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%display latex
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M = Manifold(4, 'M') print(M)
4-dimensional differentiable manifold M
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X.<t,x,th,ph> = M.chart(r't:\tau x:(0,+oo):\chi th:(0,pi):\theta ph:(0,2*pi):\phi') X
(M,(τ,χ,θ,ϕ))\left(M,({\tau}, {\chi}, {\theta}, {\phi})\right)
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g = M.lorentzian_metric('g') m0 = var('m_0', domain='real'); assume(m0>0) xs = var('xs', latex_name=r'\chi_{\rm s}', domain='real'); assume(xs>0) a = (1 - 3*sqrt(m0/(2*xs^3)) * t)^(2/3) g[0,0] = -1 g[1,1] = a^2 g[2,2] = (a*x)^2 g[3,3] = (a*x*sin(th))^2 g.display()
g=dτdτ+(312τm0χs3+1)43dχdχ+(312τm0χs3+1)43χ2dθdθ+(312τm0χs3+1)43χ2sin(θ)2dϕdϕg = -\mathrm{d} {\tau}\otimes \mathrm{d} {\tau} + {\left(-3 \, \sqrt{\frac{1}{2}} {\tau} \sqrt{\frac{m_{0}}{{\chi_{\rm s}}^{3}}} + 1\right)}^{\frac{4}{3}} \mathrm{d} {\chi}\otimes \mathrm{d} {\chi} + {\left(-3 \, \sqrt{\frac{1}{2}} {\tau} \sqrt{\frac{m_{0}}{{\chi_{\rm s}}^{3}}} + 1\right)}^{\frac{4}{3}} {\chi}^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + {\left(-3 \, \sqrt{\frac{1}{2}} {\tau} \sqrt{\frac{m_{0}}{{\chi_{\rm s}}^{3}}} + 1\right)}^{\frac{4}{3}} {\chi}^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}
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g.display_comp()
gττττ=1gχχχχ=(312τm0χs3+1)43gθθθθ=(312τm0χs3+1)43χ2gϕϕϕϕ=(312τm0χs3+1)43χ2sin(θ)2\begin{array}{lcl} g_{ \, {\tau} \, {\tau} }^{ \phantom{\, {\tau}}\phantom{\, {\tau}} } & = & -1 \\ g_{ \, {\chi} \, {\chi} }^{ \phantom{\, {\chi}}\phantom{\, {\chi}} } & = & {\left(-3 \, \sqrt{\frac{1}{2}} {\tau} \sqrt{\frac{m_{0}}{{\chi_{\rm s}}^{3}}} + 1\right)}^{\frac{4}{3}} \\ g_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & {\left(-3 \, \sqrt{\frac{1}{2}} {\tau} \sqrt{\frac{m_{0}}{{\chi_{\rm s}}^{3}}} + 1\right)}^{\frac{4}{3}} {\chi}^{2} \\ g_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & {\left(-3 \, \sqrt{\frac{1}{2}} {\tau} \sqrt{\frac{m_{0}}{{\chi_{\rm s}}^{3}}} + 1\right)}^{\frac{4}{3}} {\chi}^{2} \sin\left({\theta}\right)^{2} \end{array}
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Ric = g.ricci() print(Ric)
Field of symmetric bilinear forms Ric(g) on the 4-dimensional differentiable manifold M
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Ric.display_comp()
Ric(g)ττττ=6m062m0τχs329m0τ22χs3Ric(g)χχχχ=3223m0(32m0τ+2χs32)23χs2Ric(g)θθθθ=3223m0χ2(32m0τ+2χs32)23χs2Ric(g)ϕϕϕϕ=3223m0χ2sin(θ)2(32m0τ+2χs32)23χs2\begin{array}{lcl} \mathrm{Ric}\left(g\right)_{ \, {\tau} \, {\tau} }^{ \phantom{\, {\tau}}\phantom{\, {\tau}} } & = & -\frac{6 \, m_{0}}{6 \, \sqrt{2} \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} - 9 \, m_{0} {\tau}^{2} - 2 \, {\chi_{\rm s}}^{3}} \\ \mathrm{Ric}\left(g\right)_{ \, {\chi} \, {\chi} }^{ \phantom{\, {\chi}}\phantom{\, {\chi}} } & = & \frac{3 \cdot 2^{\frac{2}{3}} m_{0}}{{\left(-3 \, \sqrt{2} \sqrt{m_{0}} {\tau} + 2 \, {\chi_{\rm s}}^{\frac{3}{2}}\right)}^{\frac{2}{3}} {\chi_{\rm