This worksheet is based on SageManifolds (version 0.7). It implements the computation of the Simon-Mars tensor of Curzon-Chazy spacetime used in the article arXiv:1412.6542.
It is released under the GNU General Public License version 2.
(c) Claire Somé, Eric Gourgoulhon (2015)
The worksheet file in Sage notebook format is here.
We start by declaring the Curzon-Chazy spacetime as a 4-dimensional manifold:
︡935636e8-9cf2-45c3-86f7-a209171dec52︡︡{"done":true,"html":"This worksheet is based on SageManifolds (version 0.7). It implements the computation of the Simon-Mars tensor of Curzon-Chazy spacetime used in the article arXiv:1412.6542.
\nIt is released under the GNU General Public License version 2.
\n(c) Claire Somé, Eric Gourgoulhon (2015)
\nThe worksheet file in Sage notebook format is here.
\nWe start by declaring the Curzon-Chazy spacetime as a 4-dimensional manifold:
"} ︠1803ba28-d136-4f61-b8c6-d44953afbb43s︠ M = Manifold(4, 'M', latex_name=r'\mathcal{M}') print M ︡2622916e-3c8c-42b4-873c-72b160dc615c︡︡{"stdout":"4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠3111f2a1-46b1-4d01-abb9-df0f1ea167d3i︠ %htmlWe introduce the coordinates $(t,r,y,\phi)$ with $y$ related to the standard Weyl-Papapetrou coordinates $(t,r,\theta,\phi)$ by $y=\cos\theta$:
︡e9065465-23f1-4455-8502-e548f726e81e︡︡{"done":true,"html":"We introduce the coordinates $(t,r,y,\\phi)$ with $y$ related to the standard Weyl-Papapetrou coordinates $(t,r,\\theta,\\phi)$ by $y=\\cos\\theta$:
"} ︠4f46720d-1174-413b-9fb2-2b19336e0802s︠ X.We declare the only parameter of the Curzon-Chazy spacetime, which is the mass $m$ as a symbolic variable:
︡937ffb05-bf2e-450b-8ee2-270ee20ac846︡︡{"done":true,"html":"We declare the only parameter of the Curzon-Chazy spacetime, which is the mass $m$ as a symbolic variable:
"} ︠270ff48d-90f6-427b-a8a9-1f92cec3a4b2s︠ var('m') ︡8bcef82a-11ca-4950-9b57-827838195d7c︡︡{"html":"Without any loss of generality, we set $m$ to some specific value (this amounts simply to fixing some length scale):
︡7cb5c90b-7e04-4a88-8565-b3778a630806︡︡{"done":true,"html":"Without any loss of generality, we set $m$ to some specific value (this amounts simply to fixing some length scale):
"} ︠8b4799c6-af70-4081-a5be-31603671498cs︠ m = 12 ︡8878b318-7d21-4bd1-b8e5-a8945988377e︡︡{"done":true} ︠feffac48-64d2-4f0c-aedb-c38e392add7fi︠ %htmlLet us introduce the spacetime metric $g$ and set its components in the coordinate frame associated with Weyl-Papapetrou coordinates:
︡7002803a-d7ef-4301-862a-8a5ba4db5e1e︡︡{"done":true,"html":"Let us introduce the spacetime metric $g$ and set its components in the coordinate frame associated with Weyl-Papapetrou coordinates:
"} ︠339e5524-08f4-41c5-a1e8-83f986684e4fs︠ g = M.lorentz_metric('g') g[0,0] = - exp(-2*m/r) g[1,1] = exp(2*m/r-m^2*(1-y^2)/r^2) g[2,2] = exp(2*m/r-m^2*(1-y^2)/r^2)*r^2/(1-y^2) g[3,3] = exp(2*m/r)*r^2*(1-y^2) ︡182360e5-bc9d-47e7-b340-b8d3a8d597e9︡︡{"done":true} ︠17696dcb-2e27-45cd-b466-74d92fe98868s︠ g[:] ︡846dd692-0d9c-4305-8caa-d1fad46037a2︡︡{"html":"The Levi-Civita connection $\nabla$ associated with $g$:
︡12ae77d4-697f-4e54-91af-d7661238f7a7︡︡{"done":true,"html":"The Levi-Civita connection $\\nabla$ associated with $g$:
