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 Fcontains a massive objet whose mass is M, whose
radius is ro , filled by constant mass density r matter.
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 is 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.
