هندسات مترابطة وانحناء زاوي

a107

Conjugated geometries.****

We can now associate a blunt posicone and a blunt negacone. Facing each other : a portion of a sphere and a horse saddle, with opposite angular curvature + q an - q. We have a point to point correspondence (injective mapping) . On figure (39) a couple of conjugated points have been figured.

We call *conjugated geometries *two geometric structures, with a point to point link, such as the local curvature densities are opposite. This is the case for the portion of the sphere and the corresponding horse saddle. The same applies for the portion of posicone, facing a portion of negacone. Their local angular curvature densities are zero. (39)

The positive curvature, in the fold F, is entirely contained in the portion of the sphere. The portion of posicone is an euclidean surface, which is "locally flat". In the other fold F*, the conjugated fold, all the (negative) angular curvature is contained in the horse saddle. Outside, the portion of negacone is "locally flat", it contains no curvature.

Notice that given a fold you can build the other one.

General Relativity.

The basic idea is that the local content of "matter-energy" determines the local geometry, it shapes the space-time hypersurface. Notice that the composite word "matter-energy", which shows that any content determines the geometry of the universe : matter* and *radiation. In a precedent section we evoked the fact that photons contribute to (positive) curvature. Today the contribution of the cosmic background is negligible. Matter's contribution to geometry is dominant. But, in the distant past, the situation was reversed : in the Standard Model, when t < 500,000years.

Let us search some didactic model in order to figure the basic concepts of general relativity. Let us deal with steady-state systems. Consider a plane surface, without any internal stress. We can modify its geometry if we introduce local stress. We can introduce positive or negative tension (stress tensor). For example if I heat a plastic film, I will create a blister (positive curvature effect)

I can also impregnate the material with a product which, when dry, will cause local strectching (negative curvature effect).

A boiler-maker knows how to display warming and cooling to shape a metal surface, for an example, a can which has been in an accident.

Take a simple metallic tube. Let us warm it on one side and cool it on the opposite side. What will happen ?

(40)

The stress will bend the tube, as shown on figure (41).

(41)

We have introduced tensions in the metal. This is the origin of the word tensor in mathematics, resistance of material and geometry. The specialist in resistance of material will talk in terms of *stress tensor . The geometer will invoke the curvature tensor *. The specialist of general relativity will apply the basic principle :

local energy-matter content <-------> local geometry

Of course, this local energy-matter content determines the local geometry of a 4d-hypersurface. But the idea is similar.

How to write that ? Using what mathematicians call *tensors *.

It is difficult to go further in that direction, without developping a complete course of *differential geometry *. The famous Einstein equation is : (42)

**S **= c T

c is a simple constant ( called the Einstein's constant ). It depends ont the values of two other constants :

• The light velocity c.

• The constant of gravitation G.

through :

(42bis)

**S **is a geometrical tensor and takes in charge the geometrical features.

T is another tensor, that describes the local content of the universe. In this tensor you will find the matter density r and the pressure p . They are expressed as energy densities. r c2

is an energy density

But p is also an energy density. Usually one express a pressur as pascal per square meter. But a pascal per square meter is also a joule per cubic meter. A pressure is basically a volumic energy density. The fields

r (x,y,z) and p (x,y,z)

for a steady-state system, form the entry of the problem. From these scalar fields we can build the tensor T. Then the question becomes :

• What is the geometry that goes with such tensor field T (x,y,z), which satisfies the equation (42) ?

Given the local content of the Universe, the theoretician must build the local geometry of the space-time hypersurface. But, what for ?

Here one uses the second basic hypothesis :

• All the objects that compose our universe follow space-time hypersurface geodesics.

An object can be a star, a planet, an atom, a photon, an elementary particle.

Do the particles come from the field equation ? Not at all. General relativity ignores them completely. For the specialist of general relativity, the universe is a continuum, nothing else. The input functions r and p correspond to a macroscopic description of the universe. Same for the ouput : the geodesic system. For the theoretician of general relativity, the Universe is a hypersurface, nothing else. He says :

• You gave me functions r (x,y,z) and p (x,y,z). I have built for you the adequate hypersurface, which obeys the field equation. I have determined all the possible paths : the geodesic system. But I am completely unable to build particles for you. Sorry. See another department.

To sum up : the bridge between the general relativity and elementary particle world is still waiting his builder.

But the astronomer will say :

• Who cares ? Photons are supposed to follow peculiar geodesics of this hypersurface. It works : I can observe things with optical devices. Planets are also supposed to follow another kind of geodesics. It works too. I can compute their paths, predict the precession of Mercury's perihelion. There is also the gravitational lens effect.

He is right.

Few words about this gravitational effect. First of all, this image of the blunt cone is a simple didactic image. For example it cannot describe the circular paths of a planet around a star : (43)

This simply shows the limit of didactic images. But we can use this last example to illustrate the gravitational lens effect, with two geodesics :

(44)

Below, the mental, euclidean representation of space. There is a mirage effect. Instead a single object, the observer see two "gravitational mirages".