CoCalc Public FilesBHLectures / sage / Kerr_ergo_a99.ipynb
Author: Eric Gourgoulhon
Views : 108
Compute Environment: Ubuntu 18.04 (Deprecated)

# Meridional plots in Kerr spacetime for $a/m=0.99$

In [1]:
%display latex


O'Neill exponential coordinates:

In [2]:
x,y = var('x y')
r = 1/2*ln(x^2 + y^2)
costh2 = y^2/(x^2+y^2)
sinth2 = x^2/(x^2+y^2)


Function $f$ defining the ergoregion by $f<0$:

In [3]:
a = 0.99
f = r^2 -2*r + a^2*costh2
f

$\frac{1}{4} \, \log\left(x^{2} + y^{2}\right)^{2} + \frac{0.980100000000000 \, y^{2}}{x^{2} + y^{2}} - \log\left(x^{2} + y^{2}\right)$
In [4]:
ergo = region_plot(f < 0, (x,-8, 8), (y, -5, 5), incol='lightgray',
bordercol='grey',
axes_labels=[r'$\mathrm{e}^{r/m}\sin\theta$',
r'$\mathrm{e}^{r/m}\cos\theta$'])
ergo += text(r'$\mathscr{E}^+$', (1.05*e^2, 0.5*e), color='grey',
fontsize=20)
ergo += text(r'$\mathscr{E}^-$', (1.5, 0.8), color='grey',
fontsize=20)


Various remarkable surfaces:

In [5]:
Rp = exp(1 + sqrt(1-a^2))
Rm = exp(1 - sqrt(1-a^2))
Hp = circle((0,0), Rp, color='black', thickness=2) + \
text(r'$\mathscr{H}$', (0.84*Rp, 0.77*Rp), color='black', fontsize=20)
Hm = circle((0,0), Rm, color='green', thickness=2) + \
text(r'$\mathscr{H}_{\rm in}$', (0.75*Rm, 0.9*Rm), color='green', fontsize=20)
R0 = circle((0,0), 1, color='darkorange', linestyle='dotted', thickness=3) + \
text(r'$r\!=\!0$', (1.1,-1.), color='darkorange', fontsize=16)
sing = circle((1,0), 0.1, color='red', fill=True) + \
circle((-1,0), 0.1, color='red', fill=True)
rminf = circle((0,0), 0.1, edgecolor='black', facecolor='white', fill=True)
region_label = text(r'${\rm I}$', (-2.3*Rp, Rp), fontsize=20) + \
text(r'${\rm II}$', (-0.72*Rp, 0.5*Rp), fontsize=20) + \
text(r'${\rm III}$', (-0.4*Rm, 0.5*Rm), fontsize=20)
graph = ergo + Hp + Hm + R0 + sing + rminf + region_label


Carter time machine:

In [6]:
ft = (r^2+a^2)*(r^2+a^2*costh2) + 2*a^2*r*sinth2

In [7]:
tmachine = region_plot(ft < 0, (x,-2, 2), (y, -2, 2), incol='yellow', bordercol='gold')
graph += tmachine
show(graph, aspect_ratio=1)

In [8]:
graph.save("ker_ergo_a99.pdf", aspect_ratio=1)

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