Hi Imran,
On Wed, Oct 14, 2015 at 10:14 AM, Imran Ali <imranalnor...@gmail.com> wrote:
> I have implemented a SymPy program that can calculate the Riemann curvature
> tensor for a given curve element. However, I am encountering problems
> solving for the case when the curve element is the surface of a sphere
>
> \begin{align}
> ds^2 = r^2d\theta^2 + r^2 \sin^2\theta d\phi^2
> \end{align}
>
> This is obviously a 2D curve element, so the non-zero elements of the metric
> become
> \begin{align}
> g_{11} = r^2, \qquad g_{22} = r^2 \sin^2\theta.
> \end{align}
> The entries of metric are clearly a function of two variables $r$ and
> $\theta$. But the way I have created the program it treats them according to
> their differentials $d\theta$ and $d\phi$. Since $dr$ is 'zero', my metric
> is computed as
> \begin{align}
> \begin{bmatrix}
> 0 &0 &0\\
> 0 &r^2 &0\\
> 0 &0 &r^2 \sin^2\theta
> \end{bmatrix}.
> \end{align}
> The way I have coded my implementation is by asking the user for the metric
> defined as a matrix. If the matrix is 2D, then I use $u$,$v$ to represent
> the coordinates. Which in the 2D case assign $r$ as $u$ and $\theta$ as $v$.
> For 3D (with metric above), the additional value $\phi$ is assigned $w$.
If your space/surface is only 2D, then the metric tensor is a 2x2
matrix, I think it's just:
[r^2, 0]
[0, r^2 sin^2(theta)]
And that's what you need to feed into your program. Then things should
work. If you have a zero entry in the 3x3 metrix tensor, then the
coordinates are degenerate, and I guess your code can't handle it.
>
> Does anyone see my dilemma here? For 3D, I am basically trying to calculate
> the Riemann tensor for a metric with the determinant equal to zero. And for
> 2D, the $\phi$ component does not even exist.
>
> This element is important for me to test my code as this generates a
> non-zero Riemann curvature tensor. I would really appreciate any suggestions
> how I can handle this case and thereby improve my code....which fails
> completely for this case.
>
> (I posted the exact post at physics on stackexchange :
> http://physics.stackexchange.com/questions/212541/finding-the-riemann-tensor-for-the-surface-of-a-sphere-with-sympy-diffgeom#212541
> , and they gently directed my here. I have posted the code on pastebin :
> http://pastebin.com/DPxW38L0 - the problem lies in the way I have defined
> the constructor for Riemann class)
If you want to look at a working code in 4D, look here:
https://github.com/sympy/sympy/blob/master/examples/advanced/relativity.py
Then you can adapt it for a 2D case.
Ondrej
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