space-time visualization (vs. space-space visualization)

mot12345@alexandria.ucsb.edu

I have seen numerous examples (in the literature and on the Web, incl.
sci.physics.relativity) visualizing "space-space" curvature by
embedding a 2-dimensional (2-D) Schwarzschild "space-space" surface in
Euclidian 3-D space. Without a time dimension, however, those cannot
represent dynamic behavior. (Examples of "space-space" visualization
are Misner, et al., in "Gravitation" pages 614 and 837, and
Taylor/Wheeler, in "Exploring Black Holes" page 2-26, Figs. 6 and 7.
As as everywhere else, true space-time surface visualization is missing
here.)

My research has failed to come up with a visualization of a
"space-time" surface, and no one else seems to have been able to come
up with such a visualization either.

Can anyone amongst you suggest how to go about this?

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

Wolfgang
UC Santa Barbara

Igor Khavkine

On 2005-05-09, mot12345@alexandria.ucsb.edu <mot12345@alexandria.ucsb.edu>
wrote:
> I have seen numerous examples (in the literature and on the Web, incl.
> sci.physics.relativity) visualizing "space-space" curvature by
> embedding a 2-dimensional (2-D) Schwarzschild "space-space" surface in
> Euclidian 3-D space. Without a time dimension, however, those cannot
> represent dynamic behavior. (Examples of "space-space" visualization
> are Misner, et al., in "Gravitation" pages 614 and 837, and
> Taylor/Wheeler, in "Exploring Black Holes" page 2-26, Figs. 6 and 7.
> As as everywhere else, true space-time surface visualization is missing
> here.)
>
> My research has failed to come up with a visualization of a
> "space-time" surface, and no one else seems to have been able to come
> up with such a visualization either.
>
> Can anyone amongst you suggest how to go about this?
>
> Thanks for any help with this, especially any Web links, literature
> references, etc. I would particularly appreciate an example of any 2-D
> space-time surface embedded in 3-D (graph, and equations, if possible),
> if such a thing exist.

The closest thing that comes to mind is embedding a 1+1 dimensional
sheet in a 3+1 Minkoswki space such that the metric on the 1+1 sheet is
the one inherited from the ambient metric through the embedding. This
sheet could for example represent the (t,r) coordinate plane of the
space-time you want to visualize. But I haven't seen this done for any
of the commonly known space-time solutions.

On the other hand, you might want to look up Kruskal or Penrose
diagrams. These are conformal diagrams (they preserve the light cone
structure at every point) and are useful for describing global
space-time properties for common spherically and axially symmetric
space-time solutions.

Hope this helps.

Igor

carlip-nospam@physics.ucdavis.edu

mot12345@alexandria.ucsb.edu wrote:
> I have seen numerous examples (in the literature and on the Web, incl.
> sci.physics.relativity) visualizing "space-space" curvature by
> embedding a 2-dimensional (2-D) Schwarzschild "space-space" surface in
> Euclidian 3-D space. Without a time dimension, however, those cannot
> represent dynamic behavior. (Examples of "space-space" visualization
> are Misner, et al., in "Gravitation" pages 614 and 837, and
> Taylor/Wheeler, in "Exploring Black Holes" page 2-26, Figs. 6 and 7.
> As as everywhere else, true space-time surface visualization is missing
> here.)

> My research has failed to come up with a visualization of a
> "space-time" surface, and no one else seems to have been able to come
> up with such a visualization either.

Try "Spacetime Embedding Diagrams for Spherically Symmetric Black Holes,"
gr-qc/0305102.

Steve Carlip