︠09b44bae-4779-4a19-9e51-48b152801ab6as︠ %auto typeset_mode(True, display=False) ︡9b3c7e60-f84c-4502-a833-f2bf93f5a4a6︡{"auto":true}︡ ︠4b5a58ca-7d65-4b54-9b76-51bddcc37eabi︠ %html

3+1 Simon-Mars tensor in Kerr spacetime

This worksheet demonstrates a few capabilities of SageManifolds (version 0.7) in computations regarding 3+1 slicing of Kerr spacetime. In particular, it implements the computation of the 3+1 decomposition of the Simon-Mars tensor as given in the article arXiv:1412.6542.

The worksheet 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.

Spacelike hypersurface

We consider some hypersurface $\Sigma$ of a spacelike foliation $(\Sigma_t)_{t\in\mathbb{R}}$ of Kerr spacetime; we declare $\Sigma_t$ as a 3-dimensional manifold:

︡2cfe009d-ed65-4da6-b521-b6b86198f593︡︡{"done":true,"html":"

3+1 Simon-Mars tensor in Kerr spacetime

\n

This worksheet demonstrates a few capabilities of SageManifolds (version 0.7) in computations regarding 3+1 slicing of Kerr spacetime. In particular, it implements the computation of the 3+1 decomposition of the Simon-Mars tensor as given in the article arXiv:1412.6542.

\n

The worksheet is released under the GNU General Public License version 2.

\n

(c) Claire Somé, Eric Gourgoulhon (2015)

\n

The worksheet file in Sage notebook format is here.

\n

Spacelike hypersurface

\n

We consider some hypersurface $\\Sigma$ of a spacelike foliation $(\\Sigma_t)_{t\\in\\mathbb{R}}$ of Kerr spacetime; we declare $\\Sigma_t$ as a 3-dimensional manifold:

"} ︠84be549c-c9bf-41bb-9abe-4974c0493bb9︠ Sig = Manifold(3, 'Sigma', r'\Sigma', start_index=1) ︡ff40cb4b-43a7-47ae-850e-a264b9b3cb27︡︡{"done":true} ︠9d2b9b2d-0d35-43f8-afad-99729ee60cc5i︠ %html

The two Kerr parameters:

︡5078311a-5081-4599-b818-af32bd0dbc62︡︡{"done":true,"html":"

The two Kerr parameters:

"} ︠59b36c89-c44a-4ba5-8c2c-28249bac58ca︠ var('m, a') assume(m>0) assume(a>0) ︡06cb565a-776a-4a17-bf40-a1a705ba1f63︡︡{"html":"
($m$, $a$)
","done":false}︡{"done":true} ︠e19d5317-c519-4a39-a7b5-83e6a7051ca1i︠ %html

Riemannian metric on $\Sigma$

The variables introduced so far satisfy the following assumptions:

Without any loss of generality (for $m\not =0$), we may set $m=1$:

︡4ba7edda-2d4c-4bd9-99c3-c40d967b8799︡︡{"done":true,"html":"

Riemannian metric on $\\Sigma$

\n

The variables introduced so far satisfy the following assumptions:

\n\n

Without any loss of generality (for $m\\not =0$), we may set $m=1$:

"} ︠61b496c2-8d81-4323-b847-6832e351c512︠ m=1 assume(a<1) ︡09ef811d-2a3a-470d-a619-55b92064bee4︡︡{"done":true} ︠614ae45d-1158-4b89-9952-9f6e130a26bf︠ #a=1 # extreme Kerr ︡ac0d9532-dc4c-4d5b-9c43-8002a4a42b77︡︡{"done":true} ︠1a18581b-c0ee-4f55-86e1-ee0cfbe9d2f8i︠ %html

On the hypersurface $\Sigma$, we are using coordinates $(r,y,\phi)$ that are related to the standard Boyer-Lindquist coordinates $(r,\theta,\phi)$ by $y=\cos\theta$:

︡d8ac35ca-b863-4e89-99c1-305d1d384069︡︡{"done":true,"html":"

On the hypersurface $\\Sigma$, we are using coordinates $(r,y,\\phi)$ that are related to the standard Boyer-Lindquist coordinates $(r,\\theta,\\phi)$ by $y=\\cos\\theta$:

"} ︠4638a9a0-5555-4d40-a0dc-66b65766fb46︠ X. = Sig.chart(r'r:(1+sqrt(1-a^2),+oo) y:(-1,1) ph:(0,2*pi):\phi') print X ; X ︡9610a5d4-c310-4d82-9ae9-573c1d4c90c7︡︡{"stdout":"chart (Sigma, (r, y, ph))\n","done":false}︡{"html":"
$\\left(\\Sigma,(r, y, {\\phi})\\right)$
","done":false}︡{"done":true} ︠5bbab54c-21da-4704-b8bd-fae466a84867i︠ %html

Riemannian metric on $\Sigma$

The variables introduced so far obey the following assumptions:

︡9c5ecae3-b200-4c41-9784-69c3df54d0a5︡︡{"done":true,"html":"

Riemannian metric on $\\Sigma$

\n

The variables introduced so far obey the following assumptions:

"} ︠b61c25f7-c56c-4392-9873-22850dbfd7fe︠ assumptions() ︡c630ed07-9bdb-42c3-9e0c-d16e617179cf︡︡{"html":"
[$m > 0$, $a > 0$, $a < 1$, $\\text{\\texttt{r{ }is{ }real}}$, $\\text{\\texttt{y{ }is{ }real}}$, $y > \\left(-1\\right)$, $y < 1$, $\\text{\\texttt{ph{ }is{ }real}}$, ${\\phi} > 0$, ${\\phi} < 2 \\, \\pi$]
","done":false}︡{"done":true} ︠1d5af6ad-9ba6-4d26-93df-1a83d29cdef1i︠ %html

Some shortcut notations:

︡203f7130-72c5-4a8d-93cc-ae990a3eee8a︡︡{"done":true,"html":"

Some shortcut notations:

"} ︠6eff5017-aa49-4181-98a0-b5da8d9c3c72︠ rho2 = r^2 + a^2*y^2 Del = r^2 -2*m*r + a^2 AA2 = rho2*(r^2 + a^2) + 2*a^2*m*r*(1-y^2) BB2 = r^2 + a^2 + 2*a^2*m*r*(1-y^2)/rho2 ︡e88b38d4-abd3-404f-acb7-12b093ee439a︡︡{"done":true} ︠826952d6-a9c2-4fdd-8ca8-0781e9bb5fb7i︠ %html

The metric $h$ induced by the spacetime metric $g$ on $\Sigma$:

︡b82300b9-1f06-4f7b-9e8f-5e62167700b8︡︡{"done":true,"html":"

The metric $h$ induced by the spacetime metric $g$ on $\\Sigma$:

"} ︠97849366-ff98-4c3a-9663-9def0a4f1844︠ gam = Sig.riemann_metric('gam', latex_name=r'\gamma') gam[1,1] = rho2/Del gam[2,2] = rho2/(1-y^2) gam[3,3] = BB2*(1-y^2) gam.display() ︡e1aa5f05-7d3e-444c-8866-938c410ac0a9︡︡{"html":"
$\\gamma = \\left( \\frac{a^{2} y^{2} + r^{2}}{a^{2} + r^{2} - 2 \\, r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( -\\frac{a^{2} y^{2} + r^{2}}{y^{2} - 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( -\\frac{{\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{4} - a^{2} r^{2} - r^{4} - 2 \\, a^{2} r - {\\left(a^{4} - r^{4} - 4 \\, a^{2} r\\right)} y^{2}}{a^{2} y^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$
","done":false}︡{"done":true} ︠f044c37c-dc33-4762-9569-28b15c3ac763i︠ %html

A matrix view of the components w.r.t. coordinates $(r,y,\phi)$:

︡04cab3f8-4acf-42bb-b99f-77a046ced0a5︡︡{"done":true,"html":"

A matrix view of the components w.r.t. coordinates $(r,y,\\phi)$:

"} ︠72364dc1-8198-4c74-a823-dff34e258ce0︠ gam[:] ︡12167295-cbf4-485b-a426-2a8347e1afe6︡︡{"html":"
$\\left(\\begin{array}{rrr}\n\\frac{a^{2} y^{2} + r^{2}}{a^{2} + r^{2} - 2 \\, r} & 0 & 0 \\\\\n0 & -\\frac{a^{2} y^{2} + r^{2}}{y^{2} - 1} & 0 \\\\\n0 & 0 & -\\frac{{\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{4} - a^{2} r^{2} - r^{4} - 2 \\, a^{2} r - {\\left(a^{4} - r^{4} - 4 \\, a^{2} r\\right)} y^{2}}{a^{2} y^{2} + r^{2}}\n\\end{array}\\right)$
","done":false}︡{"done":true} ︠4980bda1-39e5-4c4f-8436-9cccdd4adc0bi︠ %html

Lapse function and shift vector

︡ba598a65-f1c6-4183-bba3-a7181efe4659︡︡{"done":true,"html":"

Lapse function and shift vector

"} ︠7160df27-dc1a-43c1-a9af-9ffedcbb1434︠ N = Sig.scalar_field(sqrt(Del / BB2), name='N') print N N.display() ︡5c140800-856a-4460-a7f5-1cdbd762abea︡︡{"stdout":"scalar field 'N' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} N:& \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & \\sqrt{-\\frac{a^{2} + r^{2} - 2 \\, r}{\\frac{2 \\, {\\left(y^{2} - 1\\right)} a^{2} r}{a^{2} y^{2} + r^{2}} - a^{2} - r^{2}}} \\end{array}$
","done":false}︡{"done":true} ︠43c0a490-63c8-4f4b-9217-2d20c87c0198︠ b = Sig.vector_field('beta', latex_name=r'\beta') b[3] = -2*m*r*a/AA2 # unset components are zero b.display() ︡d946e994-77f0-42d3-923d-354d8269f594︡︡{"html":"
$\\beta = \\left( -\\frac{2 \\, a r}{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\right) \\frac{\\partial}{\\partial {\\phi} }$
","done":false}︡{"done":true} ︠e651a8d3-faee-4fed-bd9c-3496624b2036i︠ %html

Extrinsic curvature of $\Sigma$

We use the formula \[ K_{ij} = \frac{1}{2N} \mathcal{L}_{\beta} \gamma_{ij} \] which is valid for any stationary spacetime:

︡5f5f8b73-ab77-4018-ac9c-c19cff3aeca7︡︡{"done":true,"html":"

Extrinsic curvature of $\\Sigma$

\n

We use the formula \\[ K_{ij} = \\frac{1}{2N} \\mathcal{L}_{\\beta} \\gamma_{ij} \\] which is valid for any stationary spacetime:

"} ︠d4f2b04c-06aa-4e72-a5cb-0a7a1ef8fe00︠ K = gam.lie_der(b) / (2*N) K.set_name('K') print K ; K.display() ︡ddbcedf1-4ed2-4ea6-b9f3-36c91c887c04︡︡{"stdout":"field of symmetric bilinear forms 'K' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$K = \\left( \\frac{{\\left(a^{3} r^{2} + 3 \\, a r^{4} + {\\left(a^{5} - a^{3} r^{2}\\right)} y^{4} - {\\left(a^{5} + 3 \\, a r^{4}\\right)} y^{2}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}}}{{\\left(a^{2} r^{4} + r^{6} + 2 \\, a^{2} r^{3} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} y^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\\right)} y^{2}\\right)} \\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} {\\left({\\left(a^{5} r + a^{3} r^{3} - 2 \\, a^{3} r^{2}\\right)} y^{3} - {\\left(a^{5} r + a^{3} r^{3} - 2 \\, a^{3} r^{2}\\right)} y\\right)}}{{\\left(a^{2} r^{4} + r^{6} + 2 \\, a^{2} r^{3} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} y^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\\right)} y^{2}\\right)} \\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\mathrm{d} y\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{{\\left(a^{3} r^{2} + 3 \\, a r^{4} + {\\left(a^{5} - a^{3} r^{2}\\right)} y^{4} - {\\left(a^{5} + 3 \\, a r^{4}\\right)} y^{2}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}}}{{\\left(a^{2} r^{4} + r^{6} + 2 \\, a^{2} r^{3} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} y^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\\right)} y^{2}\\right)} \\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + \\left( -\\frac{2 \\, \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} {\\left({\\left(a^{5} r + a^{3} r^{3} - 2 \\, a^{3} r^{2}\\right)} y^{3} - {\\left(a^{5} r + a^{3} r^{3} - 2 \\, a^{3} r^{2}\\right)} y\\right)}}{{\\left(a^{2} r^{4} + r^{6} + 2 \\, a^{2} r^{3} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} y^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\\right)} y^{2}\\right)} \\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} y$
","done":false}︡{"done":true} ︠d7d761e4-c003-4f9f-a8b2-30b923bc27aei︠ %html

Check (comparison with known formulas):

︡56536ef3-2701-4421-a8d1-4429a973c53c︡︡{"done":true,"html":"

Check (comparison with known formulas):

"} ︠7cea98ef-748e-4005-9934-4def3b42fced︠ Krp = a*m*(1-y^2)*(3*r^4+a^2*r^2+a^2*(r^2-a^2)*y^2) / rho2^2/sqrt(Del*BB2) Krp ︡60abe4fe-3323-4fa2-a0c9-dbe41c81c099︡︡{"html":"
$\\frac{{\\left({\\left(a^{2} - r^{2}\\right)} a^{2} y^{2} - a^{2} r^{2} - 3 \\, r^{4}\\right)} {\\left(y^{2} - 1\\right)} a}{{\\left(a^{2} y^{2} + r^{2}\\right)}^{2} \\sqrt{-{\\left(\\frac{2 \\, {\\left(y^{2} - 1\\right)} a^{2} r}{a^{2} y^{2} + r^{2}} - a^{2} - r^{2}\\right)} {\\left(a^{2} + r^{2} - 2 \\, r\\right)}}}$
","done":false}︡{"done":true} ︠26963ce3-bd28-4af7-9009-9ccb7544131b︠ K[1,3] - Krp ︡d564f794-2811-4444-a285-10035cefde13︡︡{"html":"
$0$
","done":false}︡{"done":true} ︠cdf83019-959f-4b5f-bbd8-b02ac8530293︠ Kyp = 2*m*r*a^3*(1-y^2)*y*sqrt(Del)/rho2^2/sqrt(BB2) Kyp ︡1d87ac59-8d68-4b53-90b9-2284a6316d71︡︡{"html":"
$-\\frac{2 \\, \\sqrt{a^{2} + r^{2} - 2 \\, r} {\\left(y^{2} - 1\\right)} a^{3} r y}{{\\left(a^{2} y^{2} + r^{2}\\right)}^{2} \\sqrt{-\\frac{2 \\, {\\left(y^{2} - 1\\right)} a^{2} r}{a^{2} y^{2} + r^{2}} + a^{2} + r^{2}}}$
","done":false}︡{"done":true} ︠b407fe08-cb42-4729-9c87-dda0255c51f9︠ K[2,3] - Kyp ︡d03c74a3-66eb-4b10-a837-404d6ae95143︡︡{"html":"
$0$
","done":false}︡{"done":true} ︠9b6a13c5-5541-49a7-9660-f6b4cc8247d2i︠ %html

