a105
| 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 on a plane, below, as shown on 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 that it will follow a geodesic of the 4d hypersurface. That's why we evoke this with geodesic crossing the grey area.
What is the meaning of this plane on 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 "cavern myth". Men are supposed to be jailed in a cavern. 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 on 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, that was quite strange for a Frenchman, by the way. Descartes was convinced that this spatial fluid pushed the heaven's bodies, and moved them along their paths. For example, if the Moon circled around the Earth, that 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 flat 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 instant distant action (through gravitational force). Later people showed that the Earth corresponded to Newton's prediction: it looked like a flat 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 on figure (26).
(26)
When you build a cone, you choose arbitrarily towards what side of the plane it will touch its conical point. On 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 angle 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 aspirins.
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, that "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 by 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 talking, 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.