I have seen numerous examples (in books and on the Web) visualizing
"space-space" curvature by embedding a 2-dimensional (2-D)
"space-space" surface in Euclidian 3-D space.

In virtually all of those examples the graphs are "mislabeled",
refering to them as "space-time" surfaces, rather than "space-space"
surfaces, which is what they really are, because they don't have a time
axis/dimension (see Taylor/Wheeler below for the lone exception --
their graph caption is correct).

For some representative "mislabeled" examples see the following
graphics:

Posted at UCSD Center for Astrophysics & Space Sciences:
http://cassfos02.ucsd.edu/physics/ph...es/sp_curv.gif
http://cassfos02.ucsd.edu/physics/ph...es/gr_geom.gif
at page
http://cassfos02.ucsd.edu/physics/ph7/GR.html

Posted at the MIT Center for Space Research:
http://space.mit.edu/IAP/2004/ORBHOLE7_small.jpg
from
http://space.mit.edu/IAP/2004/activities.html

My guess is that it's easier to show embedded space-space than embedded
space-time. Or maybe worse, perhaps it's not possible to visualize
space-time around a mass by embedding a 2-D space-time surface in
Euclidian 3-D. Or what else could be the matter?

Even Taylor and Wheeler, in "Exploring Black Holes" show only an
embedded space-space surface (page 2-26, Figs. 6 and 7), but they too
omit an embedded space-time surface.

Why am I not finding a single example of 2-D space-time embedded in
3-D?

Thanks for any help and advise, especially any Web links, book
references, etc. If such a thing exists, I would particularly
appreciate an example of a 2-D space-time surface embedded in 3-D
(graph, and equations, if possible).

Wolfgang,
Santa Barbara, CA

perhaps it's not possible to visualize space-time around a mass by
embedding a 2-D space-time surface in
Euclidian 3-D. Or what else could be the matter?

1) If one axis was temporal, then you'd have only a single, linear
space axis. This prevents illustrating, for instance, orbital motion.

2) Time is bent and stretched just like space (modulo that pesky minus
sign) , so for the purposes of illustraing stuff, there's really no
reason not to stick with space-only for simplicity.

3) The above notwithstanding, the presence of the minus sign in the
metric causes funny things to happen when the coordinate-frame is
rotated. This may not affect "rubber-sheet" GR analogies, but if I
were writing a book, that would be an additional reason to stick to 2-D
space coordinates.

4) Unless you're trying to illustrate gravitational time dialiation,
time can be added to the 2-D spatial picture in a more natural way by
imagining the model as a movie, rather than as a frame in a move or a
snapshot.

Incidentally, I find it's much easier to visualize GR in my mind if I
throw away the rubber-sheet and think of 3-D space as having variable
density, like in that Arthur C. Clarke story from the book "tales from
the white hart".

I remebered the Clarke story when I read Feynman's similar 2-D analogy
at the end of either QED (or was it "Six not-so easy pieces"?).
Rather than think of a 2-D surface warped into 3 dimensions with
gravity pulling "down" from the underside of a rubber sheet (a pretty
crude and misleading analogy), he prefers to think of a 2D surface
which is denser and certain places than others.

In the story, this guy built some kind of "force field" that made space
denser in a sphere around the machine (and the guy inside). Thus, if
you fired a rifle at him, the bullet would travel an inch or so before
dropping to the floor.

The field wouldn't collapse when he turned the machine off, and they
couldn't get food or water to him because he was effectively thousands
of miles away, though he appeared to be sitting in the same room next
to his machine.

It's hard to put into words, but If you picture a "square mile' as
taking up a smaller volume of space near a star than it does in between
galaxies, then all of a sudden something "clicks" and a lot of GR
stuff that seems wierd or paraoxical suddenly appears natural.

Gravitational lensing, for instance, becomes less magical and more like
the phenemon of refraction in a dense medium.

=[ d

perhaps it's not possible to visualize space-time around a mass by
embedding a 2-D space-time surface in
Euclidian 3-D. what else could be the matter?

1) If one axis was temporal, then you'd have only a single, linear
space axis. This prevents illustrating, for instance, orbital motion.

2) Time is bent and stretched just like space (modulo that pesky minus
sign) , so for the purposes of illustraing stuff, there's really no
reason not to stick with space-only for simplicity.

3) The above notwithstanding, the presence of the minus sign in the
metric causes funny things to happen when the coordinate-frame is
rotated. This may not affect "rubber-sheet" GR analogies, but if I
were writing a book, that would be an additional reason to stick to 2-D
space coordinates.

4) Unless you're trying to illustrate gravitational time dialiation,
time can be added to the 2-D spatial analogy in a more natural way by
imagining the model as a movie, rather than as a frame in a movie or a
snapshot.

Incidentally, I find it's much easier to visualize GR in my mind if I
throw away the rubber-sheet and think of 3-D space as having variable
density, like in that Arthur C. Clarke story from the book "tales from
the white hart".

