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Conjugate
geometries.
...We
will then associate a blunt posicone and a "blunt
negacone", comprising the same quantities of curvature,
but of opposite sign: + q
and - q.
We can place them face to face (creating along the way a "point-by-point
mapping": bi-univocal, injective ). There are then two sheets.
Let us call them F and F*. To every point of F corresponds a point of F*.
...Let us manage
so that the circular contours of the "blunt parts", carrying
curvature (positive in one sheet, negative in the other), correspond
point by point. This is illustrated by projecting the whole onto a plane. We obtain
two surfaces having conjugate curvatures.
...The
conical flanks are "non-curved", 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".
Let us note that the term energy-matter is used and not only
matter, which clearly shows that all cosmic content has an influence
on geometry, including radiation, photons (or
neutrinos). We saw above that a photon created a mini-curvature
positive in space.
...We
will first reason in stationary terms. A flat, free surface is a
surface where the tension is zero. One can modify its geometry
by creating tensions there, positive or negative (the sign is
a question of conventions). If I heat for example a plastic film
I will be able to make a blister appear there, that is to say a region
with positive curvature.
...I
can also put on the surface of a sheet of paper a product which,
by drying, will contract. The tension will make a
region with negative curvature appear.
...A
sheet-metal worker knows how to play on these tensions to deform a
sheet metal. Let us take for example a metal tube. I heat on one side,
I cool on the other. What will happen?
The tube will bend,
the heated part dilating and the cooled part contracting.
...In
so doing, we have created tensions in the
metal. This is the origin of the word tensor, in mathematics
and in geometry. The specialist in strength
of materials will speak of stress tensor . The geometer
will speak of curvature tensor .
The little experiment
above illustrates the idea:
Local energy content ----->
local geometry
...In
General Relativity, one does the same. The difference
is that this local energy-matter content determines
the geometry of a hypersurface with four dimensions and
not, as here, the geometry of a surface with two dimensions.
But the idea is similar.
...The
mathematician will then use a tensorial notation. One can hardly say more here for a non-mathematician.
But the Einstein tensor S
(we will use bold letters) corresponds to the geometry aspect.
In the Einstein equation one identifies it with another tensor
T, which describes
the energy-matter content, up to a multiplicative constant,
the "Einstein constant c
".
The famous
Einstein equation is therefore written:
S = c
T
...In
the tensor T
intervene the volume density r and
the pressure p (in fact the most general tensor T is more complex, but we will content ourselves
with this expression, usual, moreover). In a stationary configuration
one will therefore give oneself a certain distribution of density and pressure
r (x,y,z)
, p (x,y,z). With this one knows how to construct the tensor T
which thus contains all
the data of the problem. The question is then: "what is
the geometry that goes with this tensor T ,
satisfying the equation above? ".
...In
other words, the physicist, knowing the local content of the universe seeks to
determine the geometry of the universe hypersurface.
To say geometry
is to say geodesics. It is here that the second
hypothesis of General Relativity intervenes:
It is assumed that the objects which travel
in the universe
follow geodesics
of the space-time hypersurface.
By object one will mean
particles (so-called elementary particles, photons, neutrinos)
but also planets, stars, etc....
At this stage, a remark
: where are the particles in all this business?
...Answer
: the specialist in General Relativity works
in the macroscopic. The input-functions of the problem, the
volume density r and
the pressure p, correspond to a macroscopic description of the cosmic content.
Same thing for the "output". And the geometer to add:
- You have given me
functions r
(x,y,z)
and p (x,y,z), I have constructed for you the hypersurface that went with it, with its
families of geodesics. But I cannot do more. I
am in particular incapable of manufacturing particles, atoms,
etc. for you. For that, see another department...
Clearly: the bridge between
General Relativity and particle physics
is not yet built.
But the astronomer will exclaim
:
- What does it matter. This hypothesis
according to which photons follow certain geodesics of
this hypersurface works. The proof: I can make observations.
If I suppose that the planets, assimilated to point
masses, also follow geodesics of this
hypersurface, I can construct their trajectories. There are also the effects
of gravitational lensing....
He is right.
...These
gravitational lens effects, let us say a few words about them. Of course,
this image of the blunt cone is only a didactic image.
A planet which orbits circularly around a star also follows
a geodesic of space-time. Now a circle drawn
on a blunt cone is not a geodesic
:
That simply shows
the limits of didactic images, even geometric ones.
...The
photons effectively follow geodesics of the space-time
hypersurface. One will be able to use this image of the blunt cone
to illustrate it. The light rays can pass on either side of a
massive object, then converge toward the observer. If we project these geodesics,
we will obtain a mirage effect: the observer will have the impression that there
are two sources instead of one:
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