This worksheet demonstrates a few capabilities of SageManifolds (version 0.8) in computations regarding anti-de Sitter spacetime.
It is released under the GNU General Public License version 3.
(c) Eric Gourgoulhon, Michal Bejger (2015)
The corresponding worksheet file can be downloaded from here
We declare the anti-de Sitter spacetime (AdS) as a 4-dimensional differentiable manifold:
︡5a905f65-a3bd-4889-971d-c3b0ecf65cab︡︡{"done":true,"html":"This worksheet demonstrates a few capabilities of SageManifolds (version 0.8) in computations regarding anti-de Sitter spacetime.
\nIt is released under the GNU General Public License version 3.
\n(c) Eric Gourgoulhon, Michal Bejger (2015)
\nThe corresponding worksheet file can be downloaded from here
\nWe declare the anti-de Sitter spacetime (AdS) as a 4-dimensional differentiable manifold:
"} ︠2a633602-a77e-4e76-a426-8228f58c318es︠ M = Manifold(4, 'M', r'\mathcal{M}') print M ; M ︡5083962a-5a13-4813-bde9-ffafbfb9ea65︡︡{"stdout":"4-dimensional manifold 'M'\n","done":false}︡{"html":"We consider hyperbolic coordinates $(\tau,\rho,\theta,\phi)$ on $\mathcal{M}$. Allowing for the standard coordinate singularities at $\rho=0$, $\theta=0$ or $\theta=\pi$, these coordinates cover the entire spacetime manifold (which is topologically $\mathbb{R}^4$). If we restrict ourselves to regular coordinates (i.e. to considering only mathematically well defined charts), the hyperbolic coordinates cover only an open part of $\mathcal{M}$, which we call $\mathcal{M}_0$, on which $\rho$ spans the open interval $(0,+\infty)$, $\theta$ the open interval $(0,\pi)$ and $\phi$ the open interval $(0,2\pi)$. Therefore, we declare:
︡e2fc7a9f-431e-4309-bf63-4085792c8895︡︡{"done":true,"html":"We consider hyperbolic coordinates $(\\tau,\\rho,\\theta,\\phi)$ on $\\mathcal{M}$. Allowing for the standard coordinate singularities at $\\rho=0$, $\\theta=0$ or $\\theta=\\pi$, these coordinates cover the entire spacetime manifold (which is topologically $\\mathbb{R}^4$). If we restrict ourselves to regular coordinates (i.e. to considering only mathematically well defined charts), the hyperbolic coordinates cover only an open part of $\\mathcal{M}$, which we call $\\mathcal{M}_0$, on which $\\rho$ spans the open interval $(0,+\\infty)$, $\\theta$ the open interval $(0,\\pi)$ and $\\phi$ the open interval $(0,2\\pi)$. Therefore, we declare:
"} ︠d21eedde-ce55-4bf8-b0ae-60c1c842b00ds︠ M0 = M.open_subset('M_0', r'\mathcal{M}_0' ) X_hyp.The AdS metric can be defined as that induced by the immersion of $\mathcal{M}$ in $\mathbb{R}^5$ equipped with a flat pseudo-Riemannian metric of signature $(-,-,+,+,+)$. We therefore introduce $\mathbb{R}^5$ as a 5-dimensional manifold covered by canonical coordinates:
