CoCalc Shared FilesBHLectures / sage / Kerr_ergo_a50.ipynbOpen in CoCalc with one click!
Author: Eric Gourgoulhon
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Meridional plots in Kerr spacetime for a/m=0.50a/m=0.50

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%display latex

O'Neill exponential coordinates:

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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 ff defining the ergoregion by f<0f<0:

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a = 0.5 f = r^2 -2*r + a^2*costh2 f
14log(x2+y2)2+0.250000000000000y2x2+y2log(x2+y2)\frac{1}{4} \, \log\left(x^{2} + y^{2}\right)^{2} + \frac{0.250000000000000 \, y^{2}}{x^{2} + y^{2}} - \log\left(x^{2} + y^{2}\right)
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ergo = region_plot(f < 0, (x,-8, 8), (y, -7, 7), incol='lightgray', bordercol='grey', axes_labels=[r'$\mathrm{e}^{r/m}\sin\theta$', r'$\mathrm{e}^{r/m}\cos\theta$']) ergo += text(r'$\mathscr{E}^+$', (1.1*e^2, 0.5*e), color='grey', fontsize=20) #ergo += text(r'$\mathscr{E}^-$', (1.4, 0.8), color='grey', # fontsize=20)

Various remarkable surfaces:

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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'$\mathscr{H}$', (0.77*Rp, 0.77*Rp), color='black', fontsize=20) Hm = circle((0,0), Rm, color='green', thickness=2) + \ text(r'$\mathscr{H}_{\rm in}$', (1.2*Rm, 1.05*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'${\rm I}$', (-1.2*Rp, 0.8*Rp), fontsize=20) + \ text(r'${\rm II}$', (-0.5*Rp, 0.4*Rp), fontsize=20) + \ text(r'${\rm III}$', (-0.3*Rm, 0.5*Rm), fontsize=20) graph = ergo + Hp + Hm + R0 + sing + rminf + region_label

Carter time machine:

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ft = (r^2+a^2)*(r^2+a^2*costh2) + 2*a^2*r*sinth2
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tmachine = region_plot(ft < 0, (x,-8, 8), (y, -3, 3), incol='yellow', bordercol='gold') graph += tmachine show(graph, aspect_ratio=1)
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graph.save("ker_ergo_a50.pdf", aspect_ratio=1)
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