a108
| 8 |
|---|
Coordinate-invariant formalism.
This is another key-word of general relativity. We said that the work of the cosmologist was equivalent to the one which consists to predict the shape of a material, due to internal stress. Take an object, whose topology is the one of the sphere. It is a sphere made of metal. Here again we could shape it, with hot and cold air fluxes. (45)
These fluxes create stress in the metal, which modifies its shape. Of course, as the heat propagates in metal, if one stops heating and cooling, the temperature of the sphere returns to uniformity and its aspect becomes regular again. We create stress in the material, which modifies its geometry. This stress field can be described by a mathematical object called a tensor T. The geometry of the object could be calculated from a field equation, similar to Einstein's equation. (46) S = a T where a is a constant and S a geometrical tensor, which describes the geometrical features. The best way to "read" the solution would be to compute the geodesic system. We know the geodesics of the sphere, but geodesics of an egg are different. To express these geodesic we need a coordinate system. For a sphere we can use a (q,j) system : (47)
In this peculiar system of coordinates the geodesic of a sphere can be expressed into a peculiar form. For an example the curves : q = constant (meridians)
are geodesics. But the curves
j = constant (parallels) are not geodesics curves of that surface. We could define a similar system of coordinates on the surface "egg". But something is evident : The geodesic system exists independently of its mathematical representation (in a given, peculiar system of coordinates). The geodesic system is coordinate-invariant. Another example is much simpler. Consider the geodesics of a plain sheet. They are straight lines. We can describe these straight lines in cartesian coordinates : (48) We can also describe this family of geodesics in polar coordinates. Then the equations are completely different, but they refer to the same family of straight lines. These straight lines, geodesic of the plain sheet, exist independently of the chosen coordinates. They are coordinate-invariant objects. The equations are not an intrinsic attribute. Is it something that does not change when we shift from a system of coordinates to another one ? Yes : the geodesic path, between two points M1 and M2 does not change. Same thing for any line drawn on the surface. The surface, the points, the curve which joins them exist independently of the chosen coordinates. Same thing for the length of the path between M1 and M2. This is also true for a geodesic arc, which is a peculiar line joining two points : (49) By the way, this geodesic path is also an extremum path (for example the shortest one, shown there). This is similar for the space-time hypersurface, which owns its system of geodesics, also coordinate invariant. On this hypersurface a length s does exist, which belongs to the object and is independent of the chosen system of coordinates. The difficult point is that space and time are not independent quantities. We don't live in a 3d space, with points (x , y , z) . We belong to an 4d-hypersurface which is fully described by its system of geodesics. Consider two distinct points of this hypersurface M1 and M2 . Such points can be described in a given system of four coordinates :
M1 ---> (x1 , y1 , z1, t1 ) M2 ---> (x2 , y2 , z2, t2 ) These points are called *events *. We can calculate the geodesic curve which links them, if there are any. Such events are not identical. Between the two we can measure a distance s, which coordinate-invariant. This length is called :
proper time s
Assume you and I use a space ship to travel, from a point M1 to another point M2 , located in space time. s is the measure of the time on our board-watch.
You will argue : - But space exists, no ? - Be careful. This definition of what we call space and "absolute time" corresponds to an arbitrary choice. They are just some convenient way to "read" the surface, like when we wrote the straight lines equation, in a plain sheet, into two different equations. The only thing that does not change, which is coordinate-invariant, is the proper time interval Δt between two events linked by another coordinate invariant object : a geodesic line. The so-called "absolute time" t is nothing but a somewhat arbitrary chronological marker . Changing your coordinate system, you change the reading of the events. In the papers that we will present in this website you will see that this is a real problem. Anyway, you understand why physicists and mathematicians have chosen a coordinate-invariant formalism, based on tensors. Tensor-form equations are coordinate-invariant.
This is the spirit of general relativity. But, except using sophisticated hardware, it is difficult to tell you more about it.