Testing Elliptic PDE Solver with Non-Diagonal Metric

[Pablo Brubeck]

Hello, I am working with numerical relativity and spectral methods. Recently I finished a general elliptic PDE solver using spectral methods, so now I want to do Physics with it. I am interested in solving the lapse equation, which fits into this category of PDEs
$$ \nabla^2 \alpha = \alpha \left(K_{ij}K^{ij}+4\pi\left(\rho+S\right)\right), $$
where $\nabla^2$ is the Laplace-Beltrami operator. Could you please provide a non-diagonal spatial metric $\gamma_{ij}$ and a corresponding intrinsic curvature $K_{ij}$ (and maybe non-zero matter sources $\rho, S$) for which I might obtain an interesting solution for the lapse? Or could you recommend a book exercise? Also I would like a way to verify my numerical solution.

I am also willing to solve for those parameters via similar elliptic equations.

Thank you.

 

[Ambitwistor]

Hello! Congratulations on finishing your PDE solver using spectral methods. Solving the lapse equation is definitely an interesting and challenging task. To provide a non-diagonal spatial metric and corresponding intrinsic curvature, I would recommend looking into the Kerr metric, which describes the spacetime around a rotating black hole. The metric has a non-diagonal form and involves an intrinsic curvature term that depends on the black hole's spin. This would be a great exercise to test your PDE solver and see how it handles non-diagonal metrics and intrinsic curvature.

As for matter sources, you could consider adding a rotating disk of matter around the black hole, which would contribute to the energy density $\rho$ and stress tensor $S$ in your lapse equation. This would also add an interesting physical aspect to your solution.

To verify your numerical solution, you could try comparing it to known analytical solutions for the Kerr metric, such as the Boyer-Lindquist coordinates. Alternatively, you could vary the parameters (such as the black hole spin or matter distribution) and see how your solution changes, which would give you a better understanding of the behavior of the lapse equation.

I would also recommend looking into other elliptic equations that are commonly used in numerical relativity, such as the Hamiltonian and momentum constraints, to further test and validate your PDE solver.

I hope this helps and good luck with your research!