For now on, we use the expressions Krp and Kyp above for $K_{r\phi}$ and $K_{ry}$, respectively:

︡ff557ded-bed2-4cf2-81fd-bfa5fe2bf652︡︡{"done":true,"html":"

For now on, we use the expressions Krp and Kyp above for $K_{r\\phi}$ and $K_{ry}$, respectively:

"} ︠65c51c43-745d-4dbe-9a47-25990f63bd56︠ K1 = Sig.sym_bilin_form_field('K') K1[1,3] = Krp K1[2,3] = Kyp K = K1 K.display() ︡f60473d6-2bb2-45e9-ac21-d19cc49a9700︡︡{"html":"
$K = \\left( \\frac{{\\left(a^{3} r^{2} + 3 \\, a r^{4} + {\\left(a^{5} - a^{3} r^{2}\\right)} y^{4} - {\\left(a^{5} + 3 \\, a r^{4}\\right)} y^{2}\\right)} \\sqrt{a^{2} y^{2} + r^{2}}}{{\\left(a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, {\\left(a^{3} r y^{3} - a^{3} r y\\right)} \\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{{\\left(a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}}} \\right) \\mathrm{d} y\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{{\\left(a^{3} r^{2} + 3 \\, a r^{4} + {\\left(a^{5} - a^{3} r^{2}\\right)} y^{4} - {\\left(a^{5} + 3 \\, a r^{4}\\right)} y^{2}\\right)} \\sqrt{a^{2} y^{2} + r^{2}}}{{\\left(a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + \\left( -\\frac{2 \\, {\\left(a^{3} r y^{3} - a^{3} r y\\right)} \\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{{\\left(a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} y$
","done":false}︡{"done":true} ︠cfd200d2-8e32-4c1f-95c5-28ef2dbf738bi︠ %html

The type-(1,1) tensor $K^\sharp$ of components $K^i_{\ \, j} = \gamma^{ik} K_{kj}$:

︡183e3e12-a7be-4e58-95d0-f8becbc6e553︡︡{"done":true,"html":"

The type-(1,1) tensor $K^\\sharp$ of components $K^i_{\\ \\, j} = \\gamma^{ik} K_{kj}$:

"} ︠44828d1b-02b6-4080-9747-3d2ef77ed483︠ Ku = K.up(gam, 0) print Ku ; Ku.display() ︡7a09197a-d8c7-4745-969d-ba250af80870︡︡{"stdout":"tensor field of type (1,1) on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( \\frac{{\\left(a^{3} r^{2} + 3 \\, a r^{4} + {\\left(a^{5} - a^{3} r^{2}\\right)} y^{4} - {\\left(a^{5} + 3 \\, a r^{4}\\right)} y^{2}\\right)} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{{\\left(a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} y^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{2 \\, {\\left(a^{3} r y^{5} - 2 \\, a^{3} r y^{3} + a^{3} r y\\right)} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{{\\left(a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} y^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{{\\left(a^{3} r^{2} + 3 \\, a r^{4} - {\\left(a^{5} - a^{3} r^{2}\\right)} y^{2}\\right)} \\sqrt{a^{2} y^{2} + r^{2}}}{{\\left(a^{2} r^{4} + r^{6} + 2 \\, a^{2} r^{3} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} y^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\\right)} y^{2}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r + \\left( \\frac{2 \\, \\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r} a^{3} r y}{{\\left(a^{2} r^{4} + r^{6} + 2 \\, a^{2} r^{3} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} y^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\\right)} y^{2}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} y$
","done":false}︡{"done":true} ︠c10826b1-09e7-4254-9c6b-90c0d07d9b32i︠ %html

We may check that the hypersurface $\Sigma$ is maximal, i.e. that $K^k_{\ \, k} = 0$:

︡28207a9b-cd72-41f6-9cef-c475379408ba︡︡{"done":true,"html":"

We may check that the hypersurface $\\Sigma$ is maximal, i.e. that $K^k_{\\ \\, k} = 0$:

"} ︠bd95a8e2-e21e-484b-b298-081b8d40742b︠ trK = Ku.trace() print trK ︡d7447008-4320-4e7a-bce3-fcdd28d6cac4︡︡{"stdout":"scalar field on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true} ︠4b08e359-493c-43d2-a467-b59e192944c3i︠ %html

Connection and curvature

Let us call $D$ the Levi-Civita connection associated with $\gamma$:

︡74a83070-17e9-4b88-a5dd-3ab646782b06︡︡{"done":true,"html":"

Connection and curvature

\n

Let us call $D$ the Levi-Civita connection associated with $\\gamma$:

"} ︠17c01af5-baf1-4c3f-a6ba-42d7b96d2d3a︠ D = gam.connection(name='D') print D ; D ︡ceaf0cd7-2e3e-4912-8db3-36dca925a4b6︡{"stdout":"Levi-Civita connection 'D' associated with the Riemannian metric 'gam' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$D$
","done":false}︡{"done":true}︡ ︠3209d9bf-f2e1-432e-ba96-c65cb856bc33i︠ %html

The Ricci tensor associated with $\gamma$:

︡5a8b0d18-6726-4182-a4d2-2191a6526e71︡︡{"done":true,"html":"

The Ricci tensor associated with $\\gamma$:

"} ︠a205a502-c18a-4635-a1aa-0e8f85d0b45b︠ Ric = gam.ricci() print Ric ; Ric ︡a2026930-abeb-43c9-96c2-6c3955eac99c︡︡{"stdout":"field of symmetric bilinear forms 'Ric(gam)' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\mathrm{Ric}\\left(\\gamma\\right)$
","done":false}︡{"done":true} ︠f9521859-57ed-47f3-b92f-a896100b703f︠ Ric[1,1] ︡5c09301f-9d00-4400-bced-5fb9cece1307︡︡{"html":"
$-\\frac{8 \\, a^{4} r^{7} + 7 \\, a^{2} r^{9} + 2 \\, r^{11} + 5 \\, a^{6} r^{4} + 2 \\, a^{4} r^{6} - 7 \\, a^{2} r^{8} + {\\left(3 \\, a^{10} r + 3 \\, a^{6} r^{5} + a^{10} - 14 \\, a^{8} r^{2} - 11 \\, a^{6} r^{4} + 6 \\, {\\left(a^{8} + 2 \\, a^{6}\\right)} r^{3}\\right)} y^{6} + {\\left(3 \\, a^{6} - 4 \\, a^{4}\\right)} r^{5} - {\\left(9 \\, a^{10} r + 4 \\, a^{4} r^{7} + a^{10} - 30 \\, a^{8} r^{2} - 35 \\, a^{6} r^{4} - 16 \\, a^{4} r^{6} + {\\left(17 \\, a^{6} + 4 \\, a^{4}\\right)} r^{5} + 2 \\, {\\left(11 \\, a^{8} + 12 \\, a^{6}\\right)} r^{3}\\right)} y^{4} - {\\left(16 \\, a^{4} r^{7} + 5 \\, a^{2} r^{9} + 16 \\, a^{8} r^{2} + 29 \\, a^{6} r^{4} + 18 \\, a^{4} r^{6} - 7 \\, a^{2} r^{8} + {\\left(17 \\, a^{6} - 8 \\, a^{4}\\right)} r^{5} + 6 \\, {\\left(a^{8} - 2 \\, a^{6}\\right)} r^{3}\\right)} y^{2}}{3 \\, a^{2} r^{12} + r^{14} + 6 \\, a^{4} r^{9} - 2 \\, r^{13} + 4 \\, a^{6} r^{6} + {\\left(3 \\, a^{4} - 8 \\, a^{2}\\right)} r^{10} + {\\left(a^{6} - 4 \\, a^{4}\\right)} r^{8} + {\\left(a^{14} + a^{8} r^{6} - 6 \\, a^{12} r - 6 \\, a^{8} r^{5} + 3 \\, {\\left(a^{10} + 4 \\, a^{8}\\right)} r^{4} - 4 \\, {\\left(3 \\, a^{10} + 2 \\, a^{8}\\right)} r^{3} + 3 \\, {\\left(a^{12} + 4 \\, a^{10}\\right)} r^{2}\\right)} y^{8} + 4 \\, {\\left(a^{6} - 2 \\, a^{4}\\right)} r^{7} + 4 \\, {\\left(a^{6} r^{8} + a^{12} r - 5 \\, a^{6} r^{7} + {\\left(3 \\, a^{8} + 8 \\, a^{6}\\right)} r^{6} - {\\left(9 \\, a^{8} + 4 \\, a^{6}\\right)} r^{5} + {\\left(3 \\, a^{10} + 4 \\, a^{8}\\right)} r^{4} - {\\left(3 \\, a^{10} - 4 \\, a^{8}\\right)} r^{3} + {\\left(a^{12} - 4 \\, a^{10}\\right)} r^{2}\\right)} y^{6} + 2 \\, {\\left(3 \\, a^{4} r^{10} - 12 \\, a^{4} r^{9} + 2 \\, a^{10} r^{2} + 16 \\, a^{6} r^{5} + {\\left(9 \\, a^{6} + 14 \\, a^{4}\\right)} r^{8} - 2 \\, {\\left(9 \\, a^{6} + 2 \\, a^{4}\\right)} r^{7} + 3 \\, {\\left(3 \\, a^{8} - 2 \\, a^{6}\\right)} r^{6} + 3 \\, {\\left(a^{10} - 6 \\, a^{8}\\right)} r^{4} + 2 \\, {\\left(3 \\, a^{10} - 2 \\, a^{8}\\right)} r^{3}\\right)} y^{4} + 4 \\, {\\left(a^{2} r^{12} - 3 \\, a^{4} r^{9} - 3 \\, a^{2} r^{11} + 2 \\, a^{8} r^{4} + {\\left(3 \\, a^{4} + 2 \\, a^{2}\\right)} r^{10} + 3 \\, {\\left(a^{6} - 2 \\, a^{4}\\right)} r^{8} + {\\left(3 \\, a^{6} + 4 \\, a^{4}\\right)} r^{7} + {\\left(a^{8} - 6 \\, a^{6}\\right)} r^{6} + {\\left(3 \\, a^{8} - 4 \\, a^{6}\\right)} r^{5}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠d7b771da-b75b-41a0-92d9-98c9204072cf︠ Ric[1,2] ︡f675277b-86bc-4729-aecd-340179780b76︡︡{"html":"
$\\frac{{\\left(3 \\, a^{10} + 6 \\, a^{8} r^{2} + 3 \\, a^{6} r^{4} - 4 \\, a^{8} r - 8 \\, a^{6} r^{3}\\right)} y^{5} - 2 \\, {\\left(3 \\, a^{8} r^{2} + 6 \\, a^{6} r^{4} + 3 \\, a^{4} r^{6} - 2 \\, a^{8} r - 12 \\, a^{6} r^{3} - 6 \\, a^{4} r^{5}\\right)} y^{3} - {\\left(9 \\, a^{6} r^{4} + 18 \\, a^{4} r^{6} + 9 \\, a^{2} r^{8} + 16 \\, a^{6} r^{3} + 12 \\, a^{4} r^{5}\\right)} y}{a^{4} r^{8} + 2 \\, a^{2} r^{10} + r^{12} + 4 \\, a^{4} r^{7} + 4 \\, a^{2} r^{9} + 4 \\, a^{4} r^{6} + {\\left(a^{12} + a^{8} r^{4} - 4 \\, a^{10} r - 4 \\, a^{8} r^{3} + 2 \\, {\\left(a^{10} + 2 \\, a^{8}\\right)} r^{2}\\right)} y^{8} + 4 \\, {\\left(a^{6} r^{6} + a^{10} r - 2 \\, a^{8} r^{3} - 3 \\, a^{6} r^{5} + 2 \\, {\\left(a^{8} + a^{6}\\right)} r^{4} + {\\left(a^{10} - 2 \\, a^{8}\\right)} r^{2}\\right)} y^{6} + 2 \\, {\\left(3 \\, a^{4} r^{8} + 6 \\, a^{8} r^{3} - 6 \\, a^{4} r^{7} + 2 \\, a^{8} r^{2} + 2 \\, {\\left(3 \\, a^{6} + a^{4}\\right)} r^{6} + {\\left(3 \\, a^{8} - 8 \\, a^{6}\\right)} r^{4}\\right)} y^{4} + 4 \\, {\\left(2 \\, a^{4} r^{8} + a^{2} r^{10} + 3 \\, a^{6} r^{5} + 2 \\, a^{4} r^{7} - a^{2} r^{9} + 2 \\, a^{6} r^{4} + {\\left(a^{6} - 2 \\, a^{4}\\right)} r^{6}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠e10efefb-82f1-48cf-a89e-dc572f73ff43︠ Ric[1,3] ︡38d9e79e-4b23-4318-823b-a758ad3bd964︡︡{"html":"
$0$
","done":false}︡{"done":true} ︠539bbd69-d85b-4010-9b58-fad85522e1db︠ Ric[2,2] ︡e249fe63-8852-4206-94bf-eed5461e6041︡︡{"html":"
$\\frac{7 \\, a^{4} r^{7} + 5 \\, a^{2} r^{9} + r^{11} + 6 \\, a^{6} r^{4} + 4 \\, a^{4} r^{6} - 2 \\, a^{2} r^{8} + 2 \\, {\\left(3 \\, a^{10} r + 3 \\, a^{6} r^{5} - 10 \\, a^{8} r^{2} - 10 \\, a^{6} r^{4} + 2 \\, {\\left(3 \\, a^{8} + 4 \\, a^{6}\\right)} r^{3}\\right)} y^{6} + {\\left(3 \\, a^{6} - 8 \\, a^{4}\\right)} r^{5} - {\\left(9 \\, a^{10} r - a^{4} r^{7} - 34 \\, a^{8} r^{2} - 36 \\, a^{6} r^{4} - 2 \\, a^{4} r^{6} + {\\left(7 \\, a^{6} + 8 \\, a^{4}\\right)} r^{5} + {\\left(17 \\, a^{8} + 32 \\, a^{6}\\right)} r^{3}\\right)} y^{4} - 2 \\, {\\left(7 \\, a^{4} r^{7} + 2 \\, a^{2} r^{9} + 7 \\, a^{8} r^{2} + 11 \\, a^{6} r^{4} + 3 \\, a^{4} r^{6} - a^{2} r^{8} + 8 \\, {\\left(a^{6} - a^{4}\\right)} r^{5} + {\\left(3 \\, a^{8} - 8 \\, a^{6}\\right)} r^{3}\\right)} y^{2}}{a^{4} r^{8} + 2 \\, a^{2} r^{10} + r^{12} + 4 \\, a^{4} r^{7} + 4 \\, a^{2} r^{9} - {\\left(a^{12} + a^{8} r^{4} - 4 \\, a^{10} r - 4 \\, a^{8} r^{3} + 2 \\, {\\left(a^{10} + 2 \\, a^{8}\\right)} r^{2}\\right)} y^{10} + 4 \\, a^{4} r^{6} + {\\left(a^{12} - 4 \\, a^{6} r^{6} - 8 \\, a^{10} r + 4 \\, a^{8} r^{3} + 12 \\, a^{6} r^{5} - {\\left(7 \\, a^{8} + 8 \\, a^{6}\\right)} r^{4} - 2 \\, {\\left(a^{10} - 6 \\, a^{8}\\right)} r^{2}\\right)} y^{8} - 2 \\, {\\left(3 \\, a^{4} r^{8} - 2 \\, a^{10} r + 10 \\, a^{8} r^{3} + 6 \\, a^{6} r^{5} - 6 \\, a^{4} r^{7} + 2 \\, {\\left(2 \\, a^{6} + a^{4}\\right)} r^{6} - {\\left(a^{8} + 12 \\, a^{6}\\right)} r^{4} - 2 \\, {\\left(a^{10} - 3 \\, a^{8}\\right)} r^{2}\\right)} y^{6} - 2 \\, {\\left(a^{4} r^{8} + 2 \\, a^{2} r^{10} - 6 \\, a^{8} r^{3} + 6 \\, a^{6} r^{5} + 10 \\, a^{4} r^{7} - 2 \\, a^{2} r^{9} - 2 \\, a^{8} r^{2} - 2 \\, {\\left(2 \\, a^{6} + 3 \\, a^{4}\\right)} r^{6} - 3 \\, {\\left(a^{8} - 4 \\, a^{6}\\right)} r^{4}\\right)} y^{4} + {\\left(7 \\, a^{4} r^{8} + 2 \\, a^{2} r^{10} - r^{12} + 12 \\, a^{6} r^{5} + 4 \\, a^{4} r^{7} - 8 \\, a^{2} r^{9} + 8 \\, a^{6} r^{4} + 4 \\, {\\left(a^{6} - 3 \\, a^{4}\\right)} r^{6}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠37983451-81db-4d51-8cd8-63250fe23b65︠ Ric[2,3] ︡5e342b94-2a7d-4bf2-9614-42de744ba680︡︡{"html":"
$0$
","done":false}︡{"done":true} ︠bd09f7ab-5b22-4f2f-ae35-ed655a0d5125︠ Ric[3,3] ︡7e8510b3-88a0-4be3-96e7-8a29c983e597︡︡{"html":"
$\\frac{a^{4} r^{7} + 2 \\, a^{2} r^{9} + r^{11} + a^{6} r^{4} + 10 \\, a^{4} r^{6} + 13 \\, a^{2} r^{8} + 4 \\, a^{4} r^{5} + {\\left(3 \\, a^{10} r + 3 \\, a^{6} r^{5} + a^{10} - 18 \\, a^{8} r^{2} - 15 \\, a^{6} r^{4} + 2 \\, {\\left(3 \\, a^{8} + 10 \\, a^{6}\\right)} r^{3}\\right)} y^{8} - {\\left(3 \\, a^{10} r - 5 \\, a^{4} r^{7} + 2 \\, a^{10} - 38 \\, a^{8} r^{2} - 22 \\, a^{6} r^{4} + 2 \\, a^{4} r^{6} - {\\left(7 \\, a^{6} - 4 \\, a^{4}\\right)} r^{5} + {\\left(a^{8} + 60 \\, a^{6}\\right)} r^{3}\\right)} y^{6} - {\\left(3 \\, a^{4} r^{7} - a^{2} r^{9} - a^{10} + 22 \\, a^{8} r^{2} - 2 \\, a^{6} r^{4} - 14 \\, a^{4} r^{6} - 13 \\, a^{2} r^{8} + 3 \\, {\\left(3 \\, a^{6} - 4 \\, a^{4}\\right)} r^{5} + 5 \\, {\\left(a^{8} - 12 \\, a^{6}\\right)} r^{3}\\right)} y^{4} - {\\left(3 \\, a^{4} r^{7} + 3 \\, a^{2} r^{9} + r^{11} - 2 \\, a^{8} r^{2} + 10 \\, a^{6} r^{4} + 22 \\, a^{4} r^{6} + 26 \\, a^{2} r^{8} + 20 \\, a^{6} r^{3} + {\\left(a^{6} + 12 \\, a^{4}\\right)} r^{5}\\right)} y^{2}}{a^{2} r^{10} + r^{12} + 2 \\, a^{2} r^{9} + {\\left(a^{12} + a^{10} r^{2} - 2 \\, a^{10} r\\right)} y^{10} + {\\left(5 \\, a^{10} r^{2} + 5 \\, a^{8} r^{4} + 2 \\, a^{10} r - 8 \\, a^{8} r^{3}\\right)} y^{8} + 2 \\, {\\left(5 \\, a^{8} r^{4} + 5 \\, a^{6} r^{6} + 4 \\, a^{8} r^{3} - 6 \\, a^{6} r^{5}\\right)} y^{6} + 2 \\, {\\left(5 \\, a^{6} r^{6} + 5 \\, a^{4} r^{8} + 6 \\, a^{6} r^{5} - 4 \\, a^{4} r^{7}\\right)} y^{4} + {\\left(5 \\, a^{4} r^{8} + 5 \\, a^{2} r^{10} + 8 \\, a^{4} r^{7} - 2 \\, a^{2} r^{9}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠80e648fb-484a-4dac-914a-d2e503a4a4d7i︠ %html

