Manifolds Catalog¶
A catalog of manifolds to rapidly create various simple manifolds.
The current entries to the catalog are obtained by typing
manifolds.<tab>
, where <tab>
indicates pressing the tab key.
They are:
EuclideanSpace
: Euclidean spaceRealLine
: real lineOpenInterval
: open interval on the real lineSphere
: sphere embedded in Euclidean spaceTorus()
: torus embedded in Euclidean spaceMinkowski()
: 4-dimensional Minkowski spaceKerr()
: Kerr spacetime
AUTHORS:
Florentin Jaffredo (2018) : initial version
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sage.manifolds.catalog.
Kerr
(m=1, a=0, coordinates='BL', names=None)¶ Generate a Kerr spacetime.
A Kerr spacetime is a 4 dimensional manifold describing a rotating black hole. Two coordinate systems are implemented: Boyer-Lindquist and Kerr (3+1 version).
The shortcut operator
.<,>
can be used to specify the coordinates.INPUT:
m
– (default:1
) mass of the black hole in natural units (\(c=1\), \(G=1\))a
– (default:0
) angular momentum in natural units; if set to0
, the resulting spacetime corresponds to a Schwarzschild black holecoordinates
– (default:"BL"
) either"BL"
for Boyer-Lindquist coordinates or"Kerr"
for Kerr coordinates (3+1 version)names
– (default:None
) name of the coordinates, automatically set by the shortcut operator
OUTPUT:
Lorentzian manifold
EXAMPLES:
sage: m, a = var('m, a') sage: K = manifolds.Kerr(m, a) sage: K 4-dimensional Lorentzian manifold M sage: K.atlas() [Chart (M, (t, r, th, ph))] sage: K.metric().display() g = (2*m*r/(a^2*cos(th)^2 + r^2) - 1) dt*dt + 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt*dph + (a^2*cos(th)^2 + r^2)/(a^2 - 2*m*r + r^2) dr*dr + (a^2*cos(th)^2 + r^2) dth*dth + 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph*dt + (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2 dph*dph sage: K.<t, r, th, ph> = manifolds.Kerr() sage: K 4-dimensional Lorentzian manifold M sage: K.metric().display() g = (2/r - 1) dt*dt + r^2/(r^2 - 2*r) dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph sage: K.default_chart().coord_range() t: (-oo, +oo); r: (0, +oo); th: (0, pi); ph: [-pi, pi] (periodic) sage: m, a = var('m, a') sage: K.<t, r, th, ph> = manifolds.Kerr(m, a, coordinates="Kerr") sage: K 4-dimensional Lorentzian manifold M sage: K.atlas() [Chart (M, (t, r, th, ph))] sage: K.metric().display() g = (2*m*r/(a^2*cos(th)^2 + r^2) - 1) dt*dt + 2*m*r/(a^2*cos(th)^2 + r^2) dt*dr - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt*dph + 2*m*r/(a^2*cos(th)^2 + r^2) dr*dt + (2*m*r/(a^2*cos(th)^2 + r^2) + 1) dr*dr - a*(2*m*r/(a^2*cos(th)^2 + r^2) + 1)*sin(th)^2 dr*dph + (a^2*cos(th)^2 + r^2) dth*dth - 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph*dt - a*(2*m*r/(a^2*cos(th)^2 + r^2) + 1)*sin(th)^2 dph*dr + (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2 dph*dph sage: K.default_chart().coord_range() t: (-oo, +oo); r: (0, +oo); th: (0, pi); ph: [-pi, pi] (periodic)
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sage.manifolds.catalog.
Minkowski
(positive_spacelike=True, names=None)¶ Generate a Minkowski space of dimension 4.
By default the signature is set to \((- + + +)\), but can be changed to \((+ - - -)\) by setting the optional argument
positive_spacelike
toFalse
. The shortcut operator.<,>
can be used to specify the coordinates.INPUT:
positive_spacelike
– (default:True
) ifFalse
, then the spacelike vectors yield a negative sign (i.e., the signature is \((+ - - - )\))names
– (default:None
) name of the coordinates, automatically set by the shortcut operator
OUTPUT:
Lorentzian manifold of dimension 4 with (flat) Minkowskian metric
EXAMPLES:
sage: M.<t, x, y, z> = manifolds.Minkowski() sage: M.metric()[:] [-1 0 0 0] [ 0 1 0 0] [ 0 0 1 0] [ 0 0 0 1] sage: M.<t, x, y, z> = manifolds.Minkowski(False) sage: M.metric()[:] [ 1 0 0 0] [ 0 -1 0 0] [ 0 0 -1 0] [ 0 0 0 -1]
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sage.manifolds.catalog.
Torus
(R=2, r=1, names=None)¶ Generate a 2-dimensional torus embedded in Euclidean space.
The shortcut operator
.<,>
can be used to specify the coordinates.INPUT:
R
– (default:2
) distance form the center to the center of the tuber
– (default:1
) radius of the tubenames
– (default:None
) name of the coordinates, automatically set by the shortcut operator
OUTPUT:
Riemannian manifold
EXAMPLES:
sage: T.<theta, phi> = manifolds.Torus(3, 1) sage: T 2-dimensional Riemannian submanifold T embedded in the Euclidean space E^3 sage: T.atlas() [Chart (T, (theta, phi))] sage: T.embedding().display() T --> E^3 (theta, phi) |--> (X, Y, Z) = ((cos(theta) + 3)*cos(phi), (cos(theta) + 3)*sin(phi), sin(theta)) sage: T.metric().display() gamma = dtheta*dtheta + (cos(theta)^2 + 6*cos(theta) + 9) dphi*dphi