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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, twhich modifies
its shape. Of course, as the heat propagates in metal, if one stops heating
and cooling, the temperature of the sphered returns to uniformity and its
aspect becomes regular gain.
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 sytem.
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 Iuse 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 Dt
between two events linked by another coordinate invariant obect : 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.