︡70ed26de-2130-4dd1-92ac-e0a36d5b7ef1︡︡{"done":true,"html":"The AdS metric can be defined as that induced by the immersion of $\\mathcal{M}$ in $\\mathbb{R}^5$ equipped with a flat pseudo-Riemannian metric of signature $(-,-,+,+,+)$. We therefore introduce $\\mathbb{R}^5$ as a 5-dimensional manifold covered by canonical coordinates:
"} ︠e1d6267d-f2e9-40e6-af65-cf0abcb3335bs︠ R5 = Manifold(5, 'R5', r'\mathbb{R}^5') X5.The AdS immersion into $\mathbb{R}^5$ is defined as a differential mapping $\Phi$ from $\mathcal{M}$ to $\mathbb{R}^5$, by providing its expression in terms of $\mathcal{M}$'s default chart (which is X_hyp = $(\mathcal{M}_0,(\tau,\rho,\theta,\phi))$ ) and $\mathbb{R}^5$'s default chart (which is X5 = $(\mathbb{R}^5,(U,V,X,Y,Z))$ ):
︡5b500950-09bd-4fe2-83f8-90725dc4ec48︡︡{"done":true,"html":"The AdS immersion into $\\mathbb{R}^5$ is defined as a differential mapping $\\Phi$ from $\\mathcal{M}$ to $\\mathbb{R}^5$, by providing its expression in terms of $\\mathcal{M}$'s default chart (which is X_hyp = $(\\mathcal{M}_0,(\\tau,\\rho,\\theta,\\phi))$ ) and $\\mathbb{R}^5$'s default chart (which is X5 = $(\\mathbb{R}^5,(U,V,X,Y,Z))$ ):
"} ︠8a592ea4-d47d-44d9-91ae-d64f47b45666s︠ var('b') assume(b>0) Phi = M.diff_mapping(R5, [sin(b*ta)/b * cosh(rh), cos(b*ta)/b * cosh(rh), sinh(rh)/b *sin(th)*cos(ph), sinh(rh)/b *sin(th)*sin(ph), sinh(rh)/b *cos(th)], name='Phi', latex_name=r'\Phi') print Phi ; Phi.display() ︡59e4a7ef-04e4-49cf-8c54-c289481e5af4︡︡{"html":"The constant $b$ is a scale parameter. Considering AdS metric as a solution of vacuum Einstein equation with negative cosmological constant $\Lambda$, one has $b = \sqrt{-\Lambda/3}$.
Let us evaluate the image of a point via the mapping $\Phi$:
︡f0a274a1-25a6-4474-802d-9a0cb5165662︡︡{"done":true,"html":"The constant $b$ is a scale parameter. Considering AdS metric as a solution of vacuum Einstein equation with negative cosmological constant $\\Lambda$, one has $b = \\sqrt{-\\Lambda/3}$.
\nLet us evaluate the image of a point via the mapping $\\Phi$:
"} ︠aeb10b82-131e-460c-911d-649a50bdb902s︠ p = M.point((ta, rh, th, ph), name='p') ; print p ︡e3c8f0e0-8aeb-4dc1-838e-c5153527801f︡︡{"stdout":"point 'p' on 4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠0cc1137f-62b6-477c-993b-96023f713cccs︠ p.coord() ︡c12b9e30-a267-425b-9ca0-18038a158e8e︡︡{"html":"The image of $\mathcal{M}$ by the immersion $\Phi$ is a hyperboloid of one sheet, of equation $-U^2-V^2+X^2+Y^2+Z^2=-b^{-2}$. Indeed:
︡99a67af7-242e-4b5e-8ad3-7925178b2e50︡︡{"done":true,"html":"The image of $\\mathcal{M}$ by the immersion $\\Phi$ is a hyperboloid of one sheet, of equation $-U^2-V^2+X^2+Y^2+Z^2=-b^{-2}$. Indeed:
"} ︠c4865d64-a781-43f0-a0dc-7696293d839ds︠ (Uq,Vq,Xq,Yq,Zq) = q.coord() s = - Uq^2 - Vq^2 + Xq^2 + Yq^2 + Zq^2 s.simplify_full() ︡ba9a7cc3-0d3a-444e-8d5a-13feed526ed9︡︡{"html":"We may use the immersion $\Phi$ to draw the coordinate grid $(\tau,\rho)$ in terms of the coordinates $(U,V,X)$ for $\theta=\pi/2$ and $\phi=0$ (red) and $\theta=\pi/2$ and $\phi=\pi$ (green) (the brown lines are the lines $\tau={\rm const}$):
