tessel@tum.bot
Oct11-06, 02:49 PM
On Mon, 9 May 2005 mot12345@alexandria.ucsb.edu wrote:
> I have seen numerous examples (in the literature and on the Web, incl.
> sci.physics.relativity)
Hmm.... probably not a good source of information, unless you were lucky
enough to come across a recent post by sometime like Tom Roberts or Steve
Carlip.
> 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.
Fear not, many explicit embeddings of Lorentzian manifolds in higher
dimensional flat spaces E^(p,q) are known. In particular, simple local
embeddings of Friedmann models, dS and AdS models, and Schwarzschild
vacuum are available.
If you look carefully, you can find a simple example of what you call a
"space-time" embedding mentioned in MTW! Specifically, you can look (try
the index?) for an embedding of H^(1,1) in E^(1,2). This is a 1+1
dimensional reduction of the well known de Sitter "vacuum" (nonzero
Lambda). One can embed 1+1 AdS the same way; indeed, this is the same
embedding, only with "space" and "time" swapped! See
author = {S. W. Hawking and G. F. R. Ellis},
title = {The Large Scale Structure of Space-Time},
publisher = {Cambridge University Press},
year = 1973}
for a nice discussion of dS and AdS, including an embedding in E^(1,4).
You can also look for at least one past post in this group where we
discussed in great detail about a dozen (!) coordinate charts, including
at least two embeddings. Note that these are important in a much wider
context than gtr or indeed mathematical physics. For example, they arise
in studying Kleinian geometries associated with "point symmetry groups" of
such basic equations as the Laplace equation.
For an embedding of a two dimensional reduction of a Friedmann dust (one
time and one space dimension), see the semipopular book
title = {The Shape of Space: How to Visualize Surfaces and
Three-Dimensional Manifolds },
publisher = {Marcel Dekker},
series = {Monographs and textbooks in pure and applied mathematics},
volume = 96,
year = 1985}
For a precise version of this "Friedmann football" (for the "recollapsing"
dust solution or "matter dominated model" with S^3 hyperslices orthogonal
to the world lines of the dust particles) and the analogous "Friedmann
lemon" (for the radiation fluid or "radiation dominated model" with the
same symmetries), plus many more embeddings in E^(1,2), E^(1,3) and
E^(2,4) of various 1+1, 1+2, and 1+3 dimensional Lorentzian manifolds, you
can begin by working the "Coordinate Tutorial" which you can find in one
of the webpages at John Baez's website
http://math.ucr.edu/home/baez/relativity.html
BTW, it might help to point out that you should not always expect to
locally embedd a Lorentzian spacetime in E^(1,4). Sometimes an extra
dimension is needed; for example, you need at least E^(2,4) for a local
embedding of the Schwarzschild vacuum. This might explain the source of
your difficulty, if you were trying to find a local embedding of such a
Lorentzian manifold! OTH, if you were trying to embed Friedmann models,
you can obtain the S^3 hyperslice dust and radiation fluid models using
one fewer dimension.
IIRC, we've also discussed here explicit local embeddings of the various
possible three dimensional real Lie groups (c.f. "Bianchi manifolds" and
"Thurston geometries"). This can shed some light on why certain essential
Lie groups -cannot- be realized as -matrix- Lie groups. See
author = {Roger Carter and Graeme Segal and Ian MacDonald},
title = {Lectures on {L}ie Groups and {L}ie Algebras},
series = {London Mathematical Society student texts},
volume = 32,
publisher = {Cambridge University Press},
year = 1995}
Note that in case of complicated topology and such like, a -global-
embedding may require more dimensions, possibly many more dimensions.
You can look for past postings here by Rob Low mentioning an upper bound
on the least number of dimensions which is known to always suffice,
something absurd like E^(2,79). This is a Lorentzian analog of well known
embedding theorems for Riemannian manifolds (which require a positive
definite metric tensor).
> Can anyone amongst you suggest how to go about this?
The same way you'd look for a euclidean embedding. One elementary method
which often works runs as follows: fix a suitable E^(p,q) and guess a
suitable basis of tangent vectors written using a few judiciously chosen
undetermined functions. (It is a good idea to try to take advantage here
of any symmetry in the spacetime you are trying to embed.) Then compute
the induced metric on the tangent space they span to obtain the line
element for a family of submanifolds, given in terms of these undetermined
functions. Set this line element equal to the given line element for your
Lorentzian manifold (e.g. Schwarzschild vacuum in the exterior
Schwarzschild chart). Solve the resulting system of differential
equations for the undetermined functions and choose any integration
constants for convenience. See the "Coordinate Tutorial" for various
simple examples. Practice helps!
