twin universe cosmology Matter ghost matter astrophysics. 2 :
Conjugated steady state metrics. Exact solutions.
- (p1)*
Comment on this article.
Mathematically, the solution presented is without shadow points. We have simply neglected the entry pressure in the field equations, in the tensor** T**, which becomes:
which means that:
p is, dimensionally speaking, an energy density, in joules per cubic meter. rc2 also. If the medium were gaseous, this would mean, for example, that the pressure is the measure of the kinetic energy density, related to an average thermal agitation velocity . Suppose that the internal medium can be assimilated to a perfect gas. Then the matter pressure would be written:
We see that the approximation made then amounts to assuming that the thermal agitation velocity in the object is non-relativistic. This model is therefore good for describing "ordinary" stars, including stars surrounded by vacuum, with spherical symmetry, that do not rotate on themselves. This solution is different from the one previously developed and can be found described, for example, in the book of Adler, Schiffer and Bazin: Introduction to general relativity, 1975, Mac Graw Hill books. Immediately, this solution is then designed to handle a medium with non-zero pressure. We match the exterior metric with the interior metric by setting p = 0 at the surface of the star. We then obtain the metric:
It should be noted that if we then make series expansions by assuming:
the two metrics (this one and ours) converge asymptotically. Anyway, when we assume non-zero pressure, an equation of state p = p(r) is missing. But the work leads to the famous TOV equation (Tolmann, Oppenheimer, Volkov), which is a differential equation in (p, p', r) where p' denotes the spatial derivative of the pressure.
m is the function m(r):
(see the article, or the books). This equation is classically used to give a description of the interior of neutron stars, where one simply sets r = constant (on the order of 1016 g/cm3). We then obtain a differential equation giving the evolution of the pressure. Note that when the star increases its mass, which it is supposed to do at constant density, since this packing of neutrons is supposed to be incompressible, the first criticality that appears concerns the pressure, which takes an infinite value at the center, even though the star's radius is still larger than its Schwarzschild radius. Of course, we have tried to implement a similar solution for the two conjugated metrics. Physically, the problem is puzzling. In the leaf where the star is located, for example, our leaf, we have two scalar functions p(r) and r(r) which are supposed to describe the pressure field and the density in the neutron star, with r(r) = constant. In the measure where the geometry in the second leaf then follows from the equation:
S* = - c T
these elements p(r) and r(r) are then present in the second member. However, the second leaf is supposed to be empty (r* = 0) and with zero pressure (p* = 0). But the chosen structure, the system of the two coupled field equations, makes these terms contribute to the geometry of the other leaf.
When implementing the classical machinery, we find similar equations, which finally result from the classical formalism by simply changing r to - r and p to -p. We also find a TOV equation. But this differential equation must necessarily give the same solution. There cannot be two different differential equations giving p(r). However, the equation we arrive at is different. It simply corresponds to the global change:
p ---> - p r ---> - r m ---> - m
with: m ---> - m
However, the TOV differential equation is not invariant under this change and we then obtain:
(the minus sign in the denominator becomes a plus sign). Therefore, there is no solution, with non-zero pressure, at least according to this approach, inspired by the classical approach. Far from discouraging us, this finding seems to us to be an indication that the problem must be approached differently, which we will attempt in future works, devoted to the study of the approach to criticality in a neutron star. We have developed a model of the radiative era, which corresponds to the paper Geometrical Physics A, 6 , where the constants of physics are supposed to be somehow indexed on the value of the radiation pressure. When we go back before the time of decoupling, in the standard model, we arrive at conditions where not only the contribution of the pressure to the field ceases to be negligible, but where this contribution is then essentially due to radiation. This would mean that the constants of physics would depend on the electromagnetic energy density, alias radiation pressure.
Therefore, we have started an approach to the study of neutron stars, where the term:
is no longer negligible compared to r, assuming that the constants of physics (G, h, c, the neutron mass, plus other constants) depend then on the local value of the pressure (we are studying a solution supposed to be stationary, in equilibrium). As the entry into criticality of the star begins with the rise of the pressure at the center, and that in this perspective the local value of the speed of light would follow this rise, conditions where c is infinite should, according to us, go with a rupture of the space-time topology, at the heart of the star. As long as p and c remain finite, it remains hyperspherical, that is to say that one can "peel" the neutron star all the way to its center. There is always matter and we are always in the same leaf. But, and we are working in this direction, the rise of the local value of c towards an infinite value should lead to a change in topology, the geometry at the center of the star changing, with the appearance of a "hyper-toric bridge", passage between the two leaves. The matter would then flow at relativistic speed. We have considered two possible options. Either the matter input would make the star enter criticality relatively slowly (absorption of stellar wind from a companion star, for example). Then this hyper-toric bridge could lead to a quasi-stationary situation, acting like a overflow. The star would evacuate, through this passage, the excess matter it receives from its companion.
But, second option, a faster input with a more abrupt entry into criticality (for example, during the fusion of a double system, consisting of two neutron stars) the stationarity or quasi-stationarity could no longer be invoked and it would then be necessary to try to build a still speculative scenario: the rapid hyperspatial transfer of a significant part of the mass, towards the other leaf.
