)k3 − 9e2xxk2 − 9e4xxk + e6x(9x+ 4β)

+3(e2x − k

)2β3/2 sech2

(B + 3

2K3

)K17

2K15

(2√βK1 +K2

) )×

(K12

27 (e2x − k)4x2β

+

(k + e2x

)K6K18

27 (e2x − k)5x√βJ1

+2(2(k2 − 4e2xk + e4x

)√β −K1

)√2√βJ1 +K2 tanh

(B + 3

2K3

)9 (e2x − k)

3x2

+

(e2x − k

)K19 − 8e2x

√βJ1 − 4e2xK2 +

(e4x − k2

)K8

√β

3 (e2x − k)3β

)÷[

1

27 (e2x − k)4x2β

(2(k + e2x

)2xβ(

9(

(9x+ 4β)J1 − k2√β(27x+ 8β)

)k8 − 36βk7e2x

(20x2 + 6x+ 3

)J1

−4e2x((

8β2(2x− 1) + 81x(4x2 + 2x+ 1

))J1 − 2k2

√β(8(2x− 1)β2 + 9(4(x− 1)x+ 7)β + 243x

))k7

−e4x(√

β(64(95x− 27)β2 + 72(64x(6x− 1) + 15)β + 243x

(128x2 + 21

))k2 + 4J1

(648(2β − 1)x2

))k6

+4e6x(

4√β(−16(19x+ 11)β2 + 36(x(28x− 17) + 1)β + 243

(16x3 + x

))k2 + 36β

(20x2 − 78x+ 3

)J1

+J1(−8(114x+ 13)β2 + 81x(2x(18x− 5) + 1)

))k5 + 2

√βe8x

(32(43− 65x)β2 + 576(6x(4x+ 1) + 1)β

)k2

+2e8x(√

β(243x

(64x2 + 11

))k2 + J1(x(−64(9x+ β)(18x− 11β)− 405)− 108β)

)k4 + 32β2k3e10x(13− 114x)

+4e10x((

9(20x2 + 78x+ 3

)β + 81x(2x(18x+ 5) + 1)

)J1 − 4k2

√β(27(76x2 + 5

)β + 243x

(32x2 + 3

)))k3

−2e12x(√

β(32(65x+ 43)β2 − 576(6x(4x− 1) + 1)β − 243x

(64x2 + 11

))k2 + 2k2(48β(11β + 9)− 81)xJ1

+2J1(648(2β + 1)x2 + 2β(52β − 9)

))k2 + 16

√βk2e14x

(16(11− 19x)β2 + 36(x(28x+ 17) + 1)β

)+4e14x

(4k2√β(243

(16x3 + x

))−(8(2x+ 1)β2 + 9

(20x2 − 6x+ 3

)β + 81x

(4x2 − 2x+ 1

))J1

)k

+8e18x√β(8(2x+ 1)β2 + 9(4x(x+ 1) + 7)β + 243x

)k − e4x ((48β(11β − 9)− 81)x− 2β(52β + 9)) J1

+e16x(

9(9x+ 4β)J1 − k2√β(64(95x+ 27)β2 + 72(64x(6x+ 1) + 15)β + 243x

(128x2 + 21

)))−11008k5β5/2xe10x − 9e20x

√β(27x+ 8β)

)K4

)−

2√βK9 tanh

(B + 3

2K3

)9 (e2x − k)

2x2

], (31)

11

where

K15 =(k + e2x

) ((9x+ 4β)k2 − 2e2x(9x+ 2β)k + e4x(9x+ 4β)

),

K16 = 8k7e2x(81(4x+ 1)x2 + 18(19x− 13)βx+ 4(22x− 27)β2

)−(

4(9x+ 4β)(9x− 2β)k2 +√β(27x+ 16β)K1

)k6

+4e2x(

2(81(4x+ 1)x2 + 18(19x− 13)βx+ 4(22x− 27)β2

)k2 +K1

√β(x(162x+ 74β − 81)− 52β)

)k5

−e4x(

8k2(81(8x− 1)x2 − 18(2x+ 37)βx− 8(21x+ 22)β2

)−√βK1(208β + x(864x+ 800β + 729))

)k4

−8e6x((

81(4x+ 3)x2 + 18(19x+ 33)βx+ 4(43− 42x)β2)k2 + 18K1x

√β(21x+ β)

