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Conjugate 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 curvatures +q
and -q. We have a point-to-point correspondence (injective mapping). In figure (39), a pair of conjugate points is shown.
We call conjugate geometries two geometric structures,
linked point-to-point, such that their local curvature densities are opposite.
This is the case for the portion of the sphere and the corresponding horse
saddle. The same applies to a portion of a posicone facing a portion of
a 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 the posicone is a Euclidean surface,
which is "locally flat". In the other fold F*, the conjugate fold, all the
(negative) angular curvature is contained in the horse saddle. Outside, the
portion of the negacone is "locally flat"; it contains no curvature.
Note that given one fold, you can construct the other one.
General Relativity.
The basic idea is that the local content of "matter-energy"
determines the local geometry; it shapes the spacetime hypersurface. Note
that the compound term "matter-energy" indicates that any content determines
the geometry of the universe: matter and radiation. In a previous
section, we mentioned 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 < 500,000 years.
Let us seek a didactic model to illustrate the
basic concepts of general relativity. Let us consider 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 stretching (negative curvature effect).
A boilermaker knows how to use heating and cooling
to shape a metal surface, for example, a can that 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 in figure (41).
(41)
We have introduced tensions in the metal. This is the
origin of the word tensor in mathematics, relating to material resistance and geometry.
The specialist in strength of materials will speak in terms of the stress tensor. The geometer will invoke the curvature tensor. The specialist
in 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
developing a complete course of differential geometry. The famous
Einstein equation is:
(42)
S = c
T
c is a simple constant (called
the Einstein constant). It depends on the values of two other constants:
-
The speed of light c.
-
The gravitational constant G.
through:
(42bis)
S is a geometrical tensor and accounts for the
geometrical features.
T is another tensor, which describes the local content
of the universe. In this tensor, you will find the matter density ρ
and the pressure p. They are expressed as energy densities.
ρ c²
is an energy density.
But p is also an energy density. Usually, one expresses
pressure in pascals per square meter. But a pascal per square meter is also
a joule per cubic meter. Pressure is basically a volumetric energy density.
The fields
ρ(x,y,z) and p(x,y,z)
for a steady-state system, form the input of the problem.
From these scalar fields, we can build the tensor T. Then the question
becomes:
- What is the geometry associated with such a tensor field
T(x,y,z), which satisfies equation (42)?
Given the local content of the Universe, the theoretician
must construct the local geometry of the spacetime hypersurface. But, for what?
Here one uses the second basic hypothesis:
- All the objects that compose our universe follow
spacetime hypersurface geodesics.
An object can be a star, a planet, an atom, a photon, or an
elementary particle.
Do the particles come from the field equation? Not at
all. General relativity ignores them completely. For the specialist in general
relativity, the universe is a continuum, nothing else. The input functions
ρ and p correspond to
a macroscopic description of the universe. The same applies to the output: the geodesic
system. For the theoretician of general relativity, the Universe is a hypersurface,
nothing else. He says:
- You gave me functions ρ(x,y,z)
and p(x,y,z). I have
built for you the appropriate 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 general relativity and
the world of elementary particles is still waiting for its builder.
But the astronomer will say:
- Who cares? Photons are supposed to follow
particular 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.
A 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 of a single object, the observer sees two "gravitational
mirages".
