Tolman–Oppenheimer–Volkoff equations in modified Gauss–Bonnet gravity

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Tolman-Oppenheimer-Volkoff Equations in Modified

Gauss-Bonnet Gravity

D. Momeni(a) and R. Myrzakulov(a)

aEurasian International Center for Theoretical Physics and Department of General &

Theoretical Physics, Eurasian National University,

Astana 010008, Kazakhstan

Abstract

Based on a stringy inspired Gauss-Bonnet (GB) modification of classical gravity,

we constructed a model for neutron stars. We derived the modified forms of Tolman-

Oppenheimer-Volkoff (TOV) equations for a generic function of f(G) gravity. The

hydrostatic equations remained unchanged but the dynamical equations for metric

functions are modified due to the effects of GB term.

Keywords: Modified theories of gravity; Models beyond the standard models; Neutron

stars

Pacs numbers: 97.60.Jd; 12.60.-i; 04.50.Kd

1 Introduction

Observational data show that we live in an accelerating Universe at large scales [1]-

[3]. This behavior is governed beyound the classical dynamics of solar objects and even

beyound the Einstein theory for gravity as a gauge theory,so called as general relativity

(GR). So to explain this acceleration expansion at large cosmological objects we need some

kinds of modifications to the classical GR. This approach is called as modified gravity as

it is believed that it can solve the problem of acceleration without any need to exotic

fluids (for reviews see [4]-[13]). One of the simplest modifications of GR is to replace the

Ricci scalar R by an arbitrary function f(R). It is called as f(R) gravity and originally

proposed by Buchdahl [14] and recently motivated by several authors. In the context

1

of Riemmanian geometry the next higher order correction to R is called as GB term is

defined by G = R2 − RµνRµν + RµνλσR

µνλσ. Indeed it is a topological invariant term in

four dimensions and if we add it to the Ricci scalar term in Einstein gravitional action, it

has no contribution to the equations of motion. But a non minimally coupled form of this

topological term induces new features as an alternative for dark energy. This modified

f(G) gravity has been introduced as model for dark energy in [38]. The GB term plays

a very important role in cosmological models as several works have been devoted to the

interesting aspects of this term [15]-[32]. Infact this term is inspired from string theory.

In low energy limit of the stringy action the first second order term is GB. Also if we

study the most general scalar-tensor theory with second order field equation ,we observe

this GB term [33].

From other point of view, dynamics of compact objects are very important problem in

astrophysics. One of the most important objects in neutron star. Neutron stars compact

objects with radius of order 10Km[34]. But they are so massive . For a typical neutron

star, the mass is about 1.4M∗ where M∗ denotes mass of our Sun. Since it is extremely

massive and tiny in comparison to our planet,a surface gravity on this star is much higher

than Earth. The relative order of surface gravity of neutron star with respect to Earth is

about 2× 1011. One important property of neutron stars is they can be charged massive

objects. But the strength of electromagnetic fields of such stars are about some million

times stronger than the one which we have on Earth. In a very close competition with

neutron stars,we can have White Dwarfs or black holes. The main difference is order

of mass of these massive objects. Gravitational collapse is the dominant mechanism in

formaion of all these three types of massive objects.

In GR several models have been studied for neutron stars. In modified gravity f(R)

there are few refrences about neutron stars namely [35]-[37]. But in f(G) gravity there is

no work about neutron stars. In this letter we explore TOV equations for a gravitational

theory with GB term. We derived the full system of equations of motion for a spherically

symmetric static configuration with inhomogenous perfect fluid. We showed that how the

equations of motion will be changed due to GB corrections.

2

2 TOV equations in Modified Gauss-Bonnet gravity

Let us to start by the gravitational action of a generic GB model as the following [38]:

S =

∫

d4x√−g

[

R

2κ2+ f(G)

]

+ Sm , (1)

Since this action in defined by commutative connections and in the framework of a Rie-

mannian spacetime, as usual the second order scalar curvature R is defined by the Ricci

scalar. The next higher order correction due to the GB term is denoted by the algebraic

function f(G). We assume that this function is smooth enough to have higher order

derivatives fn≥2(G). It is a necesary condition for linear stability. To include the matter

fields we add a matter action Sm. The conntribution of this term to the field equations

induces the energy momentum tensor Tµν = − 2√−g

δSm

δgµν. Further more, the gravitational

coupling is defined by κ2 = 8πG. Here G is the tiny usual classical Newtonian gravita-

tional coupling. In metric formalism of modified gravity when we take metric as dynamical

variable , the equations of motion from (1) are written as the following form:

Rµν −1

2Rgµν + 8

[

Rµρνσ +Rρνgσµ −Rρσgνµ − Rµνgσρ +Rµσgνρ

+R

2(gµνgσρ − gµσgνρ)

]

∇ρ∇σfG + (GfG − f) gµν = κ2Tµν , (2)

where fGG... =dnf(G)dGn . Further more we adopted the signature of the metric gµν as (+−

−−) , and consequently ∇µVν = ∂µVν−ΓλµνVλ and Rσ

µνρ = ∂νΓσµρ−∂ρΓ

σµν+Γω

µρΓσων−Γω

µνΓσωρ

for the covariant derivative and the Riemann tensor, respectively. There is an additional

conservation law for the matter sector as ∇µTµν = 0.