)k3

+e8x(

64x(162x2 − 9βx+ 22β2

)k2 +K1

√β(x(864x+ 800β − 729)− 208β)

)k2 + 4e16x(9x+ 4β)(9x− 2β)

+8e14x(81(4x− 1)x2 + 18(19x+ 13)βx+ 4(22x+ 27)β2

)k + 4ke10x

√βK1(52β + x(162x+ 74β + 81))

−4e10x(2k2

(81(4x− 3)x2 + 18(19x− 33)βx− 4(42x+ 43)β2

))k +

√βe12x(27x+ 16β)K1

+e12x(−8k2

(81(8x+ 1)x2 + 18(37− 2x)βx+ 8(22− 21x)β2

)),

K17 = 6e4x(e−2xk2 − e2x

) (−2e8xβ3/2 + 4e6xk(27x+ 13β)

√β + k2

((9x+ β)K1 − 2k2β3/2

))− 36k2

√βe4x(6x+ β)

+e4x(

(9x+ β)K1 + 2e2xk(

2√β(27x+ 13β)k2 +K1(7β − 9x)

)) (e−2x

(k2 + e4x

)(9x+ 4β)− 2k(9x+ 2β)

)+((k2√β(9x+ 4β)(9x+ 8β)−

(162x2 + 117βx+ 16β2

)K1

)k6 + 4K1e

2x(9(28x− 3)βx+ 26(3x− 2)β2

)k5

+2e2x(

3√β(27(20x− 19)x2 + 6(82x− 93)βx+ 16(7x− 9)β2

)k2 + 2K1

(162(x+ 1)x2

))k5

−e4x((

162(16x+ 5)x2 + 27(32x− 21)βx− 16(24x+ 13)β2)K1 − 2k2

√β(81(71− 40x)x2

))k4

+2e6x(√

β(−81(20x+ 87)x2 − 18(82x+ 225)βx+ 16(27x− 43)β2

)k2 + 72K1x

(27x2 − 2βx+ β2

))k3

+e8x(

96k2x√β(135x2 + 21βx+ 14β2

)−(162(16x− 5)x2 + 27(32x+ 21)βx+ 16(13− 24x)β2

)K1

)k2

+2e10x(√

β(81(87− 20x)x2 + 18(225− 82x)βx+ 16(27x+ 43)β2

)k2 + 2K1

(162(x− 1)x2 + 9(28x+ 3)βx

))k

+6e14x√β(27(20x+ 19)x2 + 6(82x+ 93)βx+ 16(7x+ 9)β2

)k + 72k6β3/2e4x(123− 14x)x

+e12x((

162x2 + 117βx+ 16β2)K1 − 2k2

√β(81(40x+ 71)x2 + 36(14x+ 123)βx+ 16(44− 27x)β2

))+104kβ2(3x+ 2)K1 − 32k6β5/2e4x(27x+ 44)− e16x

√β(9x+ 4β)(9x+ 8β)

)sinh(2B + 3K3)

√K2 + 2K1

√β

(e2x − k)2β,

K18 = 9k4 − 16e4x(9x+ β)k2 + 2e2x(

(36x+ 20β − 9)k2 + 8J1√β)k + 2e6x(36x+ 20β + 9)k − 9e8x,

K19 = 9k2 − 2e2x(18x+ 2β + 9)k + e4x(36x+ 20β + 9).

Inserting the energy density profile into the gravitational mass formula gives

M =1

3√βC3/2

∫x3/2

(e4x(20β + 36x+ 9)− 8

√βJ1e

2x +(e2x − k

) (9k2 − 2ke2x(2β + 18x+ 9)

)√β (e2x − k)

3

+K8

√β(e2x + k

)− 4K2e

2x

√β (e2x − k)

2 +

(e2x + k

)J2K6

9 (e2x − k)5xJ1

)dx, (32)

for a 5− d hypersphere of perfect fluid.

A. Physical admissibility of the five dimensional solution

We now proceed to analyse in a qualitative sense the physical properties of our new conformally flat spacetime inEGB. The very complicated and lengthy expressions that emanate prohibit an analytical treatment of the physicstherefore we have resorted to a graphical analysis using suitably selected parameter spaces. The physical characteristics

12

FIG. 1: The pressure profile is plotted as a function of radius for 5D and 6D EGB gravity, respectively.