The scalar curvature $R = \gamma^{ij} R_{ij}$:

︡dc885a3a-c649-474a-b5ed-df46d7e58617︡︡{"done":true,"html":"

The scalar curvature $R = \\gamma^{ij} R_{ij}$:

"} ︠0908eb54-4192-46aa-9ff3-3b2de11d182e︠ R = gam.ricci_scalar(name='R') print R R.display() ︡1eacfe1c-9f93-4e4a-b259-715ef6cfbdee︡︡{"stdout":"scalar field 'R' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} \\mathrm{r}\\left(\\gamma\\right):& \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & \\frac{2 \\, {\\left(a^{6} r^{4} + 6 \\, a^{4} r^{6} + 9 \\, a^{2} r^{8} - {\\left(a^{10} - 6 \\, a^{8} r^{2} - 3 \\, a^{6} r^{4} + 8 \\, a^{6} r^{3}\\right)} y^{6} + {\\left(a^{10} - 8 \\, a^{8} r^{2} - 3 \\, a^{6} r^{4} - 6 \\, a^{4} r^{6} + 16 \\, a^{6} r^{3}\\right)} y^{4} + {\\left(2 \\, a^{8} r^{2} - a^{6} r^{4} - 9 \\, a^{2} r^{8} - 8 \\, a^{6} r^{3}\\right)} y^{2}\\right)}}{a^{4} r^{10} + 2 \\, a^{2} r^{12} + r^{14} + 4 \\, a^{4} r^{9} + 4 \\, a^{2} r^{11} + 4 \\, a^{4} r^{8} + {\\left(a^{14} + a^{10} r^{4} - 4 \\, a^{12} r - 4 \\, a^{10} r^{3} + 2 \\, {\\left(a^{12} + 2 \\, a^{10}\\right)} r^{2}\\right)} y^{10} + {\\left(5 \\, a^{8} r^{6} + 4 \\, a^{12} r - 12 \\, a^{10} r^{3} - 16 \\, a^{8} r^{5} + 2 \\, {\\left(5 \\, a^{10} + 6 \\, a^{8}\\right)} r^{4} + {\\left(5 \\, a^{12} - 8 \\, a^{10}\\right)} r^{2}\\right)} y^{8} + 2 \\, {\\left(5 \\, a^{6} r^{8} + 8 \\, a^{10} r^{3} - 4 \\, a^{8} r^{5} - 12 \\, a^{6} r^{7} + 2 \\, a^{10} r^{2} + 2 \\, {\\left(5 \\, a^{8} + 3 \\, a^{6}\\right)} r^{6} + {\\left(5 \\, a^{10} - 12 \\, a^{8}\\right)} r^{4}\\right)} y^{6} + 2 \\, {\\left(5 \\, a^{4} r^{10} + 12 \\, a^{8} r^{5} + 4 \\, a^{6} r^{7} - 8 \\, a^{4} r^{9} + 6 \\, a^{8} r^{4} + 2 \\, {\\left(5 \\, a^{6} + a^{4}\\right)} r^{8} + {\\left(5 \\, a^{8} - 12 \\, a^{6}\\right)} r^{6}\\right)} y^{4} + {\\left(10 \\, a^{4} r^{10} + 5 \\, a^{2} r^{12} + 16 \\, a^{6} r^{7} + 12 \\, a^{4} r^{9} - 4 \\, a^{2} r^{11} + 12 \\, a^{6} r^{6} + {\\left(5 \\, a^{6} - 8 \\, a^{4}\\right)} r^{8}\\right)} y^{2}} \\end{array}$
","done":false}︡{"done":true} ︠61fc0f10-8167-422f-bb80-4bc080e5be57i︠ %html

Test: 3+1 Einstein equations

Let us check that the vacuum 3+1 Einstein equations are satisfied.

We start by the contraint equations:

Hamiltonian constraint

Let us first evaluate the term $K_{ij} K^{ij}$:

︡9b489439-21c9-469f-bfdb-3e464e55523f︡︡{"done":true,"html":"

Test: 3+1 Einstein equations

\n

Let us check that the vacuum 3+1 Einstein equations are satisfied.

\n

We start by the contraint equations:

\n

Hamiltonian constraint

\n

Let us first evaluate the term $K_{ij} K^{ij}$:

"} ︠317d2fee-6809-4869-a824-4f839837318a︠ Kuu = Ku.up(gam, 1) trKK = K['_ij']*Kuu['^ij'] print trKK ; trKK.display() ︡aeaabb18-8052-49b4-ac30-16be61ce7e34︡︡{"stdout":"scalar field on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} & \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & \\frac{2 \\, {\\left(a^{6} r^{4} + 6 \\, a^{4} r^{6} + 9 \\, a^{2} r^{8} - {\\left(a^{10} - 6 \\, a^{8} r^{2} - 3 \\, a^{6} r^{4} + 8 \\, a^{6} r^{3}\\right)} y^{6} + {\\left(a^{10} - 8 \\, a^{8} r^{2} - 3 \\, a^{6} r^{4} - 6 \\, a^{4} r^{6} + 16 \\, a^{6} r^{3}\\right)} y^{4} + {\\left(2 \\, a^{8} r^{2} - a^{6} r^{4} - 9 \\, a^{2} r^{8} - 8 \\, a^{6} r^{3}\\right)} y^{2}\\right)}}{a^{4} r^{10} + 2 \\, a^{2} r^{12} + r^{14} + 4 \\, a^{4} r^{9} + 4 \\, a^{2} r^{11} + 4 \\, a^{4} r^{8} + {\\left(a^{14} + a^{10} r^{4} - 4 \\, a^{12} r - 4 \\, a^{10} r^{3} + 2 \\, {\\left(a^{12} + 2 \\, a^{10}\\right)} r^{2}\\right)} y^{10} + {\\left(5 \\, a^{8} r^{6} + 4 \\, a^{12} r - 12 \\, a^{10} r^{3} - 16 \\, a^{8} r^{5} + 2 \\, {\\left(5 \\, a^{10} + 6 \\, a^{8}\\right)} r^{4} + {\\left(5 \\, a^{12} - 8 \\, a^{10}\\right)} r^{2}\\right)} y^{8} + 2 \\, {\\left(5 \\, a^{6} r^{8} + 8 \\, a^{10} r^{3} - 4 \\, a^{8} r^{5} - 12 \\, a^{6} r^{7} + 2 \\, a^{10} r^{2} + 2 \\, {\\left(5 \\, a^{8} + 3 \\, a^{6}\\right)} r^{6} + {\\left(5 \\, a^{10} - 12 \\, a^{8}\\right)} r^{4}\\right)} y^{6} + 2 \\, {\\left(5 \\, a^{4} r^{10} + 12 \\, a^{8} r^{5} + 4 \\, a^{6} r^{7} - 8 \\, a^{4} r^{9} + 6 \\, a^{8} r^{4} + 2 \\, {\\left(5 \\, a^{6} + a^{4}\\right)} r^{8} + {\\left(5 \\, a^{8} - 12 \\, a^{6}\\right)} r^{6}\\right)} y^{4} + {\\left(10 \\, a^{4} r^{10} + 5 \\, a^{2} r^{12} + 16 \\, a^{6} r^{7} + 12 \\, a^{4} r^{9} - 4 \\, a^{2} r^{11} + 12 \\, a^{6} r^{6} + {\\left(5 \\, a^{6} - 8 \\, a^{4}\\right)} r^{8}\\right)} y^{2}} \\end{array}$
","done":false}︡{"done":true} ︠ad1676c9-380a-4cae-a544-7c1c1167dc68i︠ %html

The vacuum Hamiltonian constraint equation is \[R + K^2 -K_{ij} K^{ij} = 0 \]

︡0beba772-0a1c-4855-8118-1d4b666210f2︡︡{"done":true,"html":"

The vacuum Hamiltonian constraint equation is \\[R + K^2 -K_{ij} K^{ij} = 0 \\]

"} ︠ba4db164-097e-44da-89f1-a868db7c5863︠ Ham = R + trK^2 - trKK print Ham ; Ham.display() ︡efb42320-7299-4f60-9d69-375d816d89e6︡︡{"stdout":"scalar field 'zero' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} 0:& \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$
","done":false}︡{"done":true} ︠4ff7b00d-0321-4887-b80a-33142f2bfd1ei︠ %html

Momentum constraint

In vaccum, the momentum constraint is \[ D_j K^j_{\ \, i} - D_i K = 0 \]

︡8818cac0-2015-4627-9d33-e5007bf3a7d1︡︡{"done":true,"html":"

Momentum constraint

\n

In vaccum, the momentum constraint is \\[ D_j K^j_{\\ \\, i} - D_i K = 0 \\]

"} ︠81dad9b9-6e13-4e76-a639-0b6dab4cbab7︠ mom = D(Ku).trace(0,2) - D(trK) print mom mom.display() ︡09bc3eb8-a6b9-4af7-a1d1-685bbbf8f988︡︡{"stdout":"1-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$0$
","done":false}︡{"done":true} ︠7e38221c-0a3f-4898-956d-891dcd08bd33i︠ %html

Dynamical Einstein equations

Let us first evaluate the symmetric bilinear form $k_{ij} := K_{ik} K^k_{\ \, j}$:

︡1c522140-7406-42c6-80e3-ad3dc88457ba︡︡{"done":true,"html":"

Dynamical Einstein equations

\n

Let us first evaluate the symmetric bilinear form $k_{ij} := K_{ik} K^k_{\\ \\, j}$:

"} ︠8872f232-4bdc-44cd-b89f-a05da5f7493f︠ KK = K['_ik']*Ku['^k_j'] print KK ︡405d2451-102a-4c0b-8dd2-48245da37ec6︡︡{"stdout":"tensor field of type (0,2) on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true} ︠69c70867-35d2-4c7c-afc2-bce658084b03︠ KK1 = KK.symmetrize() KK == KK1 ︡8b3e102d-6230-4469-a534-d3c1e7101e6f︡{"html":"
$\\mathrm{True}$
","done":false}︡{"done":true}︡ ︠b304f5a5-f41c-463a-b553-3c67596eef3f︠ KK = KK1 print KK ︡5e113d0d-867c-449b-b3ae-47dcf6f60630︡{"stdout":"field of symmetric bilinear forms on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true}︡ ︠a7d97cc3-ade2-40f9-8232-17f5f49094ef︠ KK[1,1] ︡038cda65-946e-4232-a447-48fdce9df025︡{"html":"
$\\frac{a^{6} r^{4} + 6 \\, a^{4} r^{6} + 9 \\, a^{2} r^{8} - {\\left(a^{10} - 2 \\, a^{8} r^{2} + a^{6} r^{4}\\right)} y^{6} + {\\left(a^{10} + 5 \\, a^{6} r^{4} - 6 \\, a^{4} r^{6}\\right)} y^{4} - {\\left(2 \\, a^{8} r^{2} + 5 \\, a^{6} r^{4} + 9 \\, a^{2} r^{8}\\right)} y^{2}}{3 \\, a^{2} r^{12} + r^{14} + 6 \\, a^{4} r^{9} - 2 \\, r^{13} + 4 \\, a^{6} r^{6} + {\\left(3 \\, a^{4} - 8 \\, a^{2}\\right)} r^{10} + {\\left(a^{6} - 4 \\, a^{4}\\right)} r^{8} + {\\left(a^{14} + a^{8} r^{6} - 6 \\, a^{12} r - 6 \\, a^{8} r^{5} + 3 \\, {\\left(a^{10} + 4 \\, a^{8}\\right)} r^{4} - 4 \\, {\\left(3 \\, a^{10} + 2 \\, a^{8}\\right)} r^{3} + 3 \\, {\\left(a^{12} + 4 \\, a^{10}\\right)} r^{2}\\right)} y^{8} + 4 \\, {\\left(a^{6} - 2 \\, a^{4}\\right)} r^{7} + 4 \\, {\\left(a^{6} r^{8} + a^{12} r - 5 \\, a^{6} r^{7} + {\\left(3 \\, a^{8} + 8 \\, a^{6}\\right)} r^{6} - {\\left(9 \\, a^{8} + 4 \\, a^{6}\\right)} r^{5} + {\\left(3 \\, a^{10} + 4 \\, a^{8}\\right)} r^{4} - {\\left(3 \\, a^{10} - 4 \\, a^{8}\\right)} r^{3} + {\\left(a^{12} - 4 \\, a^{10}\\right)} r^{2}\\right)} y^{6} + 2 \\, {\\left(3 \\, a^{4} r^{10} - 12 \\, a^{4} r^{9} + 2 \\, a^{10} r^{2} + 16 \\, a^{6} r^{5} + {\\left(9 \\, a^{6} + 14 \\, a^{4}\\right)} r^{8} - 2 \\, {\\left(9 \\, a^{6} + 2 \\, a^{4}\\right)} r^{7} + 3 \\, {\\left(3 \\, a^{8} - 2 \\, a^{6}\\right)} r^{6} + 3 \\, {\\left(a^{10} - 6 \\, a^{8}\\right)} r^{4} + 2 \\, {\\left(3 \\, a^{10} - 2 \\, a^{8}\\right)} r^{3}\\right)} y^{4} + 4 \\, {\\left(a^{2} r^{12} - 3 \\, a^{4} r^{9} - 3 \\, a^{2} r^{11} + 2 \\, a^{8} r^{4} + {\\left(3 \\, a^{4} + 2 \\, a^{2}\\right)} r^{10} + 3 \\, {\\left(a^{6} - 2 \\, a^{4}\\right)} r^{8} + {\\left(3 \\, a^{6} + 4 \\, a^{4}\\right)} r^{7} + {\\left(a^{8} - 6 \\, a^{6}\\right)} r^{6} + {\\left(3 \\, a^{8} - 4 \\, a^{6}\\right)} r^{5}\\right)} y^{2}}$
","done":false}︡{"done":true}︡ ︠7a1fb838-f126-4837-a1d4-e9036a3d7298︠ KK[1,2] ︡124e1265-8f0b-48b2-bdb6-4c6b5290657e︡{"html":"
$\\frac{2 \\, {\\left({\\left(a^{8} r - a^{6} r^{3}\\right)} y^{5} - {\\left(a^{8} r + 3 \\, a^{4} r^{5}\\right)} y^{3} + {\\left(a^{6} r^{3} + 3 \\, a^{4} r^{5}\\right)} y\\right)}}{a^{4} r^{8} + 2 \\, a^{2} r^{10} + r^{12} + 4 \\, a^{4} r^{7} + 4 \\, a^{2} r^{9} + 4 \\, a^{4} r^{6} + {\\left(a^{12} + a^{8} r^{4} - 4 \\, a^{10} r - 4 \\, a^{8} r^{3} + 2 \\, {\\left(a^{10} + 2 \\, a^{8}\\right)} r^{2}\\right)} y^{8} + 4 \\, {\\left(a^{6} r^{6} + a^{10} r - 2 \\, a^{8} r^{3} - 3 \\, a^{6} r^{5} + 2 \\, {\\left(a^{8} + a^{6}\\right)} r^{4} + {\\left(a^{10} - 2 \\, a^{8}\\right)} r^{2}\\right)} y^{6} + 2 \\, {\\left(3 \\, a^{4} r^{8} + 6 \\, a^{8} r^{3} - 6 \\, a^{4} r^{7} + 2 \\, a^{8} r^{2} + 2 \\, {\\left(3 \\, a^{6} + a^{4}\\right)} r^{6} + {\\left(3 \\, a^{8} - 8 \\, a^{6}\\right)} r^{4}\\right)} y^{4} + 4 \\, {\\left(2 \\, a^{4} r^{8} + a^{2} r^{10} + 3 \\, a^{6} r^{5} + 2 \\, a^{4} r^{7} - a^{2} r^{9} + 2 \\, a^{6} r^{4} + {\\left(a^{6} - 2 \\, a^{4}\\right)} r^{6}\\right)} y^{2}}$
","done":false}︡{"done":true}︡ ︠3f712f0b-3dac-4751-bc8d-04db2c459d7e︠ KK[1,3] ︡47c4dd80-0a90-4466-bb31-6e6fc9dad5d9︡{"html":"
$0$
","done":false}︡{"done":true}︡ ︠40b58407-7b26-4a15-895c-b42af626085e︠ KK[2,2] ︡74c2f362-26cb-4dda-b3db-3bd4110a07ee︡{"html":"
$-\\frac{4 \\, {\\left({\\left(a^{8} r^{2} + a^{6} r^{4} - 2 \\, a^{6} r^{3}\\right)} y^{4} - {\\left(a^{8} r^{2} + a^{6} r^{4} - 2 \\, a^{6} r^{3}\\right)} y^{2}\\right)}}{a^{4} r^{8} + 2 \\, a^{2} r^{10} + r^{12} + 4 \\, a^{4} r^{7} + 4 \\, a^{2} r^{9} + 4 \\, a^{4} r^{6} + {\\left(a^{12} + a^{8} r^{4} - 4 \\, a^{10} r - 4 \\, a^{8} r^{3} + 2 \\, {\\left(a^{10} + 2 \\, a^{8}\\right)} r^{2}\\right)} y^{8} + 4 \\, {\\left(a^{6} r^{6} + a^{10} r - 2 \\, a^{8} r^{3} - 3 \\, a^{6} r^{5} + 2 \\, {\\left(a^{8} + a^{6}\\right)} r^{4} + {\\left(a^{10} - 2 \\, a^{8}\\right)} r^{2}\\right)} y^{6} + 2 \\, {\\left(3 \\, a^{4} r^{8} + 6 \\, a^{8} r^{3} - 6 \\, a^{4} r^{7} + 2 \\, a^{8} r^{2} + 2 \\, {\\left(3 \\, a^{6} + a^{4}\\right)} r^{6} + {\\left(3 \\, a^{8} - 8 \\, a^{6}\\right)} r^{4}\\right)} y^{4} + 4 \\, {\\left(2 \\, a^{4} r^{8} + a^{2} r^{10} + 3 \\, a^{6} r^{5} + 2 \\, a^{4} r^{7} - a^{2} r^{9} + 2 \\, a^{6} r^{4} + {\\left(a^{6} - 2 \\, a^{4}\\right)} r^{6}\\right)} y^{2}}$
","done":false}︡{"done":true}︡ ︠5c493e09-b4ac-4376-8ce1-19a2c476c2ed︠ KK[2,3] ︡e4a543f7-f941-42bc-ac62-080a9ab4a27e︡{"html":"
$0$
","done":false}︡{"done":true}︡ ︠fab9b616-5f14-4b40-a49b-19fc3c34334b︠ KK[3,3] ︡bddd69ef-8ccb-4c6e-b3bc-a858afb56233︡{"html":"
$\\frac{a^{6} r^{4} + 6 \\, a^{4} r^{6} + 9 \\, a^{2} r^{8} + {\\left(a^{10} - 6 \\, a^{8} r^{2} - 3 \\, a^{6} r^{4} + 8 \\, a^{6} r^{3}\\right)} y^{8} - 2 \\, {\\left(a^{10} - 7 \\, a^{8} r^{2} - 3 \\, a^{6} r^{4} - 3 \\, a^{4} r^{6} + 12 \\, a^{6} r^{3}\\right)} y^{6} + {\\left(a^{10} - 10 \\, a^{8} r^{2} - 2 \\, a^{6} r^{4} - 6 \\, a^{4} r^{6} + 9 \\, a^{2} r^{8} + 24 \\, a^{6} r^{3}\\right)} y^{4} + 2 \\, {\\left(a^{8} r^{2} - a^{6} r^{4} - 3 \\, a^{4} r^{6} - 9 \\, a^{2} r^{8} - 4 \\, a^{6} r^{3}\\right)} y^{2}}{a^{2} r^{10} + r^{12} + 2 \\, a^{2} r^{9} + {\\left(a^{12} + a^{10} r^{2} - 2 \\, a^{10} r\\right)} y^{10} + {\\left(5 \\, a^{10} r^{2} + 5 \\, a^{8} r^{4} + 2 \\, a^{10} r - 8 \\, a^{8} r^{3}\\right)} y^{8} + 2 \\, {\\left(5 \\, a^{8} r^{4} + 5 \\, a^{6} r^{6} + 4 \\, a^{8} r^{3} - 6 \\, a^{6} r^{5}\\right)} y^{6} + 2 \\, {\\left(5 \\, a^{6} r^{6} + 5 \\, a^{4} r^{8} + 6 \\, a^{6} r^{5} - 4 \\, a^{4} r^{7}\\right)} y^{4} + {\\left(5 \\, a^{4} r^{8} + 5 \\, a^{2} r^{10} + 8 \\, a^{4} r^{7} - 2 \\, a^{2} r^{9}\\right)} y^{2}}$
","done":false}︡{"done":true}︡ ︠ee2db939-20a9-4463-bd03-64db501bb5cci︠ %html

In vacuum and for stationary spacetimes, the dynamical Einstein equations are \[ \mathcal{L}_\beta K_{ij} - D_i D_j N + N \left( R_{ij} + K K_{ij} - 2 K_{ik} K^k_{\ \, j}\right) = 0 \]

︡6a07ec3a-47bc-4610-afa8-76bbfc7b0d9a︡︡{"done":true,"html":"

In vacuum and for stationary spacetimes, the dynamical Einstein equations are \\[ \\mathcal{L}_\\beta K_{ij} - D_i D_j N + N \\left( R_{ij} + K K_{ij} - 2 K_{ik} K^k_{\\ \\, j}\\right) = 0 \\]

"} ︠17fc9998-3d4b-4d7b-a9a4-98e2ee26623d︠ dyn = K.lie_der(b) - D(D(N)) + N*( Ric + trK*K - 2*KK ) print dyn dyn.display() ︡f65ab329-507f-41c3-87f5-99beac7fd870︡︡{"stdout":"tensor field of type (0,2) on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$0$
","done":false}︡{"done":true} ︠3ff63e3c-d8da-45ff-a26e-fb86ef4ba8b4i︠ %html

Hence, we have checked that all the vacuum 3+1 Einstein equations are fulfilled.

Electric and magnetic parts of the Weyl tensor

The electric part is the bilinear form $E$ given by \[ E_{ij} = R_{ij} + K K_{ij} - K_{ik} K^k_{\ \, j} \]

︡ac7706b3-2c17-47a5-9166-a6121cc294d5︡︡{"done":true,"html":"

Hence, we have checked that all the vacuum 3+1 Einstein equations are fulfilled.

