This worksheet demonstrates a few capabilities of SageManifolds (version 0.8) in computations regarding 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 de Sitter spacetime as a 4-dimensional differentiable manifold:
︡1a79dab5-14b4-46b6-bf98-37f372689817︡︡{"done":true,"html":"This worksheet demonstrates a few capabilities of SageManifolds (version 0.8) in computations regarding 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 de Sitter spacetime as a 4-dimensional differentiable manifold:
"} ︠24d5fa0f-8346-40be-b1f9-d290e19c7eb7s︠ M = Manifold(4, 'M', r'\mathcal{M}') print M ; M ︡85f6bb30-509d-428e-bd30-179bddf5418b︡︡{"stdout":"4-dimensional manifold 'M'\n","done":false}︡{"html":"We consider hyperspherical coordinates $(\tau,\chi,\theta,\phi)$ on $\mathcal{M}$. Allowing for the standard coordinate singularities at $\chi=0$, $\chi=\pi$, $\theta=0$ or $\theta=\pi$, these coordinates cover the entire spacetime manifold (which is topologically $\mathbb{R}\times\mathbb{S}^3$). If we restrict ourselves to regular coordinates (i.e. to consider only mathematically well defined charts), the hyperspherical coordinates cover only an open part of $\mathcal{M}$, which we call $\mathcal{M}_0$, on which $\chi$ spans the open interval $(0,\pi)$, $\theta$ the open interval $(0,\pi)$ and $\phi$ the open interval $(0,2\pi)$. Therefore, we declare:
︡59173d33-0274-40c4-bb0f-e92bd7ab4c55︡︡{"done":true,"html":"We consider hyperspherical coordinates $(\\tau,\\chi,\\theta,\\phi)$ on $\\mathcal{M}$. Allowing for the standard coordinate singularities at $\\chi=0$, $\\chi=\\pi$, $\\theta=0$ or $\\theta=\\pi$, these coordinates cover the entire spacetime manifold (which is topologically $\\mathbb{R}\\times\\mathbb{S}^3$). If we restrict ourselves to regular coordinates (i.e. to consider only mathematically well defined charts), the hyperspherical coordinates cover only an open part of $\\mathcal{M}$, which we call $\\mathcal{M}_0$, on which $\\chi$ spans the open interval $(0,\\pi)$, $\\theta$ the open interval $(0,\\pi)$ and $\\phi$ the open interval $(0,2\\pi)$. Therefore, we declare:
"} ︠f0299b37-e526-4d0c-b43c-40e69a814df5s︠ M0 = M.open_subset('M_0', r'\mathcal{M}_0') X_hyp.The de Sitter metric can be defined as that induced by the embedding of $\mathcal{M}$ into a 5-dimensional Minkowski space, i.e. $\mathbb{R}^5$ equipped with a flat Lorentzian metric. We therefore introduce $\mathbb{R}^5$ as a 5-dimensional manifold covered by canonical coordinates:
︡fca2182f-7629-4410-a727-14864325a0f7︡︡{"done":true,"html":"The de Sitter metric can be defined as that induced by the embedding of $\\mathcal{M}$ into a 5-dimensional Minkowski space, i.e. $\\mathbb{R}^5$ equipped with a flat Lorentzian metric. We therefore introduce $\\mathbb{R}^5$ as a 5-dimensional manifold covered by canonical coordinates:
"} ︠d0b1e74d-4c91-4322-b795-80a0a7ca4195s︠ R5 = Manifold(5, 'R5', r'\mathbb{R}^5') X5.The embedding of $\mathcal{M}$ 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 = $(\mathcal{M}_0,(\tau,\chi,\theta,\phi))$ ) and $\mathbb{R}^5$'s default chart (which is X5 = $(\mathbb{R}^5,(T,W,X,Y,Z))$ ):
︡f221c66e-b9dc-45fc-80de-7cc4b93060e5︡︡{"done":true,"html":"The embedding of $\\mathcal{M}$ 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 = $(\\mathcal{M}_0,(\\tau,\\chi,\\theta,\\phi))$ ) and $\\mathbb{R}^5$'s default chart (which is X5 = $(\\mathbb{R}^5,(T,W,X,Y,Z))$ ):
"} ︠fd1bf3f9-e4b2-48c7-9421-c203fc254746s︠ var('b') Phi = M.diff_mapping(R5, [sinh(b*ta)/b, cosh(b*ta)/b * cos(ch), cosh(b*ta)/b * sin(ch)*sin(th)*cos(ph), cosh(b*ta)/b * sin(ch)*sin(th)*sin(ph), cosh(b*ta)/b * sin(ch)*cos(th)], name='Phi', latex_name=r'\Phi') print Phi ; Phi.display() ︡19ad33d3-b254-4a94-8456-7fea8eb685e4︡︡{"html":"The constant $b$ is a scale parameter. Considering de Sitter metric as a solution of vacuum Einstein equation with positive cosmological constant $\Lambda$, one has $b = \sqrt{\Lambda/3}$.
Let us evaluate the image of a point via the mapping $\Phi$:
︡3b910702-3d26-4988-8552-562ed75d4786︡︡{"done":true,"html":"The constant $b$ is a scale parameter. Considering de Sitter metric as a solution of vacuum Einstein equation with positive cosmological constant $\\Lambda$, one has $b = \\sqrt{\\Lambda/3}$.
\nLet us evaluate the image of a point via the mapping $\\Phi$:
"} ︠db99fc29-1ac0-4e02-a922-56c5f7380704s︠ p = M.point((ta, ch, th, ph), name='p') ; print p ︡67fda435-17c3-46a5-9440-b32c0aefd7d6︡︡{"stdout":"point 'p' on 4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠840a88f5-a4ee-4e85-978e-e2a4da3f4d88s︠ p.coord() ︡4816f7f7-7ae9-4456-80b0-3234d342f6ed︡︡{"html":"The image of $\mathcal{M}$ by $\Phi$ is a hyperboloid of one sheet, of equation $-T^2+W^2+X^2+Y^2+Z^2=b^{-2}$. Indeed:
︡169d759a-c5c2-40f9-b675-4523d50a1ce2︡︡{"done":true,"html":"The image of $\\mathcal{M}$ by $\\Phi$ is a hyperboloid of one sheet, of equation $-T^2+W^2+X^2+Y^2+Z^2=b^{-2}$. Indeed:
"} ︠5a77bbbb-4503-4804-876a-6726eec509efs︠ (Tq,Wq,Xq,Yq,Zq) = q.coord() s = -Tq^2 + Wq^2 + Xq^2 + Yq^2 + Zq^2 s.simplify_full() ︡317e2113-669c-48cb-aadf-138c4878c917︡︡{"html":"We may use the embedding $\Phi$ to draw the coordinate grid $(\tau,\chi)$ in terms of the coordinates $(W,X,T)$ for $\theta=\pi/2$ and $\phi=0$ (red) and $\theta=\pi/2$ and $\phi=\pi$ (green) (the brown lines are the lines $\tau={\rm const}$):
︡a7f58c7b-5e1f-47aa-af3e-2279d975a032︡︡{"done":true,"html":"We may use the embedding $\\Phi$ to draw the coordinate grid $(\\tau,\\chi)$ in terms of the coordinates $(W,X,T)$ for $\\theta=\\pi/2$ and $\\phi=0$ (red) and $\\theta=\\pi/2$ and $\\phi=\\pi$ (green) (the brown lines are the lines $\\tau={\\rm const}$):
"} ︠c8924865-aa83-486a-ae6d-1c263d1c8c42s︠ graph1 = X_hyp.plot(X5, mapping=Phi, ambient_coords=(W,X,T), fixed_coords={th:pi/2, ph:0}, nb_values=9, color={ta:'red', ch:'brown'}, thickness=2, max_value=2, parameters={b:1}, label_axes=False) graph2 = X_hyp.plot(X5, mapping=Phi, ambient_coords=(W,X,T), fixed_coords={th:pi/2, ph:pi}, nb_values=9, color={ta:'green', ch:'brown'}, thickness=2, max_value=2, parameters={b:1}, label_axes=False) show(set_axes_labels(graph1+graph2,'W','X','T'), aspect_ratio=1) ︡3e3373c9-2c1d-42e5-9471-4e765e0afbef︡︡{"done":false,"file":{"uuid":"f97f4212-d46d-4ad8-b8f7-4979d61774e6","filename":"f97f4212-d46d-4ad8-b8f7-4979d61774e6.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠b0c32c4b-e08c-47d9-bf00-638b410c1ef7i︠ %htmlFirst, we introduce on $\mathbb{R}^5$ the Minkowski metric $h$:
︡46a2101e-2eb3-452a-a3c7-2e6277ec8e18︡︡{"done":true,"html":"First, we introduce on $\\mathbb{R}^5$ the Minkowski metric $h$:
"} ︠00290c1e-af84-4a35-b444-9921f139b586s︠ h = R5.lorentz_metric('h') h[0,0], h[1,1], h[2,2], h[3,3], h[4,4] = -1, 1, 1, 1, 1 h.display() ︡af70c54a-d38e-4285-882f-890adf08aa0c︡︡{"html":"As mentioned above, the de Sitter metric $g$ on $\mathcal{M}$ is that induced by $h$, i.e.$g$ is the pullback of $h$ by the mapping $\Phi$:
︡66015e29-a443-46e0-8133-268c82b58bf5︡︡{"done":true,"html":"As mentioned above, the de Sitter metric $g$ on $\\mathcal{M}$ is that induced by $h$, i.e.$g$ is the pullback of $h$ by the mapping $\\Phi$:
"} ︠1f2033e1-fd99-4876-91ea-3e1fb4b77cccs︠ g = M.metric('g') g.set( Phi.pullback(h) ) ︡5d3bf181-75f6-4d24-896b-327a885a0652︡︡{"done":true} ︠82611c22-c1f0-4f6f-ab01-af664add70abi︠ %htmlThe expression of $g$ in terms of $\mathcal{M}$'s default frame is found to be
︡11d729b4-efa3-4795-a072-228ccba36891︡︡{"done":true,"html":"The expression of $g$ in terms of $\\mathcal{M}$'s default frame is found to be
"} ︠c075b76d-dea7-4bba-b39c-74b3e3411526s︠ g.display() ︡89f32c3a-8bd9-4a33-b1fd-1b9f6ac2dd91︡︡{"html":"The Riemann tensor of $g$ is
︡e8de7ecc-1688-4aa6-b25d-db7e096eaa46︡︡{"done":true,"html":"The Riemann tensor of $g$ is
"} ︠58358f7f-dfb7-410b-bbc3-f6e10a948911s︠ Riem = g.riemann() print Riem Riem.display() ︡203d7e6d-6c97-4f85-8c46-e44419a1739d︡︡{"stdout":"tensor field 'Riem(g)' of type (1,3) on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"The Ricci tensor:
︡9b0779bd-c621-4774-a809-0eea8fa77737︡︡{"done":true,"html":"The Ricci tensor:
"} ︠d2c5757d-d8b7-4412-8c18-406d3f39447bs︠ Ric = g.ricci() print Ric Ric.display() ︡97caaef6-9f5d-4e88-af8b-554d237f19f0︡︡{"stdout":"field of symmetric bilinear forms 'Ric(g)' on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"The Ricci scalar:
︡00dc28fa-4415-4526-b0dd-0fb71cc1ac8b︡︡{"done":true,"html":"The Ricci scalar:
"} ︠6125f2a0-de34-4f20-a144-86a36f361010s︠ R = g.ricci_scalar() print R R.display() ︡272d51d7-4d0a-44a0-bde0-ce5479f29f9f︡︡{"stdout":"scalar field 'r(g)' on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"We recover the fact that de Sitter 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 $R^i_{\ \, jlk} = -\frac{R}{6} g_{j[k} \delta^i_{\ \, l]}$:
︡c547389d-1313-4bfc-b621-3ac084e7f338︡︡{"done":true,"html":"We recover the fact that de Sitter 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 $R^i_{\\ \\, jlk} = -\\frac{R}{6} g_{j[k} \\delta^i_{\\ \\, l]}$:
"} ︠a2ae7c4d-1e8d-4810-b9d0-1c65db56478ds︠ 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 ︡fce48102-e28d-4b9d-bd49-b116e28bdff6︡︡{"html":"We may also check that de Sitter metric is a solution of the vacuum Einstein equation with (positive) cosmological constant:
︡9e290803-38a7-47ba-a869-2b5004801949︡︡{"done":true,"html":"We may also check that de Sitter metric is a solution of the vacuum Einstein equation with (positive) cosmological constant:
"} ︠66c1f5d7-c9a1-4fce-98d7-ddb66ea8800as︠ Lambda = 3*b^2 Ric - 1/2*R*g + Lambda*g == 0 ︡34bf530c-9823-409d-a5ab-7d5d367d11cd︡︡{"html":"