FIG. 2: Variation of density as a function of the radial coordinate for compact star. The labels of the curves are the same asgiven in Fig. 1.

of the 5-d hypersphere are displayed in the left panels of each of the figures while the 6-d plots appear on theright. In the case of the 5-d plots we have selected the parameters as follows after extensive empirical fine-tuning:A = 4, B = 1, C = 0.06, k = 0.6, β = 0.4 and α = 0.01.

It is worth noting at this point that the specific models we are presenting are only valid for the choices of parameterspace in both 5 and 6 dimensions. There are a number of parameters that are moving parts in the system so it is notpossible to say that the solution is generic. We have exhibited the existence of at least one physically well behavedmodel for the general EGB spacetime we have found in 5 and 6D. Researchers working in simulations may utiliseour general solution and incorporate data pertaining to known structures to make an identification with observedphenomena such as neutron stars, white dwarfs, cold planets and so on.

(i) From Fig. 1 it may be observed that the pressure decreases smoothly towards the surface layer of the star. Thepressure vanishes for some finite radius corresponding approximately to x = 0.01 which defines the boundaryof the star. This immediately suggests that the model may represent a compact star.

(ii) The density profile plotted as a function of the radial coordinate x in Fig. 2 reveals that the density is amonotonically decreasing function as one moves from the stellar center towards the stellar boundary. This isconsidered to be physically reasonable.

(iii) In order to prevent superluminal sound speeds within the stellar fluid it is required that the condition 0 ≤ dpdρ ≤ 1

be satisfied throughout the stellar interior. From Fig. 3 we observe that this criterion is met everywhere exceptat the centre of the star. In order for our model to obey causality requirements we must make an incisionfrom r = 0 to some finite radius r = r0, thus removing this portion of the solution. We can then replace thisportion with a well–behaved solution from the center (r = 0) to r = r0 and match it to our present solutionfrom r = r0 to r = R where R is the boundary of the star. In this way we can effectively create core-envelope

13

FIG. 3: Variation of speed of sound with radius.

FIG. 4: Variation of energy conditions with radius.

models of compact objects in 5D EGB gravity which to our knowledge do not appear in the literature. We donot pursue this aspect now however we indicate a work-around the presence of the undesirable central singularity.

(iv) The physical admissibility of our solutions require that the following energy conditions, viz., (i)weak energycondition (WEC), (ii) strong energy condition (SEC) and (iii) dominant energy condition (DEC) hold true ateach interior point of the star. These conditions are equivalent to

WEC : ρ− p ≥ 0, SEC : ρ+ p ≥ 0, DEC : ρ+ 4p ≥ 0. (33)

It is clear from Fig. 4. that all three energy conditions are satisfied for the 5-d model we have developed andfor our choice of parametric space.

(v) In the absence of anisotropy or dissipation it is well-known that there has no upper mass limit if the EoS hasan adiabatic index Γ > 4/3 where

Γ =p+ ρ

p

dp

dρ, (34)

characterising stable regions within the fluid sphere. This result has been extended to include pressureanisotropy, viscosity and heat flow by Chan and co-workers [55, 56]. They showed that dissipation in particular,renders the system less unstable by diminishing the total mass entrapped inside the fluid sphere. Fig. 5 revealsan interesting peculiarity in the 5D stability index. The central portion of the star appears to be more stablethan regions greater than some finite radius away from the center. This behaviour is unphysical in a realisticstellar model as the central density is highest here and would expect more heating and thus thermodynamicallyunstable regions. As one moves away to the cooler regions, heat generation would drop-off thus leading to

14

FIG. 5: Variation of adiabatic index with radius.

FIG. 6: Mass as a function of the radial coordinate for compact star. In both cases the parameters are given in Fig. 1.

stable regions. The adiabatic index points to the fact that our solutions would be suitable in describing acore-envelope model of a compact object in EGB gravity.

(vi) The mass profile for the 5D model is plotted in Fig. 6. We observe that the mass function increases as onetends away from the centre of the stellar configuration with m(0) = 0 which is expected as more mass iscontained within larger concentric shells.

(vii) A star remains in static equilibrium under the forces namely, gravitational force and hydrostatics force, if thesummation of two active forces on the system is zero. Along these directions, a wide class of astrophysicalsolutions (including wormholes/compact stars) have been studied ( see, e.g., Refs. [64–67]). This condition isformulated mathematically via,

−dpdx− y

y(ρ+ p) = 0, or Fh + Fg = 0, (35)

where the first term represents the hydrodynamic force (Fh) and the second term corresponds to gravitationalforce (Fg), respectively. The variation of the forces is shown in Fig. 7 and our system is stable in terms of theequilibrium of forces.

The analysis above suggests that the conformally flat stellar model we have derived does exhibit pleasing physicalbehaviour consistent with the elementary requirements. In contrast with the other conformally flat solution theSchwarzschild interior, this solution displays a variable energy density and consequently does not suffer from thepathology of an infinite sound speed.

15

FIG. 7: Variation of hydrodynamic (Fh) and gravitational (Fg) forces versus the radial coordinate x.

FIG. 8: Pressure, energy density and speed of sound versus radial pulsation x + εx for ε of the order 0.1, 0.01 and 0.001,respectively. The labels of the curves are the same as given in Fig. 1.

VI. SIX DIMENSIONAL CASE

We now turn our attention to the remaining critical dimension in EGB namely the six dimensional case. The EGBfield equations for a six dimensional metric

ds2 = −e2νdt2 + e2λdr2 + r2dΩ2, (36)

16

where dΩ2 = dθ2 + sin2 θdφ2 + sin2 θ sin2 φdψ2 + sin2 θ sin2 φ sin2 ψdη2 (2) when expanded amount to the system

ρ =1

e4λr4[(4r3e2λ − 48αr(1− re2λ))λ′ − 6r2e2λ(1− e2λ) + 12α(e2λ − 1)2

], (37)

p =1

e4λr4[(1− e2λ)(6r2e2λ − 48αrν′ + 12αe2λ − 12α) + 4r3e2λν′

], (38)

p =1

e4λr2((12α(e2λ − 1) + r2e2λ)(ν′′ + (ν′)2 − ν′λ′) + 24αν′λ′

)+

1

e4λr3((3r2e2λ + 12α(e2λ − 1))(ν′ − λ′) + 3re2λ(1− e2λ)

), (39)

in the canonical spherical coordinates (xa). The nonlinearity in the system (37)–(39) has now greatly increasedbecause of the presence of the EGB coupling parameter α and further dynamical terms that were suppressed in 5-d.

The system (37)–(39) may be converted to the equivalent form

12βx(Z − 1)Z − 4x2Z − 6x(Z − 1) + 3β(Z − 1)2

x2=

ρ

C, (40)

(24βx(1− Z) + 8x2

)Zy + (Z − 1)(6x+ 3β(1− Z))y

x2y=

p

C, (41)

4x2Z (x+ 3β[1− Z]) y + 2x(x2Z + 3β

[(1− 3Z)Zx− 2Z(1− Z)

])y

+3 (β(1− Z) + x)(Zx− Z + 1

)y = 0. (42)

through the coordinate change x = Cr2, e2ν = y2(x) and e−2λ = Z(x). Note that (42) is the equation of pressureisotropy in six-dimensional EGB theory and we have redefined β = 4αC. It has been expressed as a linear secondorder differential equation in y (if Z is a known quantity) which is the great advantage of this coordinate choice. Thecondition of pressure of isotropy may also be expressed as[

x2(2xy + 3y) + 3βx(2xy + y − (6xy + y)Z)]Z

−3β[4x2y − 4xy − y

]Z2 +

[x(4x2y − 3y) + 6β(2x2y − 2xy − y)

]Z + 3(x+ β)y = 0 (43)

a nonlinear first order differential in Z (if y is a known quantity). We observe that equation (40) is easily integratedif we assume that the density is constant, ie., ρ = ρ0. Integration of (40) yields

Z(x) = 1 +x

3β

1±

√9c1β2

x52

+3βρ0 + 5C

5C

, (44)

where c1 is a constant of integration. Note that (44) when written in the form

Z = 1 +x

3β± 1

3β

√9c1β2

x1/2+

(1 +

3βρ05C

)x2 (45)

reveals a singularity at the centre x = 0 unlike the 5-dimensional case which was singularity-free. This point furtherillustrates the manifest different physics inherent in 5 and 6 dimensional EGB gravity. Note that the six dimensionalvacuum solution of Boulware and Deser [20] is regained from result (44) by setting the constant density ρ0 = 0 andis given by

Z(x) = 1 +x

3β

1±

√1 +

9c1β2

x52

. (46)

Bogdanos et al [57] have analysed the 6–dimensional case in EGB and demonstrated the validity of Birkhoff’s theoremfor this order.

