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.

However, I have not found even a single visualization of a
corresponding
"space-time" surface anywhere.

A curved 2-D space-time surface (1 time axis plus 1 space axis)
embedded in 3-D could visualize dynamic behavior, which mere a mere
space-space slice cannot.

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.

And even Misner, et al., in "Gravitation" pages 614 and 837, show only
an
embedded space-space surface and NO 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. I would particularly appreciate an example of any 2-D
space-time surface embedded in 3-D (graph, and equations, if possible),
f such a thing exists, .

Wolfgang,
Santa Barbara, CA

wrote:
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.

However, I have not found even a single visualization of a
corresponding "space-time" surface anywhere.

_Relativity Visualized_ by Lewis Epstein has a bunch of them. I don't know
whether they're correct. The author is a bit cranky, but it's an interesting
book in any case.

It's not clear to me that this kind of embedding even makes sense. The point
of an embedding is that it preserves the metric, and you obviously can't
preserve a 1+1 metric in 3+0 space. You could embed in 2+1 dimensions and
then identify that with 3+0, or you could somehow Euclidify the 1+1 metric
and then embed in 3+0, but I think these would lack the intuitive appeal of
a space-space embedding. That may be why none of the books do it. It's also
possible that the obvious choices of 1+1 surface can't be embedded even with
one of the above tricks. The fact that a space-space slice of the
Schwarzschild solution is embeddable is something of an accident.

(Epstein's approach, by the way, is to embed a surface parameterized by
coordinate position and *proper* time, so that coordinate time works out as
the Euclidean path length (dt^2 = ds^2 + dx^2). I told you it was an
interesting book.)

-- Ben