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Alan
science forum addict
Joined: 03 May 2005
Posts: 51
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 Posted: Fri Jul 14, 2006 6:52 pm Post subject:
Visualizing a curved space
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I have a 2D Riemann metric tensor with coordinates (x,y) in a half-plane
-Infinity < x < Infinity, y > 0. I can calculate the geodesics.
I am looking for suggestions on algorithms for visualizations
of the geometry. (Ideally, something I can implement in Mathematica).
The space is somewhat similar to the Poincare hyperbolic
half-plane, in that the metric has the form
ds^2 = a(y) dx^2 + b(y) dy^2
I am a real novice in differential geometry, so the more elementary
the suggestions, the better.
Thanks!
alan |
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Oh No
science forum addict
Joined: 06 Apr 2006
Posts: 82
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| Posted: Sat Jul 15, 2006 5:29 am Post subject:
Re: Visualizing a curved space
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Thus spake Alan <info@optioncity.REMOVETHIS.net>
| Quote: | I have a 2D Riemann metric tensor with coordinates (x,y) in a half-plane
-Infinity < x < Infinity, y > 0. I can calculate the geodesics.
I am looking for suggestions on algorithms for visualizations
of the geometry. (Ideally, something I can implement in Mathematica).
The space is somewhat similar to the Poincare hyperbolic
half-plane, in that the metric has the form
ds^2 = a(y) dx^2 + b(y) dy^2
I am a real novice in differential geometry, so the more elementary
the suggestions, the better.
Thanks!
alan
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You could draw a field of orthogonal pairs of unit vectors (or vectors
of fixed length). They will need to be stretched by factors 1/a and 1/b
at different parts of the map. Then you can think of the geometry by
inverting the stretch in your mind's eye.
Regards
--
Charles Francis
substitute charles for NotI to email |
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David Park
science forum beginner
Joined: 29 Apr 2005
Posts: 46
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| Posted: Sat Jul 15, 2006 5:29 am Post subject:
Re: Visualizing a curved space
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"Alan" <info@optioncity.REMOVETHIS.net> wrote in message
news:LYSdnUfK_voKgyrZnZ2dnUVZ_qWdnZ2d@adelphia.com...
| Quote: | I have a 2D Riemann metric tensor with coordinates (x,y) in a half-plane
-Infinity < x < Infinity, y > 0. I can calculate the geodesics.
I am looking for suggestions on algorithms for visualizations
of the geometry. (Ideally, something I can implement in Mathematica).
The space is somewhat similar to the Poincare hyperbolic
half-plane, in that the metric has the form
ds^2 = a(y) dx^2 + b(y) dy^2
I am a real novice in differential geometry, so the more elementary
the suggestions, the better.
Thanks!
alan
|
Dear Alan,
If you are willing to invest a little money, $150 or $180 to buy the
required Mathematica packages then you could easily investigate and
visualize cases such as you describe. And if you wanted to you could join a
small group (3 persons at present) working on learning these subjects.
The packages are described on the Mathematica page of my web site below.
They are the Tensorial 4.0 package for tensor calculus and the DrawGraphics
package for making diagrams and animations. The ExtendUnits package might
also be useful for implementing geometric units when calculating practical
cases and for working with units in general.
Tensorial 4.0 is fully documented with Help pages for each command and many
examples. The first extended example in the Help is called Flat and Curved.
The curved portion is based on Problem 8.12 from Hartle's Gravitation text.
It investigates the metric
dS^2 == (dx^2 + dy^2)/y^2
The example generates the geodesic equations and plots the geodesics.
It also calculates the Riemann tensor, the Ricci tensor and the scalar
curvature.
On the Mathematica page there is a pdf document called
FullSpectrumPhysics.pdf that you could look at to get some idea of how these
packages work with Mathematica. There is also a pdf document on the
Tensorial page that illustrates using Tensorial 4.0 on Exercise [22.32] in
Penrose's 'Road to Reality', calculating the Laplacian operator on the
surface of a sphere.
David Park
djmp@earthlink.net
http://home.earthlink.net/~djmp/ |
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Greg Egan
science forum addict
Joined: 01 May 2005
Posts: 75
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| Posted: Sat Jul 15, 2006 5:29 am Post subject:
Re: Visualizing a curved space
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In article <LYSdnUfK_voKgyrZnZ2dnUVZ_qWdnZ2d@adelphia.com>, Alan
<info@optioncity.REMOVETHIS.net> wrote:
| Quote: | I have a 2D Riemann metric tensor with coordinates (x,y) in a half-plane
-Infinity < x < Infinity, y > 0. I can calculate the geodesics.
I am looking for suggestions on algorithms for visualizations
of the geometry. (Ideally, something I can implement in Mathematica).
The space is somewhat similar to the Poincare hyperbolic
half-plane, in that the metric has the form
ds^2 = a(y) dx^2 + b(y) dy^2
I am a real novice in differential geometry, so the more elementary
the suggestions, the better.