17

Equation (43) is an Abel differential equation of the second kind and admits only a few exact solutions for certaincases. Inserting (19) into the pressure isotropy equation gives(

xZ − Z + 1)(−3β√Z sinh f(x) + (3β + 2x) cosh f(x)− 6βZ cosh f(x)

)= 0 (47)

after factorization. As for the 5 dimensional case, the Schwarzschild interior solution emanates from the vanishing ofthe first factor on the left hand side of (47). The other factor generates the condition

tanh f(x) =2x+ 3β(1− 2Z)

3β√Z

(48)

which has a remarkably similar form similar to that for the 5 dimensional case despite the presence of extra terms inthe pressure isotropy equation. Following the same procedures as in the previous section the general solution of (48)has the form

Z =(e4x + h2)(15β + 8x)− 2h(9β + 8x)e2x

24β (e2x − h) 2±√

(e2x + h) 2 ((e4x + h2)(27β + 16x)− 2h(21β + 16x)e2x)

8√

3β (e2x − h) 2(49)

where h is a constant of integration. Once again the form of the temporal potential may be obtained via equation(19) albeit in terms of an unresolved integral.

We calculate the physical quantities relevant to our investigation. Not unexpected, these expressions are unwieldyon account of the long expressions for the gravitational potentials. The energy density is given by

ρ

C=

[4L0x

(−9k3

(√L3 (k + e2x)

2 −√βk2

)+ k2e2x

(√βk2(40β + 72x− 9) + (27− 16β)

√L3 (k + e2x)

2

)−ke4x

(6√βk2(−4β + 12x+ 3) + (16β + 27)

√L3 (k + e2x)

2

)+ 3e6x

(2√βk2(4β − 12x+ 3)

+3

√L3 (k + e2x)

2

)+√βke8x(40β + 72x+ 9)− 9

√βe10x

)]× 1

x2+

(−L0) 2

27βx2 (e2x − k)4

+2x(

4β(k2 − 4ke2x + e4x

)− 9x

(e2x − k

)2+ 2√βL1

)3βx2 (e2x − k)

2

/27β(e2x − k

)5√L3 (k + e2x)

2

x2

−4

9(k2−2ke2x(2x+1)+e4x(4x+1))

β + L5√β√L3(k+e2x)

2− 4ke2x + 20e4x

9 (e2x − k)2 − 4L0e

2x

9β (e2x − k)3

, (50)

while the isotropic particle pressure assumes the form

p

C=

L0

(4βk2 + 9k2x+ 2

√β

√L3 (k + e2x)

2 − 2ke2x(8β + 9x) + e4x(4β + 9x)

)27βx2 (e2x − k)

4

+1

27x2 (e2x − k)2

[8

√9β (k2 − 2ke2x + e4x) + L0

β (e2x − k)2

(2βk2 − 3k2x+

√β

√L3 (k + e2x)

2

+2ke2x(3x− 4β) + e4x(2β − 3x))(√9β (k2 − 2ke2x + e4x) + L0

β (e2x − k)2 + L4

)]. (51)

18

The expressions below will assist us analyse the energy conditions.

ρ− pC

=[2(

12√βk6L1

(−4β2 + 81x2 + 17βx

)− 2k5e2x

(k2(270β(8x+ 5)x2 − 3645x3 + 48β2(23x+ 32)x

+16β3(4x+ 9))− 8√βL1

(27(2x− 9)x2 + β2(33− 4x) + 6β(5x+ 6)x

))+2k4e4x

(3k2

(720β(2x+ 5)x2 − 2835x3 + 8β2(199− 44x)x− 32β3(10x+ 1)

)+2√βL1

(27(45− 32x)x2 + 4β2(104x− 33) + 9β(64x− 49)x

))+ 6k3e6x

(k2(90β(8x− 57)x2 + 2835x3

+16β2(23x− 84)x+ 16β3(1− 20x))

+ 32√βL1x

(−6β2 + 27x2 − 29βx

))−4√βk2e8x

(32√βk2x(9x− β)(15x− 2β) + L1

(27(32x+ 45)x2 − 4β2(104x+ 33)− 9β(64x+ 49)x

))+2ke10x

(3k2

(90β(8x+ 57)x2 − 2835x3 + 16β2(23x+ 84)x− 16β3(20x+ 1)

