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Conjugate Geometries.
...We will then associate a dull posicone and a "dull negacone," possessing the same amounts of curvature but with opposite signs: +q and -q. We can place them face to face (in the process creating a "point-by-point mapping": bijective, injective). There are then two sheets. Let us call them F and F*. To every point on F corresponds a point on F*.
...Let us arrange things so that the circular contours of the "dull parts," which carry curvature (positive in one sheet, negative in the other), correspond point by point. We illustrate this by projecting everything onto a plane. We obtain two surfaces having conjugate curvatures.
...The conical flanks are "uncurved"; they are elements of Euclidean surfaces. We will say that at every point of these surfaces the local curvature is zero. The spherical cap and the saddle correspond point by point. Their curvatures are opposite.
General Relativity.
...The starting point is the idea that the geometry of the cosmos is determined by its content in "energy-matter." Note that we use the term energy-matter rather than just matter, which clearly shows that any cosmic content influences geometry, including radiation, photons (or neutrinos). As we saw earlier, a photon creates a small positive curvature in space.
...We will first reason in a stationary context. A flat, free surface is one where tension is zero. We can modify its geometry by introducing tensions, positive or negative (the sign is a matter of convention). For example, if I heat a plastic film, I could cause a bulge to appear, i.e., a region with positive curvature.
...I could also apply a substance to the surface of a sheet of paper that, as it dries, contracts. The resulting tension will produce a region with negative curvature.
...A metalworker knows how to manipulate these tensions to deform a sheet. Take, for example, a metal tube. I heat one side and cool the other. What will happen?
The tube will bend, the heated side expanding and the cooled side contracting.
...In doing so, we have created tensions in the metal. This is the origin of the word tensor, in mathematics and geometry. The specialist in material strength will speak of the stress tensor. The geometer will speak of the curvature tensor.
The small experiment above illustrates the idea:
Local energy content -----> local geometry
...In General Relativity, we do the same. The difference is that this local energy-matter content determines the geometry of a four-dimensional hypersurface, not, as here, the geometry of a two-dimensional surface. But the idea is similar.
...The mathematician will then use a tensorial notation. It is not possible to say much more here for a non-mathematician. But Einstein's tensor S (we will use bold letters) corresponds to the geometric aspect. In Einstein's equation, it is identified with another tensor T, which describes the energy-matter content, up to a multiplicative constant, the "Einstein constant c."
Thus, Einstein's famous equation is written:
S = c T
...In the tensor T, the mass density ρ and pressure p appear (in fact, the most general tensor T is more complex, but we will content ourselves with this usual expression). In a stationary configuration, we will thus be given a certain distribution of density and pressure ρ(x,y,z), p(x,y,z). With this, we know how to construct the tensor T, which thus contains all the data of the problem. The question then becomes: "What geometry corresponds to this tensor T, satisfying the above equation?"
...In other words, the physicist, knowing the local content of the universe, seeks to determine the geometry of the spacetime hypersurface.
Geometry implies geodesics. This is where the second assumption of General Relativity comes into play:
We assume that objects moving through the universe
follow geodesics of the spacetime hypersurface.
By "object" we mean particles (elementary particles, photons, neutrinos), but also planets, stars, etc.
At this stage, a remark: where are the particles in all this?
...Answer: the specialist in General Relativity works at the macroscopic level. The input functions of the problem, the mass density ρ and pressure p, correspond to a macroscopic description of cosmic content. The same applies to the "output." And the geometer adds:
- You gave me functions ρ(x,y,z) and p(x,y,z); I have constructed the corresponding hypersurface, with its families of geodesics. But I cannot do more. In particular, I am unable to produce particles, atoms, etc. For that, consult another service...
In short: the bridge between General Relativity and particle physics has not yet been built.
But the astronomer will say:
- What does it matter? The assumption that photons follow certain geodesics of this hypersurface works. Proof: I can make observations. If I assume that planets, treated as point masses, also follow geodesics of this hypersurface, I can construct their trajectories. There are also gravitational lensing effects...
He is right.
...Let us say a few words about these gravitational lensing effects. Of course, this image of the dull cone is merely a didactic illustration. A planet orbiting circularly around a star also follows a geodesic of spacetime. But a circle drawn on a dull cone is not a geodesic:
This merely illustrates the limitations of didactic images, even when they are geometric.
...Photons indeed follow geodesics of the spacetime hypersurface. We may use this image of the dull cone to illustrate it. Light rays can pass on either side of a massive object, then converge toward the observer. If we project these geodesics, we obtain a mirage effect: the observer will have the impression of seeing two sources instead of one:
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