| 10 |
Space
of representation.
...We
had seen that a cylinder was a developable surface. Now take a sheet of paper. It is a flat, Euclidean surface. Its
geodesics are straight lines. Draw a few straight lines on this
sheet, then crumple it.
...If
you could rigidify this crumpled flat surface, you would realize
that this operation has in no way modified the distribution of
its geodesics, which you could draw again with your
adhesive tape. You have simply played with the mode of representation
of this plane in its three-dimensional embedding space.
A less
complicated way of proceeding consists in transforming a sheet
of metal into...corrugated iron:
Geodesics
: unchanged.
...Geometric
objects exist independently of the way
in which we represent them to ourselves, independently of their space
of representation .
...We
are supposed to inhabit a "four-dimensional hypersurface":
space-time. General Relativity consists in
trying to construct its geometry, as the solution of a field
equation, then in "reading" this geometry,
by analyzing the geodesics of the hypersurface. It is obvious
that there is no longer any question then of a space of representation. To do
this one would need to have a vision in five dimensions, which we do not
possess.
...In
practice, we use coordinates which are those of Euclidean space,
of projection. Let us imagine that we are looking for a geometric solution
suitable for describing space-time in the vicinity of a massive body
and inside it. We will assume that the system
possesses spherical symmetry. Moreover we will assume
that this system is stationary (or quasi-stationary).
...We
will then use spherical coordinates (r ,
q , j) . In two dimensions, we will have only two of them and our symmetry
will be circular. We will then use the system of polar
coordinates of the plane:
...This
model of the blunt body is a 2d didactic image of a
stationary solution which actually exists in General Relativity
and which was invented by the Austrian Schwarzschild,
in 1917, as a particular solution of the "Einstein equation"
:
S = c T
already presented above. This solution is
clever and subtle. As regards calculation, it is not simple to construct.
This precision to try to dispel a myth: that of an Einstein,
a genius isolated in the world of his time, peopled with ignoramuses.
...From
this solution one then shows that there exist, around a mass
possessing spherical symmetry, plane geodesics
, situated in planes, and one knows how to calculate their form
: r = f (q). These trajectories (or at least their
projection in our mental space of representation, Euclidean) are
"quasi Keplerian" and Kepler's laws then appear as
an approximation, when the mass which creates this geometry
(in the Newtonian vision, this "force") remains moderate, that is
to say when the local curvature, within this mass, remains weak.
...This
solution is one of the pivots of General Relativity
and, although this cannot be evoked by simple
didactic images like those we offer the reader, it is this one which
allows predicting and calculating for example the advance of the perihelion
of Mercury. Einstein used this solution to explain this effect,
already known, and at the same time reaped all the laurels
of what was henceforth called "Einstein's theory". Why
did Schwarzschild not exploit his discovery himself? Because
he absolutely insisted on enlisting and leaving for the trenches,
where he was gassed and died shortly afterwards.
...Moreover
one is not very sure that this famous
Einstein equation is really his. Apparently it would have been
suggested to him by the great mathematician Hilbert. Einstein
also did not welcome with enthusiasm the later discovery
of the Russian Friedmann, who discovered, for his part, the non-stationary
solution of the field equation allowing the evolution
of the universe to be described. Same thing, in 1921 for the work of the young mathematician
Kaluza, whose work, rediscovered, now constitutes the starting
point of superstring theory. These things are scientifically
of little interest and in no way diminish the value of Einstein,
but show that the sporting spirit does not necessarily go with the scientific
value of an individual.
In the solution developed
by Schwarzschild, technically, space is in two parts. Inside
the star the density of matter r is
assumed constant. The energy-matter tensor T,
which depends on it, is also non-zero. Outside
r and
T are zero.
...This
composite geometry is therefore a solution of two different
equations, with or without a second member. The density of matter
presents a discontinuity at the surface of the star (this is
also the case for the pair "interior" Schwarzschild solution
and "exterior" Schwarzschild solution. In this case the star is
a sphere of constant density and this falls abruptly
to zero at the surface of the star). But the continuity
of the geodesics can nevertheless be ensured,
through mathematical conditions of which the image has been
given above (truncated cone-spherical cap junction).
...When
the mass becomes large and the curvature effects pronounced, the
trajectories then deviate more clearly from the Keplerian model,
for example in the vicinity of a neutron star. Below
the advance of the perihelion around such a star (around the sun
the trajectory ellipse of Mercury advances by 0.15 degree per century).
...The
formula and the program which allow these trajectories to be calculated have
moreover nothing complicated about them. We will give them one day on this site,
for the curious.
...At
the moment we are laying a few geometric markers with a view to
later discussions, while recalling that the models indicated
are only indicative in character. .
Summary of
article Summary of
Science Home
Page
Number of consultations of this page since July 1, 2004
: