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...We are not made to live like brutes, but to follow the paths of virtue and knowledge (Dante, The Inferno).
(We are not made to live like fools, but to follow the paths of virtue and knowledge. Dante: The Inferno).
*** Introduction.
...We aim to present new ideas in cosmology, astrophysics, and theoretical physics.
The ideas underlying these works are relatively simple and can be expressed in three different ways.
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In technical language for specialists.
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In a schematic form.
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In a popularized form.
...There are various ways to popularize scientific concepts. In most cases, popularizers offer only verbal images designed to stimulate the reader's imagination. But, in the end, the reader has learned little.
...By fully exploiting imagery—meaning geometry, which is deeply connected with physics and all kinds of sciences—we can ensure the reader gains a double benefit. Analogies help the reader grasp the broad outlines of a scientific approach. At the same time, the reader will also acquire real geometric knowledge, particularly in 2D geometry.
A first schematic idea: Newton and anti-Newton.
To be schematic means to focus only on one aspect of a model.
...Cosmic structures are organized by a fundamental force: gravity. If cosmic objects did not attract each other, nothing—or almost nothing—would happen. Galaxies would not form, stars would not form either, and we would not be here to discuss it. For instance, if gravity were repulsive, it would keep all particles maximally separated, and the cosmos would remain shapeless, hence ultimately very boring.
Newton's law of universal gravitation is:
...G is the gravitational constant. m and m' are two masses, and d is the distance between them. As will be seen later, the model we propose consists in making two types of masses interact, which we call either "ordinary masses" (matter in the classical sense—what galaxies, stars, and atoms are made of) and "phantom masses," m and m*, so named because they are fundamentally unobservable optically. The interaction scheme is then as follows:
Ordinary masses attract each other according to Newton's law.
"Phantom masses" also attract each other, according to the same law.
However, we assume that ordinary matter and phantom matter repel each other, according to "anti-Newton":
...At this stage, one could place "point masses" in a computer and observe what happens when these attractive and repulsive forces act.
...The physicist would immediately say: this is equivalent to introducing negative masses m* < 0. Indeed, if we reverse the sign of one of the masses in Newton's law:
the force changes sign.
...If we reverse the signs of both masses, the force does not change sign. Thus, one might be tempted to conclude that the above model amounts to assuming that in the universe two types of masses coexist—positive and negative—implying the existence of particles with negative energy E = mc². The possibility of negative masses was considered by mathematician Jean-Marie Souriau in 1970, via group theory. See his book "Structure of Dynamical Systems," Dunod, pages 197 to 200.
...The problem with objects having negative energy is, among other things: what happens when they meet objects with positive energy?
...Recall that antimatter has positive mass and positive energy. When a matter particle meets an antimatter particle, they produce photons with positive energy. But when a particle with positive mass m > 0 and positive energy E > 0 meets a particle with mass -m and energy E* = -E, the result would be..... nothing.
Moreover, the physicist would object that we have never observed any particle with negative mass.
...Thus ends the schematic presentation. As will be seen later, the system actually functions "as if phantom masses were negative," according to Newton and anti-Newton, but the model is not so simplistic. To move beyond this schematic view, one must become familiar with a certain number of geometric concepts.
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