)+8√βL1

(27(2x+ 9)x2 − β2(4x+ 33) + 6β(5x− 6)x

))− 6e12x

(k2(−32β3 − 45(32β + 63)x3

+16β(22β + 225)x2 + 8β2(40β + 199)x)

+ 2√βL1

(−4β2 + 81x2 + 17βx

))−3k8(4β + 9x)

(8β2 + 45x2 + 20βx

)− 2ke14x

(270β(8x− 5)x2 + 3645x3 + 48β2(23x− 32)x

+16β3(4x− 9))

+ 3e16x(4β + 9x)(8β2 + 45x2 + 20βx

))]/ 9x2

(e2x − k

)2243βL3x4 (e2x − k)

7(k + e2x)− 8cL4

√L3−2

√βL1

β(e2x−k)2(L2 +

√βL1

) , (52)

ρ+ p

C=[8(−k6L1

(4β2 + 27x2 + 51βx

)+ 4k5e2x

(L1

(27(2x+ 1)x2 − β2(4x+ 13) + 6β(5x+ 3)x

)−√βk2

(27(10x+ 7)x2 + 2β2(4x− 27) + 6β(23x− 11)x

))+ k4e4x

(L1

(−27(32x+ 5)x2

+52β2(8x+ 1) + 9β(64x+ 1)x)− 4√βk2

(−108(5x+ 2)x2 + 8β2(15x+ 11) + 3β(44x+ 85)x

))+4k3e6x

(√βk2

(27(10x− 3)x2 + 2β2(43− 60x) + 6β(23x+ 39)x

)+ 12L1x

(−6β2 + 27x2 − 29βx

))−k2e8x

(32√βk2x(9x− β)(15x− 2β) + L1

(27(32x− 5)x2 + 52β2(1− 8x) + 9β(1− 64x)x

))+4ke10x

(√βk2

(27(10x+ 3)x2 − 2β2(60x+ 43) + 6β(23x− 39)x

)+ L1

(27(2x− 1)x2

+β2(13− 4x) + 6β(5x− 3)x))

+ e12x(

4√βk2

(108(5x− 2)x2 + 8β2(11− 15x) + 3β(85− 44x)x

)+L1

(4β2 + 27x2 + 51βx

))− 2√βk8(β − 12x)(4β + 9x)− 4

√βke14x

(27(10x− 7)x2

+2β2(4x+ 27) + 6β(23x+ 11)x)

+ 2√βe16x(β − 12x)(4β + 9x)

)]/8L4

(√βL1 + L2

)√L3−2

√βL1

β(e2x−k)2

9x2 (e2x − k)2 + 27

√βx2

(e2x − k

)5L5, (53)

19

ρ+ 5p

C=

40L4

(√βL1 + L2

)√L3−2

√βL1

β(e2x−k)2

9x2 (e2x − k)2 −

[4(

6√βk6L1

(−4β2 + 189x2 + 85βx

)+ 2k5e2x

(k2(54β(20x+ 17)x2

+3645x3 + 24β2(23x− 97)x+ 8β3(4x− 99))

+ 4√βL1

(−27(2x+ 21)x2 + β2(4x+ 105) + 6β(3− 5x)x

))−2k4e4x

(3k2

(144β(5x− 19)x2 + 2835x3 − 4β2(44x+ 653)x− 160β3(x+ 2)

)+√βL1

(−27(32x+ 105)x2 + 4β2(104x+ 105) + 9β(64x+ 101)x

))− 6k3e6x

(k2(328β3 + 45(8β − 63)x3

+2β(92β + 2403)x2 + 40β2(57− 4β)x)

+ 16√βL1x

(−6β2 + 27x2 − 29βx

))+2√βk2e8x

(32√βk2x(9x− β)(15x− 2β) + L1

(420β2 − 9(64β + 315)x2 + 864x3 + β(909− 416β)x

))−2ke10x

(3k2

(−328β3 + 45(8β + 63)x3 + 2β(92β − 2403)x2 − 40β2(4β + 57)x

)+4√βL1

(27(2x− 21)x2 + β2(105− 4x) + 6β(5x+ 3)x

))− 6e12x

(k2(144β(5x+ 19)x2 − 2835x3

+4β2(653− 44x)x− 160β3(x− 2))

