Jean-Pierre Petit
Lambda Laboratory
...In the section devoted to MHD, we have seen that it could be possible, using disk-shaped MHD aerodynes, to cruise at hypersonic velocity at low altitude without creating sonic booms or turbulence—a completely noiseless flight.
...Second question: Is interstellar travel possible?
...Classical answer: No, due to the constraints of special relativity.
...A solution suggested by O'Neill: Humans could travel to other stars if they accept that only their distant descendants could reach these other systems. It would have to be a one-way journey, with no possibility of return, which implies huge spacecraft, as large as major terrestrial cities, carrying grass, trees, animals—everything. The modern version of Noah's Ark. Source of energy: Hydrogen collected along the way, combined via fusion processes. Source of materials: Asteroids.
...Poetic...
...Of course: There would be no possibility of communicating with those who remain on Earth. I am skeptical. Moreover, I think that if we were to build such a monster and take place in it, upon reaching another distant planet, orbiting another star and inhabited by human creatures, upon landing, these people would say:
- Glad to meet you. We expected you. Your descendants warned us twenty thousand years ago. You know, there are more modern ways to travel now.
...I wouldn't take the risk of being so ridiculous. So, can we think about something fairly different?
...The reader may have a look at the papers on my website, devoted to theoretical cosmology. Recent works will be presented in Marseille, France, in June 2001, at the international meeting on astrophysics and cosmology, entitled "Where is the matter?", organized by the Laboratoire d'Astrophysique de Marseille (to which I belong).
1 - Twin universe geometry.
...The concept of the twin universe was first introduced by Andrei Sakharov in 1967 ([1], [2], [3], [4]). Later, I published two papers in the French Comptes Rendus de l'Académie des Sciences de Paris ([5] and [6]), unaware of Sakharov's previous works. The underlying geometric structure corresponds to a two-point bundle. Give the fold of this bundle the metric structure (g, g*), where g and g* are Riemannian metrics with signatures (+ - - -).
Fig. 1: Twin universe: a two-point bundle with Riemannian metric structure (g, g*).
...We obtain a point-to-point mapping linking two "conjugate points" M and M*, which can be described by the same system of coordinates {µi}. Let F and F* be the two folds composing the bundle. With the two metrics, we can build systems of geodesics, but since F and F* are disconnected, the two families of geodesics are disconnected. As a conclusion, if these metrics yield null-geodesics and if one assumes that light travels along them in both folds, any structure in a given fold will be geometrically invisible from the other one.
...In classical General Relativity, one considers a single fold, associated with the field equation (Einstein equation):
(1)
S = c T - L g
where S is a geometrical tensor, c is the Einstein constant, T is the energy-matter tensor, and L is the so-called, puzzling cosmological constant, introduced by the French mathematician Élie Cartan.
...Consider the following coupled system of field equations:
(2)
S = c (T - T*)
(3)
S* = c (T* - T)
from which we immediately get:
(4)
S* = - S
Note that this definitively does not imply g* = - g.
...The Newtonian approximation gives the following Poisson equation:
(5)
Δφ = 4πG(ρ - ρ*)
In this new model:
- Matter attracts matter, through Newton's law.
- Twin matter attracts twin matter, through Newton's law.
- Matter and twin matter repel each other through an "anti-Newton law."
What about the classical local check of General Relativity?
...The solar system is a very dense portion of the universe. In the adjacent portion of the twin fold, twin matter is pushed away. Then the system is very close to:
(6)
S = c T
(7)
S* = - T
...Equation (6) reduces to the Einstein equation, so that all classical verifications hold. What about gravitons? Which path do they follow? The answer is composed of two arguments:
- Field equations provide a macroscopic description of the universe, which ignores the existence of particles and just gives systems of geodesics.
- By the way: What is a graviton?
2 - The question of the repulsive power of vacuum. An alternative answer.
...When we look at equation (2), we see that T* acts like a "cosmological constant." It represents the "repulsive power of the twin universe," which can play a role in non-steady coupled solutions. The assumption of homogeneity and isotropy gives the Riemannian metrics the well-known Robertson-Walker form, as follows:
(8)
(9)
...The radial distances between conjugate points (same u, an adimensional "radial distance," with respect to an arbitrary point) are not automatically equal:
(10)
r = R u .......................r* = R*u
Write adimensional coordinates, where t is the time-marker.
(11)
{ t, u, θ, φ }
... { u, θ, φ } are classical spherical coordinates. Remember that a field equation is coordinate-invariant. The choice of coordinates remains free in each fold, where we can define different cosmic times:
(12)
t ...and... t*
These variables are linked to the adimensional variable t through:
(13)
t = T t ............t* = T* t
where T and T* are characteristic time scales. Introducing adimensional proper times s and s*:
(14)
s = cT s .........s* = - c*T* s*
we transform the two metrics into their adimensional forms, introducing adimensional scale factors R(t) and R*(t), through:
(15)
R = cT R
R* = c*T* R*
(16)
(17)
...We put the field equations into their adimensional forms, using:
(18)
ρ = ρ₀ w
ρ* = ρ*₀ w*
p = p₀ p
p* = p*₀ p*
Following, these tensors, written in their adimensional forms:
(19)
At the end, we get four coupled second-order differential equations (instead of two, in the classical approach):
(20)
(21)
(22)
(23)
...We need some additional hypotheses. Assume that the two universes have "parallel lives" during their radiative epoch, i.e.:
w(t) = w*(t),
which imposes negative curvature indices (k = k* = -1). After decoupling, we neglect the pressure terms (dust universes):
(24-a)
(24-b)
(24-c)
(24-d)
from which we immediately get:
(25-a)
(25-b)
Introducing mass conservation in both folds:
(26)
ρ R³ = constant ρ* R*³ = constant
the system becomes:
(27-a)
(27-b)
...Notice that R = R* gives R" = R*" = 0. On the other hand, if the two universes were "fully coupled," i.e., R*/R = constant, this peculiar solution would correspond to Friedmann models, with "parallel evolutions." But we consider that they are coupled by the gravitational field, through (27-a) and (27-b), which shows that linear expansion is unstable. If, for example, R > R*, then R" > 0 and R*" < 0. The system can be solved numerically. The typical solution corresponds to Figure 2.
Fig. 2: The evolution of the scale parameters of the universe and the twin universe.
...We see that this system of two universes interacting through gravitational force is unstable. If one universe goes faster, pushed by its twin, the other one slows down. The observed acceleration of our universe is then caused by the "repulsive power of its twin universe." The histories of the two differ. Ours is cooler and more rarefied. The twin is warmer and denser.
3 - Other observational confirmations.
...The twin universe theory offers a lot of observational confirmations. See the papers on the website, and references [5], [6], and [7]. The action of repulsive twin matter on the matter of galaxies explains the "missing mass effect" and the flatness of the corresponding rotation curve at a distance:
Fig. 3: Galaxy confined by surrounding (geometrically invisible) twin matter.
Fig. 4: Corresponding rotation curve.
..