Thanks!
alan
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One thing you might like to try is looking for a surface in
three-dimensional Euclidean space, such that the induced metric on that
surface is your two-dimensional Riemannian metric.
Suppose you have a surface in R^3 which is parameterised by your two
coordinates x and y, with the point (x,y) on the surface having R^3
Cartesian coordinates:
(x,y) -> (F(x,y), G(x,y), H(x,y))
The R^3 Cartesian coordinates of the tangent vectors are then:
@_x -> (@F/@x, @G/@x, @H/@x)
@_y -> (@F/@y, @G/@y, @H/@y)
The metric induced on the surface by the Euclidean metric on R^3 can be
found just by taking the ordinary dot products of these vectors with
themselves, and with each other:
ds^2 = [(@F/@x)^2 + (@G/@x)^2 + (@H/@x)^2] dx^2
+ 2 [(@F/@x)(@F/@y) + (@G/@x)(@G/@y) + (@H/@x)(@H/@y)] dx dy
+ [(@F/@y)^2 + (@G/@y)^2 + (@H/@y)^2] dy^2
Now, in your case you want the metric:
| Quote: | ds^2 = a(y) dx^2 + b(y) dy^2
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so you want to look for solutions F(x,y), G(x,y), H(x,y) to the partial
differential equations:
(@F/@x)^2 + (@G/@x)^2 + (@H/@x)^2 = a(y)
(@F/@x)(@F/@y) + (@G/@x)(@G/@y) + (@H/@x)(@H/@y) = 0
(@F/@y)^2 + (@G/@y)^2 + (@H/@y)^2 = b(y)
One simple example would be a(y)=R^2 (sin y)^2, b(y)=R^2. Then one
solution would be:
F(x,y) = R sin y cos x
G(x,y) = R sin y sin x
H(x,y) = R cos y
Obviously the details will depend on your own a(y) and b(y). If you
can't find closed-form solutions, you might be able to find solutions
numerically. (My own experience with Mathematica 4 numerically solving
PDEs is that you should take a 2-week holiday while it churns away, and
the final result is as likely as not to be an error message saying that
it couldn't achieve the required precision, but you might have a better
version or faster hardware ...) |
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tessel@um.bot
science forum addict
Joined: 20 Sep 2005
Posts: 60
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| Posted: Wed Jul 19, 2006 6:35 am Post subject:
Re: Visualizing a curved space
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On Sat, 15 Jul 2006, David Park wrote:
| Quote: | If you are willing to invest a little money, $150 or $180 to buy the
required Mathematica packages then you could easily investigate and
visualize cases such as you describe. And if you wanted to you could
join a small group (3 persons at present) working on learning these
subjects.
They are the Tensorial 4.0 package for tensor calculus and the
DrawGraphics package for making diagrams and animations. The ExtendUnits
package might also be useful for implementing geometric units when
calculating practical cases and for working with units in general.
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To the OP: I am not familiar with Tensorial but I think I can suggest an
alternative approach which will probably be more convenient and will
certainly be significantly cheaper, at least for readers who are
registered university students or who already own Mathematica or Maple.
For investigating specific (semi)-Riemannian manifolds, you will almost
certainly get more bang for your time/money with a mature, powerful and
FREE Maple package (compatible with Maple 9.5-10.0)
http://grtensor.org/
I am not affiliated in any way with the GRTensorII team, BTW, but I have
used this software extensively and know that it is convenient and very
easy to learn and use. In his recent book, Eric Poisson also sings the
praises of GRTensorII.
Some specific comments: GRTensorII fully supports the convenient use of NP
formalism and frame fields (orthonormal bases, rather than coordinate
bases) for tensor computations. Even for working with coordinate bases,
the built in Maple tensor package is incredibly clunky by comparision.
Datasharing with other Maple packages can sometimes be a bit problematic
with GRTensorII, but I might be able to help with specific problems of
this nature. For example, computing Killing vectors is a breeze with
GRTensorII and the casesplit command from Maple's built-in PDETools
package.
In fact, as I see it, there is only one possible caveat: Maple itself is
not cheap. But registered students can get it (legally) for a hundred
dollars or so in the US, so for students Maple plus GRTensorII is
definitely the way to go. Similarly, Mathematica is not cheap---unless
you are a registered university student. (Last time I checked, at least.)
For working with completely general Lorentzian manifolds, e.g. symbolic
index gymnastics, other packages may offer significant advantages, but
these tend to be more specialized.
For Mathematica users unwilling to try Maple (not that there is much new
syntax or many new commands to learn if you just want to start using
GRTensorII), there is GRTensorM, an older and somewhat less capable
counterpart of GRTensorII developed by the same team (come to think of it,
GRTensorM is so old that it might not work with the most recent versions
of Mathematica). Another good choice for Mathematica users might be the
package
http://www.math.washington.edu/~lee/Ricci/
which is mature, reliable, well documented, and free. I am slightly
acquainted with Jack Lee but have no affiliation with his software.
HTH!
"T. Essel" |
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