We suppose that the metric of the neutron star is static-spherically symmetric with

coordinates xµ = (ct, r, θ, ϕ) in the following form:

ds2 = c2e2φdt2 − e2λdr2 − r2(dθ2 + sin2 θdϕ2). (3)

For the matter field, we suppose that the non zero components of energy momentum

tensor are T νµ = diag(ρc2,−p,−p,−p). We insert (3) in (2) and by using theformula given

in appendix, the diagonal components of (2) for (µ, ν) = (ct, ct) and (µ, ν) = (r, r) are

3

given by the following equations:

− 1

r2(2rλ′ + e2λ − 1) + 8e−2λ

(

fGG(G′′ − 2λ′G′) + fGGG(G

′)2)(1− e2λ

r2− 2(φ′′ + φ′2)

)

(4)

+(GfG − f)e2λ = κ2ρc2e2λ.

− 1

r2(2rφ′ − e2λ + 1)− (GfG − f)e2λ = κ2pe2λ. (5)

We need an additional equation for metric functions and f(G). We use the trace of (2) .

It gives us:

R + 8Gρσ∇ρ∇σfG − 4(GfG − f) = −κ2(ρc2 − p). (6)

Or with metric (3) it reads as the follows:

2(

φ′′ + φ′2 − φ′λ′ +2

r(φ′ − λ′) +

1− e2λ

r2

)

(7)

+8e−2λ(2φ′

r+

1− e2λ

r2

)(

fGG(G′′ − 2λ′G′) + fGGG(G

′)2)

+ 4(GfG − f)e2λ = κ2e2λ(ρc2 − 3p).

The hydrostatic (continuty equation) follows from the equation , ∇µTµν for ν = r. We

obtain:

dp

dr= −(p+ ρc2)φ′. (8)

The continuty equation is satisfied identically for ν = t. Note that when f(G) = G ,then

(5,5) reduce to the GR equations trivially.

To have proceed in TOV equations, we replace the metric function with the following

expression in terms of the gravitational mass function M = M(r):

e−2λ = 1− 2GM

c2r=⇒ GdM

c2dr=

1

2

[

1− e−2λ(1− 2rλ′)]

. (9)

The plan is to rewrite (5,5,8) in terms of {dp

dr, dM

dr, ρ} in a dimensionless form. For this

purpose,we introduce the following set of the dimensionless parameters,

M → mM⋆, r → rgr, ρ → ρM⋆

r3g, p → pM⋆c

2

r3g, G → G

r4g. (10)

4

Here rg =GM⋆

c2= 1.47473Km. From now we use the dimensionless parameters. Continuty

equation converts to the following :

dp

dr= −(p+ ρ)φ′. (11)

But (9) converts to the following:

dλ

dr=

m

r31− r

mdmdr

2mr− 1

. (12)

Using the (11,12) in (5) we obtain:

2

r

1− 2mr

p+ ρc2+

2m

r3− r2g(GfG − f) = 8πp. (13)

For (rr) equation we obtain:

− 2

r2dm

dr+ 8r2g(1−

2m

r)2[

fGG

(

r2gG′′ − r3g

2m

r21− r dm

dr2mr− 1

G′)

+ r2gfGGGG′2]

(14)

×[

− 2m/r3

1− 2mr

+ 2d

dr

(dp

dr

p+ ρ

)

− 2(

dp

dr

p+ ρ

)2]

+ (GfG − f) = 8πρ.

And the last equation for trace, in the dimensionless parameters reads:

2(1− 2m

r)(

− d

dr(

dp

dr

p+ ρ) + (

dp

dr

p+ ρ)2 +

m

r31− r

mdmdr

2mr− 1

(dp

dr

p+ ρ)− 2m/r3

1− 2mr

)

(15)

+8[

− 2dp

dr

r(p+ ρ)− 2m/r3

1− 2mr

][

fGG

(

r2gG′′ − r3g

2m

r21− r dm

dr2mr− 1

G′)

+ r2gfGGGG′2]

4(GfG − f) = 8π(ρ− 3p).

