# Visualizing vector fields lines in Kerr

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.

• I am speaking only of the visualization aspects here - I have been plotting geodesics in the Kerr metric for some time now. Firstly the easy part, I just use the conversion on this page en.wikipedia.org/wiki/Boyer%E2%80%93Lindquist_coordinates. For the visualization, I use something like "gnuplot -p -e "set style data vector; splot 'vector_fields.data' using 1:2:3:4:5:6" where the format is x,y,z,dx,dy,dz, if that helps. Commented Mar 30, 2023 at 13:47
• Do you mean the equations for $x,y,z$? Aren't those only for the case $M\to 0$? Commented Mar 30, 2023 at 18:39
• On closer inspection, the lower case "m" is not defined anywhere. If they really mean "M", then that statement (M -> 0) is nonsense since "a" is defined as J/M, where M is the mass. Looks like a bit of a mess TBH, but I have always used that transformation for non-zero "M"! I'll take another look tomorrow. Commented Mar 30, 2023 at 19:10
• OK I've just checked Visser's "The Kerr Spacetime", arxiv.org/abs/0706.0622. Looks like the wiki page has garbled equations 59 with 60-62, but the (M -> 0) part is only applicable to the derivation of the coordinate transformation. I think this is the right conclusion, what do you think? (EDIT haha that is exactly the reference quoted in the wiki page!) Commented Mar 30, 2023 at 19:16
• OK, now I see what you were asking: x, y, z are the coordinates, and dx, dy, dz are the field components. My other comments are still valid though. Commented Mar 30, 2023 at 21:13

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.