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General Relativity and curvature.
...We have said that matter curved space, determined the geometry of the universe, of the "universal hypersurface". But, in General Relativity, curvature is either positive or zero. In our environment we see concentrations of mass: the sun, the planets, the stars, etc... Between them, something that we equate with a vacuum. But does this vacuum exist?
...The physicist's vacuum is what remains when matter has been removed. But it is not nothing. The most thorough vacuum is always populated with photons. Question: do photons create a curvature, in the universe?
...One would be tempted to answer "no", since photons are supposed to have a zero mass. But it is a matter of their "inertial mass". Do they have a "gravific mass", contributing to the gravitational field?
Before speaking of photons, let us speak of anti-matter. A moment ago we made a surface with two conical points.
...Mechanically, if you made the object, you must have arranged the two conical elements in the same direction. But you could have proceeded differently:
...But a cone is a cone, whether it directs its "point" upwards or downwards. If you build this strange object and you trace geodesics on it, with your adhesive tape, you will arrive at the same result. These two conical points S1 and S2 are indeed points of concentrated positive curvature.
...If one equates curvature and mass, it is always the didactic image of the geometry in the vicinity of two positive point masses.
...It is not such a bad image of the matter-antimatter duality and it makes us touch with our finger one thing: antimatter has a positive mass. Like matter, it contributes to creating locally a positive curvature.
...Matter and antimatter can, on meeting, annihilate each other to give radiation, photons. And vice-versa. One can therefore give a didactic image of the photon by bringing the two vertices S1 and S2 together. Then you make your two conical elements by joining A and B, C and D.
...Incidentally this model suggests that the photon is its antiparticle. Indeed one can no longer say in which direction the point of the cone is oriented.
...How can one subject cardboard to such contortions? But we will do many others of this kind later on. The fact remains that if you trace a triangle of geodesics around the point where you will have made your two conical points converge, you will find a positive excess over the Euclidean sum
...The photon, as a result of this annihilation, of this conjunction of matter and antimatter, positively curves space.
...At the stage where we are, everything is positive: masses, curvature, energy. What geometry would be created by a negative mass? If such masses existed, they would create a local negative curvature. This brings us to speak of negacones.
The negacones.
...To make a classical cone, a "posicone", one removed a sector corresponding to an angle q and joined edge to edge. Here we are going to do the opposite. We are going to make a cut with scissors in our cardboard sheet and on the contrary fit into it a sort of flat wedge, of angle q.
...We have shown, on the right, a triangle made of geodesics. The sum is this time less than the Euclidean sum, by an angle q. We will say that the point S is a point of concentration of negative curvature. With a round edge, we would have:
...Of course, if the triangle made by geodesics does not contain the point S, the sum will be equal to p. The "flank" of this negacone is Euclidean, contains no curvature. This one, negative, is concentrated at S.
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