Those who are specialists in GR may go through the paper and express their valued opinions. This paper on which the no black holes paper is based appears in the Research Gate.
Schwarzschild Interior Solution with Λ
Nalin de Silva Devadithya
Present Address: 1695, 24th Avenue NE, Issaquah, WA 98029, US
Email: nalindesilva48@gmail.com
ABSTRACT
The Schwarzschild de Sitter solution for the exterior of a spherical mass distribution is well known. In this paper a solution is found for the interior of a spherical mass distribution of constant density with the cosmological constant.
INTRODUCTION
The well-known exterior solution of a spherical mass distribution, with the cosmological constant can be written as
(1.1)
where m = GM / c2 and Λ is the cosmological constant.
We derive an interior solution for a spherical distribution of matter that merges with the above solution at the boundary.
The interior Schwarzschild solution without the cosmological constant is given by
ds^2 = [3/2(1-r02/R2)1/2 - 1/2(1-r2/R2)1/2]2 c2 dt2 - (1-r2/R2)-1dr2 - r2(dθ2 + sin2θdΦ2) (1.2)
where R2 = 3c2/8πGρ0 and ρ0 is a constant.
Let M = (4/3)πr03ρ0 , r0 being the radius of the spherical distribution and M being the mass of the spherical distribution of matter and radiation. It is clear that R2 = r03 /2m, where m = GM /c2.
The solution (1.2) is obtained by considering the field equations
R^μν-1/2 R ̅g^μν =〖κ T〗^μν (1.3)
where Rμυ , gμυ, are the Ricci tensor, metric tensor and Tμυ the energy momentum tensor of ordinary matter and radiation given by Tμυ = (ρ0 + p/c^2 ) ( dxᶮ)/ds dxᶹ/ds – ( p/c^2 ) gμυ,
( R) ̅ is the Ricci scalar, and κ =(-8πG)/c^2 . We follow the sign convention given in Adler et al [1] .
As has been noted (see for example Adler et al [1]) the constant ρ0 does not imply that the density of the distribution of matter and radiation is a constant but the mass M is given by M = (4/3)πr03ρ0. In arriving at the solution (1.2) the boundary condition p =0 when r=r0 has been used. At r = r0 , the solution merges with the Schwarzschild exterior solution
d〖s 〗^2= (1-2m/r)c^2 dt^2-(dr^2)/(1-2m/r)-r^(2 ) (dϴ^(2 )+ sin^2 ϴ dφ^(2 )) where m = G M / c2
FIELD EQUATIONS WITH Λ
We now consider the field equations
R^μν-1/2 R ̅g^μν =〖κ T〗^μν-Λg^(μν ) (2.1)
We define Λ^' equal to (Λc^2)/8πG in order that Λ^' has same dimensions as density ρ[ML^(-3) ] and could be compared with ρ. Since κ =(-8πG)/c^2 , (2.1) can be written as
R^μν-1/2 R ̅g^μν =〖κ ( T〗^μν+ Λ^' g^μν)
=κ [(ρ0 + p/c^2 )dxᶮ/ds dxᶹ/ds – ( p/c^2 ) gμυ + Λ^' g^μν] (2.2)
For a stationary space time (2.2) can be written as
R^μν-1/2 R ̅g^μν = κ [(ρ0 + Λ^')g^00 –( p/c^2 - Λ^') g αα ] where α = 1,2,3
This implies that Λ^' contributes positively to the density and negatively to the pressure.
As Hemantha and de Silva [2, 3] have shown the FLRW metrics give rise to the equations
-(3a ̇^2)/(a^2 c^2 )-3k/a^2 = κρ-Λ = κ (ρ + Λ^' )
k/a^2 +(a ̇^2+2aa ̈)/(a^2 c^2 ) = κp/c^2 +Λ = κ (p/c^2 - Λ^')
demonstrating the contribution of Λ^' to the density positively and to the pressure negatively in the FLRW metrics. Here a is the scalar factor.