\n\n

Electric and magnetic parts of the Weyl tensor

\n

The electric part is the bilinear form $E$ given by \\[ E_{ij} = R_{ij} + K K_{ij} - K_{ik} K^k_{\\ \\, j} \\]

"} ︠cf814373-8a19-403e-91b0-4444001e9a2f︠ E = Ric + trK*K - KK print E ︡e3592b51-0fea-4559-b74c-abd9a558de3e︡︡{"stdout":"field of symmetric bilinear forms on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true} ︠05a19d28-b080-45c3-9dd9-8ea29328fd5d︠ E[1,1] ︡9d627dd5-1b08-4c9d-ab64-770c63085c34︡︡{"html":"
$-\\frac{3 \\, a^{4} r^{3} + 5 \\, a^{2} r^{5} + 2 \\, r^{7} - 2 \\, a^{2} r^{4} + 3 \\, {\\left(a^{6} r + a^{4} r^{3} - 2 \\, a^{4} r^{2}\\right)} y^{4} - {\\left(9 \\, a^{6} r + 16 \\, a^{4} r^{3} + 7 \\, a^{2} r^{5} - 6 \\, a^{4} r^{2} - 2 \\, a^{2} r^{4}\\right)} y^{2}}{2 \\, a^{2} r^{8} + r^{10} + 2 \\, a^{4} r^{5} - 2 \\, r^{9} + {\\left(a^{4} - 4 \\, a^{2}\\right)} r^{6} + {\\left(a^{10} + a^{6} r^{4} - 4 \\, a^{8} r - 4 \\, a^{6} r^{3} + 2 \\, {\\left(a^{8} + 2 \\, a^{6}\\right)} r^{2}\\right)} y^{6} + {\\left(3 \\, a^{4} r^{6} + 2 \\, a^{8} r - 8 \\, a^{6} r^{3} - 10 \\, a^{4} r^{5} + 2 \\, {\\left(3 \\, a^{6} + 4 \\, a^{4}\\right)} r^{4} + {\\left(3 \\, a^{8} - 4 \\, a^{6}\\right)} r^{2}\\right)} y^{4} + {\\left(3 \\, a^{2} r^{8} + 4 \\, a^{6} r^{3} - 4 \\, a^{4} r^{5} - 8 \\, a^{2} r^{7} + 2 \\, {\\left(3 \\, a^{4} + 2 \\, a^{2}\\right)} r^{6} + {\\left(3 \\, a^{6} - 8 \\, a^{4}\\right)} r^{4}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠07d79df2-25ef-4535-b709-30fd9496bcd3︠ E[1,1].factor() ︡21fad2bf-ae21-4478-a134-dbf0e99043fc︡︡{"html":"
$-\\frac{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} - 3 \\, a^{4} - 5 \\, a^{2} r^{2} - 2 \\, r^{4} + 2 \\, a^{2} r\\right)} {\\left(3 \\, a^{2} y^{2} - r^{2}\\right)} r}{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} + r^{2}\\right)}^{2} {\\left(a^{2} + r^{2} - 2 \\, r\\right)}}$
","done":false}︡{"done":true} ︠54032e8f-63ee-4e4a-aa22-a428f50e8414︠ E[1,2] ︡ba3aeef0-b230-4f99-b5fa-ae8abed0342d︡︡{"html":"
$\\frac{3 \\, {\\left({\\left(a^{6} + a^{4} r^{2}\\right)} y^{3} - 3 \\, {\\left(a^{4} r^{2} + a^{2} r^{4}\\right)} y\\right)}}{a^{2} r^{6} + r^{8} + 2 \\, a^{2} r^{5} + {\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} y^{6} + {\\left(3 \\, a^{6} r^{2} + 3 \\, a^{4} r^{4} + 2 \\, a^{6} r - 4 \\, a^{4} r^{3}\\right)} y^{4} + {\\left(3 \\, a^{4} r^{4} + 3 \\, a^{2} r^{6} + 4 \\, a^{4} r^{3} - 2 \\, a^{2} r^{5}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠5ef1d8bc-1672-41da-8b57-c9e56540cd38︠ E[1,2].factor() ︡b5c93c54-1dce-460c-880b-7d3a028dc777︡︡{"html":"
$\\frac{3 \\, {\\left(a^{2} y^{2} - 3 \\, r^{2}\\right)} {\\left(a^{2} + r^{2}\\right)} a^{2} y}{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} + r^{2}\\right)}^{2}}$
","done":false}︡{"done":true} ︠8315a02d-d4e6-4c9c-b71f-dbe8cd649a08︠ E[1,3] ︡a0eb4bfa-d0c3-4441-8a74-458c6bfca7b1︡︡{"html":"
$0$
","done":false}︡{"done":true} ︠716f096c-c111-4a9f-b4ae-d52aeb6447a0︠ E[2,2] ︡4759b1c4-5c04-434e-9db2-054bbb8c5760︡︡{"html":"
$-\\frac{3 \\, a^{4} r^{3} + 4 \\, a^{2} r^{5} + r^{7} - 4 \\, a^{2} r^{4} + 6 \\, {\\left(a^{6} r + a^{4} r^{3} - 2 \\, a^{4} r^{2}\\right)} y^{4} - {\\left(9 \\, a^{6} r + 14 \\, a^{4} r^{3} + 5 \\, a^{2} r^{5} - 12 \\, a^{4} r^{2} - 4 \\, a^{2} r^{4}\\right)} y^{2}}{{\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} y^{8} - a^{2} r^{6} - r^{8} - 2 \\, a^{2} r^{5} - {\\left(a^{8} - 2 \\, a^{6} r^{2} - 3 \\, a^{4} r^{4} - 4 \\, a^{6} r + 4 \\, a^{4} r^{3}\\right)} y^{6} - {\\left(3 \\, a^{6} r^{2} - 3 \\, a^{2} r^{6} + 2 \\, a^{6} r - 8 \\, a^{4} r^{3} + 2 \\, a^{2} r^{5}\\right)} y^{4} - {\\left(3 \\, a^{4} r^{4} + 2 \\, a^{2} r^{6} - r^{8} + 4 \\, a^{4} r^{3} - 4 \\, a^{2} r^{5}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠a2894889-74a5-42d9-908f-929074ddfb63︠ E[2,2].factor() ︡821d36db-1a21-4db5-8a91-c44c2ebc86b2︡︡{"html":"
$-\\frac{{\\left(2 \\, a^{4} y^{2} + 2 \\, a^{2} r^{2} y^{2} - 4 \\, a^{2} r y^{2} - 3 \\, a^{4} - 4 \\, a^{2} r^{2} - r^{4} + 4 \\, a^{2} r\\right)} {\\left(3 \\, a^{2} y^{2} - r^{2}\\right)} r}{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} + r^{2}\\right)}^{2} {\\left(y + 1\\right)} {\\left(y - 1\\right)}}$
","done":false}︡{"done":true} ︠e579c42d-dd37-43ba-af25-fffdf340a5da︠ E[2,3] ︡6f364657-bab8-4970-a06b-d6762d98b050︡︡{"html":"
$0$
","done":false}︡{"done":true} ︠132ce405-ea9d-49db-90f3-8027633e0853︠ E[3,3] ︡90f92853-7829-4296-9e65-aad1cbb3f9bd︡︡{"html":"
$\\frac{a^{2} r^{5} + r^{7} + 3 \\, {\\left(a^{6} r + a^{4} r^{3} - 2 \\, a^{4} r^{2}\\right)} y^{6} + 2 \\, a^{2} r^{4} - {\\left(3 \\, a^{6} r + a^{4} r^{3} - 2 \\, a^{2} r^{5} - 12 \\, a^{4} r^{2} - 2 \\, a^{2} r^{4}\\right)} y^{4} - {\\left(2 \\, a^{4} r^{3} + 3 \\, a^{2} r^{5} + r^{7} + 6 \\, a^{4} r^{2} + 4 \\, a^{2} r^{4}\\right)} y^{2}}{a^{8} y^{8} + 4 \\, a^{6} r^{2} y^{6} + 6 \\, a^{4} r^{4} y^{4} + 4 \\, a^{2} r^{6} y^{2} + r^{8}}$
","done":false}︡{"done":true} ︠30354fc1-40f5-48d0-b2cc-7441eaaae245︠ E[3,3].factor() ︡b64f0a4a-e90d-4325-9a12-52668d2a6d11︡︡{"html":"
$\\frac{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(3 \\, a^{2} y^{2} - r^{2}\\right)} r {\\left(y + 1\\right)} {\\left(y - 1\\right)}}{{\\left(a^{2} y^{2} + r^{2}\\right)}^{4}}$
","done":false}︡{"done":true} ︠01df12ce-7e69-4da6-9fa1-3f7dc951dbc1i︠ %html

The magnetic part is the bilinear form $B$ defined by \[ B_{ij} = \epsilon^k_{\ \, l i} D_k K^l_{\ \, j}, \]

where $\epsilon^k_{\ \, l i}$ are the components of the type-(1,2) tensor $\epsilon^\sharp$, related to the Levi-Civita alternating tensor $\epsilon$ associated with $\gamma$ by $\epsilon^k_{\ \, l i} = \gamma^{km} \epsilon_{m l i}$. In SageManifolds, $\epsilon$ is obtained by the command volume_form() and $\epsilon^\sharp$ by the command volume_form(1) (1 = 1 index raised):

︡1f80d208-d412-4a2e-8692-19753259c598︡︡{"done":true,"html":"

The magnetic part is the bilinear form $B$ defined by \\[ B_{ij} = \\epsilon^k_{\\ \\, l i} D_k K^l_{\\ \\, j}, \\]

\n

where $\\epsilon^k_{\\ \\, l i}$ are the components of the type-(1,2) tensor $\\epsilon^\\sharp$, related to the Levi-Civita alternating tensor $\\epsilon$ associated with $\\gamma$ by $\\epsilon^k_{\\ \\, l i} = \\gamma^{km} \\epsilon_{m l i}$. In SageManifolds, $\\epsilon$ is obtained by the command volume_form() and $\\epsilon^\\sharp$ by the command volume_form(1) (1 = 1 index raised):

"} ︠674dfd51-b625-4e29-8371-2d651acdf078︠ eps = gam.volume_form() print eps ; eps.display() ︡01aefa02-cb61-49b4-9dd5-cfe41ec4e472︡︡{"stdout":"3-form 'eps_gam' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\epsilon_{\\gamma} = \\left( \\frac{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} y^{2} + r^{2}}}{\\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\mathrm{d} r\\wedge \\mathrm{d} y\\wedge \\mathrm{d} {\\phi}$
","done":false}︡{"done":true} ︠0695b835-2ef2-4e21-9fbe-a8580a3715a1︠ epsu = gam.volume_form(1) print epsu ; epsu.display() ︡524188bd-778d-472d-b146-1f171340e1f5︡︡{"stdout":"tensor field of type (1,2) on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( \\frac{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{\\sqrt{a^{2} y^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{\\sqrt{a^{2} y^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} y + \\left( \\frac{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} {\\left(y^{2} - 1\\right)}}{\\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} {\\left(y^{2} - 1\\right)}}{\\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + \\left( -\\frac{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} {\\left(a^{2} y^{2} + r^{2}\\right)}^{\\frac{3}{2}}}{{\\left({\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{4} - a^{2} r^{2} - r^{4} - 2 \\, a^{2} r - {\\left(a^{4} - r^{4} - 4 \\, a^{2} r\\right)} y^{2}\\right)} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} y + \\left( \\frac{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} {\\left(a^{2} y^{2} + r^{2}\\right)}^{\\frac{3}{2}}}{{\\left({\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{4} - a^{2} r^{2} - r^{4} - 2 \\, a^{2} r - {\\left(a^{4} - r^{4} - 4 \\, a^{2} r\\right)} y^{2}\\right)} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} r$
","done":false}︡{"done":true} ︠2db4d8d3-3906-489f-abca-26956ebdd5fa︠ DKu = D(Ku) B = epsu['^k_li']*DKu['^l_jk'] print B ︡e5e750a8-2c44-43df-862d-ccda936901d2︡︡{"stdout":"tensor field of type (0,2) on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true} ︠b1a44a23-c766-49dc-80b1-8d3b8ac05ddfi︠ %html

Let us check that $B$ is symmetric:

︡e9d8749d-cb3f-4bc6-a24c-779b8fbe9ea4︡︡{"done":true,"html":"

Let us check that $B$ is symmetric:

"} ︠61436395-df39-42ba-811d-6e23d1a78ef1︠ B1 = B.symmetrize() B == B1 ︡acaafa69-c505-4c46-b865-6eab3865f204︡︡{"html":"
$\\mathrm{True}$
","done":false}︡{"done":true} ︠a08fb8b2-a518-4fca-a33e-a1afe3a43986i︠ %html

Accordingly, we set

︡35a4107a-2474-4cad-88c8-eefa765fd575︡︡{"done":true,"html":"

