twin universe cosmology Matter ghost matter astrophysics. 2: Conjugated steady state metrics. Exact solutions. (p7)
Conclusion.
Studying a model based on two coupled field equations, referring to a two-fold structure, we have shown that non-homogeneous, steady-state exact solutions do exist and we have constructed them. A 2D didactic model was provided to illustrate the concept of conjugated geometries and induced geometry. The geodesic analysis confirms the results obtained via Newtonian approximation.
References.
[1] Petit J.P.: The missing mass effect. Il Nuovo Cimento B Vol. 109 July 1994, pp. 697-710
[2] Petit J.P.: Twin Universe Cosmology. Astrophysics and Space Science. Astr. And Sp. Sc. 226: 273-307, 1995
[3] J.P. Petit & P. Midy: Repulsive dark matter. Geometrical Physics A, 3, pp. 221-237, 1998.
[4] J.P. Petit & P. Midy: Matter ghost-matter astrophysics. 1: The geometrical framework. The matter era and the Newtonian approximation. Geometrical Physics A, 4, pp., 1998.
[5] J.P. Petit & P. Midy: Repulsive dark matter. Geometrical physics A, 3, Feb. 1998.
[6] J.P. Petit & P. Midy: Matter ghost-matter astrophysics. 1: The matter era and the Newtonian approximation. Geometrical physics A, 4, March 1998.
[7] R. Adler, M. Bazin & M. Schiffer: Introduction to General Relativity. Mac Graw Hill Book Company, 1965.
Acknowledgements:
This work is supported by the French CNRS and by the A. Dreyer Brevets et Développement company.
Deposited in sealed envelope at the Académie des Sciences de Paris, 1998.
Commentary on this article.
Mathematically, the solution presented has no singularities. We have simply neglected the pressure term in the field equations within the stress-energy tensor T, which then becomes:
which means that:
p is, dimensionally speaking, an energy density, in joules per cubic meter. So is rc². If the medium were gaseous, this would mean, for example, that pressure measures the kinetic energy density associated with a mean thermal agitation velocity . Suppose the interior medium can be approximated as an ideal gas. Then the matter pressure would be written as:
We see that the approximation made amounts to assuming that the thermal agitation velocity within the object is non-relativistic. Therefore, this model is suitable for describing "ordinary" stars, including spherically symmetric stars surrounded by vacuum, which do not rotate.
This solution differs from the previously developed one, which can be found, for example, in the book by Adler, Schiffer, and Bazin: Introduction to General Relativity, 1975, Mac Graw Hill Books. This earlier solution was explicitly designed to handle a medium with non-zero pressure. The matching between the exterior and interior metrics is achieved by setting p = 0 at the star's surface. We then obtain the metric:
It should be noted that if we perform a series expansion under the assumption:
the two metrics (this one and ours) asymptotically converge. In any case, when non-zero pressure is assumed, an equation of state p = p(r) is missing. Nevertheless, the work leads to the famous TOV equation (Tolmann, Oppenheimer, Volkov), a differential equation in (p, p', r), where p' denotes the spatial derivative of pressure.
m is the function m(r):
(see the article or the referenced works). This equation is classically used to describe the interior of neutron stars, where one simply assumes r = constant (on the order of 10¹⁶ g/cm³). One then obtains a differential equation describing the pressure evolution. Note that as the star's mass increases—supposedly at constant density, since the neutron packing is assumed incompressible—the first critical point encountered concerns the pressure, which becomes infinite at the center, even though the star's radius remains larger than its Schwarzschild radius.
Of course, we attempted to implement a similar solution for the two conjugated metrics. Physically, the problem is perplexing. In the sheet where the star is located—say, our sheet F—we have two scalar functions p(r) and r(r), intended to describe the pressure and density fields within the neutron star, with r(r) = constant. Given that the geometry in the second sheet follows from the equation:
S* = -c T
these quantities p(r) and r(r) appear in the right-hand side. Yet the second sheet is supposed to be empty (r* = 0) and have zero pressure (p* = 0). However, the chosen structure—the system of two coupled field equations—implies that these terms contribute to the geometry of the other sheet.
When applying the classical machinery, we recover similar equations, ultimately derived from the classical formalism by simply replacing r with -r and p with -p. We also obtain a TOV-type equation. But this differential equation must necessarily yield the same solution. There cannot be two different differential equations giving p(r). However, the equation we arrive at is different. It corresponds simply to the global transformation:
p → -p
r → -r
m → -m
with m → -m.
But the TOV differential equation is not invariant under this transformation, and we then obtain:
(the minus sign in the denominator becomes a plus sign).
Thus, there is no solution with non-zero pressure, at least within this approach inspired by classical methods. Far from discouraging us, this result seems to indicate that the problem must be approached differently—something we will attempt in future work dedicated to studying the approach to criticality in neutron stars. We have developed a model of the radiation era, corresponding to the paper Geometrical Physics A, 6, where physical constants are supposed to be effectively indexed on the value of radiation pressure. When going back before the decoupling epoch in the standard model, one indeed reaches conditions where not only does the pressure's contribution to the gravitational field cease to be negligible, but this contribution is then essentially due to radiation. This would imply that physical constants depend on electromagnetic energy density, i.e., radiation pressure. Therefore, we have begun an approach to studying neutron stars where the term:
is no longer negligible compared to r, assuming that physical constants (G, h, c, neutron mass, and other constants) depend on the local value of pressure (we are studying a stationary, equilibrium solution). Since the onset of criticality in the star begins with a rise in central pressure, and from this perspective the local speed of light would follow this increase, conditions where c becomes infinite should, in our view, be accompanied by a breakdown of spacetime topology at the star's core. As long as p and c remain finite, the geometry remains hyperspherical, meaning one can "peel" the neutron star down to its center. There is always matter, and one remains in the same sheet. However, and we are actively pursuing this line of thought, the rise of the local value of c toward infinity should trigger a topological change, with the geometry at the star's center transforming, possibly leading to the emergence of a "hypertoroidal bridge," a passage between...