This may imply that Λ^' contributes to the dark matter in the form of density and to the dark energy in the form of pressure. We also note that
6a ̈/a = κ (ρ + 3〖p/c^2 -2 Λ〗^' ) i.e. a ̈/a = (-4πG)/(3c^2 ) (ρ + 3〖p/c^2 -2 Λ〗^' ) (2.4)
This equation is similar to the equation one obtains in Newtonian gravitation in the case of a spherical distribution of matter of density ρ. ρ + 3〖p/c^2 -2 Λ〗^' can be considered as an effective density.
SOLUTION WITH Λ
We follow Adler et al [1] in deriving the Schwarzschild interior solution with Λ. The detailed calculation is not given here but replacing ρ0 and p/c^2 in Adler et al [1] respectively by ρ0 + Λ^' and p/c^2 - Λ^' , with the boundary condition that the pressure is zero at the boundary r = r 0 , one obtains
ds^2 = [3/2 {( ρ0 / (ρ0 + Λ^')} (1-r02/R2)1/2 - (1 )/2 {( ρ0 -2Λ^')/( ρ0 +Λ^' )}(1-r2/R2)1/2]2 c2 dt2 - (1-r2/R2)-1dr2
- r2(dθ2 + sin2θdΦ2) (3.1)
where R2 is now given by R2 =3c2/8πG (ρ0 +Λ^')
As expected, (3.1) reduces to (1.2) when Λ^'=0 with R2 =3c2/8πG ρ0 .
At r = r 0 , (3.1) becomes
ds^2 = (1-r02/R2)c2 dt2 - (1-r2/R2)-1dr2 - r2(dθ2 + sin2θdΦ2) (3.2)
Since R2 = 3c2/8πG (ρ0 +Λ^') , 1- r02/R2 = 1 - 8πG (ρ0 +Λ^') r02/ 3c2 = 1- 2m/r0 - 8πG Λ^' r02/ 3c2
= 1- 2m/r0 - 1/3 Λ r02
(3.2) takes the form
ds^2 = (1- 2m/r0 - 1/3 Λ r02) c2 dt2 - (1- 2m/r0 - 1/3 Λ r02)-1dr2 - r2(dθ2 + sin2θdΦ2) the Schwarzschild de Sitter metric at r = r 0.
When ρ0 = 0, (3.1) becomes ds^2 = (1-r2/R2)c2 dt2 - (1-r2/R2)-1dr2 - r2(dθ2 + sin2θdΦ2) (3.3)
Since R2 =3c2/8πG Λ^') now and Λ^' is equal to (Λc^2)/8πG, (3.3) is transformed to
ds^2= (1 - 1/3 Λ r2) c2 dt2 - (1- 1/3 Λ r2)-1dr2 - r2(dθ2 + sin2θdΦ2) (3.4)
the well - known de Sitter space time, in the absence of matter and radiation.
It should be noted that the solution (3.1) is valid only if r02 < R2. Since R2 =3c2/8πG (ρ0 +Λ^'), the solution is valid only if r02 < 3c2/8πG (ρ0 +Λ^'). That is only r0 >2m1,
where m1 =4πG (ρ0 +Λ^') r03/ 3c2.
If ρ0 =0, the solution is valid only if r02 < 3c2/8πGΛ^' = 3/Λ as shown by (3.1). The Schwarzschild interior solution is valid only if r02 < 3c2/8πG ρ0 i.e. if, r0> 2m.
REFERENCES
1.Adler, R., Bazin, M., Schiffer, M., (1965) Introduction to General Relativity,
United States of America: McGraw-Hill, Inc.
2. Hemantha M.D.P., de Silva Nalin, (2003), Annual Research Symposium University of Kelaniya, 2003, 61
3. Hemantha M.D.P., de Silva Nalin, (2004), Annual Research Symposium University of Kelaniya, 2004, 55