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Representative space.
We have seen, in a preceeding section, that a cylinder
can be put flat. Now, take a piece of paper, a flat sheet. It is an euclidean
surface. You can draw geodesic on it. Now crumple it.
(64)
If you could make this crumpled surface rigid and draw
geodesics on it, with sticky tape, you would meet again the same system !
The surface wasn't really changed. If some inhabitant lives in such a "flatland"
he perhaps can't realize the crumpling process. Everything would stay normal
for him, what life is like for him today : following geodesic of his 2d space
time surface, for exemple.
Crumpling the sheet, you have just modified the representative
system, the way the 2d surface imbeded 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 Iwas in 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 would
have drawn a geodesic before operation, we would have found that the geodesic
system would be unchanged.
But, to tell the truth, Inot really sure that this kind
of guy knows what a geodesic is, from a mathematical point of view, of course.
Any person who build baskets uses geodesing lines, naturally.
Iremember Iwas a teacher in basketry, at a holiday camp,
near by Burlington and lake Champlain, Vermont ... many years ago.
Keep in mind that geometrical objects have their own existence
and properties, which are independent from the way you figure it, in a higher
dimensions' number space. Crumpled or not, a sheet is a sheet, i.e. an 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 conviced that Ilive in a higher dimensions's 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
Iam skeptical about it, but that is another subject.
Admit you live in a 4d hypersuface and follow the geodesics
of this space-time, that 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. Personnaly,
Ithink that, if God exists, he must live in a ten dimensional hyperworld.
The subsequent arguments will be developped in Geometrical Physics B, and
come from group theory.
Does God owns a group structure ?
Practically, the specialist of general relativy 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 a 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 return to our 2d didactic image, to the blunt cone.
It is suposed do 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, in that time, Albert was
not a lonely genious, lost on a desert island. Many people think that the
great German mathematician Hilberth 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 immediatly :
- 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 univers 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 Ihad of known that Universe was non steady, Iwould
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. Iam 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, outiside, r and
T are null.
(68) 
This is a composite geometry. In 3d the
mass density shows an abupt 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 linkeage 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 eliptic paths is 0.15
° per century.
(69) 
Someday we will include the formulas and
program which makes 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