+√βL1

(−4β2 + 189x2 + 85βx

))− 3k8(4β + 9x)

×(4β2 + 45x2 + 68βx

)+ 2ke14x

(792β3 + 135(8β − 27)x3 + 6β(92β − 153)x2 + 8β2(4β + 291)x

)+3e16x(4β + 9x)

(4β2 + 45x2 + 68βx

))× 27βL3x

2(e2x − k

)5 (k + e2x

)], (54)

For simplicity of notations we have usedL21 = k4(4β + 9x) + 4βk3e2x − 18k2e4xx+ 4βke6x + e8x(4β + 9x),

L2 = k2(2β − 3x) + 2ke2x(3x− 4β) + e4x(2β − 3x),L3 = k2(4β + 9x)− 2ke2x(2β + 9x) + e4x(4β + 9x)L5 = k3(4β + 9x)− 9k2e2xx− 9ke4xx+ e6x(4β + 9x),

L4 = tanh

(B + 1

2

∫ x0

3β(e2x−k)x√L3−2(e2x+k)

√βL3

dx

).

We have elected not to include the expressions for the adiabatic stability parameter Γ, mass profile as well as thesound-speed squared as these expressions are extremely long. Nevertheless, we have plotted the behaviour of thesequantities and have studied them carefully in what follows.

A. Physical analysis of the 6-d hypersphere

For the purposes of graphical plots in 6-d the following parameter values were obtained for pleasing physicalbehaviour: A = 2, B = 1, C = 0.1, k = −0.9 and β = 8. Observations of the right panels of each of the figures from1 to 6 reveal interesting properties of the 6-d hypersphere.

(i) The boundedness of the solution is revealed in Fig. 1 which illustrates that the pressure decreases monotonicallyfrom the centre of the star and vanishes at the boundary. This qualifies the model as a potential compact starmodel suitable for applicability to neutron stars or cold planets.

(ii) The density plotted in Fig. 2 increases sharply at the centre of the configuration and drops off smoothlytowards the surface of the stellar surface. This behaviour is not inconsistent with stellar structure dynamicsas the interior processes are not known. Importantly the density remains positive definite within the distribution.

(iii) The causality criterion 0 ≤ vs = dpdρ ≤ 1 holds everywhere inside the fluid sphere as observed in Fig. 3. We

observe that the sound speed decreases from the centre outwards attaining a minimum value at the surface ofthe star. This could be due to phase transitions leading to different equations of state in different regions of thestar [58].

(iv) The (i)weak energy condition (WEC), (ii) strong energy condition (SEC) and (iii) dominant energy condition(DEC) given by

WEC : ρ− p ≥ 0, SEC : ρ+ p ≥ 0, DEC : ρ+ 5p ≥ 0. (55)

20

FIG. 9: Pressure, energy density and speed of sound versus radial pulsation x + εx for ε of the order 0.1, 0.01 and 0.001,respectively. The labels of the curves are the same as given in Fig. 1.

should hold everywhere inside the star. We observe that all energy conditions are satisfied throughout theinterior of the star for the 6-d hypersphere.

(v) The stability index plotted in Fig. 5 shows that the surface layers are more stable than the central regions.This is expected as the density is highest at the center making energy-generating processes such as nuclearfusion more efficient leading to more heat generation here. The heated core is likely to be more unstable thanthe surface of the star. The requirement Γ > 4

3 is always satisfied as depicted in Fig. 5.

(vi) From Fig. 6 it may be observed that the mass function increases as one moves away from the centre of thestellar configuration with m(0) = 0 which is expected as more mass is contained within larger concentric shells.

(vii) Using the expression (35) the modified form of the TOV equation has been plotted in Fig. 7. We observe thatthe system is completely stable under the equilibrium of the different forces, i.e., Fh + Fg = 0, with respect tothe radial coordinate x.

The plots above demonstrate the viability of the new 6-d conformally flat solution in representing realistic stellardistributions. The persistent central singularity may be eliminated by the inclusion of a suitable nonsingular core andits attendant matching requirements between the different layers.

