Re: two questions on the bend of the space



Hola, Kara,


> Is it more or less uniform that bend?


It is definitely not uniform, and it's not as simple as the radius of a
bend.

You normally think of curvature of a surface as being the change in
orientation of the surface per unit length as you go along a path. The
change of orientation is typically measured in radians ( 1 radian ~
57°). This sort of curvature has the dimensions of 1/length. For a
sphere of radius r the orientation changes by 2*pi when you go all
around the circumference 2*pi*r of the sphere, so the curvature is 1/r,
i.e. the reciprocal of the radius of the bend.

The orientation is a property of the surface in the space in which it
is embedded, and it is not itself in the surface. The sort of curvature
you meet in relativity is different. It is to do with behavior
completely within the surface. Take a direction within the surface, and
move it so that it stays within the surface, keeping as constant a
direction as is possible. When you come back to your starting point, it
will not necessarily be parallel to its initial direction.

For example, suppose you are at the North Pole and have a horizontal
arrow pointing towards Spain. Carry it, keeping it horizontal and as
close to the original direction as possible. First go towards Spain,
then head past down Africa to the South Pole. All the way, you are
travelling in the direction of the arrow. When you arrive at the South
Pole, the arrow will point towards New Zealand. Now head back to the
North Pole, but this time along a path perpendicular to your arrow, via
Mexico City. All the way back north, the arrow is pointing to the west,
to your left. When you get back to the North Pole, the arrow is
pointing towards New Zealand. You have kept it as parallel to itself as
is consistent with keeping it horizontal, but it has rotated 180°.

For small closed paths, this change in orientation is proportional to
the area within the closed loop. This sort of curvature is what is
dealt with in General Relativity and has dimensions of
1/area=1/length^2. It was first treated by Gauss. For a sphere of
radius r the Gaussian curvature is equal to 1/r^2.


> Is it known or can you calculate the radius of the bend of the space?


It's not as simple as that. See:

http://casa.colorado.edu/~ajsh/schwp.html
http://www.astro.ku.dk/~cramer/RelViz/text/geom_web/node3.html


Cheers,

Zigoteau.

.

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