twin universe cosmology Matter matter ghost astrophysics. 2: Conjugated steady state metrics. Exact solutions. (p3)
Now consider the associated internal solution, for the metric g*. It corresponds to the field equation :
(40) S* = - c T
We choose the metric form :
(41)
ds2 = en c2 dt2 - [ el* dr2 + r2 (dq2 + sin2q dj2) ]
Then the differential equations system is :
(42)
(43)
(44)
Notice the minus signs in the right hands.
We first add (42) and (43) and get :
(45)
Since c is negative it implies that l*' + n*' is negative or zero. From the system (42) + (43) + (44) we get :
(46)
(47)
(48)
Write :
(49)
Combining to (46) :
(50)
Identical to (27).
(47) can be solved :
(51)
or :
(52)
Introduce :
(53)
we get :
(54)
B being a constant. Then the internal metric g* becomes :
(55)
The link with the corresponding external solution is ensured if :
(56)
and our internal metric g* becomes :
(57)
