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Possible problems
posed by a choice of coordinates.
...We
are going to evoke the risks incurred by imposing a coordinate system
on a geometric solution, by expressing
this solution in this particular coordinate system:
it still has to be adequate. When we look at the solution
above, assuming that this geometry is a solution of a
field equation, the recourse to a coordinate system
(r , q) presupposed that the topology was
"locally spherical", in two dimensions, of course. That is to
say that inside any circle "centered on this hypothetical geometric
center one could always inscribe
a smaller circle, until this circle becomes a point. Mathematically
one would say that any circle of radius r delimits a "contractile cell".
...In
3d, locally, the universe would be "like Russian dolls". Inside
a sphere one could always inscribe a sphere of
smaller area. In 3d, it is a locally spherical topology.
Can it be
otherwise?
Yes, if the topology
of the surface is "locally toric". In 2d this gives this:
...Remark:
the object of the figure above is a 2d surface in the sense that it takes
two parameters to locate the position of a point on it. In this
sense a curve is a "one-dimensional surface". When he speaks
of the circle, the geometer will use the expression "sphere
S1", that is to say "one-dimensional sphere": it
takes only a single parameter, the abscissa, to locate a point
on a curve, a one-dimensional object. The sphere S2, the "ordinary"
sphere and the circle, the sphere S1 have something in common:
they are "closed" objects (a concept then borrowed from
topology).
...This
number of quantities serving to define the position of a
point in a space is precisely the definition of
the dimension of this space. Thus one will consider space
time (x,y,z,t) as a four-dimensional hypersurface, because it
takes four quantities to define a point in it, called an "event".
End of this remark
on the concept of dimension.
...One
must keep one thing in mind. The geometer who constructs
a particular solution of a field equation is blind,
he cannot see the geometric object that he obtains.
He can only explore it, through its geodesics,
by describing these in a particular coordinate
system. The polar coordinates of a moment ago
corresponded to the intersection of the surface by a family of coaxial
cylinders:
and by a family of
planes passing through the common axis of these.
In 3d it would be
the intersection of space by a family of concentric spheres.
...But
what happens if one cuts the surface presenting this sort of
tubular bridge by a family of concentric cylinders? As long as the cylinders
cut the surface, it is fine. But when their perimeter
becomes smaller than that of the "throat circle" these sections become
... imaginary curves. Let p be the perimeter of the throat
circle. Let us associate it with a length Rg such that p = 2 p
Rg .
...It
is clear that any cylinder of the family such that r < Rg does
not cut the surface. When the geometer becomes interested
in the appearance of the geodesics of the surface for r <
Rg he will find imaginary geometric objects.
...When
one seeks the intersection of two points with a line, corresponding for
example to x = xo one finds two real values, for y, when
the line effectively cuts the circle. Otherwise these values are purely
imaginary.
...If
a man, who explores a surface in darkness, without being able to
perceive its shape , who does not know that the topology of this one is
locally toric , he could be extremely disconcerted.
The surface can be located with the help of two families
of curves:
...Each
curve is defined by a parameter. A point M , at the intersection
of these two curves, is indeed defined by two quantities (a,b),
the two values of the curves passing through M.
...The
first is constituted by circles which are not
geodesics of the surface (except the throat circle), the second
by geodesics, of hyperbolic appearance, orthogonal to
these circles. The hyperbolic curves evoke plunging trajectories,
which allow passing from one sheet to another sheet.
One can obviously have the same situation for a 3d space,
locally hypertoric. The circles will be replaced by a family
of spheres, among which one will find a throat sphere
, of minimal area. The lines constituting the trajectories orthogonal to
this family of spheres constitute plunging trajectories allowing
passing through this hypertoric tunnel and reappearing in
another 3d sheet (or foliation).
...This
remark is not gratuitous. We will have the opportunity to return to it when we put
the black hole model under examination. Indeed, in this model,
when one penetrates "inside the horizon
sphere" the mass of a particle becomes... purely imaginary (that and many
other things as well). One is then entitled to wonder if one is still
in the space-time hypersurface. The particular choice of coordinates
(t , r , q
, j)
which implies a locally hyperspherical topology
(existence of a radial coordinate r, capable of taking values below the radius of the
horizon sphere, of the Schwarzschild sphere) would it be relevant?
A well-known astrophysicist
wrote, a few years ago:
- We now know much more
about the interior of black holes.
But black holes,
if they exist, do they have an interior, or do they correspond to
a locally hypertoric topology?
...One
sees all that a choice of coordinate system can induce.
The geometric solution exists. It possesses geodesics.
But we only know how to "read" all this by projecting into our mental space
of representation: a Euclidean space time, which is not even
relativistic. To choose a coordinate system is to choose
a system of reading, a system of projection.
...Like Plato's characters, we can
only observe shadows on a "Euclidean screen". It still remains to choose
the right lens of the "projection system".
...
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