| 5 |
Trajectories, paths.
The basic idea of general relativity is to
assimilate objects' trajectories to geodesic lines, whatever the object is
: a planet, an atom. We are going to illustrate this concept, using 2d surfaces.
Of course, it's just a didactic image, for the 4d geodesic system is fairly
different (in fact, frankly ugly).
(23)
Nonsense! It is a 4d system!
I know, Jean-Marie, I know. This is just a didactic image...
We can draw geodesic lines on a blunt cone. See figure
(24). Then we can project it onto a plane, below, as shown in the figure.
(24) and (24')
This is supposed to evoke the paths of particles in a
portion of space where a mass concentration is present (grey area). One crosses
the grey area. It means that it runs inside matter. Is it possible?
A neutrino interacts so rarely with matter that it can
cross the sun. Doing so, it will follow a geodesic of the 4d hypersurface.
That's why we evoke this with geodesics crossing the grey area.
What is the meaning of this plane onto which we project
our geodesics? It is nothing but our mental representation of the universe.
We think it is Euclidean. As the objects do not follow straight lines, we
think the bending of their paths is due to "forces". When a comet approaches
the sun, turns, and goes back, we think that it is due to the attractive gravitational
action of the sun on it. But this is due to the curvature of space. The comet
follows a geodesic of space-time. In this 4d world it goes "straight". Everything
goes straight, matter, light.
Many centuries ago Plato invented his cave myth. Men
are supposed to be jailed in a cave. Outside is the "reality". Inside they
only see the dancing shadow of this reality, projected on the wall. Similarly,
our mental representation of the world is ... the wall onto which a more sophisticated
4d structure is projected.
General relativity and curvature.
We said, above, that matter curved space, shaped the universe's
geometry, the shape of "the 4d hypersurface called Universe". In classical
general relativity, the local curvature is positive or zero.
We assimilate stars, planets, atoms to positive curvature
concentrations (we will see further what is a negative curvature).
Between stars, planets, atoms, is something we call "void".
But does the void exist?
For a physicist the void, the vacuum, is what you get
when you remove all matter.
But can space exist without matter? Newton thought it
could exist. He was the inventor of the void. The French philosopher Descartes
held the opposite position. He thought that some cosmic fluid did exist, between
planets. He imagined the universe like a cup of tea, which was quite strange
for a Frenchman, by the way. Descartes was convinced that this spatial fluid
pushed the heavenly bodies, and moved them along their paths. For example,
if the Moon circled around the Earth, it was because it was caught in some sort of fluid
vortex, surrounding our planet.
If the passage of the moon caused tidal effects, in oceans,
that was how, following Descartes, the moon pushed the ocean, through a sort
of fluid cushion. He thought the Earth was shaped like an elongated ellipsoid.
Newton had the opposite opinion. He thought that the Earth
was shaped like a flattened ellipsoid, due to a centrifugal force. But Newton was
also an alchemist. You know the French: very conventional. They refused Newton's
idea for a long time. Voltaire liked Newton's ideas. He fought for them, and
finally won. The cosmic fluid of Professor Descartes became some sort of phantasm,
while the void of Professor Newton gained the status of solid reality.
Newton completed his vision, introducing the concept of
instantaneous action at a distance (through gravitational force). Later people showed
that the Earth corresponded to Newton's prediction: it looked like a flattened
ellipsoid.
So, Newton was right and Descartes was wrong.
Well, things are now not quite so simple. First, the gravitational
action is not instantaneous. The gravitational field propagates at the velocity
of light. Then the void is not so empty as we thought centuries ago.
It is the fate of science. Some ideas are right at some
periods, then false in some sense, in another, then right in another one.
And so on. It oscillates like a pendulum.
Consider a vacuum pump, a very efficient one. Conceptually
it is a simple cylinder, with a piston. Initially the volume is zero. Then
we pull the piston. The junction between the cylinder and the piston is so
good that no molecule, atom, any particle can get in. We think that we have
created a perfect void. See figure (24 bis).
(24 bis)
But immediately the wall of the pump emits radiation, thermal
radiation, i.e. photons corresponding to infra-red rays. These photons occupy
this "perfect void", where the pressure is not strictly zero, for there is
a radiation pressure, weak, but non-null.
What is a photon? One says they have zero mass. Then,
how is the curvature inside the pump? Is it zero? Is it a zero curvature
density portion of space?
In the next section, we build a surface with two conical
points. See figure 25.
(25)
You take a piece of paper, scissors. You make two cuts
and join the segments:
S1A and S1 B
S2 C and S2 D
But you can do it differently, as shown in figure (26).
(26)
When you build a cone, you choose arbitrarily towards
what side of the plane it will touch its conical point. In figure (25) you
have automatically chosen the same side, the same orientation, for the two
conical points. In (26) these orientations are opposite.
But a conical point is a conical point, whatever the direction
it points to. If you draw geodesic lines, with such a conical point inside,
you will get an angular excess, corresponding to that concentrated angular
curvature. See figure (27).
(27)
If you draw a triangle made with geodesic lines, which
contains the two points S1 and S2, the sum of the angles
will be 180° + q1
- q2 .
What does all this mean?
It is a good didactic image for matter anti-matter duality.
Both have positive mass. Both create a local positive space curvature.
But they are... different. All that will be explained in details in Geometrical
Physics B, papers 1 to 4. But... don't forget your bottle of aspirin.
Matter and anti-matter have different geometries. They
differ in their "additional dimensions".
Matter plus anti-matter gives light, photons. So that
we may consider that a photon corresponds to two grains of matter and anti-matter,
glued together.
You can build such a strange surface with the two conical
points S1 and S2, tending each towards the other. See
figure (28).
(28)
The object is symmetrical, which "explains" why the photon
is identical to its antiparticle.
You can draw a triangle with three geodesics. The sum
is 180° plus 2q,
the small angle featuring the mass (same mass for the two components, matter
and anti-matter).
(29)
Then the photon gives a positive space curvature. Our
universe is supposed to be a mixture of masses and photons. Both contribute
to its local curvature. What we call void is composed of joint cosmic radiation
photons (what physicists call a "blackbody"). Here a blackbody corresponding
to a "cosmic oven" with a 2.7° K absolute temperature.
Then, according to classical general relativity concepts,
between mass concentrations, space is slightly bent, due to the presence of
photons. Strictly speaking, if one wants to figure a mass concentration, with
no matter around, he should draw:
(29 bis)
What could be the impact of negative masses on geometry?
If these masses do exist, they should create local negative
curvature density.
