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Representative space.
We have seen, in a preceding section, that a cylinder can be flattened. Now, take a piece of paper, a flat sheet. It is an Euclidean surface. You can draw geodesics on it. Now crumple it. (64)
If you could make this crumpled surface rigid and draw geodesics on it, with sticky tape, you would again find the same system! The surface wasn't really changed. If some inhabitant lived in such a "flatland", he might not realize the crumpling process. Everything would remain normal for him, as it is today: following the geodesics of his 2D space-time surface, for example.
Crumping the sheet, you have simply modified the representative system, that is, the way the 2D surface is embedded in 3D Euclidean space.
A simpler modification is to transform a flat sheet of metal into a corrugated surface. See figure (65) (65)
Many years ago I was in a big market of Addis Abeba, Ethiopia. There, metal is rare. You find factories where young men transform corrugated iron surfaces into flat plates, using a simple hammer. If one of them had drawn a geodesic before the operation, we would have found that the geodesic system would remain unchanged.
But, to tell the truth, I am not really sure that this kind of person knows what a geodesic is, from a mathematical point of view, of course. Any person who builds baskets naturally uses geodesic lines.
I remember I was a teacher in basketry, at a holiday camp, near Burlington and Lake Champlain, Vermont... many years ago.
Keep in mind that geometric objects have their own existence and properties, independent of the way you represent them in a space with a higher number of dimensions. Crumpled or not, a sheet is a sheet, that is, an Euclidean surface.
We are supposed to live in a 4D hypersurface. We all live the same way, in principle. But my wife Claire, who is a very charming person, is convinced that I live in a space with more dimensions (five, according to her). This sometimes causes communication difficulties when I am somewhere in my personal fifth dimension.
But do women really live in a 4D hypersurface? Sometimes I doubt it, but that is another question.
Assume you live in a 4D hypersurface and follow the geodesics of this space-time, like an ox following its furrow.
Now suppose you are God. You want a complete representation of this 4D hypersurface. Then you need at least one more dimension. Personally, I think that, if God exists, he must live in a ten-dimensional hyperworld. The following arguments will be developed in Geometrical Physics B, and come from group theory.
Does God have a group structure?
Practically, the specialist of general relativity calculates a solution of a field equation (Einstein's solution). Then he examines the geodesic system. They are "straight 4D-lines". In space-time, as you follow geodesics, the general order is:
- Go straight! Don't turn left or right.
You obey, simply because you cannot do anything else. Turning is nonsense in space-time. Everything, everyone "goes straight".
But things, paths, trajectories, seem curved to our 3D eyes. We read them in our mental space representation. We face the wall of Plato's cave, looking at dancing 3D shadows.
Let us return to our 2D didactic image, to the blunt cone. It is supposed to represent space near a mass concentration (grey area). We assume it corresponds to a steady state.
We may use spherical coordinates (r, q, j) as space markers (in 3D). In 2D we have only two: (r, q).
Then we can project the figure on a plane and use the same set of polar coordinates. See next figure.
(66)
As said above, the blunt cone surface is a crude didactic model, suggesting a particular solution of the Einstein field equation
(67) S = c T
built in 1917 by Schwarzschild. This is a brilliant and clever work. Just to say that, at that time, Albert was not a lonely genius, lost on a desert island. Many people think that the great German mathematician Hilbert invented the "Einstein equation". Some others have suggested that Mrs. Einstein could have contributed efficiently to the building of special relativity, which naturally comes from the works of Poincaré and Lorentz (if you look at Einstein's works, you will see that he rarely mentions other people).
Schwarzschild's solution is a milestone of general relativity. One uses it to compute the relativistic trajectories of planets around the sun, demonstrating the precession of Mercury's perihelion.
Anyone would say immediately:
- Why didn't Schwarzschild calculate that himself?
There was a very good reason for it: He was dead.
Schwarzschild was a patriot and insisted on going to the front in 1917. There he was gassed and died later. Einstein continued the work, which became "the Einstein theory".
It was a steady-state solution. Later, Einstein tried to build a model of the universe, where curvature could be identified with energy-matter content. But, at that time, nobody knew that the universe was non-steady. Albert tried to build a steady-state model, but things did not go well. Then he visited Elie Cartan, a great French mathematician, who recommended adding a constant in the field equation, which is what Einstein did.
Then a Russian glider pilot named Friedmann invented a non-steady solution. Around the same time, Edwin Hubble discovered the red shift and the non-steady features of the universe. Einstein was very disappointed and said:
- If I had known that the universe was non-steady, I would have found the solution before Friedmann!
As the Lacedaemonians used to say.
But that story did not end there. Initially, Friedmann had built the cyclic solution, one of the three that compose the "Friedmann models".
Einstein remained silent for years. Then, after Friedmann's death, he published the "Einstein-de Sitter model", "the parabolic Friedmann solution".
Later, a young Polish researcher named Kaluza submitted a paper to "Professor Einstein", which was refused for more than a year. Kaluza complained to Einstein, who replied:
- You should look more closely at this theory. I am skeptical...
Many years later, Kaluza's idea (adding a fifth dimension to space-time) became the starting point of advanced works (including the superstring approach). See Geometrical Physics B.
Well, Albert was not such a sportsman...
Let us return to the steady-state 3D model corresponding to space-time geometry around the sun. The calculation gives geodesics located in planes. If the curvature effect is moderate, and the velocity small compared to the speed of light c, their projection, in a representative Euclidean space-time, corresponds to quasi-Keplerian paths and Kepler's laws. We can ignore time and represent these geodesics in planes, using polar coordinates.
r = f (q).
In the Schwarzschild solution, there are actually two linked "metric solutions", as shown in figure (68). Inside the "massive body", the mass density r is supposed to be constant. There, the energy-matter T tensor is non-zero. But outside, r and T are zero.
(68)
This is a composite geometry. In 3D, the mass density shows a sudden discontinuity at the surface (assumed to be a sphere) of the "mass concentration". This is similar to the discontinuity of the angular curvature density on the surface (non-zero in the grey area, zero outside). The boundary becomes a S1 sphere, that is, a... circle.
In 4D, the mathematical link can be established to ensure the continuity of the geodesic lines. This is similar to the linking portion of a sphere or a posicone.
When the mass becomes significant (which cannot be described by our crude 2D didactic model), the closed paths are no longer elliptical.
See figure (69). This drawing corresponds to the trajectory of a spacecraft around a neutron star.
The trajectory of Mercury around the sun is similar, but the precession of the perihelion of the elliptical paths is 0.15° per century.
(69)
Someday we will include the formulas and program which make it possible to play with that problem. It is not very difficult.
Some mathematical information at the end of this page and the next one, if you want you can go directly to page 13