"} ︠4a6b4362-055d-4411-b0c1-2a3abed0cd96s︠ nab = g.connection() ; print nab ︡f3377238-adcc-4c6c-a6a2-fedcf3dae183︡︡{"stdout":"Levi-Civita connection 'nabla_g' associated with the Lorentzian metric 'g' on the 4-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠231038fb-407e-4f00-b0cf-cc1f2cc62d80i︠ %htmlAs a check, we verify that the covariant derivative of $g$ with respect to $\nabla$ vanishes identically:
︡d11c60c2-4227-4165-a627-960b10364405︡︡{"done":true,"html":"As a check, we verify that the covariant derivative of $g$ with respect to $\\nabla$ vanishes identically:
"} ︠86fafcc8-3e4c-415d-bbf4-3eb3581c8e99s︠ nab(g).display() ︡2f714436-7d19-4c89-b8cd-7fc81d881599︡︡{"html":"The default vector frame on the spacetime manifold is the coordinate basis associated with Weyl-Papapetrou coordinates:
︡f6934852-b902-42c7-ac8b-e9734b558f68︡︡{"done":true,"html":"The default vector frame on the spacetime manifold is the coordinate basis associated with Weyl-Papapetrou coordinates:
"} ︠69c49352-64d2-45fa-b62e-990cb6b0be70s︠ M.default_frame() is X.frame() ︡c4d10f85-157e-4a65-8829-ca625350157e︡︡{"html":"Let us consider the first vector field of this frame:
︡d2a73911-e77e-4a91-b325-d84da42bfb03︡︡{"done":true,"html":"Let us consider the first vector field of this frame:
"} ︠3e28fa6c-6c81-4688-9410-ab1342fac770s︠ xi = X.frame()[0] ; xi ︡eeca8213-e611-42ff-8c73-f69cbc812854︡︡{"html":"The 1-form associated to it by metric duality is
︡f2412a87-e9b6-494b-949a-9cfde8f1889c︡︡{"done":true,"html":"The 1-form associated to it by metric duality is
"} ︠b13eff10-5177-4e09-b8ed-56df8120f9cbs︠ xi_form = xi.down(g) xi_form.set_name('xi_form', r'\underline{\xi}') print xi_form ; xi_form.display() ︡5a0de45d-c07c-4a7f-a8f1-ec32f2cfaa90︡︡{"stdout":"1-form 'xi_form' on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"Its covariant derivative is
︡544b41b5-e9a6-461f-8c8c-f7b68e3dc292︡︡{"done":true,"html":"Its covariant derivative is
"} ︠6c682d8f-c76b-4c83-9f8e-5e8787d563e5s︠ nab_xi = nab(xi_form) print nab_xi ; nab_xi.display() ︡2fa9da89-d866-4b61-adaf-4b4973a8dd6c︡︡{"stdout":"tensor field 'nabla_g xi_form' of type (0,2) on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"Let us check that the Killing equation is satisfied:
︡38fc22da-cea1-48f5-9d23-6488a95463d3︡︡{"done":true,"html":"Let us check that the Killing equation is satisfied:
"} ︠db43d384-d650-41a0-bd3a-5b282f0b398ds︠ nab_xi.symmetrize().display() ︡20d75325-82ab-4f70-84ef-96ba5a49f8c1︡︡{"html":"Equivalently, we check that the Lie derivative of the metric along $\xi$ vanishes:
︡f15611d1-f25c-42c3-9d56-302e861acffa︡︡{"done":true,"html":"Equivalently, we check that the Lie derivative of the metric along $\\xi$ vanishes:
"} ︠2915bd52-250f-49e5-b6f7-eb6ddf7043a9s︠ g.lie_der(xi).display() ︡37250085-b0fd-45a9-a036-40af18d12546︡︡{"html":"Thank to Killing equation, $\nabla_g \underline{\xi}$ is antisymmetric. We may therefore define a 2-form by $F := - \nabla_g \xi$. Here we enforce the antisymmetry by calling the function antisymmetrize() on nab_xi:
︡133cbf1f-a46a-4a61-a181-6a23e68976de︡︡{"done":true,"html":"Thank to Killing equation, $\\nabla_g \\underline{\\xi}$ is antisymmetric. We may therefore define a 2-form by $F := - \\nabla_g \\xi$. Here we enforce the antisymmetry by calling the function antisymmetrize() on nab_xi:
"} ︠22cfd51c-8d9f-4a82-9e8a-6bf4be0bb9bcs︠ F = - nab_xi.antisymmetrize() F.set_name('F') print F F.display() ︡e9b5a673-de5b-4c7b-89db-fac2159b3153︡︡{"stdout":"2-form 'F' on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"We check that
︡596dfe6b-6cb5-4c81-927a-6a6a1965d720︡︡{"done":true,"html":"We check that
"} ︠8eb2af33-491c-4cc2-b763-11a9531571eds︠ F == - nab_xi ︡0bca1476-3c54-41ac-9f8e-bfeb6e3e3ad8︡︡{"html":"The squared norm of the Killing vector is
︡9a0bb746-4e61-4214-b479-a10edcd7e8d9︡︡{"done":true,"html":"The squared norm of the Killing vector is
"} ︠9d92bc13-1b45-4485-aaed-2c31eab275bcs︠ lamb = - g(xi,xi) lamb.set_name('lambda', r'\lambda') print lamb lamb.display() ︡c5c27d3c-ac6e-4531-b205-faf8cca3484d︡︡{"stdout":"scalar field 'lambda' on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"Instead of invoking $g(\xi,\xi)$, we could have evaluated $\lambda$ by means of the 1-form $\underline{\xi}$ acting on the vector field $\xi$:
︡576b3c7a-9b2d-4047-9a39-2c789b1c7e41︡︡{"done":true,"html":"Instead of invoking $g(\\xi,\\xi)$, we could have evaluated $\\lambda$ by means of the 1-form $\\underline{\\xi}$ acting on the vector field $\\xi$:
"} ︠09bc1d06-c311-469a-b633-d1a8473d0a36s︠ lamb == - xi_form(xi) ︡29ccf1d2-c5bc-499c-a88e-46f193e41d90︡︡{"html":"or we could have used index notation in the form $\lambda = - \xi_a \xi^a$:
︡06610cad-bb11-4e69-ba6c-9951a6ef8020︡︡{"done":true,"html":"or we could have used index notation in the form $\\lambda = - \\xi_a \\xi^a$:
"} ︠4650bae9-0aee-433d-85ef-bc8acf29ecfbs︠ lamb == - ( xi_form['_a']*xi['^a'] ) ︡dbea00d6-53ff-4de8-928b-d2b70b4b013e︡︡{"html":"The Riemann curvature tensor associated with $g$ is
︡c96c2df1-6394-4111-ac09-ae5c015a11c0︡︡{"done":true,"html":"The Riemann curvature tensor associated with $g$ is
"} ︠76494bc0-4157-4cce-bea7-b84735e46ad6s︠ Riem = g.riemann() print Riem ︡f852dbfe-a387-4d3d-b31b-a9d3e2bdf27c︡︡{"stdout":"tensor field 'Riem(g)' of type (1,3) on the 4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠1483e628-5eb3-4e94-9786-7e63e4da2783i︠ %htmlThe component $R^0_{\ \, 101} = R^t_{\ \, rtr}$ is
︡86c2f379-a904-4023-8b7a-b2e5df7b22ac︡︡{"done":true,"html":"The component $R^0_{\\ \\, 101} = R^t_{\\ \\, rtr}$ is
"} ︠58b949e7-68b5-46f9-96e8-73ac7fe4c438s︠ Riem[0,1,0,1] ︡d773a48f-ddc2-4273-bc70-38bf4b4f580b︡︡{"html":"while the component $R^2_{\ \, 323} = R^y_{\ \, \phi y \phi}$ is
︡8119a8f8-caae-424e-9be6-46289fb79db5︡︡{"done":true,"html":"while the component $R^2_{\\ \\, 323} = R^y_{\\ \\, \\phi y \\phi}$ is
"} ︠043c646c-018b-40e7-9325-251b073ccdc6s︠ Riem[2,3,2,3] ︡ea253272-a8b1-4d85-b78b-c4f568461421︡︡{"html":"The Ricci tensor:
︡e8b8dcd3-f71c-48da-a971-2984c3b4a16c︡︡{"done":true,"html":"The Ricci tensor:
"} ︠7efc26a4-95e0-4a0a-8488-95a609f6dadds︠ Ric = g.ricci() print Ric ︡2e8692c4-4994-4889-8183-219f642bb06c︡︡{"stdout":"field of symmetric bilinear forms 'Ric(g)' on the 4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠f4decf4d-7e4e-427a-8646-8c0765f3963ci︠ %htmlLet us check that the Curzon-Chazy metric is a solution of the vacuum Einstein equation:
︡85f26087-ed2c-45b6-8459-42499885a8a9︡︡{"done":true,"html":"Let us check that the Curzon-Chazy metric is a solution of the vacuum Einstein equation:
"} ︠2fbb5277-4260-4cd1-a452-d23c05658089s︠ Ric.display() ︡76e307ea-7d24-4c5d-a35d-ad958ef9e152︡︡{"html":"The Weyl conformal curvature tensor is
The Weyl conformal curvature tensor is
Let us exhibit two of its components $C^0_{\ \, 123}$ and $C^0_{\ \, 101}$:
︡08a3a85a-e35d-4887-8e5e-1ab781f78ee8︡︡{"done":true,"html":"Let us exhibit two of its components $C^0_{\\ \\, 123}$ and $C^0_{\\ \\, 101}$:
"} ︠58ecc34b-56a6-4e32-9cf7-18a53d821b70s︠ C[0,1,2,3] ︡1b17118a-e3ea-40dd-b50e-ac0524820a45︡︡{"html":"To form the Mars-Simon tensor, we need the fully covariant (type-(0,4) tensor) form of the Weyl tensor (i.e. $C_{\alpha\beta\mu\nu} = g_{\alpha\sigma} C^\sigma_{\ \, \beta\mu\nu}$); we get it by lowering the first index with the metric:
︡eb21c6cb-a0cc-402e-b416-805b1fd2eb88︡︡{"done":true,"html":"To form the Mars-Simon tensor, we need the fully covariant (type-(0,4) tensor) form of the Weyl tensor (i.e. $C_{\\alpha\\beta\\mu\\nu} = g_{\\alpha\\sigma} C^\\sigma_{\\ \\, \\beta\\mu\\nu}$); we get it by lowering the first index with the metric:
"} ︠46eaca62-1da4-411c-aef4-b393e8d19542s︠ Cd = C.down(g) print Cd ︡43550040-3398-4536-bbe8-cd79f15ca2e6︡︡{"stdout":"tensor field of type (0,4) on the 4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠91beb1ac-68b6-45df-8b75-e9931783e53di︠ %htmlThe (monoterm) symmetries of this tensor are those inherited from the Weyl tensor, i.e. the antisymmetry on the last two indices (position 2 and 3, the first index being at position 0):
︡0390f0b6-4668-44a9-b052-afa79f5960bb︡︡{"done":true,"html":"The (monoterm) symmetries of this tensor are those inherited from the Weyl tensor, i.e. the antisymmetry on the last two indices (position 2 and 3, the first index being at position 0):
"} ︠758abda1-1f05-4173-b229-4d17d681b022s︠ Cd.symmetries() ︡768f9ccd-499c-46ed-80d5-069da2b343d0︡︡{"stdout":"no symmetry; antisymmetry: (2, 3)\n","done":false}︡{"done":true} ︠8fe9131c-168f-4c22-8a48-f6524d73bbf5i︠ %htmlActually, Cd is also antisymmetric with respect to the first two indices (positions 0 and 1), as we can check:
︡6f202def-76de-4005-9cdd-fd3ca98c9679︡︡{"done":true,"html":"Actually, Cd is also antisymmetric with respect to the first two indices (positions 0 and 1), as we can check:
"} ︠15c1917a-3d8d-4022-a8ea-30db1846d95as︠ Cd == Cd.antisymmetrize(0,1) ︡7895e81f-55d8-470e-a319-c8ac146c50b0︡︡{"html":"To take this symmetry into account explicitely, we set
︡3dee89f6-f934-4f19-8163-9ab318b410f5︡︡{"done":true,"html":"To take this symmetry into account explicitely, we set
"} ︠baefe6c3-4a39-4454-99fd-947d06f53551s︠ Cd = Cd.antisymmetrize(0,1) ︡f2008525-098d-4ad4-a6cd-148094ab6de7︡︡{"done":true} ︠0f19039d-dc7c-4bce-a1a0-2b3b620ae648i︠ %htmlHence we have now
︡9499f578-75c3-40fe-adb1-fdc4b46f8caa︡︡{"done":true,"html":"Hence we have now
"} ︠a1590dd4-04f2-41f6-a54a-f714308f278es︠ Cd.symmetries() ︡2b47835d-2c6c-46de-9545-c15ac8410dd8︡︡{"stdout":"no symmetry; antisymmetries: [(0, 1), (2, 3)]\n","done":false}︡{"done":true} ︠925d54d3-47c8-4b66-b8ad-de1acfcd6108i︠ %htmlThe Simon-Mars tensor with respect to the Killing vector $\xi$ is a rank-3 tensor introduced by Marc Mars in 1999 (Class. Quantum Grav. 16, 2507). It has the remarkable property to vanish identically if, and only if, the spacetime $(\mathcal{M},g)$ is locally isometric to a Kerr spacetime.