s}}^{2}} \\ \mathrm{Ric}\left(g\right)_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & \frac{3 \cdot 2^{\frac{2}{3}} m_{0} {\chi}^{2}}{{\left(-3 \, \sqrt{2} \sqrt{m_{0}} {\tau} + 2 \, {\chi_{\rm s}}^{\frac{3}{2}}\right)}^{\frac{2}{3}} {\chi_{\rm s}}^{2}} \\ \mathrm{Ric}\left(g\right)_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & \frac{3 \cdot 2^{\frac{2}{3}} m_{0} {\chi}^{2} \sin\left({\theta}\right)^{2}}{{\left(-3 \, \sqrt{2} \sqrt{m_{0}} {\tau} + 2 \, {\chi_{\rm s}}^{\frac{3}{2}}\right)}^{\frac{2}{3}} {\chi_{\rm s}}^{2}} \end{array}
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Ric.display()
Ric(g)=(6m062m0τχs329m0τ22χs3)dτdτ+3223m0(32m0τ+2χs32)23χs2dχdχ+3223m0χ2(32m0τ+2χs32)23χs2dθdθ+3223m0χ2sin(θ)2(32m0τ+2χs32)23χs2dϕdϕ\mathrm{Ric}\left(g\right) = \left( -\frac{6 \, m_{0}}{6 \, \sqrt{2} \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} - 9 \, m_{0} {\tau}^{2} - 2 \, {\chi_{\rm s}}^{3}} \right) \mathrm{d} {\tau}\otimes \mathrm{d} {\tau} + \frac{3 \cdot 2^{\frac{2}{3}} m_{0}}{{\left(-3 \, \sqrt{2} \sqrt{m_{0}} {\tau} + 2 \, {\chi_{\rm s}}^{\frac{3}{2}}\right)}^{\frac{2}{3}} {\chi_{\rm s}}^{2}} \mathrm{d} {\chi}\otimes \mathrm{d} {\chi} + \frac{3 \cdot 2^{\frac{2}{3}} m_{0} {\chi}^{2}}{{\left(-3 \, \sqrt{2} \sqrt{m_{0}} {\tau} + 2 \, {\chi_{\rm s}}^{\frac{3}{2}}\right)}^{\frac{2}{3}} {\chi_{\rm s}}^{2}} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \frac{3 \cdot 2^{\frac{2}{3}} m_{0} {\chi}^{2} \sin\left({\theta}\right)^{2}}{{\left(-3 \, \sqrt{2} \sqrt{m_{0}} {\tau} + 2 \, {\chi_{\rm s}}^{\frac{3}{2}}\right)}^{\frac{2}{3}} {\chi_{\rm s}}^{2}} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}
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Ric[0,0]
6m062m0τχs329m0τ22χs3-\frac{6 \, m_{0}}{6 \, \sqrt{2} \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} - 9 \, m_{0} {\tau}^{2} - 2 \, {\chi_{\rm s}}^{3}}
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G = Ric - 1/2*g.ricci_scalar() * g G.set_name('G') print(G)
Field of symmetric bilinear forms G on the 4-dimensional differentiable manifold M
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G.display_comp()
Gττττ=12m062m0τχs329m0τ22χs3\begin{array}{lcl} G_{ \, {\tau} \, {\tau} }^{ \phantom{\, {\tau}}\phantom{\, {\tau}} } & = & -\frac{12 \, m_{0}}{6 \, \sqrt{2} \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} - 9 \, m_{0} {\tau}^{2} - 2 \, {\chi_{\rm s}}^{3}} \end{array}
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u = M.vector_field('u') u[0] = 1 u.display()
u=τu = \frac{\partial}{\partial {\tau} }
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g(u,u).display()
g(u,u):MR(τ,χ,θ,ϕ)1\begin{array}{llcl} g\left(u,u\right):& M & \longrightarrow & \mathbb{R} \\ & \left({\tau}, {\chi}, {\theta}, {\phi}\right) & \longmapsto & -1 \end{array}
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u_form = u.down(g) print(u_form)
1-form on the 4-dimensional differentiable manifold M
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u_form.display()
dτ-\mathrm{d} {\tau}
