## Null curves in time

See also the first post and the other posts in this category. Reading them in the order posted might be a good idea.

Archimerged has been playing with gnuplot. (He no longer uses Mathematica, being a supporter of free/libre software. Sage is a nice free computer algebra package, and it uses gnuplot for graphics.) He got reminded of gnuplot because some images on Wikipedia were made with it. It has a very clunky interface, like VMS, which it was modeled after. (VMS was about the fifth operating system Ark learned). But Ark spent time log ago learning gnuplot, so it’s easier for him than to learn a 3D modeling package like k3d, which he intends to learn, but its faster not to.

Anyway, here we have a nice colored plot of the outgoing null curves of the metric Ark models his electron after (Ark isn’t dropping the name of the metric or giving its usual interpretation):

The third dimension is time and each color is a different principal null curve. You can tell these curves are outgoers because they start at small diameter and as time increases (upward) they move outward from the circle at the center. All of these curves are at the same latitude of the sphere. Note how they are straightening out.

Thinking in 4D is not easy, and intuitions can be misleading. In the end, you have to prove things in algebra to be sure the geometry is right. Ark is presently trying to figure out how to smoothly deform those curves so they all go in the same direction. That’s fairly easy. The hard part is the fact that there is another set of them (the ingoers) to deal with. Maybe it can’t be done. Ark hopes not, but it wouldn’t be the first time. See also the blog motto.

For each pair of distinct points p and q, either pNq, or pTq. Relative to p, q is neither here nor there (lightlike, null), or else q is elsewhere (there, spacelike). Every curve on the surface is either the null curve, or it is spacelike. If pNq, then either pBq, or pAq. One is before the other, but no points off a given null curve are before or after any of the points on the null curve.

Another one of these surfaces fits just above (after) this one, but rotated about the axis a little. That is, Ark’s electron is always sending these null curves out somewhere. But another diagram is needed… Off to make it.

Here is the code which produced that plot. Ark used gimp to convert the postscript to png.
``` set size ratio -1 0.5, 0.5 unset zeroaxis unset xtics unset ytics unset ztics unset border unset colorbox r(a,b) = sqrt(a**2+b**2) mod2pi(a) = a - 2*pi*floor(a/(2*pi)) phi(a,b) = (r(a,b) > 1 ? mod2pi(atan2(a,b) - 1/(r(a,b) - 1)) : 0) T(a,b) = Tr(r(a,b)) Tr(r) = (r > 1 ? r - 2/(r - 1) + 2*log(r - 1) : -100) set parametric set terminal table set output "3D.table" set samples 600 set isosamples 600 splot [0:2*pi] [1/1.05:1/16.0] cos(u)/v, sin(u)/v, T(cos(u)/v,sin(u)/v) set terminal postscript color enhanced solid linewidth 0.5 portrait set output "3D.ps" set cbrange [0:2*pi] set palette model HSV functions gray,0.75,1 set view 60,0 set pm3d at s explicit corners2color c1 splot "3D.table" using 1:2:3:(phi(\$1,\$2)) with pm3d notitle ```