Let us evaluate the Simon-Mars tensor by following the formulas given in Mars' article. The starting point is the self-dual complex 2-form associated with the Killing 2-form $F$, i.e. the object $\mathcal{F} := F + i \, {}^* F$, where ${}^*F$ is the Hodge dual of $F$:
︡17b5a493-acb5-4073-bc2a-96400c102f0d︡︡{"done":true,"html":"The Simon-Mars tensor with respect to the Killing vector $\\xi$ is a rank-3 tensor introduced by Marc Mars in 1999 (Class. Quantum Grav. 16, 2507). It has the remarkable property to vanish identically if, and only if, the spacetime $(\\mathcal{M},g)$ is locally isometric to a Kerr spacetime.
\nLet us evaluate the Simon-Mars tensor by following the formulas given in Mars' article. The starting point is the self-dual complex 2-form associated with the Killing 2-form $F$, i.e. the object $\\mathcal{F} := F + i \\, {}^* F$, where ${}^*F$ is the Hodge dual of $F$:
"} ︠7a102757-b9be-4c0f-8638-0d15e8c5ef88s︠ FF = F + I * F.hodge_star(g) FF.set_name('FF', r'\mathcal{F}') print FF ; FF.display() ︡4e55643c-431a-4547-9a5c-3c5696a332b1︡︡{"stdout":"2-form 'FF' on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"Let us check that $\mathcal{F}$ is self-dual, i.e. that it obeys ${}^* \mathcal{F} = -i \mathcal{F}$:
︡bced88c5-5daf-4004-82d4-4c0b15033cc3︡︡{"done":true,"html":"Let us check that $\\mathcal{F}$ is self-dual, i.e. that it obeys ${}^* \\mathcal{F} = -i \\mathcal{F}$:
"} ︠5e2f3527-310a-4062-9eae-528ae95f37bas︠ FF.hodge_star(g) == - I * FF ︡1bc050e3-585e-4ee7-be5d-3852ccb3a52c︡︡{"html":"Let us form the right self-dual of the Weyl tensor as follows
\[\mathcal{C}_{\alpha\beta\mu\nu} = C_{\alpha\beta\mu\nu} + \frac{i}{2} \epsilon^{\rho\sigma}_{\ \ \ \mu\nu} \, C_{\alpha\beta\rho\sigma}\]
where $\epsilon^{\rho\sigma}_{\ \ \ \mu\nu}$ is associated to the Levi-Civita tensor $\epsilon_{\rho\sigma\mu\nu}$ and is obtained by
︡aed89548-0f67-48f8-b77e-294349d88aa0︡︡{"done":true,"html":"Let us form the right self-dual of the Weyl tensor as follows
\n\\[\\mathcal{C}_{\\alpha\\beta\\mu\\nu} = C_{\\alpha\\beta\\mu\\nu} + \\frac{i}{2} \\epsilon^{\\rho\\sigma}_{\\ \\ \\ \\mu\\nu} \\, C_{\\alpha\\beta\\rho\\sigma}\\]
\nwhere $\\epsilon^{\\rho\\sigma}_{\\ \\ \\ \\mu\\nu}$ is associated to the Levi-Civita tensor $\\epsilon_{\\rho\\sigma\\mu\\nu}$ and is obtained by
"} ︠bb658110-7345-40a2-b658-98936027e3bds︠ eps = g.volume_form(2) # 2 = the first 2 indices are contravariant print eps eps.symmetries() ︡cb4fc43d-b15a-441d-a8ae-47c5e4ba845d︡︡{"stdout":"tensor field of type (2,2) on the 4-dimensional manifold 'M'\n","done":false}︡{"stdout":"no symmetry; antisymmetries: [(0, 1), (2, 3)]\n","done":false}︡{"done":true} ︠477a0f9a-a88a-462d-bc34-7006cbda2605i︠ %htmlThe right self-dual Weyl tensor is then:
︡65f4dd9d-273d-4fb2-b282-e0b0de7daa6c︡︡{"done":true,"html":"The right self-dual Weyl tensor is then:
"} ︠db4cf275-b502-421d-94f5-70935d24e810s︠ CC = Cd + I/2*( eps['^rs_..']*Cd['_..rs'] ) CC.set_name('CC', r'\mathcal{C}') ; print CC ︡122da1fb-e054-4c0c-906c-784b854bcd4b︡︡{"stdout":"tensor field 'CC' of type (0,4) on the 4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠1ec16684-4603-4b88-b334-41a10e99f45ds︠ CC.symmetries() ︡2081b252-d331-456f-9c4c-cee58fd6fa59︡︡{"stdout":"no symmetry; antisymmetries: [(0, 1), (2, 3)]\n","done":false}︡{"done":true} ︠854cd536-b52c-4e6a-8f3b-c13b47439b9bs︠ CC[0,1,2,3] ︡3351f10e-4482-4ade-9af6-568a74322fdb︡︡{"html":"The Ernst 1-form $\sigma_\alpha = 2 \mathcal{F}_{\mu\alpha} \, \xi^\mu$ (0 = contraction on the first index of $\mathcal{F}$):
︡c4a4917c-1041-414a-a412-06f533a8a2cd︡︡{"done":true,"html":"The Ernst 1-form $\\sigma_\\alpha = 2 \\mathcal{F}_{\\mu\\alpha} \\, \\xi^\\mu$ (0 = contraction on the first index of $\\mathcal{F}$):
"} ︠32c25cbd-191e-4ca8-93e6-1d72b73a49d8s︠ sigma = 2*FF.contract(0, xi) ︡6ed24176-fba7-4c8c-ba25-3b0741885ab0︡︡{"done":true} ︠e9208b93-a5dd-4c9a-b09c-95e03d42024ai︠ %htmlInstead of invoking the function contract(), we could have used the index notation to denote the contraction:
︡be646c73-e4f8-42a7-9ee2-702c1a1fa595︡︡{"done":true,"html":"Instead of invoking the function contract(), we could have used the index notation to denote the contraction:
"} ︠6f028a1e-a9ec-44c2-8b64-68f5e3a962c6s︠ sigma == 2*( FF['_ma']*xi['^m'] ) ︡08f574c2-4b3d-4067-98b1-f18f45ae48dc︡︡{"html":"The symmetric bilinear form $\gamma = \lambda \, g + \underline{\xi}\otimes\underline{\xi}$:
︡6617d998-eb57-47c0-8c7e-99154be001a2︡︡{"done":true,"html":"The symmetric bilinear form $\\gamma = \\lambda \\, g + \\underline{\\xi}\\otimes\\underline{\\xi}$:
"} ︠669eda9e-b73d-41e9-ab14-e6e5a881251es︠ gamma = lamb*g + xi_form * xi_form gamma.set_name('gamma', r'\gamma') print gamma ; gamma.display() ︡57c4e84e-d65c-48a9-90ff-509d104cedae︡︡{"stdout":"field of symmetric bilinear forms 'gamma' on the 4-dimensional manifold 'M'\n","done":false}︡{"html":"The first part of the Simon-Mars tensor is
\[ S^{(1)}_{\alpha\beta\gamma} = 4 \mathcal{C}_{\mu\alpha\nu\beta} \, \xi^\mu \, \xi^\nu \, \sigma_\gamma\]
︡baf45f01-bb16-449c-87a6-d257919ee466︡︡{"done":true,"html":"The first part of the Simon-Mars tensor is
\n\\[ S^{(1)}_{\\alpha\\beta\\gamma} = 4 \\mathcal{C}_{\\mu\\alpha\\nu\\beta} \\, \\xi^\\mu \\, \\xi^\\nu \\, \\sigma_\\gamma\\]
"} ︠f63b538a-d79c-4f55-a9a8-9f14d040348fs︠ S1 = 4*( CC.contract(0,xi).contract(1,xi) ) * sigma print S1 ︡0aeaf5c9-4233-4f02-a2ca-e1c5afbe09cf︡︡{"stdout":"tensor field of type (0,3) on the 4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠9d2d4629-8445-4964-a0cd-a0bd7e6d102ci︠ %htmlThe second part is the tensor