In the Schwarzschild example, note that since the Schwarzschild chart is
only valid on the exterior region, you can't expect to obtain a global
embedding without further work. But you can try obtaining a local
embedding of the Painleve chart (valid down to r = 0) or a global chart
such as the Kruskal/Szekeres chart.
> 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.
If you have any questions about the "Coordinate Tutorial" or the other
resources I mentioned, ask again, since we certainly know lots of explicit
examples of embeddings of interesting spacetimes.
You can also try the standard monograph
author = {D. Kramer and H. Stephani and E. Herlt and M. MacCallum},
title = {Exact Solutions of {E}instein's Field Equations},
publisher = {Cambridge University Press},
edition = {second},
series = {Cambridge monographs on mathematical physics},
volume = 6,
year = 2003}
which devotes a chapter to various embedding problems, for example
determining the most general vacuum spacetimes which can be embedded with
fewer dimensions than required by the generic Lorentzian spacetime (for
which, IIRC, E^(2,4) will work -locally-).
I should warn you that embeddings have so far not proven very useful in
gtr. However, we have pointed out on several previous occasions that
suitable "polynomial approximations" to embedded manifolds are suceptible
to Puiseaux expansions near algebraic singularities, and one can try to
use this to analyze curvature and other geometric singularities. IIRC, in
the "Coordinate Tutorial" this is carried out for the well-known Painleve
chart covering one half of the "maximal analytic extension of the
Schwarzschild vacuum" (i.e. the spacetime described by the
Kruskal/Szekeres chart). This is interesting because the spatial slices
there are each isometric to E^3, so the picture we obtain is analogous to
the "cusp" curve in a typical tangent developable surface, as in "surface
theory" in classical differential geometry, except of course that now we
have nonzero intrinsic curvature.
Hmmm... if memory serves, tangent developable surfaces are discussed in
the very nice little UG level textbook
author = {Richard S. Millman and George D. Parker},
title = {Elements of Differential Geometry},
publisher = {Prentice-Hall},
year = 1977}
Re the "local versus global disinction" in differential geometry, the
discussion in this book of Milnor's theorem about plane curves may be
helpful.
"T. Essel"
> I have seen numerous examples (in the literature and on the Web, incl.
> sci.physics.relativity)
Hmm.... probably not a good source of information, unless you were lucky
enough to come across a recent post by sometime like Tom Roberts or Steve
Carlip.
> 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.
Fear not, many explicit embeddings of Lorentzian manifolds in higher
dimensional flat spaces E^(p,q) are known. In particular, simple local
embeddings of Friedmann models, dS and AdS models, and Schwarzschild
vacuum are available.
If you look carefully, you can find a simple example of what you call a
"space-time" embedding mentioned in MTW! Specifically, you can look (try
the index?) for an embedding of H^(1,1) in E^(1,2). This is a 1+1
dimensional reduction of the well known de Sitter "vacuum" (nonzero
Lambda). One can embed 1+1 AdS the same way; indeed, this is the same
embedding, only with "space" and "time" swapped! See
author = {S. W. Hawking and G. F. R. Ellis},
title = {The Large Scale Structure of Space-Time},
publisher = {Cambridge University Press},
year = 1973}
for a nice discussion of dS and AdS, including an embedding in E^(1,4).
You can also look for at least one past post in this group where we
discussed in great detail about a dozen (!) coordinate charts, including
at least two embeddings. Note that these are important in a much wider
context than gtr or indeed mathematical physics. For example, they arise
in studying Kleinian geometries associated with "point symmetry groups" of
such basic equations as the Laplace equation.
For an embedding of a two dimensional reduction of a Friedmann dust (one
time and one space dimension), see the semipopular book
title = {The Shape of Space: How to Visualize Surfaces and
Three-Dimensional Manifolds },
publisher = {Marcel Dekker},
series = {Monographs and textbooks in pure and applied mathematics},
volume = 96,
year = 1985}
For a precise version of this "Friedmann football" (for the "recollapsing"
dust solution or "matter dominated model" with S^3 hyperslices orthogonal
to the world lines of the dust particles) and the analogous "Friedmann
lemon" (for the radiation fluid or "radiation dominated model" with the
same symmetries), plus many more embeddings in E^(1,2), E^(1,3) and
E^(2,4) of various 1+1, 1+2, and 1+3 dimensional Lorentzian manifolds, you
can begin by working the "Coordinate Tutorial" which you can find in one
of the webpages at John Baez's website
http://math.ucr.edu/home/baez/relativity.html
BTW, it might help to point out that you should not always expect to
locally embedd a Lorentzian spacetime in E^(1,4). Sometimes an extra
dimension is needed; for example, you need at least E^(2,4) for a local
embedding of the Schwarzschild vacuum. This might explain the source of
your difficulty, if you were trying to find a local embedding of such a
Lorentzian manifold! OTH, if you were trying to embed Friedmann models,
you can obtain the S^3 hyperslice dust and radiation fluid models using
one fewer dimension.
