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Invariance through
change of coordinates.
...Here is
a key concept of General Relativity, which is
not easy to present. We have said that seeking a "cosmological
solution", stationary or non-stationary, amounts to constructing
a four-dimensional hypersurface that is a "solution of the field
equation".
...Let us
take for example a sheet-metal object that has the topology of the sphere.
It is "a sheet-metal sphere". Here again one readily imagines
that this surface could be deformed by heating and cooling
it in places. For example, by heating at one point and cooling the
antipodal region, one would transform this sphere into an egg. An
egg is an object that has the topology of the sphere, but which is a
surface of variable curvature.
...By
heating in one place and cooling in another, we will
create stresses in the metal. Of course, since
this material is conductive, if one stopped heating and cooling,
the temperature would become uniform and the object would return
to its spherical shape. What matters is that one manages to create
a stationary situation with a non-uniform temperature field.
This field generates stresses and one could concretize these
stresses in the form of a mathematical object T
called a tensor.
Something describes
the geometry of the object. It is called a metric. It is from this second mathematical object that one can
:
-
Calculate the geometric
tensor S -
Calculate the geodesics
of the surface.
The geometry
of this surface could be calculated from an equation
analogous to Einstein's equation, of the style:
S = a T
where a is a
constant. By knowing a priori the temperature field in the
sheet metal, hence the stress tensor, one could deduce
its geometry. The best way to "read" this geometry
would then be to analyze the system of geodesics.
We know those of the sphere (its "great circles"). The
geodesics of an egg are different.
...To
describe these geodesics we will need to define
a system of coordinates on the surface. For the sphere,
one could take the classic azimuth-site system.
...In
this particular system of coordinates, the geodesics
of the sphere would correspond to certain equations.
On this sphere
the curves q
= Cte represent the family of geodesics passing
through two points. On the other hand the curves j
= Cte (parallels)
are not geodesics of the surface.
...One
could also define an analogous system of coordinates
and write the equations of the geodesics of the
"egg" surface. But one immediately notices an essential thing: the
geodesics of the surface are independent of the coordinates
that one chooses to describe them , just as the points
of a sphere, or of an egg, exist, independently of the system
of coordinates used to locate them.
...Likewise,
on a plane, one can represent points in Cartesian coordinates
or in polar coordinates. The straight lines of the plane
are geodesics.
A straight line can be
described in two systems of coordinates:
...It
is the same geodesic, with two totally different descriptions. The straight lines of the plane exist independently
of the manner in which one describes them, of the choice of coordinates
used. And one can imagine... an infinity of them.
...So,
what is intrinsic? Answer: the length s measured
on a straight line (or along any curvilinear contour). Between two
points M1 and M2 of a surface the path of minimal
length is a geodesic.
...Likewise
the distance separating two points, on a geodesic
of the "sphere" or "egg" objects is also a quantity that
is independent of the system of coordinates chosen.
If one takes two points M1 and M2 on a surface and
traces the geodesic arc joining them, the length s measured
along this arc will be the same, whatever the system of
coordinates used to locate the points.
...It
is the same for the four-dimensional hypersurface that we
call "universe". It possesses its system of geodesics,
also invariant through change of coordinates. We do not
live in a space (x , y , z , t) with position coordinates
and a time coordinate, but in a four-dimensional hypersurface
that can be entirely described by its network
of geodesics. On these geodesics there exists
a length s that is also invariant through change of coordinates.
The points of this hypersurface are no longer points of space, but
points of a space-time hypersurface. They are called events
. Two distinct events are therefore separated
by something that is called s. But what is it then?
It is proper
time .
...A
geodesic trajectory in this space-time hypersurface
separates two events M1 and M2. All that I can
say is that if I had used a vehicle to make
this journey in space-time, a lapse of
time s would have elapsed, on my on-board clock.
A
choice of coordinates consists in locating the points of
space-time by space coordinates (x , y , z) and a time coordinate
t . But since this choice is arbitrary this space and this time have
no intrinsic existence. They are only ways of "reading"
the surface, of traveling over it. Sole constraint: depending on the hypothesis
made, one can move only along geodesics
and on these, the only reliable thing to which one can
cling, is the "elapsed proper time" s, and not this time
t , a simple system of chronological marking (chronological
marker).
For each choice of the
system of coordinates, a different system of
reading events, of phenomena.
...Physicists have therefore sought a formalism that is independent
of the choice of coordinates. It is the essence of the tensorial formalism. One cannot say more on this subject, at the risk of entering into relatively complex technical
details.
The problem of singularities.
On a sphere,
the choice of classic angular coordinates introduces two polar
singularities.
Impossible to map
a sphere without introducing this kind of polar singularity.
...Note
that one can map a sphere with a single
singularity. One creates on the sphere a first family of
curves (circles) by cutting it with planes, as indicated below
:
Then a second family
:
Outside this single
singularity, no problem. If one looked at this sphere
from the other side, one would see this:
...Outside
the single singularity S, one locates the points without
difficulty. But the values of the parameters a
and b defining this mesh
singularity S are .... anything...
...Yet
a sphere is not geometrically, intrinsically
singular. Turn a billiard ball in all directions, or an egg, you
will discover no singular point in it.
These
singularities have therefore been created by
the choice of coordinates.
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