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General Relativity and Curvature.
...We have said that matter curves space, determining the geometry of the universe, of the "hypersurface of the universe." But in General Relativity, curvature is either positive or zero. In our environment, we observe concentrations of mass: the Sun, planets, stars, etc... Between them, something we assimilate to empty space. But does this vacuum really exist?
...The physicist's vacuum is what remains after matter has been removed. But it is not nothing. The most complete vacuum is always filled with photons. Question: do photons create curvature in the universe?
...One might be tempted to answer "no," since photons are supposed to have zero mass. But this refers to their "inertial mass." Do they possess a "gravitational mass," contributing to the gravitational field?
Before discussing photons, let us talk about antimatter. Earlier, we constructed a surface with two conical points.
...Mechanically, if you built the object, you probably placed the two conical elements in the same direction. But you could have done it differently:
...A cone is a cone, whether its "tip" points upward or downward. If you construct this strange object and trace geodesics on it using your adhesive tape, you will reach the same result. These two conical points S1 and S2 are indeed points of concentrated positive curvature.
...If we equate curvature with mass, this is still a didactic image of geometry near two positive point masses.
...This is not such a bad image of matter-antimatter duality, and it brings us to a crucial point: antimatter has positive mass. Like matter, it contributes locally to positive curvature.
...Matter and antimatter can, upon meeting, annihilate each other to produce radiation, photons. And vice versa. Thus, we can give a didactic image of the photon by bringing the two tips S1 and S2 together. Then you construct your two conical elements by joining A to B, and C to D.
...Along the way, this model suggests that the photon is its own antiparticle. Indeed, one can no longer say in which direction the cone's tip is oriented.
...How can one subject cardboard to such contortions? But we will perform even more such manipulations later on. In any case, if you draw a geodesic triangle around the point where you have brought the two conical points together, you will find a positive excess compared to the Euclidean sum.
...The photon, as a result of this annihilation, this conjunction of matter and antimatter, curves space positively.
...At this stage, everything is positive: mass, curvature, energy. What would be the geometry created by negative mass? If such masses existed, they would create a local negative curvature. This leads us to discuss negative cones.
Negative Cones.
...To make a standard cone, a "posicone," we removed a sector corresponding to an angle q and joined the edges together. Here, we will do the opposite. We will cut into our cardboard sheet and instead insert a flat wedge of angle q.
...On the right, we have drawn a triangle made of geodesics. This time, the sum is less than the Euclidean sum by an angle q. We will say that point S is a point of concentrated negative curvature. With a rounded edge, we would have:
...Of course, if the geodesic triangle does not contain point S, the sum will be equal to π. The "side" of this negative cone is Euclidean, containing no curvature. The curvature, negative, is concentrated at point S.
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