Visualizing vector fields lines in Kerr Newman

Question

I want to visualise an test EM field in Kerr spacetime (I want to plot the integral field lines). The field is a test field in the sense, that it doesn't change the background (metric).

Could anyone point me to some more technical literature or some Mathematica/python code that would help me?

What I thought I would do: I have the field $F^{\alpha \beta}$ in Boyer-Lindquist coordinates. I then took the ZAMO tetrad (observer family) $e^\alpha_a$ with $e^\mu_0 = u^\mu$ is the for velocity normalized to $$u_\alpha u^\alpha = -1.$$ I then projected the EM tensor onto the tetrad $$ F^{\alpha \beta} e_\alpha^a e_\beta^a = F^{ab} $$ and from this I took the standard definition of electric field $$ E^\alpha = -\frac{1}{2} F^{\alpha \beta }u_\beta \implies E^a = -\frac{1}{2} F^{a0} $$ So now I have the electric field with respects to the ZAMO tetrad.

But how do i visualize this? The tetrad is still expressed in Boyer-Lindquist coordinates (which are like spheriodial coordinates). Do I have to transform into Kerr-Schild coordinates which reduce to cartesian (minkowski) in the flat-spacetime limit?

I hope it's obvious from the context, but I used Greek letters $\alpha \beta$ to denote the indices with respect to coordinates and Latin letters $ab$ to denote indices with respect to tetrad.

For example I would like to plot something like figure 2 and figure 3 in this article.

Edit: The question can also be reduced to - do I always need to transform to cartesian like coordinates when plotting fields on curved background.

Answer 1

Nataa a asked: "I want to visualise an test EM field in Kerr spacetime [...] But how do i visualize this? [...] Could anyone point me to some more technical literature or some Mathematica/python code that would help me?"

The Mathematica code for the $\rm \{\lambda,z\}$-plot

is at kerr.newman.yukterez.net at 17) streamplot, also see doi:10.1103/physreva.36.5118

For the vertical and horizontal magnetic and electric field lines we have

$\rm M_z=Q \ Im[(z-ia)/\sigma] \ \ \ \ \ \ \ \ \ M_{\lambda}=Q \ Im[\lambda/\sigma]$

$\rm E_z=Q \ Re[(z-ia)/\sigma] \ \ \ \ \ \ \ \ \ \ E_{\lambda}=Q \ Re[\lambda/\sigma]$

$\rm \sigma=[\lambda^2+(z-ia)^2]^{3/2} \ \ \ \ \ \ \ \ \ \ \lambda=\sqrt{x^2+y^2}$

In a streamplot you'd plot $\rm \{M_z, M_{\lambda}\}$ as vector, and in a contourplot its magnitude.