A being living in a hyperbolic universe would be totally unable to perceive only through their senses, who lives in an area seen as curious from outside. Within the Poincaré disk, to then be moved to the edge of the disc, will go shrinking along with all their measuring instruments. Everything, absolutely everything, including himself, will go to contract in the same proportion. If traveling several thousand miles that be, seen from outside the circle of investment by a superior being able to live beyond this universe , it has become ten times smaller, I do not know, because everything around it has become ten times smaller. Their rulers to measure will be ten times smaller, like houses, cars and trees around it. According to him, believe you live in a perfectly flat universe, where the inhabitants are from outside as follows:
But we, that somehow we are superior beings to him able to see what he can not see, we realize exactly what is happening. However, there is a way in which he may realize that you live in a universe where space is not Euclidean but hyperbolic, and this form will not come through the senses but through his intellect. In a hyperbolic universe, the sum of the angles of a triangle is always less than two right angles (180 degrees). The experimental confirmation of this fact is what will let you know who lives in a hyperbolic universe. And if the sum of the angles of a triangle was greater than 180 degrees, then you know you live in a universe elliptical. An old story tells us that unconfirmed Gauss, the prince of mathematics, he tried to do just that, measuring as accurately as possible the internal angles of a triangle and add them huge, but the experimental results have been inconclusive. Even with a triangle the size of our solar system, may not have detected a difference in their smallness. This is precisely one of the dilemmas that confront modern astronomy. If the universe is a flat universe, Euclidean, then any direction will extend to infinity, what is called an open universe . But if the universe, by the enormous amount of matter we know it contains matter whose gravitational pull can significantly distort the spatial curvature of the Universe, is a elliptical universe, then it is a finite universe, infinitely great, but after all finite, we call a closed universe in which no matter in what direction we move in a straight line on an intergalactic spaceship capable of maintaining its forward unaltered course, sooner or later return behind the starting point, as well as when we move over the surface of the earth by one of its meridians.
The Poincaré disk creation, with the inhabitants of the universe hyperbolic unable to realize through their senses of living in a world so strange, surely had something to do with a philosophical discourse of the Poincaré who became one of the most famous philosophical puzzles of all time: the bending night.
Suppose that during the previous night, while everyone was sleeping, everything in the universe doubled in size, everything in the universe is twice as large today than it was last night. We now ask: is there any way in which we learn what happened? The first thing that comes to mind is that something so huge it would certainly be easily detectable, it would leave a mark extremely obvious. But this argument comes down to us when we realize that absolutely everything has doubled in size, including the rules and measuring tapes we use to measure. We could not use anything to measure the increase in size of all, because the tools we use to measure also increased in size in the same proportion. We could not use the yardstick , the bar made of an alloy of platinum-iridium kept in a suburb of Paris, because it also doubled its size. And the new basis for measuring length in the metric system, defined as 1,650,763.73 wavelengths of orange light emitted in a vacuum under an electric shock by the gas krypton-86 (which makes unnecessary travel to Paris to consult the standard meter) does any good, because fluorescent tubes containing krypton gas was twice as large than it was last night, like the atoms of krypton gas in them. The orbits of electrons around the nucleus of the atoms of krypton gas have doubled in size, which is why their light has a wavelength that is twice of what it was last night, so the retinas of our eyes twice their normal size will still see the light emitted from the same orange color.
We went outside and everything around it has also doubled its size. The car in which we move is twice as big as last night, so we took the car without detect what has happened. The streets and roads have doubled in size, so we will continue walking the same distance as usual. Since everything in the universe has doubled in size, occur to us that to see the stars from the sky during the night we will watch with less intensity, since they are now much further away from us twice the distance we had from them night before folding. But that does not serve us anything, because our retinas have enlarged twice and are twice as sensitive to light coming from those stars who now have double the size they were yesterday. Nothing around us, it seems, will give us the slightest clue of what happened while we slept.
Since Poincaré gave the hypotheses about what would be our absolute inability to realize what happened while we slept, that doubling the size of everything, including our inability to realize that everything can be increasing size now while my readers are reading this exhibition, for several decades it was assumed that something so monumental be completely out of our perceptions, even the same Poincaré postulated that could not speak of such a change was real in the sense that, being beyond our ability to identify, was out of the reality of our lives.
