With all todays talk about the shape of space and all that I thought people
may find this usefull. It is from a mailing list dedicated to Astrophysics.
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Some rambling about curvature...
What forces a space to be flat? If a space is curved does that
automatically and necessarily require that it exist as a subspace of some
higher dimensional flat space? To answer this, think back to high school
Euclidean geometry... You'll remember that the flatness of Euclidean
(mathematical) space was forced onto space. To put it differently, it
required a separate postulate or axiom --the parallel lines postulate. The
parallel lines postulate says that two lines which start out parallel will
always remain parallel. Without that forced structure, nothing requires
space to be flat, whether real physical space, or an axiomatic mathematical
space.
Most of the textbook examples of curved spaces use a simple "embedding" in
flat 3d space as an easy way to illustrate the concept of a curved space.
These pedagogic spaces, unfortunately for people trying to understand
general relativity and cosmology, do not distinguish between "intrinsic
curvature" and "extrinsic curvature". A space is curved intrinsically when
measurements of geometric relationships solely confined to that space allow
one to detect curvature locally. Extrinsic curvature is the curvature of a
subspace measured through geometric relationships in a higher dimensional
space that surrounds it.
How about a circle? Is it intrinsically curved? The circle itself
constitutes a 1-dimensional space. Embedded in the 2-dimensional plane, it
certainly has curvature. You can measure that *extrinsic* curvature by
giving the radius of the circle. Smaller radius means greater curvature, so
we can define the extrinsic curvature to be 1/R. But what about intrinsic
curvature? Is a circle intrinsically curved? Can you perform measurements
based only on the points that are part of the circle and detect the fact
that it is a circle and not a line? Clearly, no. There's nothing about
distance measurements on a circle that are different from distance
measurements on a line. You can't detect the curvature locally. Now if
you're thinking ahead a bit, you might say, 'but I can detect the fact that
the circle is curved by travelling all the way around it and returning to
my starting point'. That's not curvature --it's global topology. Whether a
space is closed or open or has connections between distant points is not
necessarily related in any direct way with curvature.
How about a cylinder? Is the 2-dimensional space defined by the surface of
a cylinder flat or curved? Extrinsically, it's clearly curved, but
measurements made within that 2-dim space will detect no curvature. This is
closely related to the fact that you can take a flat piece of paper and
roll it up into a cylinder without stretching the paper in any way. If you
draw a triangle on the surface of a cylinder, the angles will still add up
to 180 degrees. If you start out any two lines initially parallel and
transport them parallel to their own tangent vectors, they remain parallel
forever. A cylinder is a 2-dim flat space. Incidentally, this space is also
closed, and it is widely seen in a more obviously flat representation:
video games with left-right wrap-around are cylindrical spaces.
A 2-dimensional sphere has both intrinsic and extrinsic curvature. When you
embed it in a flat 3-space, you automatically get an ordinary-looking
curved surface. But more significantly, when you perform geometric
measurements on the surface of the sphere, you can detect its intrinsic
curvature. Start out two lines parallel to each other (think of lines of
longitude starting out parallel at the equator). When you follow their
straightest possible motion, you discover that the lines eventually
converge and meeet. That's the clearest indication that a 2-dim sphere has
intrinsic curvature. Note that 2-dim "ants" confined to such a surface
could easily perform this experiment. Next, suppose we lay out a triangle
using straightest possible lines (also shortest possible) for the sides of
the triangle. Adding up the angles, we would find that the sum is always
greater than 180 degrees.
Finally, what about the universe?? Does the fact that it is curved require
that it be embedded in some higher dimensional space. Does the existence of
curvature in three spatial dimensions (and one time dimension) imply that
the universe is a subspace of some higher dimensional space? In short,
absolutely not. The universe does not "curve into" anything. It is
intrinsically curved. Furthermore, the universe does not "expand into"
anything. The geometry of the universe is what it is. Forcing it into a
flat higher dimensional space can be a nice visualization tool on occasion,
but it is not required by either math or physics.