I remebered the Clarke story when I read Feynman's similar 2-D analogy
at the end of either QED (or was it "Six not-so easy pieces"?).
Rather than think of a 2-D surface warped into 3 dimensions with
gravity pulling "down" from the underside of a rubber sheet (a pretty
crude and misleading analogy), he prefers to think of a 2D surface
which is denser and certain places than others.

In Clarke's story, this guy built some kind of "force field" that made
space denser in a sphere around the machine (and the guy inside).
Thus, if you fired a rifle at him, the bullet would travel an inch or
so before dropping to the floor. I think he had to communicate by
radio.

The field wouldn't collapse when he turned the machine off, and they
couldn't get food or water to him because he was effectively thousands
of miles away, though he appeared to be sitting in the same room next
to his machine.

It's hard to put into words, but If you picture a "cubic mile' as
taking up a smaller volume of space near a star than it does in between
galaxies, then all of a sudden something "clicks" and a lot of GR
stuff that seems wierd or paraoxical suddenly appears natural.

Gravitational lensing, for instance, becomes less magical and more like
the phenemon of refraction in a dense medium.

=[ d

PS to any other Clarke fans:

1) he's 90 and in a wheelchair, but still furiously productve

2) he was knighted by the queen, so now everyone refers to him as "Sir
Arthur".

3) he DID survive the tsunami that washed over his home of sri lanka

4) he's paying big, big money from his own resources to help the people
there, and he released an eloquent press statement referring to life
per-se, and the world as a planet, and urging other pople in the world
to help too.

Rather than another "rubber sheet" analogy, I was hoping to find
something like the space-time equivalent of Figs. 6 and 7 (Space
geometry for a plane) in Taylors/Wheeler "Exploring Black Holes...",
see at

http://www.eftaylor.com/pub/chapter2.pdf
see page 2-26.

Thanks,
Wolfgang

wrote:
Why am I not finding a single example of 2-D space-time embedded in
3-D?

Because it is not possible to do so isometrically, and that is what
these diagrams are doing. That is, for the 2-d spatial slice of a
manifold embedded in 3-d Euclidean space, the intrinsic metric of the
2-d slice is the same as the Euclidean metric of the 3-d space in which
it is embedded. But for a 2-d space-time slice that simply is not
possible, because of the minus sign in the metric for the time component.


Tom Roberts

Thanks, Tom. I had considered that.

But then I was thinking that it is possible to draw a flat space-time
diagram with various world lines in it -- in fact, that's done all the
time in special relativity.

Now wouldn't it be possible to piece togeter a curved 2-d surface out
of flat space-time pieces to approximate the real thing, and thus
visualize the curved space-time?

Actually, in another of Wheller's books, Journey into Gravity and
Spacetime, page 25, they are doing just that, if only qualitatively.
They are piecing together a 2-d automobile fender surface with bits of
free-float reference frames (just a silly visualization example).

Wolfgang

Tom Roberts wrote:
wrote:
Why am I not finding a single example of 2-D space-time embedded in
3-D?

Because it is not possible to do so isometrically, and that is what
these diagrams are doing. That is, for the 2-d spatial slice of a
manifold embedded in 3-d Euclidean space, the intrinsic metric of the

2-d slice is the same as the Euclidean metric of the 3-d space in
which
it is embedded. But for a 2-d space-time slice that simply is not
possible, because of the minus sign in the metric for the time
component.


Tom Roberts

wrote:
But then I was thinking that it is possible to draw a flat space-time
diagram with various world lines in it -- in fact, that's done all the
time in special relativity.

Now wouldn't it be possible to piece togeter a curved 2-d surface out
of flat space-time pieces to approximate the real thing, and thus
visualize the curved space-time?

No. On that flat space-time diagram, along the 45-degree lines the
interval is zero. So in an isometric embedding into a Euclidean space
every point of that light-cone "X" would have to be a single point. And
for an instant later, the corresponding X would also have to all be a
single point. And for a small distance to the side again the entire "X"
would have to be a single point. Etc. And yet the regions not on the X
have nonzero interval and are therefore different points. All that
simply cannot be done.

What you can do is a non-isometric embedding. This is what the
space-time diagrams are. Basically you apply Minkowski coordinates onto
the spacetime, and then plot a 2-d slice of spacetime on a 2-d Euclidean
plane via the mapping (x,t) = (X,Y). But like any 2-d map of the
earth's surface, distance on such a space-time diagram does not
correspond to interval in spacetime (except in certain specific cases).


Actually, in another of Wheller's books, Journey into Gravity and
Spacetime, page 25, they are doing just that, if only qualitatively.
They are piecing together a 2-d automobile fender surface with bits of
free-float reference frames (just a silly visualization example).

I have not seen that book, but from your description that is a spatial
surface. Sure this can be done for well-behaved spatial surfaces. But
the geometrical structure of SR is just plain different.


Tom Roberts