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rho = function('rho') T = rho(t,x)* (u_form * u_form) T.set_name('T') print(T)
Field of symmetric bilinear forms T on the 4-dimensional differentiable manifold M
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T.display()
T=ρ(τ,χ)dτdτT = \rho\left({\tau}, {\chi}\right) \mathrm{d} {\tau}\otimes \mathrm{d} {\tau}
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E = G - 8*pi*T E.set_name('E') print(E)
Field of symmetric bilinear forms E on the 4-dimensional differentiable manifold M
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E.display()
E=(4(122πm0τχs32ρ(τ,χ)18πm0τ2ρ(τ,χ)4πχs3ρ(τ,χ)+3m0)62m0τχs329m0τ22χs3)dτdτE = \left( -\frac{4 \, {\left(12 \, \sqrt{2} \pi \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} \rho\left({\tau}, {\chi}\right) - 18 \, \pi m_{0} {\tau}^{2} \rho\left({\tau}, {\chi}\right) - 4 \, \pi {\chi_{\rm s}}^{3} \rho\left({\tau}, {\chi}\right) + 3 \, m_{0}\right)}}{6 \, \sqrt{2} \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} - 9 \, m_{0} {\tau}^{2} - 2 \, {\chi_{\rm s}}^{3}} \right) \mathrm{d} {\tau}\otimes \mathrm{d} {\tau}
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E.display_comp()
Eττττ=4(122πm0τχs32ρ(τ,χ)18πm0τ2ρ(τ,χ)4πχs3ρ(τ,χ)+3m0)62m0τχs329m0τ22χs3\begin{array}{lcl} E_{ \, {\tau} \, {\tau} }^{ \phantom{\, {\tau}}\phantom{\, {\tau}} } & = & -\frac{4 \, {\left(12 \, \sqrt{2} \pi \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} \rho\left({\tau}, {\chi}\right) - 18 \, \pi m_{0} {\tau}^{2} \rho\left({\tau}, {\chi}\right) - 4 \, \pi {\chi_{\rm s}}^{3} \rho\left({\tau}, {\chi}\right) + 3 \, m_{0}\right)}}{6 \, \sqrt{2} \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} - 9 \, m_{0} {\tau}^{2} - 2 \, {\chi_{\rm s}}^{3}} \end{array}
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E[0,0]
4(122πm0τχs32ρ(τ,χ)18πm0τ2ρ(τ,χ)4πχs3ρ(τ,χ)+3m0)62m0τχs329m0τ22χs3-\frac{4 \, {\left(12 \, \sqrt{2} \pi \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} \rho\left({\tau}, {\chi}\right) - 18 \, \pi m_{0} {\tau}^{2} \rho\left({\tau}, {\chi}\right) - 4 \, \pi {\chi_{\rm s}}^{3} \rho\left({\tau}, {\chi}\right) + 3 \, m_{0}\right)}}{6 \, \sqrt{2} \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} - 9 \, m_{0} {\tau}^{2} - 2 \, {\chi_{\rm s}}^{3}}
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eq = (-E[0,0]/4).expr().numerator() == 0 eq
122πm0τχs32ρ(τ,χ)18πm0τ2ρ(τ,χ)4πχs3ρ(τ,χ)+3m0=012 \, \sqrt{2} \pi \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} \rho\left({\tau}, {\chi}\right) - 18 \, \pi m_{0} {\tau}^{2} \rho\left({\tau}, {\chi}\right) - 4 \, \pi {\chi_{\rm s}}^{3} \rho\left({\tau}, {\chi}\right) + 3 \, m_{0} = 0
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solve(eq, rho(t,x))
[ρ(τ,χ)=3m02(62πm0τχs329πm0τ22πχs3)]\left[\rho\left({\tau}, {\chi}\right) = -\frac{3 \, m_{0}}{2 \, {\left(6 \, \sqrt{2} \pi \sqrt{m_{0}} {\tau} {\chi_{\rm s}}^{\frac{3}{2}} - 9 \, \pi m_{0} {\tau}^{2} - 2 \, \pi {\chi_{\rm s}}^{3}\right)}}\right]
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