Accordingly, we set

"} ︠06f33e08-d8c3-45ae-b09b-d2d16053a2cc︠ B = B1 B.set_name('B') print B ︡4608dd73-6d9a-4e6c-9510-2227ee499a19︡︡{"stdout":"field of symmetric bilinear forms 'B' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true} ︠1e0bdc27-af6a-4c48-b5be-7aec68559641︠ B[1,1] ︡892eec22-334e-470d-b21c-7fa30a6f65e1︡︡{"html":"
$-\\frac{{\\left(a^{7} + a^{5} r^{2} - 2 \\, a^{5} r\\right)} y^{5} - {\\left(3 \\, a^{7} + 8 \\, a^{5} r^{2} + 5 \\, a^{3} r^{4} - 2 \\, a^{5} r - 6 \\, a^{3} r^{3}\\right)} y^{3} + 3 \\, {\\left(3 \\, a^{5} r^{2} + 5 \\, a^{3} r^{4} + 2 \\, a r^{6} - 2 \\, a^{3} r^{3}\\right)} y}{2 \\, a^{2} r^{8} + r^{10} + 2 \\, a^{4} r^{5} - 2 \\, r^{9} + {\\left(a^{4} - 4 \\, a^{2}\\right)} r^{6} + {\\left(a^{10} + a^{6} r^{4} - 4 \\, a^{8} r - 4 \\, a^{6} r^{3} + 2 \\, {\\left(a^{8} + 2 \\, a^{6}\\right)} r^{2}\\right)} y^{6} + {\\left(3 \\, a^{4} r^{6} + 2 \\, a^{8} r - 8 \\, a^{6} r^{3} - 10 \\, a^{4} r^{5} + 2 \\, {\\left(3 \\, a^{6} + 4 \\, a^{4}\\right)} r^{4} + {\\left(3 \\, a^{8} - 4 \\, a^{6}\\right)} r^{2}\\right)} y^{4} + {\\left(3 \\, a^{2} r^{8} + 4 \\, a^{6} r^{3} - 4 \\, a^{4} r^{5} - 8 \\, a^{2} r^{7} + 2 \\, {\\left(3 \\, a^{4} + 2 \\, a^{2}\\right)} r^{6} + {\\left(3 \\, a^{6} - 8 \\, a^{4}\\right)} r^{4}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠73d83983-3ca8-4405-bdd1-003f776f05ae︠ B[1,1].factor() ︡c6a5fe01-1347-4c1b-bc38-e92ed545c525︡︡{"html":"
$-\\frac{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} - 3 \\, a^{4} - 5 \\, a^{2} r^{2} - 2 \\, r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} - 3 \\, r^{2}\\right)} a y}{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} + r^{2}\\right)}^{2} {\\left(a^{2} + r^{2} - 2 \\, r\\right)}}$
","done":false}︡{"done":true} ︠e345ce8e-7b90-46e1-be75-63420974c658︠ B[1,2] ︡22b15eb4-3208-4c23-9116-52e10cf3424b︡︡{"html":"
$\\frac{3 \\, {\\left(a^{3} r^{3} + a r^{5} - 3 \\, {\\left(a^{5} r + a^{3} r^{3}\\right)} y^{2}\\right)}}{a^{2} r^{6} + r^{8} + 2 \\, a^{2} r^{5} + {\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} y^{6} + {\\left(3 \\, a^{6} r^{2} + 3 \\, a^{4} r^{4} + 2 \\, a^{6} r - 4 \\, a^{4} r^{3}\\right)} y^{4} + {\\left(3 \\, a^{4} r^{4} + 3 \\, a^{2} r^{6} + 4 \\, a^{4} r^{3} - 2 \\, a^{2} r^{5}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠69668f51-15e9-4be7-bce3-7eda5863b91f︠ B[1,2].factor() ︡5e7dd18b-1219-4c7e-a487-d3ea6eda75b0︡︡{"html":"
$-\\frac{3 \\, {\\left(3 \\, a^{2} y^{2} - r^{2}\\right)} {\\left(a^{2} + r^{2}\\right)} a r}{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} + r^{2}\\right)}^{2}}$
","done":false}︡{"done":true} ︠4cded374-908c-4a1b-8ae3-215f4ed12629︠ B[1,3] ︡493a886e-158b-4172-a48f-68a55055aa9a︡︡{"html":"
$0$
","done":false}︡{"done":true} ︠7dbac6a9-ad88-4851-970b-d270c1bdb65a︠ B[2,2] ︡b384df84-1615-43ac-86b6-81152f84c8fa︡︡{"html":"
$-\\frac{2 \\, {\\left(a^{7} + a^{5} r^{2} - 2 \\, a^{5} r\\right)} y^{5} - {\\left(3 \\, a^{7} + 10 \\, a^{5} r^{2} + 7 \\, a^{3} r^{4} - 4 \\, a^{5} r - 12 \\, a^{3} r^{3}\\right)} y^{3} + 3 \\, {\\left(3 \\, a^{5} r^{2} + 4 \\, a^{3} r^{4} + a r^{6} - 4 \\, a^{3} r^{3}\\right)} y}{{\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} y^{8} - a^{2} r^{6} - r^{8} - 2 \\, a^{2} r^{5} - {\\left(a^{8} - 2 \\, a^{6} r^{2} - 3 \\, a^{4} r^{4} - 4 \\, a^{6} r + 4 \\, a^{4} r^{3}\\right)} y^{6} - {\\left(3 \\, a^{6} r^{2} - 3 \\, a^{2} r^{6} + 2 \\, a^{6} r - 8 \\, a^{4} r^{3} + 2 \\, a^{2} r^{5}\\right)} y^{4} - {\\left(3 \\, a^{4} r^{4} + 2 \\, a^{2} r^{6} - r^{8} + 4 \\, a^{4} r^{3} - 4 \\, a^{2} r^{5}\\right)} y^{2}}$
","done":false}︡{"done":true} ︠26952506-3ebe-4c7c-883c-e3a90b87d1b6︠ B[2,2].factor() ︡3b6dfd14-f09d-4382-81b7-1ebc1d3680dd︡︡{"html":"
$-\\frac{{\\left(2 \\, a^{4} y^{2} + 2 \\, a^{2} r^{2} y^{2} - 4 \\, a^{2} r y^{2} - 3 \\, a^{4} - 4 \\, a^{2} r^{2} - r^{4} + 4 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} - 3 \\, r^{2}\\right)} a y}{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} + r^{2}\\right)}^{2} {\\left(y + 1\\right)} {\\left(y - 1\\right)}}$
","done":false}︡{"done":true} ︠b11028a0-17ec-4843-bd64-bd950ace96ba︠ B[2,3] ︡5155ff35-adfe-41a0-850d-436046021653︡︡{"html":"
$0$
","done":false}︡{"done":true} ︠18bfc1a4-f07c-4f7a-946e-46fdb3fd8485︠ B[3,3] ︡59ad0a7b-7e52-4300-8d37-95f16b4d9c78︡︡{"html":"
$\\frac{{\\left(a^{7} + a^{5} r^{2} - 2 \\, a^{5} r\\right)} y^{7} - {\\left(a^{7} + 3 \\, a^{5} r^{2} + 2 \\, a^{3} r^{4} - 4 \\, a^{5} r - 6 \\, a^{3} r^{3}\\right)} y^{5} + {\\left(2 \\, a^{5} r^{2} - a^{3} r^{4} - 3 \\, a r^{6} - 2 \\, a^{5} r - 12 \\, a^{3} r^{3}\\right)} y^{3} + 3 \\, {\\left(a^{3} r^{4} + a r^{6} + 2 \\, a^{3} r^{3}\\right)} y}{a^{8} y^{8} + 4 \\, a^{6} r^{2} y^{6} + 6 \\, a^{4} r^{4} y^{4} + 4 \\, a^{2} r^{6} y^{2} + r^{8}}$
","done":false}︡{"done":true} ︠60d4e024-f7f7-45f8-9f5a-c2a14d1c9bda︠ B[3,3].factor() ︡e803c2dc-cc05-4804-aceb-6ab4ea1a0c96︡︡{"html":"
$\\frac{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} - 3 \\, r^{2}\\right)} a {\\left(y + 1\\right)} {\\left(y - 1\\right)} y}{{\\left(a^{2} y^{2} + r^{2}\\right)}^{4}}$
","done":false}︡{"done":true} ︠28842427-ee08-422b-a6fc-5866ce00daa3i︠ %html

3+1 decomposition of the Simon-Mars tensor

We follow the computation presented in arXiv:1412.6542. We start by the tensor $E^\sharp$ of components $E^i_ {\ \, j}$:

︡7a4089ee-7168-4a11-934f-594538a15a0e︡︡{"done":true,"html":"

3+1 decomposition of the Simon-Mars tensor

\n

We follow the computation presented in arXiv:1412.6542. We start by the tensor $E^\\sharp$ of components $E^i_ {\\ \\, j}$:

"} ︠607a905a-4348-436b-b293-451f5b063439︠ Eu = E.up(gam, 0) print Eu ︡4ad4e2b5-774f-4238-a25f-aaa9e54dd7de︡︡{"stdout":"tensor field of type (1,1) on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true} ︠b7470fc0-b0b2-44de-aa74-61bbb5fe1713i︠ %html

Tensor $B^\sharp$ of components $B^i_{\ \, j}$:

︡1cadc884-9c13-4c65-83d0-f6dc6dfbfda3︡︡{"done":true,"html":"

Tensor $B^\\sharp$ of components $B^i_{\\ \\, j}$:

"} ︠cc568af4-fc68-4d34-b982-eb70f5bfab65︠ Bu = B.up(gam, 0) print Bu ︡59ab115b-ab72-4143-89b2-840671bdd980︡︡{"stdout":"tensor field of type (1,1) on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true} ︠6060a92d-30cb-4c25-ac50-c748465f2fd5i︠ %html

1-form $\beta^\flat$ of components $\beta_i$ and its exterior derivative:

︡c7e8c14b-5d36-4b30-81d7-82f796c0d890︡︡{"done":true,"html":"

1-form $\\beta^\\flat$ of components $\\beta_i$ and its exterior derivative:

"} ︠c1e39782-e9d6-4575-8659-6ea2fb2f2bdc︠ bd = b.down(gam) xdb = bd.exterior_der() print xdb ; xdb.display() ︡adbf964e-763b-49ed-8f5b-31904a6eccc2︡︡{"stdout":"2-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( \\frac{2 \\, {\\left(a^{3} y^{4} + a r^{2} - {\\left(a^{3} + a r^{2}\\right)} y^{2}\\right)}}{a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}} \\right) \\mathrm{d} r\\wedge \\mathrm{d} {\\phi} + \\left( \\frac{4 \\, {\\left(a^{3} r + a r^{3}\\right)} y}{a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}} \\right) \\mathrm{d} y\\wedge \\mathrm{d} {\\phi}$
","done":false}︡{"done":true} ︠8422e208-7e7b-40ee-9d8b-1b2bb08cc9b5i︠ %html

Scalar square of shift $\beta_i \beta^i$:

︡09fedbe0-4e7a-47c2-8619-a1cfe1c34c5e︡︡{"done":true,"html":"

Scalar square of shift $\\beta_i \\beta^i$:

"} ︠cb4dddb5-e002-43b9-8ef2-27e6f9db2116︠ b2 = bd(b) print b2 ; b2.display() ︡9eb17180-d1c0-4656-a89a-ca33d9e9ed45︡︡{"stdout":"scalar field on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} & \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & -\\frac{4 \\, {\\left(a^{2} r^{2} y^{2} - a^{2} r^{2}\\right)}}{a^{2} r^{4} + r^{6} + 2 \\, a^{2} r^{3} + {\\left(a^{6} + a^{4} r^{2} - 2 \\, a^{4} r\\right)} y^{4} + 2 \\, {\\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\\right)} y^{2}} \\end{array}$
","done":false}︡{"done":true} ︠9fd6b9ee-6b5c-42fd-b412-d0a002679b59i︠ %html

Scalar $Y = E(\beta,\beta) = E_{ij} \beta^i \beta^j$:

︡2b7e151d-db2e-4bbe-8271-f1e7622b8330︡︡{"done":true,"html":"

Scalar $Y = E(\\beta,\\beta) = E_{ij} \\beta^i \\beta^j$:

"} ︠362de9a3-07d1-4399-8ab4-befc61ec3f4b︠ Ebb = E(b,b) Y = Ebb print Y ; Y.display() ︡e4749bd4-f9c2-401a-a093-e1ff6b597315︡︡{"stdout":"scalar field on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} & \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & \\frac{4 \\, {\\left(3 \\, a^{4} r^{3} y^{4} + a^{2} r^{5} - {\\left(3 \\, a^{4} r^{3} + a^{2} r^{5}\\right)} y^{2}\\right)}}{a^{2} r^{10} + r^{12} + 2 \\, a^{2} r^{9} + {\\left(a^{12} + a^{10} r^{2} - 2 \\, a^{10} r\\right)} y^{10} + {\\left(5 \\, a^{10} r^{2} + 5 \\, a^{8} r^{4} + 2 \\, a^{10} r - 8 \\, a^{8} r^{3}\\right)} y^{8} + 2 \\, {\\left(5 \\, a^{8} r^{4} + 5 \\, a^{6} r^{6} + 4 \\, a^{8} r^{3} - 6 \\, a^{6} r^{5}\\right)} y^{6} + 2 \\, {\\left(5 \\, a^{6} r^{6} + 5 \\, a^{4} r^{8} + 6 \\, a^{6} r^{5} - 4 \\, a^{4} r^{7}\\right)} y^{4} + {\\left(5 \\, a^{4} r^{8} + 5 \\, a^{2} r^{10} + 8 \\, a^{4} r^{7} - 2 \\, a^{2} r^{9}\\right)} y^{2}} \\end{array}$
","done":false}︡{"done":true} ︠09f9075e-aaad-474b-822a-4fb0139f2ba1︠ Ebb.function_chart().factor() ︡1dc672e6-38b7-475c-a1c7-29c06c5b223b︡︡{"html":"
$\\frac{4 \\, {\\left(3 \\, a^{2} y^{2} - r^{2}\\right)} a^{2} r^{3} {\\left(y + 1\\right)} {\\left(y - 1\\right)}}{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} + r^{2}\\right)}^{4}}$
","done":false}︡{"done":true} ︠9191b9e9-fbbf-42b1-b82d-58d870f7ec2b︠ Ebb.display() ︡b39bf33a-b049-41ab-a408-8ac7c9c4998d︡︡{"html":"
$\\begin{array}{llcl} & \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & \\frac{4 \\, {\\left(3 \\, a^{2} y^{2} - r^{2}\\right)} a^{2} r^{3} {\\left(y + 1\\right)} {\\left(y - 1\\right)}}{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} + r^{2}\\right)}^{4}} \\end{array}$
","done":false}︡{"done":true} ︠334cb764-7dff-4b54-bd62-b3212bda887bi︠ %html

Scalar $\bar Y = B(\beta,\beta) = B_{ij}\beta^i \beta^j$:

︡5c7769cd-e03b-4541-8f27-532c38215b9a︡︡{"done":true,"html":"

Scalar $\\bar Y = B(\\beta,\\beta) = B_{ij}\\beta^i \\beta^j$:

"} ︠647d7b74-3c91-4031-a3c2-53124195d2f6︠ Bbb = B(b,b) Y_bar = Bbb print Y_bar ; Y_bar.display() ︡739d0050-3f11-4764-96c7-46902faf3500︡︡{"stdout":"scalar field 'B(beta,beta)' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} B\\left(\\beta,\\beta\\right):& \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & \\frac{4 \\, {\\left(a^{5} r^{2} y^{5} + 3 \\, a^{3} r^{4} y - {\\left(a^{5} r^{2} + 3 \\, a^{3} r^{4}\\right)} y^{3}\\right)}}{a^{2} r^{10} + r^{12} + 2 \\, a^{2} r^{9} + {\\left(a^{12} + a^{10} r^{2} - 2 \\, a^{10} r\\right)} y^{10} + {\\left(5 \\, a^{10} r^{2} + 5 \\, a^{8} r^{4} + 2 \\, a^{10} r - 8 \\, a^{8} r^{3}\\right)} y^{8} + 2 \\, {\\left(5 \\, a^{8} r^{4} + 5 \\, a^{6} r^{6} + 4 \\, a^{8} r^{3} - 6 \\, a^{6} r^{5}\\right)} y^{6} + 2 \\, {\\left(5 \\, a^{6} r^{6} + 5 \\, a^{4} r^{8} + 6 \\, a^{6} r^{5} - 4 \\, a^{4} r^{7}\\right)} y^{4} + {\\left(5 \\, a^{4} r^{8} + 5 \\, a^{2} r^{10} + 8 \\, a^{4} r^{7} - 2 \\, a^{2} r^{9}\\right)} y^{2}} \\end{array}$
","done":false}︡{"done":true} ︠56d9a5a9-b715-4b6d-9a17-489eca11b6b5︠ Bbb.function_chart().factor() ︡e95342c0-38f3-4775-93b9-b90ce8915cad︡︡{"html":"
$\\frac{4 \\, {\\left(a^{2} y^{2} - 3 \\, r^{2}\\right)} a^{3} r^{2} {\\left(y + 1\\right)} {\\left(y - 1\\right)} y}{{\\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \\, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \\, a^{2} r\\right)} {\\left(a^{2} y^{2} + r^{2}\\right)}^{4}}$
","done":false}︡{"done":true} ︠73d5a2f5-ff95-4563-8be7-52eb2230dc52i︠ %html

1-form of components $Eb_i = E_{ij} \beta^j$:

︡d7d3086a-000d-46c7-a48d-c7e3cf4c3164︡︡{"done":true,"html":"

1-form of components $Eb_i = E_{ij} \\beta^j$:

"} ︠20970f45-9277-45bb-a61e-8e4a13abd725︠ Eb = E.contract(b) print Eb ; Eb.display() ︡10b556bd-304f-422d-ae6e-212472259e0e︡︡{"stdout":"1-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( -\\frac{2 \\, {\\left(3 \\, a^{3} r^{2} y^{4} + a r^{4} - {\\left(3 \\, a^{3} r^{2} + a r^{4}\\right)} y^{2}\\right)}}{a^{8} y^{8} + 4 \\, a^{6} r^{2} y^{6} + 6 \\, a^{4} r^{4} y^{4} + 4 \\, a^{2} r^{6} y^{2} + r^{8}} \\right) \\mathrm{d} {\\phi}$
","done":false}︡{"done":true} ︠2f61da78-4d3c-451f-82ba-1c2f7f3fc826i︠ %html

Vector field of components $Eub^i = E^i_{\ \, j} \beta^j$:

︡194a32f9-7fd3-43de-9102-3e6f175c8d13︡︡{"done":true,"html":"

Vector field of components $Eub^i = E^i_{\\ \\, j} \\beta^j$:

"} ︠67c98ece-d19d-4f4a-b9a2-4fcb8aba5715︠ Eub = Eu.contract(b) print Eub ; Eub.display() ︡cfd240a4-abf3-427f-b211-63b59921e626︡︡{"stdout":"vector field on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( \\frac{2 \\, {\\left(3 \\, a^{3} r^{2} y^{2} - a r^{4}\\right)}}{a^{2} r^{8} + r^{10} + 2 \\, a^{2} r^{7} + {\\left(a^{10} + a^{8} r^{2} - 2 \\, a^{8} r\\right)} y^{8} + 2 \\, {\\left(2 \\, a^{8} r^{2} + 2 \\, a^{6} r^{4} + a^{8} r - 3 \\, a^{6} r^{3}\\right)} y^{6} + 6 \\, {\\left(a^{6} r^{4} + a^{4} r^{6} + a^{6} r^{3} - a^{4} r^{5}\\right)} y^{4} + 2 \\, {\\left(2 \\, a^{4} r^{6} + 2 \\, a^{2} r^{8} + 3 \\, a^{4} r^{5} - a^{2} r^{7}\\right)} y^{2}} \\right) \\frac{\\partial}{\\partial {\\phi} }$
","done":false}︡{"done":true} ︠940e5fe9-f680-4968-93e3-f4584bea1785i︠ %html

1-form of components $Bb_i = B_{ij} \beta^j$:

︡281f53af-5482-476d-982e-f71886c6ee53︡︡{"done":true,"html":"

1-form of components $Bb_i = B_{ij} \\beta^j$:

"} ︠8219df1b-765c-4750-8ae5-8b7c5e67f6cf︠ Bb = B.contract(b) print Bb ; Bb.display() ︡1c418593-d0c1-4b63-a1e3-20a3c04f4f9e︡︡{"stdout":"1-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( -\\frac{2 \\, {\\left(a^{4} r y^{5} + 3 \\, a^{2} r^{3} y - {\\left(a^{4} r + 3 \\, a^{2} r^{3}\\right)} y^{3}\\right)}}{a^{8} y^{8} + 4 \\, a^{6} r^{2} y^{6} + 6 \\, a^{4} r^{4} y^{4} + 4 \\, a^{2} r^{6} y^{2} + r^{8}} \\right) \\mathrm{d} {\\phi}$
","done":false}︡{"done":true} ︠4843321e-237b-4af7-b716-e9c681acf6e5i︠ %html

Vector field of components $Bub^i = B^i_{\ \, j} \beta^j$:

︡2ae5d5e7-9511-48fb-9b63-9a51df2f6f72︡︡{"done":true,"html":"

Vector field of components $Bub^i = B^i_{\\ \\, j} \\beta^j$:

"} ︠4ef9d9c6-1937-49db-8b17-cdaaf1f078ba︠ Bub = Bu.contract(b) print Bub ; Bub.display() ︡eaf79566-0b74-4e3a-923d-c7685a80aefd︡︡{"stdout":"vector field on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( \\frac{2 \\, {\\left(a^{4} r y^{3} - 3 \\, a^{2} r^{3} y\\right)}}{a^{2} r^{8} + r^{10} + 2 \\, a^{2} r^{7} + {\\left(a^{10} + a^{8} r^{2} - 2 \\, a^{8} r\\right)} y^{8} + 2 \\, {\\left(2 \\, a^{8} r^{2} + 2 \\, a^{6} r^{4} + a^{8} r - 3 \\, a^{6} r^{3}\\right)} y^{6} + 6 \\, {\\left(a^{6} r^{4} + a^{4} r^{6} + a^{6} r^{3} - a^{4} r^{5}\\right)} y^{4} + 2 \\, {\\left(2 \\, a^{4} r^{6} + 2 \\, a^{2} r^{8} + 3 \\, a^{4} r^{5} - a^{2} r^{7}\\right)} y^{2}} \\right) \\frac{\\partial}{\\partial {\\phi} }$
","done":false}︡{"done":true} ︠8e2d7d3d-631d-49cb-a392-e61754c872b6i︠ %html

Vector field of components $Kub^i = K^i_{\ \, j} \beta^j$:

︡620898ad-674e-4e56-a69d-77532c33a9fa︡︡{"done":true,"html":"

Vector field of components $Kub^i = K^i_{\\ \\, j} \\beta^j$:

"} ︠b3d87328-b575-464a-9711-30d81afeae91︠ Kub = Ku.contract(b) print Kub ; Kub.display() ︡185c8249-176a-454b-bb0a-c91e66e85a2c︡︡{"stdout":"vector field on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( -\\frac{2 \\, {\\left(a^{4} r^{3} + 3 \\, a^{2} r^{5} + {\\left(a^{6} r - a^{4} r^{3}\\right)} y^{4} - {\\left(a^{6} r + 3 \\, a^{2} r^{5}\\right)} y^{2}\\right)} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{{\\left(a^{2} r^{6} + r^{8} + 2 \\, a^{2} r^{5} + {\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} y^{6} + {\\left(3 \\, a^{6} r^{2} + 3 \\, a^{4} r^{4} + 2 \\, a^{6} r - 4 \\, a^{4} r^{3}\\right)} y^{4} + {\\left(3 \\, a^{4} r^{4} + 3 \\, a^{2} r^{6} + 4 \\, a^{4} r^{3} - 2 \\, a^{2} r^{5}\\right)} y^{2}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} y^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial r } + \\left( -\\frac{4 \\, {\\left(a^{4} r^{2} y^{5} - 2 \\, a^{4} r^{2} y^{3} + a^{4} r^{2} y\\right)} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{{\\left(a^{2} r^{6} + r^{8} + 2 \\, a^{2} r^{5} + {\\left(a^{8} + a^{6} r^{2} - 2 \\, a^{6} r\\right)} y^{6} + {\\left(3 \\, a^{6} r^{2} + 3 \\, a^{4} r^{4} + 2 \\, a^{6} r - 4 \\, a^{4} r^{3}\\right)} y^{4} + {\\left(3 \\, a^{4} r^{4} + 3 \\, a^{2} r^{6} + 4 \\, a^{4} r^{3} - 2 \\, a^{2} r^{5}\\right)} y^{2}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} y^{2} + r^{2}}} \\right) \\frac{\\partial}{\\partial y }$
","done":false}︡{"done":true} ︠ba1998a8-abaf-47b0-8717-24dab42f5666︠ T = 2*b(N) - 2*K(b,b) print T ; T.display() ︡d5b5007f-6569-4cab-ac80-b1f6f5c808ce︡︡{"stdout":"scalar field 'zero' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} 0:& \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$
","done":false}︡{"done":true} ︠d9e10acf-7e22-4cb9-93ee-d2897be4bf91︠ Db = D(b) # Db^i_j = D_j b^i Dbu = Db.up(gam, 1) # Dbu^{ij} = D^j b^i bDb = b*Dbu # bDb^{ijk} = b^i D^k b^j T_bar = eps['_ijk']*bDb['^ikj'] print T_bar ; T_bar.display() ︡73e22058-f7e7-44db-b556-a14e105f0026︡︡{"stdout":"scalar field on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} & \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$
","done":false}︡{"done":true} ︠afbdb425-7d4b-4528-9fa8-b98f56576a2d︠ epsb = eps.contract(b) print epsb epsb.display() ︡185e2046-c45f-4d53-bc71-47b975a9941c︡︡{"stdout":"2-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( -\\frac{2 \\, \\sqrt{a^{2} y^{2} + r^{2}} a r}{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}} \\right) \\mathrm{d} r\\wedge \\mathrm{d} y$
","done":false}︡{"done":true} ︠64b8461a-a086-4138-be1e-5431c314eb01︠ epsB = eps['_ijl']*Bu['^l_k'] print epsB ︡d5a59133-5892-4b77-b403-00043b93e4a0︡︡{"stdout":"tensor field of type (0,3) on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true} ︠aee7e77e-8f79-49a8-88a4-84e3ae5e0ccf︠ epsB.symmetries() ︡f3020fb0-a84a-4dc0-b4f2-cf77359a3f2a︡︡{"stdout":"no symmetry; antisymmetry: (0, 1)\n","done":false}︡{"done":true} ︠61335d9d-1324-4ef4-bea7-2c157a7a63d2︠ epsB[1,2,3] ︡c93328d2-b9c7-4ee6-ba2f-03b2e0668ea3︡︡{"html":"
$-\\frac{{\\left(a^{3} y^{3} - 3 \\, a r^{2} y\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} \\sqrt{a^{2} y^{2} + r^{2}}}{{\\left(a^{6} y^{6} + 3 \\, a^{4} r^{2} y^{4} + 3 \\, a^{2} r^{4} y^{2} + r^{6}\\right)} \\sqrt{a^{2} + r^{2} - 2 \\, r}}$
","done":false}︡{"done":true} ︠95494885-5531-4cb6-86ca-20a4d4c2852c︠ Z = 2*N*( D(N) -K.contract(b)) + b.contract(xdb) print Z ; Z.display() ︡5ce1e032-c1e7-4f58-b813-abf29929dadf︡︡{"stdout":"1-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( -\\frac{2 \\, {\\left(a^{2} y^{2} - r^{2}\\right)}}{a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}} \\right) \\mathrm{d} r + \\left( \\frac{4 \\, a^{2} r y}{a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}} \\right) \\mathrm{d} y$
","done":false}︡{"done":true} ︠3a5a8e6d-2688-44b3-ac21-86593d6971cf︠ DNu = D(N).up(gam) A = 2*(DNu - Ku.contract(b))*b + N*Dbu Z_bar = eps['_ijk']*A['^kj'] print Z_bar ; Z_bar.display() ︡be256df8-aadc-493e-9fc2-d678c65d29a8︡︡{"stdout":"1-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( \\frac{4 \\, a r y}{a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}} \\right) \\mathrm{d} r + \\left( \\frac{2 \\, {\\left(a^{3} y^{2} - a r^{2}\\right)}}{a^{4} y^{4} + 2 \\, a^{2} r^{2} y^{2} + r^{4}} \\right) \\mathrm{d} y$
","done":false}︡{"done":true} ︠33729059-f630-4414-bc89-ea26f31f7f99︠ # Test: Dbdu = D(bd).up(gam,1).up(gam,1) # (Db)^{ij} = D^i b^j A = 2*b*(DNu - Ku.contract(b)) + N*Dbdu Z_bar0 = eps['_ijk']*A['^jk'] # NB: '^jk' and not 'kj' Z_bar0 == Z_bar ︡5aa76a42-2db9-4771-ac82-f6fa92850f08︡{"html":"
$\\mathrm{True}$
","done":false}︡{"done":true}︡ ︠b6e3e48b-af2a-4c2e-b573-ba83cff8c4f0︠ W = N*Eb + epsb.contract(Bub) print W ; W.display() ︡ee621daa-853b-4ae1-b436-11e4daa8c308︡{"stdout":"1-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( -\\frac{2 \\, {\\left(3 \\, a^{3} r^{2} y^{4} + a r^{4} - {\\left(3 \\, a^{3} r^{2} + a r^{4}\\right)} y^{2}\\right)} \\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{{\\left(a^{8} y^{8} + 4 \\, a^{6} r^{2} y^{6} + 6 \\, a^{4} r^{4} y^{4} + 4 \\, a^{2} r^{6} y^{2} + r^{8}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}}} \\right) \\mathrm{d} {\\phi}$
","done":false}︡{"done":true}︡ ︠771543c7-1bfa-473c-ba60-adf71e7ea04e︠ W_bar = N*Bb - epsb.contract(Eub) print W_bar ; W_bar.display() ︡952f52e6-6798-4717-9cd6-131339117005︡{"stdout":"1-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\left( -\\frac{2 \\, {\\left(a^{4} r y^{5} + 3 \\, a^{2} r^{3} y - {\\left(a^{4} r + 3 \\, a^{2} r^{3}\\right)} y^{3}\\right)} \\sqrt{a^{2} y^{2} + r^{2}} \\sqrt{a^{2} + r^{2} - 2 \\, r}}{{\\left(a^{8} y^{8} + 4 \\, a^{6} r^{2} y^{6} + 6 \\, a^{4} r^{4} y^{4} + 4 \\, a^{2} r^{6} y^{2} + r^{8}\\right)} \\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}}} \\right) \\mathrm{d} {\\phi}$
","done":false}︡{"done":true}︡ ︠570411c9-8ad2-4e45-9275-4f6a3a3f3dae︠ W[3].factor() ︡e05f8ac2-29dc-43f7-bb02-c616b47fe105︡{"html":"
$-\\frac{2 \\, {\\left(3 \\, a^{2} y^{2} - r^{2}\\right)} \\sqrt{a^{2} + r^{2} - 2 \\, r} a r^{2} {\\left(y + 1\\right)} {\\left(y - 1\\right)}}{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} {\\left(a^{2} y^{2} + r^{2}\\right)}^{\\frac{7}{2}}}$
","done":false}︡{"done":true}︡ ︠86b99dc2-5bb0-4d19-a941-8fdd5e374fb8︠ W_bar[3].factor() ︡5a31fb24-cd43-4cf4-930d-6471334d4415︡{"html":"
$-\\frac{2 \\, {\\left(a^{2} y^{2} - 3 \\, r^{2}\\right)} \\sqrt{a^{2} + r^{2} - 2 \\, r} a^{2} r {\\left(y + 1\\right)} {\\left(y - 1\\right)} y}{\\sqrt{a^{2} r^{2} + r^{4} + 2 \\, a^{2} r + {\\left(a^{4} + a^{2} r^{2} - 2 \\, a^{2} r\\right)} y^{2}} {\\left(a^{2} y^{2} + r^{2}\\right)}^{\\frac{7}{2}}}$
","done":false}︡{"done":true}︡ ︠25729eff-c7fe-43a3-b4db-94a5dcf6fefa︠ M = - 4*Eb(Kub - DNu) - 2*(epsB['_ij.']*Dbu['^ji'])(b) print M ; M.display() ︡6deafad5-fc77-4eca-a49a-6c47eb1841fe︡{"stdout":"scalar field 'zero' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} 0:& \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$
","done":false}︡{"done":true}︡ ︠003a6fb3-4a7a-4fbf-90df-a4624a92ac3c︠ M_bar = 2*(eps.contract(Eub))['_ij']*Dbu['^ji'] - 4*Bb(Kub - DNu) print M_bar ; M_bar.display() ︡8c7641db-e421-4cd7-bcba-7ee29ebe96a4︡{"stdout":"scalar field 'zero' on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"html":"
$\\begin{array}{llcl} 0:& \\Sigma & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, y, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$
","done":false}︡{"done":true}︡ ︠477bff0d-5ce0-478b-9b4b-0ce62170b14a︠ A = epsB['_ilk']*b['^l'] + epsB['_ikl']*b['^l'] + Bu['^m_i']*epsb['_mk'] - 2*N*E xdbE = xdb['_kl']*Eu['^k_i'] L = 2*N*epsB['_kli']*Dbu['^kl'] + 2*xdb['_ij']*Eub['^j'] + 2*xdbE['_li']*b['^l'] \ + 2*A['_ik']*(Kub - DNu)['^k'] print L ︡bb2fe841-7717-446f-b608-e0a97f74ca8f︡{"stdout":"1-form on the 3-dimensional manifold 'Sigma'\n","done":false}︡{"done":true}︡ ︠9bf8432f-3403-490b-a3a1-d34791c23ad4︠ L[1] ︡de44302f-f0e5-48f3-9a7d-145f53749ffd︡{"html":"
$-\\frac{8 \\, {\\left(5 \\, a^{4} r y^{4} - 10 \\, a^{2} r^{3} y^{2} + r^{5}\\right)}}{a^{10} y^{10} + 5 \\, a^{8} r^{2} y^{8} + 10 \\, a^{6} r^{4} y^{6} + 10 \\, a^{4} r^{6} y^{4} + 5 \\, a^{2} r^{8} y^{2} + r^{10}}$
","done":false}︡{"done":true}︡ ︠da8f1a63-a19c-4419-bd21-8182ee2effe4︠ L[1].factor() ︡12ff9928-4d3f-4cfa-a599-59f9b3ad3942︡{"html":"
$-\\frac{8 \\, {\\left(5 \\, a^{4} y^{4} - 10 \\, a^{2} r^{2} y^{2} + r^{4}\\right)} r}{{\\left(a^{2} y^{2} + r^{2}\\right)}^{5}}$
","done":false}︡{"done":true}︡ ︠517ee502-c483-4e27-a402-24ae4cebcc55︠ L[2] ︡346a5df9-1405-40f9-8cbb-5c22e81da150︡{"html":"
$-\\frac{8 \\, {\\left(a^{6} y^{5} - 10 \\, a^{4} r^{2} y^{3} + 5 \\, a^{2} r^{4} y\\right)}}{a^{10} y^{10} + 5 \\, a^{8} r^{2} y^{8} + 10 \\, a^{6} r^{4} y^{6} + 10 \\, a^{4} r^{6} y^{4} + 5 \\, a^{2} r^{8} y^{2} + r^{10}}$
","done":false}︡{"done":true}︡ ︠8b78f027-8fb2-4bb0-bd6b-50ef825b4046︠ L[2].factor() ︡38e38e6b-80be-4ca5-922c-a67dfb81b540︡{"html":"
$-\\frac{8 \\, {\\left(a^{4} y^{4} - 10 \\, a^{2} r^{2} y^{2} + 5 \\, r^{4}\\right)} a^{2} y}{{\\left(a^{2} y^{2} + r^{2}\\right)}^{5}}$
","done":false}︡{"done":true}︡ ︠9a227ce7-6664-4dd3-8383-e20d800956e8︠ L[3] ︡399addb5-13d1-4836-94a7-c8508b5264de︡{"html":"
$0$
","done":false}︡{"done":true}︡ ︠fae2e4cb-e7d2-4b50-acc3-35fad0d13c69︠ N2pbb = N^2 + b2 V = N2pbb*E - 2*(b.contract(E)*bd).symmetrize() + Ebb*gam + 2*N*(b.contract(epsB).symmetrize()) print V ︡9b5316ca-a1b3-4465-934e-b95ae8610fe0︡ ︠36e875d4-ff52-4b93-9b3d-03f8cc0d28fa︠ V[1,1] ︡f18e7847-a45e-45a4-a67f-83b3c6a65be0︡ ︠28eb1f39-2956-485d-bcbb-cd757c4bb4b8︠ V[1,1].factor() ︡6274c73d-fa81-4b8c-bcaf-ba65e629164e︡ ︠8c5b8523-f8c8-40ce-9b69-028ad07ad193︠ V[1,2] ︡af2f6f89-3911-4bbe-bd25-882309f9d51e︡ ︠1444d871-4ca0-4855-9cd8-139b849af87c︠ V[1,2].factor() ︡936c20b1-d0dc-4306-9fed-691e576cf0b0︡ ︠d79ed938-b11e-4f19-b10e-0a9ad8c9ae14︠ V[1,3] ︡ec07ff16-1108-4d49-9bfc-5ea829efdc4a︡ ︠856b5783-5861-49a2-a527-8ff9af4143bd︠ V[2,2] ︡fe544ca1-1a8f-4d8a-bc83-052a8c60fc2b︡ ︠ba127333-531a-4c05-a3ab-da57f84015c7︠ V[2,2].factor() ︡bbe24518-43df-40fe-92b1-0dcd1b9b563c︡ ︠bd061366-5cd7-42d2-964d-b59507aedec3︠ V[2,3] ︡f13e5c20-4b19-4b3d-8b15-c6e4215f51ab︡ ︠6f022b45-e11c-4d73-b84d-b08417433d59︠ V[3,3] ︡764dbcff-ec06-4059-9537-dd00a1cb69e6︡ ︠743d866f-7960-442b-b4e2-93b8265d3b80︠ V[3,3].factor() ︡0d6e8466-b04b-4e21-9e6c-f2a2ef784cb1︡ ︠5bff1b95-f374-48ef-8f1f-a012b7b0eb48︠ beps = b.contract(eps) V_bar = N2pbb*B - 2*(b.contract(B)*bd).symmetrize() + Bbb*gam \ -2*N*(beps['_il']*Eu['^l_j']).symmetrize() print V_bar ︡f0edf217-213b-4faa-94f5-e07352a74b66︡ ︠1caafe43-c04a-40e3-b9cf-b2b5ffc36e4a︠ V_bar[1,1] ︡bbe7e22e-ef6a-43dd-8df8-1792bb9d8f38︡ ︠b9ced511-7502-4932-96bc-37fd7340da23︠ V_bar[1,1].factor() ︡1b9d7463-0a29-4ac9-8252-c46464d3f74d︡ ︠db362083-b00e-4016-9530-3464ce61549c︠ V_bar[1,2] ︡e2588a98-947c-4f4b-99db-468779e4efea︡ ︠1f80c140-986f-49b7-9803-9dfc42a580ff︠ V_bar[1,2].factor() ︡6755aca4-ffca-490e-8660-6c0379ff5972︡ ︠7efe97be-ed07-4c62-85c5-f6c08cdd762e︠ V_bar[1,3] ︡8357fa6c-2f7d-42c8-bc6b-ae6e3cf8494b︡ ︠0578c639-4304-4d9b-90b8-d9d67f74e54f︠ V_bar[2,2] ︡f37f5f82-543f-4413-8607-f76cc29132c9︡ ︠271cfbcf-7898-4422-9876-9098b36f1da4︠ V_bar[2,2].factor() ︡dcd128ec-eba9-44b2-aaa8-2e4bfb9e694f︡ ︠19a4e20d-b135-46ff-a257-013b079c5972︠ V_bar[2,3] ︡0ab9cb59-e101-4580-803c-54addbc8762d︡ ︠392f3e29-7a24-48b3-8e65-e47ceb49f3c4︠ V_bar[3,3] ︡035c3f8b-d117-4d4f-86c8-946adc818732︡ ︠63f6ac33-afbf-41c0-a0a8-39287b07fc30︠ V_bar[3,3].factor() ︡e2ab466e-2066-4a30-b0e0-c388579722ef︡ ︠814240ed-93db-406f-8014-27f5243abb2e︠ G = (N^2 - b2)*gam + bd*bd print G ︡0934763b-75a8-4ed9-809b-f8a66b84a6b5︡ ︠d7e9c4fd-9e86-4af8-a601-956d0f072ba8︠ G.display() ︡d85c5063-bbc6-4aeb-bf15-01c1a3e65ee3︡ ︠b8a357a6-8ef3-4bc5-87a6-8090302e0920i︠ %html

