twin universe cosmology Matter ghost matter astrophysics. 2:
Conjugated steady state metrics. Exact solutions. (p2)
3) Schwarzschild-like coupled internal exact solutions.
Consider the case where the fold F* is empty and the Fold F contains a massive object whose mass is M, whose radius is ro, filled by constant mass density r.
It corresponds to the set of equations :
(12)
S = c T
(13) *S = - **c T
with T* = 0. In the classical theory, one derives the internal Schwarzschild solution, giving the T tensor the form :
(14)
The chosen metric form is :
(15)
ds² = en c² dt² - [ el dr² + r² ( dq² + sin²q dj²) ]
In the second members of the differential equations, coming from the field equation we find terms :
(16)
The second corresponds to the contribution of the pressure to the field. It can be neglected for moderate pressures. In the case of a gas it corresponds to the approximation << c, the first being the thermal velocity. If the body is a solid (planet), it means that the pressure contribution is weak, that cannot be asserted if the object is a neutron star. We are going to deal, in the following, with the justified physical assumption :
(17)
Then the differential equation can be written into the simpler form :
(18)
(19)
(20)
c being the Einstein's constant :
(21)
We first add (18) and (19) and get :
(22)
Since c is negative it implies that l' + n' is positive or zero. From the system (18) + (19) + (20) we get :
(23)
(24)
(25)
Write :
(26)
Combining to (23) :
(27)
m(r) is a length, Schwarzschild length like. We refind the status of M(r) as a geometric mass.
(24) can be solved. Write :
(28)
or :
(29)
Introduce :
(30)
we get :
(31)
A being a constant. Then the internal metric becomes :
(32)
When r = ro, the external metric becomes :
(33)
or :
(34)
or :
(35)
The link with the external metric is ensured if :
(36)
Our (p » 0) internal metric solution becomes :
(37)
Notice that we perform expansions into series, according to :
(38)
our internal metric and the classical non zero pressure one [7] :
(39)
fit asymptotically.