IIRC, we've also discussed here explicit local embeddings of the various
possible three dimensional real Lie groups (c.f. "Bianchi manifolds" and
"Thurston geometries"). This can shed some light on why certain essential
Lie groups -cannot- be realized as -matrix- Lie groups. See
author = {Roger Carter and Graeme Segal and Ian MacDonald},
title = {Lectures on {L}ie Groups and {L}ie Algebras},
series = {London Mathematical Society student texts},
volume = 32,
publisher = {Cambridge University Press},
year = 1995}
Note that in case of complicated topology and such like, a -global-
embedding may require more dimensions, possibly many more dimensions.
You can look for past postings here by Rob Low mentioning an upper bound
on the least number of dimensions which is known to always suffice,
something absurd like E^(2,79). This is a Lorentzian analog of well known
embedding theorems for Riemannian manifolds (which require a positive
definite metric tensor).
> Can anyone amongst you suggest how to go about this?
The same way you'd look for a euclidean embedding. One elementary method
which often works runs as follows: fix a suitable E^(p,q) and guess a
suitable basis of tangent vectors written using a few judiciously chosen
undetermined functions. (It is a good idea to try to take advantage here
of any symmetry in the spacetime you are trying to embed.) Then compute
the induced metric on the tangent space they span to obtain the line
element for a family of submanifolds, given in terms of these undetermined
functions. Set this line element equal to the given line element for your
Lorentzian manifold (e.g. Schwarzschild vacuum in the exterior
Schwarzschild chart). Solve the resulting system of differential
equations for the undetermined functions and choose any integration
constants for convenience. See the "Coordinate Tutorial" for various
simple examples. Practice helps!
In the Schwarzschild example, note that since the Schwarzschild chart is
only valid on the exterior region, you can't expect to obtain a global
embedding without further work. But you can try obtaining a local
embedding of the Painleve chart (valid down to r = 0) or a global chart
such as the Kruskal/Szekeres chart.
> 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.
If you have any questions about the "Coordinate Tutorial" or the other
resources I mentioned, ask again, since we certainly know lots of explicit
examples of embeddings of interesting spacetimes.
You can also try the standard monograph
author = {D. Kramer and H. Stephani and E. Herlt and M. MacCallum},
title = {Exact Solutions of {E}instein's Field Equations},
publisher = {Cambridge University Press},
edition = {second},
series = {Cambridge monographs on mathematical physics},
volume = 6,
year = 2003}
which devotes a chapter to various embedding problems, for example
determining the most general vacuum spacetimes which can be embedded with
fewer dimensions than required by the generic Lorentzian spacetime (for
which, IIRC, E^(2,4) will work -locally-).
I should warn you that embeddings have so far not proven very useful in
gtr. However, we have pointed out on several previous occasions that
suitable "polynomial approximations" to embedded manifolds are suceptible
to Puiseaux expansions near algebraic singularities, and one can try to
use this to analyze curvature and other geometric singularities. IIRC, in
the "Coordinate Tutorial" this is carried out for the well-known Painleve
chart covering one half of the "maximal analytic extension of the
Schwarzschild vacuum" (i.e. the spacetime described by the
Kruskal/Szekeres chart). This is interesting because the spatial slices
there are each isometric to E^3, so the picture we obtain is analogous to
the "cusp" curve in a typical tangent developable surface, as in "surface
theory" in classical differential geometry, except of course that now we
have nonzero intrinsic curvature.
Hmmm... if memory serves, tangent developable surfaces are discussed in
the very nice little UG level textbook
author = {Richard S. Millman and George D. Parker},
title = {Elements of Differential Geometry},
publisher = {Prentice-Hall},
year = 1977}
Re the "local versus global disinction" in differential geometry, the
discussion in this book of Milnor's theorem about plane curves may be
helpful.
"T. Essel"