Gallery

Here are some figures produced with SageManifolds:

  • Stereographic coordinates on the 2-sphere:
    sphere_S2_stereo

    This represents the coordinate grids of the two stereographic charts on the 2-dimensional sphere: in red (resp. green), stereographic coordinates from the North pole N (resp. South pole S). This figure is produced with the method plot() of SageManifolds charts. The corresponding worksheet is here.

  • Vector frame of stereographic coordinates on the 2-sphere:
    sphere_stereo_frame

    This represents the vector frame \(\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right)\) on the 2-dimensional sphere, which is associated with the stereographic coordinates \((x,y)\) from the North pole. The vector field \(\frac{\partial}{\partial x}\) (resp. \(\frac{\partial}{\partial y}\)) is represented in blue (resp. in red). This figure is produced with the method plot() of SageManifolds vector fields. The corresponding worksheet is here.

  • A curve on the 2-sphere:
    sphere_loxodrome

    This represents a loxodrome of the sphere with its tangent vector field. This figure is produced with the methods DifferentiableCurve.plot() and VectorField.plot(). The corresponding worksheet is here.

  • Boy surface:
    boy_surface

    This represents the coordinate grids of three charts (red, green and blue) covering the real projective plane \(\mathbb{RP}^2 \), the latter being immersed in the Euclidean space \(\mathbb{R}^3\) by the Apéry Map (Boy surface). This figure is produced with the method plot() of SageManifolds charts. The corresponding worksheet is here.

  • Kruskal diagram of Schwarzschild spacetime:
    Kruskal_diag

    This represents the coordinate grid of the standard Schwarzschild-Droste coordinates \((t,r,\theta,\varphi)\) in terms of the Kruskal-Szekeres coordinates \((U,V,\theta,\varphi)\). The figure is drawn at a fixed value of \((\theta,\varphi)\). The solid lines are lines \(t=\)const, while the dashed ones are lines \(r=\)const. The black hole interior (resp. exterior) is depicted in green (resp. red). The thick yellow line represents the central singularity, located at \(r=0\), and the black line depicts the event horizon. This figure is produced with the method plot() of SageManifolds charts. The corresponding worksheet is here.

  • Carter-Penrose diagram of Schwarzschild spacetime:
    Carter-Penrose_diag

    This is a global view of Schwarzschild spacetime obtained by plotting two charts of standard Schwarzschild-Droste coordinates \((t,r,\theta,\varphi)\) in terms of compactified coordinates \((\hat{T},\hat{X},\theta,\varphi)\): one chart for the region \(\mathscr{M}_{\rm I}\cup\mathscr{M}_{\rm II}\) and the other one for the region \(\mathscr{M}_{\rm III}\cup\mathscr{M}_{\rm IV}\). \(\mathscr{H}\) is the black hole event horizon. The figure is drawn at a fixed value of \((\theta,\varphi)\). The solid lines are lines \(t=\)const, while the dashed ones are lines \(r=\)const. This figure is produced with the method plot() of SageManifolds charts. The corresponding worksheet is here.

  • Simon-Mars scalar in Tomimatsu-Sato spacetime:
    Tomimatsu-Sato_Simon-Mars_scal

    This figure shows the isocontours of the logarithm of the real part of the Simon-Mars scalar defined in arXiv:1412.6542 in a meridional plane of the \(\delta=2\) Tomimatsu-Sato spacetime. The plane is spanned by the Weyl-Lewis-Papapetrou coordinates \((\rho,z)\). The Simon-Mars scalar diverges at the ring singularity of Tomimatsu-Sato spacetime located at \((\rho,z) \simeq (1.140, 0)\). The black line marks the boundary of the ergoregion. This is Fig. 12 of the article arXiv:1412.6542. The corresponding worksheet is here.

  • Horizon formation in Vaidya spacetime:
    vaidya_horizons

    This represents the gravitational collapse of a radiative shell (in yellow) in Vaidya's spherically symmetric solution of the Einstein equation. The green curves are radial null geodesics, the red curve marks the trapping horizon and the black one the event horizon. The corresponding worksheet is here.