\[ S^{(2)}_{\alpha\beta\gamma} = - \gamma_{\alpha\beta} \, \mathcal{C}_{\rho\gamma\mu\nu} \, \xi^\rho \, \mathcal{F}^{\mu\nu}\]
which we compute by using the index notation to denote the contractions:
︡61791bd7-f551-43cf-b47a-ef359de6bb8e︡︡{"done":true,"html":"The second part is the tensor
\n\\[ S^{(2)}_{\\alpha\\beta\\gamma} = - \\gamma_{\\alpha\\beta} \\, \\mathcal{C}_{\\rho\\gamma\\mu\\nu} \\, \\xi^\\rho \\, \\mathcal{F}^{\\mu\\nu}\\]
\nwhich we compute by using the index notation to denote the contractions:
"} ︠e0f2eb8c-8cd6-49db-873e-b45c291d9352︠ FFuu = FF.up(g) xiCC = CC['_.r..']*xi['^r'] S2 = gamma * ( xiCC['_.mn']*FFuu['^mn'] ) print S2 ︡0e36a184-28f6-47de-88ca-32eee7b7cc05︡{"stdout":"tensor field of type (0,3) on the 4-dimensional manifold 'M'\n","done":false}︡{"done":true}︡{"done":true}︡ ︠d8ff07e2-fe36-4e71-9904-08d6f0054880︠ S2.symmetries() ︡1416a672-7b92-4617-9d67-11c4fbb9631f︡︡{"stdout":"symmetry: (0, 1); no antisymmetry\n","done":false}︡{"done":true} ︠ca35e021-2c8e-492f-a2aa-432f579617c1i︠ %htmlThe Mars-Simon tensor with respect to $\xi$ is obtained by antisymmetrizing $S^{(1)}$ and $S^{(2)}$ on their last two indices and adding them:
We use the index notation for the antisymmetrization:
︡f8c3f5ac-15b3-42d9-b7c1-34c6aef93557︡︡{"done":true,"html":"The Mars-Simon tensor with respect to $\\xi$ is obtained by antisymmetrizing $S^{(1)}$ and $S^{(2)}$ on their last two indices and adding them:
\nWe use the index notation for the antisymmetrization:
"} ︠b27bbe0f-9525-4ee0-87be-c9bc30a1073as︠ S1A = S1['_a[bc]'] S2A = S2['_a[bc]'] ︡8347f731-1851-4ebb-9b2a-f2ebae326bf3︡{"done":true}︡ ︠dd395504-381a-4a7a-b7f8-565e53214b3ei︠ %htmlAn equivalent writing would have been (the last two indices being in position 1 and 2):
︡7a98bd3f-1463-426c-8156-c70725c92c59︡︡{"done":true,"html":"An equivalent writing would have been (the last two indices being in position 1 and 2):
"} ︠fd8f0020-b5ec-48a2-89d9-63d5cb706dfds︠ # S1A = S1.antisymmetrize(1,2) # S2A = S2.antisymmetrize(1,2) ︡91e3094d-2ae1-4777-99c0-11a407b48308︡{"done":true}︡ ︠662447f7-c086-4b67-8000-7ffd53f8f0fai︠ %htmlThe Simon-Mars tensor is
︡354630bd-8da4-4596-ae55-dd119a1b1c8c︡︡{"done":true,"html":"The Simon-Mars tensor is
"} ︠8f95ff1a-8dd4-4768-9eb4-57cf99c14de6s︠ S = S1A + S2A S.set_name('S') ; print S S.symmetries() ︡2bf34ce4-bd76-4499-ad32-dd036ed2f36e︡︡{"stdout":"tensor field 'S' of type (0,3) on the 4-dimensional manifold 'M'\n","done":false}︡{"stdout":"no symmetry; antisymmetry: (1, 2)\n","done":false}︡{"done":true} ︠ef933719-67cd-4ec7-adb5-b9d6cb5f0864s︠ S.display() ︡9b5c0219-4745-4050-88e8-6922c20e9e84︡︡{"html":"Hence the Simon-Mars tensor is not zero: the Curzon-Chazy spacetime is not locally isomorphic to the Kerr spacetime.