21

B. Stability under radial pulsations

We combine our comments on the question of whether the models developed are stable under radial pulsations.Historically, stability was analysed extensively by Chandrasekhar [59] who developed a test for stability under per-turbations of the pressure and energy density for a linearised system of Einstein’s equations. This scheme is onlyapplicable to the Einstein’s equations and not directly useful in EGB theory. It would be useful to carry out thecomputations of Chandrasekhar within higher curvature gravity. Bardeen et al devised a catalogue of methods tostudy the normal modes of radial pulsations of stellar models. A normal mode of radial pulsation for an equilibriumsolution has the form δr = ξ(r) exp(iσt) where ξ is a trial function and the eigenvalue σ is the frequency. Additionallyit is assumed, for this process that, the Lagrangian change of the pressure vanishes at the boundary and that the trialfunction is finite at the centre.

It is worth noting that stability can take on different meanings depending on the context. Ample warnings arecontained in the work of Bardeen et al [60]. Kokkotas and Ruoff [61] make use of a numerical technique and give twoapproaches to studying stability: the shooting method and the method of finite differences. In the context of EGBgravity the stability of scalarized black holes has been considered by Silva et al [62]. Since there is equivalent criteriaof the Chandrasekhar integral condition for EGB gravity, we shall analyse our exact solution under radial pulsationsby perturbing the radial coordinate x as x+ εx, where ε is a very small quantity in relation to the stellar radius, in theenergy density, pressure and sound speed. It will be immediately apparent that for small values of ε the dynamicalquantities converge to the original shapes. We provide plots where ε is of the order 0.1, 0.01, 0.001 in order to confirmthat our solution is indeed stable under radial pulsations. This follows the same spirit as the well known Lyapunovstability of curves and mimics the approach of Herrera [55, 63] in discussing the concept of cracking in anisotropicspheres.

Fig. 8 and 9 depicts the energy density, pressure and sound speed profiles for the 5 and 6 dimensional conformallyflat EGB models under pulsations of the type x → x + εx for three different orders of magnitude of ε namely0.1, 0.01, 0.001 and each plot showing increments up to seven curves. In other words, for each profile we have plottedseven increments to check the trend: for example, for the order of magnitude 0.001 we show the curves for ε =0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007. It is clear that in all cases as ε decreases the perturbed curves convergetoward the equilibrium curve ε = 0. This is the case also for larger orders of magnitude ε = 0.01 and 0.1. This maybe understood to mean that these dynamical quantities are stable under radial pulsations. It must be borne in mindthat we have achieved an exact solution in closed form only for Z whereas the form for y is expressed as an integral.Consequently, we had to resort to numerical techniques thus necessitating the check whether small disturbances ofthe system generate severe deviations from the equilibrium position.

VII. CONCLUSION

We have thoroughly investigated conformally flat static isotropic spacetimes in Einstein–Gauss–Bonnet spacetimes.Our conclusions may be summarised as follows:

(i) The interior Schwarzschild metric generates a constant density hypersphere in EGB, however, the hypothesis of aconstant density results in a generalized Schwarzschild solution. In the 5 dimensional case the general solutionis completely free of singularities at the stellar centre but a coordinate central singularity in the 6 dimensionalcase is unavoidable.

(ii) The assumption of constant density leads to the generalised Schwarzschild solutions for 5 and 6 dimensions asmentioned in (i). It has been verified by direct substitution that the generalised Schwarzschild solutions for 5and 6 dimensions result in the conformal flatness equation being identically satisfied. So the general constantdensity metric is indeed conformally flat. Conversely, assuming conformal flatness generates two branches ofsolutions one of which is Schwarzschild and the other is a new solution which does not yield constant density.The temporal gravitational potential is expressed as an integral, however, with the aid of numerical techniques itis possible to analyse the physical properties of the solution graphically. Suitable parameter spaces are exhibiteddemonstrating that the models in 5 and 6 dimensions satisfy basic physical requirements demanded of stellardistributions.

(iii) Finally, the Schwarzschild metric is known to be conformally flat. However, the assumption of conformal flatnessleads to two branches of solutions only one of which is Schwarzschild. As mentioned in (ii) above, the non-Schwarzschild solution is shown to be reasonably behaved in both 5 and 6 dimensions.

When subjected to an elementary test of stability with respect to radial pulsations, both five and six dimensionalmodels converged to the equilibrium position verifying its stability. The new conformally flat solution did not exhibit

22

the defect of an infinite sound speed as its counterpart the Schwarzschild metric does in standard Einstein gravity.Moreover, a variable density profile was admitted as compared to the incompressible Schwarzschild sphere.

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