-FER
may find this usefull. It is from a mailing list dedicated to Astrophysics.
============================================================================
=============================
Some rambling about curvature...
What forces a space to be flat? If a space is curved does that
automatically and necessarily require that it exist as a subspace of some
higher dimensional flat space? To answer this, think back to high school
Euclidean geometry... You'll remember that the flatness of Euclidean
(mathematical) space was forced onto space. To put it differently, it
required a separate postulate or axiom --the parallel lines postulate. The
parallel lines postulate says that two lines which start out parallel will
always remain parallel. Without that forced structure, nothing requires
space to be flat, whether real physical space, or an axiomatic mathematical
space.
Most of the textbook examples of curved spaces use a simple "embedding" in
flat 3d space as an easy way to illustrate the concept of a curved space.
These pedagogic spaces, unfortunately for people trying to understand
general relativity and cosmology, do not distinguish between "intrinsic
curvature" and "extrinsic curvature". A space is curved intrinsically when
measurements of geometric relationships solely confined to that space allow
one to detect curvature locally. Extrinsic curvature is the curvature of a
subspace measured through geometric relationships in a higher dimensional
space that surrounds it.
How about a circle? Is it intrinsically curved? The circle itself
constitutes a 1-dimensional space. Embedded in the 2-dimensional plane, it
certainly has curvature. You can measure that *extrinsic* curvature by
giving the radius of the circle. Smaller radius means greater curvature, so
we can define the extrinsic curvature to be 1/R. But what about intrinsic
curvature? Is a circle intrinsically curved? Can you perform measurements
based only on the points that are part of the circle and detect the fact
that it is a circle and not a line? Clearly, no. There's nothing about
distance measurements on a circle that are different from distance
measurements on a line. You can't detect the curvature locally. Now if
you're thinking ahead a bit, you might say, 'but I can detect the fact that
the circle is curved by travelling all the way around it and returning to
my starting point'. That's not curvature --it's global topology. Whether a
space is closed or open or has connections between distant points is not
necessarily related in any direct way with curvature.
How about a cylinder? Is the 2-dimensional space defined by the surface of
a cylinder flat or curved? Extrinsically, it's clearly curved, but
measurements made within that 2-dim space will detect no curvature. This is
closely related to the fact that you can take a flat piece of paper and
roll it up into a cylinder without stretching the paper in any way. If you
draw a triangle on the surface of a cylinder, the angles will still add up
to 180 degrees. If you start out any two lines initially parallel and
transport them parallel to their own tangent vectors, they remain parallel
forever. A cylinder is a 2-dim flat space. Incidentally, this space is also
closed, and it is widely seen in a more obviously flat representation:
video games with left-right wrap-around are cylindrical spaces.
A 2-dimensional sphere has both intrinsic and extrinsic curvature. When you
embed it in a flat 3-space, you automatically get an ordinary-looking
curved surface. But more significantly, when you perform geometric
measurements on the surface of the sphere, you can detect its intrinsic
curvature. Start out two lines parallel to each other (think of lines of
longitude starting out parallel at the equator). When you follow their
straightest possible motion, you discover that the lines eventually
converge and meeet. That's the clearest indication that a 2-dim sphere has
intrinsic curvature. Note that 2-dim "ants" confined to such a surface
could easily perform this experiment. Next, suppose we lay out a triangle
using straightest possible lines (also shortest possible) for the sides of
the triangle. Adding up the angles, we would find that the sum is always
greater than 180 degrees.
Finally, what about the universe?? Does the fact that it is curved require
that it be embedded in some higher dimensional space. Does the existence of
curvature in three spatial dimensions (and one time dimension) imply that
the universe is a subspace of some higher dimensional space? In short,
absolutely not. The universe does not "curve into" anything. It is
intrinsically curved. Furthermore, the universe does not "expand into"
anything. The geometry of the universe is what it is. Forcing it into a
flat higher dimensional space can be a nice visualization tool on occasion,
but it is not required by either math or physics.
-FER