︡39b31ebf-53bf-45b8-abb3-97f89e2cfcd9︡︡{"done":true,"html":"We may use the immersion $\\Phi$ to draw the coordinate grid $(\\tau,\\rho)$ in terms of the coordinates $(U,V,X)$ for $\\theta=\\pi/2$ and $\\phi=0$ (red) and $\\theta=\\pi/2$ and $\\phi=\\pi$ (green) (the brown lines are the lines $\\tau={\\rm const}$):
"} ︠3badfed4-1d46-45d3-af08-fe161cd77e48s︠ graph1 = X_hyp.plot(X5, mapping=Phi, ambient_coords=(V,X,U), fixed_coords={th:pi/2, ph:0}, ranges={ta:(0,2*pi), rh:(0,2)}, nb_values=9, color={ta:'red', rh:'brown'}, thickness=2, parameters={b:1}, label_axes=False) graph2 = X_hyp.plot(X5, mapping=Phi, ambient_coords=(V,X,U), fixed_coords={th:pi/2, ph:pi}, ranges={ta:(0,2*pi), rh:(0,2)}, nb_values=9, color={ta:'green', rh:'brown'}, thickness=2, parameters={b:1}, label_axes=False) show(set_axes_labels(graph1+graph2,'V','X','U'), aspect_ratio=1) ︡cfd04a31-eb20-4ca1-9657-1c854c696351︡︡{"done":false,"file":{"uuid":"4f38eed0-94dd-4cd5-82b4-be310e21c94e","filename":"4f38eed0-94dd-4cd5-82b4-be310e21c94e.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠b5639770-b567-4a4e-a6be-58ffcf28ce79i︠ %htmlFirst, we introduce on $\mathbb{R}^5$ the flat pseudo-Riemannian metric $h$ of signature $(-,-,+,+,+)$:
︡ecac02bd-0e14-4705-977c-a949c0b186ae︡︡{"done":true,"html":"First, we introduce on $\\mathbb{R}^5$ the flat pseudo-Riemannian metric $h$ of signature $(-,-,+,+,+)$:
"} ︠8a3b5ec3-b0dd-494c-b9aa-26016286931as︠ h = R5.metric('h') h[0,0], h[1,1], h[2,2], h[3,3], h[4,4] = -1, -1, 1, 1, 1 h.display() ︡1c178614-c8b2-4551-851f-34ca4b60da12︡︡{"html":"As mentionned above, the AdS metric $g$ on $\mathcal{M}$ is that induced by $h$, i.e.$g$ is the pullback of $h$ by the mapping $\Phi$:
︡ed4b8e15-88b4-46ba-a93c-4bfea38c4c97︡︡{"done":true,"html":"As mentionned above, the AdS metric $g$ on $\\mathcal{M}$ is that induced by $h$, i.e.$g$ is the pullback of $h$ by the mapping $\\Phi$:
"} ︠c5f26bed-9f06-4b61-ba7c-bc9a35acd29cs︠ g = M.metric('g') g.set( Phi.pullback(h) ) ︡15202338-535a-4844-860a-64e73b2e74fc︡︡{"done":true} ︠c486f9e2-5fd7-4677-a0fa-1b795d4029b8i︠ %htmlThe expression of $g$ in terms of $\mathcal{M}$'s default frame is found to be
︡545ef86b-4fed-4376-b1e0-5adb78c03f0d︡︡{"done":true,"html":"The expression of $g$ in terms of $\\mathcal{M}$'s default frame is found to be
"} ︠7247c07a-a7b8-473d-9690-adcb3e66ae0as︠ g.display() ︡822f7fb0-fcc6-4ffa-bf3b-ad2eb4f4d616︡︡{"html":"The Riemann tensor of $g$ is
︡b4222e04-49b3-4332-a146-8facff56183f︡︡{"done":true,"html":"The Riemann tensor of $g$ is
"} ︠83537f0d-4951-4dda-9033-740d3280c668s︠ Riem = g.riemann() print Riem Riem.display() ︡812a4f44-0861-44db-a796-840309dc872e︡︡{"stdout":"tensor field 'Riem(g)' of type (1,3) on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"The Ricci tensor:
︡e0917f8e-a4da-4d51-a410-2cea9cd26a25︡︡{"done":true,"html":"The Ricci tensor:
"} ︠e9cdb083-6903-420e-84ef-6c2b6eea71d8s︠ Ric = g.ricci() print Ric Ric.display() ︡985fee6a-ddef-4ce7-a62f-0fa33328df6c︡︡{"stdout":"field of symmetric bilinear forms 'Ric(g)' on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"The Ricci scalar:
︡59536501-aff2-433b-9420-d8f238259a76︡︡{"done":true,"html":"The Ricci scalar:
"} ︠834781c5-1b1e-458a-bd94-cdd172bdf9b5s︠ R = g.ricci_scalar() print R R.display() ︡cd527551-3a3a-4cf9-af8a-1e9866700469︡︡{"stdout":"scalar field 'r(g)' on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"We recover the fact that AdS spacetime has a constant curvature. It is indeed a maximally symmetric space. In particular, the Riemann tensor is expressible as
\[ R^i_{\ \, jlk} = \frac{R}{n(n-1)} \left( \delta^i_{\ \, k} g_{jl} - \delta^i_{\ \, l} g_{jk} \right) \]
where $n$ is the dimension of $\mathcal{M}$: $n=4$ in the present case. Let us check this formula here, under the form $R^i_{\ \, jlk} = -\frac{R}{6} g_{j[k} \delta^i_{\ \, l]}$:
︡bc6922c8-6cbc-469e-8fd2-86fa017c1d32︡︡{"done":true,"html":"We recover the fact that AdS spacetime has a constant curvature. It is indeed a maximally symmetric space. In particular, the Riemann tensor is expressible as
\n\\[ R^i_{\\ \\, jlk} = \\frac{R}{n(n-1)} \\left( \\delta^i_{\\ \\, k} g_{jl} - \\delta^i_{\\ \\, l} g_{jk} \\right) \\]
\nwhere $n$ is the dimension of $\\mathcal{M}$: $n=4$ in the present case. Let us check this formula here, under the form $R^i_{\\ \\, jlk} = -\\frac{R}{6} g_{j[k} \\delta^i_{\\ \\, l]}$:
"} ︠6598f855-0628-4bbd-badf-bdc433de081cs︠ delta = M.tangent_identity_field() Riem == - (R/6)*(g*delta).antisymmetrize(2,3) # 2,3 = last positions of the type-(1,3) tensor g*delta ︡d15f4c98-b08b-41e1-90df-2e86187fae5a︡︡{"html":"We may also check that AdS metric is a solution of the vacuum Einstein equation with (negative) cosmological constant:
︡dcefef53-e31d-4ab6-b21d-25cbcfb3d205︡︡{"done":true,"html":"We may also check that AdS metric is a solution of the vacuum Einstein equation with (negative) cosmological constant:
"} ︠0c8cf710-fd36-4a3f-a183-40eca418f3eas︠ Lambda = -3*b^2 Ric - 1/2*R*g + Lambda*g == 0 ︡323adf3c-c1cf-4837-84f8-f0659ec04f57︡︡{"html":"Let us introduce spherical coordinates $(\tau,r,\theta,\phi)$ on the AdS spacetime via the coordinate change \[ r = \frac{\sinh(\rho)}{b} \]
︡21439943-49d9-43c4-9825-c90d7e83cede︡︡{"done":true,"html":"Let us introduce spherical coordinates $(\\tau,r,\\theta,\\phi)$ on the AdS spacetime via the coordinate change \\[ r = \\frac{\\sinh(\\rho)}{b} \\]
"} ︠6b7ff19b-9ebe-426e-8634-52b0abf04187s︠ X_spher.The expression of the metric tensor in the new coordinates is
︡fa34227f-45c4-4614-ad10-336ff2341fbc︡︡{"done":true,"html":"The expression of the metric tensor in the new coordinates is
"} ︠8fafb558-7030-4bef-a616-e4e60dce5886s︠ g.display(X_spher.frame(), X_spher) ︡b72a4c3d-83db-4789-8462-245695c978f0︡︡{"html":"