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Representative space.
We have seen, in a preceding section, that a cylinder can be laid flat. Now, take a piece of paper, a flat sheet. It is a 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 encounter the same system again! The surface wasn't really changed. If some inhabitant lives in such a "flatland," he perhaps cannot realize the crumpling process. Everything would stay normal for him, what life is like for him today: following geodesics of his 2d space-time surface, for example.
By crumpling the sheet, you have merely modified the representative system, 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 the big market of Addis Ababa, Ethiopia. There, metal is rare. You find workshops 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 be unchanged.
But, to tell the truth, I am not really sure that this kind of guy knows what a geodesic is, from a mathematical point of view, of course. Any person who builds baskets uses geodesic lines, naturally.
I remember I was a basketry teacher, at a holiday camp, near Burlington and Lake Champlain, Vermont ... many years ago.
Keep in mind that geometrical objects have their own existence and properties, which are independent of the way you figure them, in a higher-dimensional number space. Crumpled or not, a sheet is a sheet, i.e. a Euclidean surface.
We are supposed to live in a 4d hypersurface. We all are supposed to live the same, by the way. But my wife Claire, who is a very charming person, is convinced that I live in a higher-dimensional space (five, she thinks). It induces, sometimes, some difficulties for communication when I am staying somewhere in my personal fifth dimension.
But do females really live in a 4d-hypersurface? Sometimes I am skeptical about it, but that is another subject.
Admit you live in a 4d hypersurface and follow the geodesics of this space-time, which you follow like an ox following its furrow.
Now suppose you are God. You want a full 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 subsequent arguments will be developed in Geometrical Physics B, and come from group theory.
Does God own a group structure?
Practically, the specialist in general relativity computes 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, just because you cannot do something else. Turning is nonsense, in space-time. Everything, everybody "goes straight."
But things, paths, trajectories, look bent for our 3d eyes. We read it in our mental space representation. We face the wall of Plato's cavern, looking at dancing 3d-shadows.
Let us return to our 2d didactic image, to the blunt cone. It is supposed to figure space at the vicinity of a mass-concentration (grey area). We assume it corresponds to 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 peculiar 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's equation." Some others have suggested that madam Einstein could have contributed efficiently to the building of special relativity, which comes naturally 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, evidencing the precession of Mercury's perihelion.
Anyone would say immediately:
- Why did Schwarzschild not calculate that by 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 job, 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 to energy-matter content. But, at this time, nobody knew that universe was non-steady. Albert tried to build a steady state model, but things were not so good. Then he visited Elie Cartan, a great French mathematician, who recommended to add a constant in the field equation, which is what Einstein did.
Then a Russian glider pilot named Friedmann invented a non-steady solution. About the same period Edwin Hubble discovered the red shift and non-steady features of the Universe. Einstein was very disappointed and said:
- If I had known that Universe was non steady, I would have found the solution before Friedmann!
If, the Lacedemonian used to say.
But that story didn't end there. Initially, Friedmann had built the cyclic solution, one of the three which compose the "Friedmann models."
Einstein kept silent during years. Then, after Friedman's death, he published the "Einstein-de Sitter's model," "the parabolic Friedmann solution."
Later a young Polish researcher named Kaluza submitted a paper to "Professor Einstein," refused to publish it for more than a year. Kaluza complained to Einstein, who replied:
- You should look closer to 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 sport......
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 with respect to light velocity c, their projection, in a representative euclidean space-time, corresponds to quasi keplerian paths and Kepler's laws. We can forget time and figure these geodesic in planes, using polar coordinates.
r = f (q).
In the Schwarzschild solution, there is, in fact two linked "metric solutions," as evoked on 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 null.
(68) 
This is a composite geometry. In 3d the mass density shows an abrupt discontinuity at the surface (supposed 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, null outside. The border becomes a S1 sphere, i.e. a ... circle).
In 4d the mathematical link can be built, in order to ensure the continuity of the geodesic lines. This is similar to the linkage portion of a sphere-portion of a posicone.
When the mass becomes important (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 elliptic paths is 0.15° per century.
(69) 
Someday we will include the formulas and program which makes it possible to play with that problem. It is not very difficult.
Some mathematical information on the end of this page and on the next one, if you want you can go directly to page 13