However, more recently, there have been philosophers who have questioned the idea that a doubling of the universe is undetectable night. And they have done something using mathematics in their pride have left out: the use physics as a means to verify or justify experimentally the reality and consequences of those ideas purely theoretical. According to these philosophers of art, a doubling night would have severe consequences that would be easily detectable in several ways. Among these philosophers we quote Brian Ellis and George Schlesinger, who in his works published in 1962 and in 1964 set out the reasons why a doubling all night long could not hide from us. Schlesinger, using the law of universal gravitation formulated by Isaac Newton (the bodies are drawn in direct proportion to the product of their masses and inversely as the square of the distance between their centers of gravity), says that after a night doubling the gravity of the Earth would be a quarter of what it was before, because the radius of the earth would have doubled in size while the bulk would remain the same. Doubling the radius of the Earth without duplicating its mass takes us on an "inverse square of distance, to the gravitational attraction falls by half but not the fourth. Suddenly, we all feel much lighter than before, as if we were walking on the surface of the moon.
And there were other ways of measuring the bending night. Although the change would not be detectable by using a scale, because the gravitational pull in both arms of the balance would fall identically, the change would be detectable using mercury barometers such as those used in the past to measure atmospheric pressure. In these barometers, the height of mercury column is what we measured pressure, so when we say that at sea level atmospheric pressure is 760 millimeters of mercury which are actually saying is that in a column of mercury it will reach a height of 760 mm. This height depends on three factors: air pressure, the density of mercury, and the force of gravity. Under normal atmospheric pressure varies very little. The atmospheric pressure would be eight times lower after the night bending, since all the volumes increase due to the cube of the distance. The density of mercury would also be eight times greater, so that these effects would cancel each other. But as gravity would be the fourth of what it was before, the mercury would rise to a height four times greater than normal amount, which in our barometers that measure twice what would be detected as measured by a height twice height they reached last night. Thus, here we not only have a way of confirming that night there was an increase in the size of everything, but even magnitude of the increase. Applying the doubling
night to other situations, Schlesinger was equally interesting conclusions. Specifically, they showed that a pendulum clock would measure any time an amount greater by a factor of 1.4142 (the square root of two). Assuming other fundamental laws of physics remain unchanged, the bending night there also manifest their effects. One is the law of conservation of angular momentum . Assuming that after the night bending remains a conservation of angular momentum, then the Earth's rotation would diminish and the days longer than they were before. Another fundamental law hitherto remains unchanged is the law of conservation of energy , which can be applied to the entire universe before and after a nighttime bending. In planetary theory Niels Bohr on the atom, it is considered that the electrons are whirling around the nucleus and are held in orbit by the electrical attraction between Bohr assumed that the negative charge (-) of the electrons orbiting around the nucleus with positive charge ( +). A night doubling everything in the universe require that the hydrogen atoms (the most abundant element in the universe) the electrons circling the atom are now at a greater distance from the atomic nucleus by a factor of 2. Carry an electron from its orbit to a new orbit farther from the nucleus required to provide energy to carry out the separation process (Bohr in physics, this energy is supplied by a quantum of light energy is absorbed by the atom, which allows the electron to "jump" to a more distant orbit.) For all hydrogen atoms in the universe (and all other atoms of other elements which are planets, asteroids and galaxies) measured twice as measuring yesterday, you will need to provide a prodigious amount of energy, which has to come from somewhere so it can continue to fulfill the principle of conservation of energy. This would come at the expense of a cooling of the Universe .
The overnight doubling of the universe would bring many consequences for many physical phenomena, it would be impossible not to detect it through our senses. It would be the best event confirmed in human history.
Returning to the Euclidean geometry and non-Euclidean geometries, although these geometries can be built in a completely axiomatic on the blackboard, without leaving for anything outside the home, without seeing what happens outside, as something disconnected from physical reality Nature around us, the philosophical conundrum Poincaré legacy and how it has been solved physics tells us that, however much they be repugnant to many mathematicians the idea, it might be an inseparable part of what mathematics can be able to describe in relation to the real world. Einstein himself made a mistake when it believed that everything could be derived without carrying out any experiment (one famous anecdote tells us that when someone asked Einstein where his laboratory, he took the pen from his lapel and said pointing to her "Here"). In so doing, closing the door, Einstein himself was deprived of many important contributions that could be carried out especially in the area of \u200b\u200bmodern quantum mechanics, with which he maintained a personal conflict to the end of his days .
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