First we form the "square" of the Simon-Mars tensor:
︡a6405f51-0dc2-4be2-afc5-f909a8525980︡︡{"done":true,"html":"Hence the Simon-Mars tensor is not zero: the Curzon-Chazy spacetime is not locally isomorphic to the Kerr spacetime.
\nFirst we form the \"square\" of the Simon-Mars tensor:
"} ︠d58cf0a3-8fee-4f2a-af43-81508ec19ab5s︠ Su = S.up(g) print Su ︡e8bed40b-22e2-4d50-afb5-bd6172e626a7︡︡{"stdout":"tensor field of type (3,0) on the 4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠77baa4e8-44b6-4fcf-b1c9-131d4b25518ds︠ SS = S['_ijk']*Su['^ijk'] print SS ︡70cf86b5-74a7-459c-ab00-dfba7848e19b︡︡{"stdout":"scalar field on the 4-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠7b9c8f9f-134e-4300-8faf-174f98dfadc1s︠ SS.display() ︡401ec652-fa3e-40c7-ba8b-c7925a3a6148︡︡{"html":"Then we take the real and imaginary part of this compex scalar field. Because this spacetime is spherically symmetric, we expect that the imaginary part vanishes.
︡d81580c5-c286-403f-9698-a555a6d61e6d︡︡{"done":true,"html":"Then we take the real and imaginary part of this compex scalar field. Because this spacetime is spherically symmetric, we expect that the imaginary part vanishes.
"} ︠8118580e-74f1-4349-af57-871c9bbc0bfbs︠ SS1 = real(SSE) ; SS1 ︡d9d0e797-4b10-4d32-97bd-9ba76bb0daa6︡︡{"html":"Furthermore we scale those scalars by the ADM mass of the Curzon-Chazy spacetime, which corresponds to $m$:
︡59552600-c2dc-4b18-81eb-5c9624ec04b5︡︡{"done":true,"html":"Furthermore we scale those scalars by the ADM mass of the Curzon-Chazy spacetime, which corresponds to $m$:
"} ︠958dc778-a055-4555-8495-6f20a53546dds︠ SS1ad = m^6*SS1 ; SS1ad ︡84d644e1-39d2-4ba5-ab89-993c6f8faef2︡︡{"html":"And we take the log of this quantity
︡1ea06801-20f2-481a-9cf2-54c9dfed5f99︡︡{"done":true,"html":"And we take the log of this quantity
"} ︠c63f6678-b8f8-419a-bbe5-40bd0bf5365fs︠ lSS1ad = log(SS1ad,10) ; lSS1ad ︡69bc1874-c335-4c4b-845b-6e67f16335b1︡︡{"html":"Then we plot the value of this quantity as a function of $\rho = x = r \sqrt{1-y^2}$ and $z = r y$, thereby producing Figure 10 of arXiv:1412.6542:
︡db3ff6e8-44b4-4e41-ab27-ebbe22cade42︡︡{"done":true,"html":"Then we plot the value of this quantity as a function of $\\rho = x = r \\sqrt{1-y^2}$ and $z = r y$, thereby producing Figure 10 of arXiv:1412.6542:
"} ︠9ada7c8d-da95-4c50-8c60-6862e6f24d4es︠ var('x z') lSS1xzad = lSS1ad.subs(r=sqrt(x^2+z^2), y = z/sqrt(x^2+z^2)).simplify_full() ; lSS1xzad ︡e002722a-c122-4094-8148-0406d20d252d︡︡{"html":"