3+1 decomposition of the real part of the Simon-Mars tensor

We follow Eqs. (77)-(80) of arXiv:1412.6542:

︡fb0a4169-dc13-48a3-9127-95a3d7674d4d︡︡{"done":true,"html":"

3+1 decomposition of the real part of the Simon-Mars tensor

\n

We follow Eqs. (77)-(80) of arXiv:1412.6542:

"} ︠3c9fc088-db94-4dd7-b4c2-f985c6d7d1b3︠ S1 = (4*(V*Z - V_bar*Z_bar) + G*L).antisymmetrize(1,2) print S1 ︡b128004d-6084-4cfd-9621-1cd4e134cb10︡ ︠56120dc6-7cf4-4385-97fc-5777b86d5251︠ S1.display() ︡10957b9d-5221-4ea2-896a-1cfba53e1f81︡ ︠8a338161-ed4c-4912-ab03-b8283230eb8b︠ S2 = 4*(T*V - T_bar*V_bar - W*Z + W_bar*Z_bar) + M*G - N*bd*L print S2 ︡10d1967f-b36d-41eb-b1c9-1efbe245de8e︡ ︠dd0dc96a-ddf5-4514-98e2-cf9153bb54ef︠ S2.display() ︡a24a524c-25ab-44bf-9207-cfffb19485c3︡ ︠72a2e746-a220-448b-a74d-3e662bab24ec︠ S3 = (4*(W*Z - W_bar*Z_bar) + N*bd*L).antisymmetrize() print S3 ︡52313bdd-b4cf-48b8-88c4-fd72ca7b871b︡ ︠d42c7760-616e-4b0a-89cd-680fe7836ea6︠ S3.display() ︡d3e09649-630c-4a15-8864-5ff584988e15︡ ︠f52259ba-c130-40fd-828c-b4a60fa6765b︠ S2[3,1] == -2*S3[3,1] ︡93a5c4aa-aefd-4522-bb36-8730f141fc86︡ ︠b44eec44-705a-4992-bd5f-5c2df75a6dc7︠ S2[3,2] == -2*S3[3,2] ︡74d42208-a73f-475b-bc53-63e8bea41d4a︡ ︠f7d37dd2-1ccb-4eb3-99b3-6a4cdc2097de︠ S4 = 4*(T*W - T_bar*W_bar) -4*(Y*Z - Y_bar*Z_bar) + N*M*bd - b2*L print S4 ︡0d456e35-8472-46be-8d8e-90a3b729a703︡ ︠4c291119-be2c-4097-acf2-8bff25f70550︠ S4.display() ︡77e67d9f-e8ba-49b9-8a8e-cbbaaca17039︡ ︠59cb7ba7-a8fa-4e8d-8b79-f0cd75120670i︠ %html

Hence all the tensors $S^1$, $S^2$, $S^3$ and $S^4$ involved in the 3+1 decomposition of the real part of the Simon-Mars are zero, as they should since the Simon-Mars tensor vanishes identically for the Kerr spacetime.

3+1 decomposition of the imaginary part of the Simon-Mars tensor

We follow Eqs. (82)-(85) of arXiv:1412.6542.

︡0d8b800c-4deb-4d46-8f54-5382d592f06b︡︡{"done":true,"html":"

Hence all the tensors $S^1$, $S^2$, $S^3$ and $S^4$ involved in the 3+1 decomposition of the real part of the Simon-Mars are zero, as they should since the Simon-Mars tensor vanishes identically for the Kerr spacetime.

\n\n

3+1 decomposition of the imaginary part of the Simon-Mars tensor

\n\n

We follow Eqs. (82)-(85) of arXiv:1412.6542.

"} ︠b2294127-700d-4f27-ac29-ca57edd8f961︠ epsE = eps['_ijl']*Eu['^l_k'] print epsE ︡6c2f3ec9-519a-46ae-87b7-e3c4b9284c7f︡ ︠c2f582be-c42e-4c23-9e1c-e7f4b206becc︠ A = - epsE['_ilk']*b['^l'] - epsE['_ikl']*b['^l'] - Eu['^m_i']*epsb['_mk'] - 2*N*B xdbB = xdb['_kl']*Bu['^k_i'] L_bar = - 2*N*epsE['_kli']*Dbu['^kl'] + 2*xdb['_ij']*Bub['^j'] + 2*xdbB['_li']*b['^l'] \ + 2*A['_ik']*(Kub - DNu)['^k'] print L_bar ︡0f4312b5-7130-4c44-83a7-330dbf30304f︡ ︠0259c407-a876-44af-a487-3edec06a0ef7︠ L_bar.display() ︡54f3f837-dc1d-4f82-a733-ccb2ac4d7efa︡ ︠9b6ed1af-7d62-4b42-9bee-3542a030a1ec︠ S1_bar = (4*(V*Z_bar + V_bar*Z) + G*L_bar).antisymmetrize(1,2) print S1_bar ︡ceb18aa4-e8ff-436b-825f-d1b927c50603︡ ︠0630a1b8-2011-491a-89bf-4e5e0d7d5186︠ S1_bar.display() ︡a8964877-9944-4f5b-aa28-3c744a5b1953︡ ︠dc773b83-4e3f-4716-a859-4391813d1082︠ S2_bar = 4*(T_bar*V + T*V_bar) - 4*(W*Z_bar + W_bar*Z) + M_bar*G - N*bd*L_bar print S2_bar ︡90475a77-bee2-4d55-b15e-4789256768a5︡ ︠b0bd0884-9dfa-4878-be63-9c15663185b3︠ S2_bar.display() ︡73288756-1499-4000-9fa3-e3826dc00aa8︡ ︠7f863fa2-1c02-435d-80e2-4985e259940b︠ S3_bar = (4*(W*Z_bar + W_bar*Z) + N*bd*L_bar).antisymmetrize() print S3_bar ︡76267f48-f14a-42b3-9677-d560ce664092︡ ︠751d4356-f751-4c6b-9842-075af92627a3︠ S3_bar.display() ︡81c8b9cb-94d2-448d-b840-402c9fd556f2︡ ︠922f18d7-2736-46bf-96ae-8f63852c7a07︠ S4_bar = 4*(T_bar*W + T*W_bar - Y*Z_bar - Y_bar*Z) + M_bar*N*bd - b2*L_bar print S4_bar ︡06645487-f889-4cac-aa5b-d538a62cdfce︡ ︠eb9575a5-251d-49dd-a30d-a9de92aabf27︠ S4_bar.display() ︡a4f3d75f-287b-449e-9568-05980564c2d5︡ ︠2a501510-96b8-476c-9b4c-1a2434a2fe7ai︠ %html

Hence all the tensors ${\bar S}^1$, ${\bar S}^2$, ${\bar S}^3$ and ${\bar S}^4$ involved in the 3+1 decomposition of the imaginary part of the Simon-Mars are zero, as they should since the Simon-Mars tensor vanishes identically for the Kerr spacetime.

︡7c200938-3469-4dfa-9eab-b113bd9ad5a2︡︡{"done":true,"html":"

Hence all the tensors ${\\bar S}^1$, ${\\bar S}^2$, ${\\bar S}^3$ and ${\\bar S}^4$ involved in the 3+1 decomposition of the imaginary part of the Simon-Mars are zero, as they should since the Simon-Mars tensor vanishes identically for the Kerr spacetime.

"} ︠132af374-83ba-4dfc-b0e7-54dd036514da︠ ︡6bb8bc42-83fc-4875-a72b-db77320250e1︡