3 Neutron star models in f(G) gravity

We mention here that we need to speciefy the equation of state of the neutron star.

The model has a realistic but so complex form so called as SLy [39] and FPS [40] models:

ζ =a1 + a2ξ + a3ξ

3

1 + a4ξf(a5(ξ − a6)) + (a7 + a8ξ)f(a9(a10 − ξ))+

+(a11 + a12ξ)f(a13(a14 − ξ)) + (a15 + a16ξ)f(a17(a18 − ξ)),

5

where

ζ = log(P/dyncm−2) , ξ = log(ρ/gcm−3) , f(x) =1

exp(x) + 1.

The coefficients ai can be found in[34]. Also a linear model is proposed as quark matter

model for neutron stars.

We should write the f(G) = ǫh(G) and write the perturbation solutions as the follow-

ing for different quantities [41],[42]:

ρ = ρ0 + ǫρ1 + ..., p = p0 + ǫp1 + ...,

m = m0 + ǫm1 + ..., G = G0 + ǫG1 + ..., (16)

Where we assume that all functions {ρ0, p0, m0, G0} satisfythe standard TOV equations

in zeroth order of ǫ. One can solve the system of the differential equations (13,15,16)

numerically.

4 Conclusion

Neutron stars are massive,compact astrophysical objects. They can be modelled as

spherically symmetric stars with a speciefic equation of state. In general relativity several

models or neutron stars have been investigated. Specially when we adopted a linear

equation of state,the model has been studied as quark matter star. In modified gravity of

f(R) gravity,this problem has been solved perfectly using an empirical equation of state.

In this work we formulated neutron stars in a Gauss-Bonnet modified gravity,so called

as f(G) gravity in which G denotes the topological invariant term of four dimensional

spacetime. We derived the full set of equations of motion for spherically symmetric

metric filled by cosmic fluid. We also derived the modified Tolman-Oppenheimer-Volkoff

equation. This equation shows the dynamical behavior of the model in terms of the

mass,pressure and energy density. This set of equations is dimensionless. We should

specify the model for f(G). A way to find the solutions is to think about the modified

part as a perturbation. So, we are able to perform perturbations for all variables. But

6

a more realistic approach is to solve equations numerically which is beyound our current

letter.

5 APPENDIX

In this appendix we present different geometrical quantities as we used in this let-

ter. For metric (3) the following non zero components of the symmetric connection are

obtained:

Γ112 = φ, Γ2

11 = φ′e2φ−2λ, Γ222 = λ′, Γ2

23 = −re−2λ, (17)

Γ244 = −r sin2 θe−2λ, Γ3

23 =1

r, Γ3

44 = − sin θ cos θ, Γ424 =

1

r, Γ4

34 = cot θ. (18)

So, the non vanishing components of Einstein tensor read:

G11 = −e2φ−2λ(2λ′

r+

e2λ − 1

r2), (19)

G22 = −2φ′

r+

e2λ − 1

r2(20)

G33 = −re−2λ(

φ′ − λ′ + r(φ′′ + φ′2)− rφ′λ′)

(21)

G44 = − sin2 θG33. (22)

For Riemann tensor we obtain:

R1212 = e2φ(φ′λ′ − φ′′ − φ′2), R1313 = −re2φ−2λφ′, R1414 = sin2θR1313, (23)

R2323 = −rλ′, R2424 = sin2 θR2323, R3434 = sin2 θr2e−2λ(1− e2λ). (24)

Thus the Ricci scalar is as the following:

R = −2e−2λ(

φ′′ + φ′2 − φ′λ′ + 2φ′ − λ′

r+

1− e2λ

r2

)

. (25)

The GB term reads:

−e4λG

4=

3(1− e2λ)2

r4+

8e2λφ′2

r2+ 3λ′2φ′2 + 24

φ′λ′

r2+ 6φ′′φ′2 +

8e2λφ′′

r2(26)

−6λ′φ′3 + 3φ′4 +6λ′2

r2+ 3φ′′2 − 2

φ′2

r2− 6λ′φ′φ′′ − 8

e2λφ′λ′

r2− 8

φ′′

r2.

7

To derive the hydrostatic equation we use the following equation:

∇µTµ2 = ∂µT

µ2 + Γµ

µσTσ2 − Γσ

µ2Tµσ = 0, ∂µ(−pδµ,2)− pΓµ

µ2 + pΓaa2 − ρc2Γ1

12 = 0,

−dp

dr− (p+ ρc2)Γ1

12 = 0 =⇒ dp

dr= −(p+ ρc2)φ′.

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