Contact Us!
CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutSign UpSign In

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.

| Download
Project: Lifshitz BH
Views: 126
Tweet
Share
Share

%typeset_mode True
M = Manifold(5, 'M')

print M
5-dimensional differentiable manifold M 
X.<v,x,y1,y2,r> = M.chart('v x y1:y_1 y2:y_2 r')
X
(M,(v,x,y1,y2,r))\displaystyle \left(M,(v, x, {y_1}, {y_2}, r)\right)


g = M.lorentzian_metric('g')
nu = var('nu', latex_name=r'\nu', domain='real')
f = function('f')(v, r)
g[0,0] = -exp(2*nu*r)*f
g[0,4] = exp(nu*r)
g[1,1] = exp(2*nu*r)
g[2,2] = exp(2*r)
g[3,3] = exp(2*r)
g.display()
g=−e(2 νr)f(v,r)dv⊗dv+e(νr)dv⊗dr+e(2 νr)dx⊗dx+e(2 r)dy1⊗dy1+e(2 r)dy2⊗dy2+e(νr)dr⊗dv\displaystyle g = -e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \mathrm{d} v\otimes \mathrm{d} v + e^{\left({{\nu}} r\right)} \mathrm{d} v\otimes \mathrm{d} r + e^{\left(2 \, {{\nu}} r\right)} \mathrm{d} x\otimes \mathrm{d} x + e^{\left(2 \, r\right)} \mathrm{d} {y_1}\otimes \mathrm{d} {y_1} + e^{\left(2 \, r\right)} \mathrm{d} {y_2}\otimes \mathrm{d} {y_2} + e^{\left({{\nu}} r\right)} \mathrm{d} r\otimes \mathrm{d} v
g[:]
(−e(2 νr)f(v,r)000e(νr)0e(2 νr)00000e(2 r)00000e(2 r)0e(νr)0000)\displaystyle \left(\begin{array}{rrrrr} -e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) & 0 & 0 & 0 & e^{\left({{\nu}} r\right)} \\ 0 & e^{\left(2 \, {{\nu}} r\right)} & 0 & 0 & 0 \\ 0 & 0 & e^{\left(2 \, r\right)} & 0 & 0 \\ 0 & 0 & 0 & e^{\left(2 \, r\right)} & 0 \\ e^{\left({{\nu}} r\right)} & 0 & 0 & 0 & 0 \end{array}\right)
Riem = g.riemann()
print Riem
Tensor field Riem(g) of type (1,3) on the 5-dimensional differentiable manifold M 
Riem[0,0,0,4]
−ν2e(νr)f(v,r)−32 νe(νr)∂ f∂r−12 e(νr)∂2 f∂r2\displaystyle -{{\nu}}^{2} e^{\left({{\nu}} r\right)} f\left(v, r\right) - \frac{3}{2} \, {{\nu}} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2}
Ric = g.ricci()
print Ric
Field of symmetric bilinear forms Ric(g) on the 5-dimensional differentiable manifold M 
Ric.display()
Ric(g)=(2 (ν2+ν)e(2 νr)f(v,r)2+(2 ν+1)e(2 νr)f(v,r)∂ f∂r−12 (ν+2)e(νr)∂ f∂v+12 e(2 νr)f(v,r)∂2 f∂r2)dv⊗dv+(−2 (ν2+ν)e(νr)f(v,r)−(2 ν+1)e(νr)∂ f∂r−12 e(νr)∂2 f∂r2)dv⊗dr+(−2 (ν2+ν)e(2 νr)f(v,r)−νe(2 νr)∂ f∂r)dx⊗dx+(−2 (ν+1)e(2 r)f(v,r)−e(2 r)∂ f∂r)dy1⊗dy1+(−2 (ν+1)e(2 r)f(v,r)−e(2 r)∂ f∂r)dy2⊗dy2+(−2 (ν2+ν)e(νr)f(v,r)−(2 ν+1)e(νr)∂ f∂r−12 e(νr)∂2 f∂r2)dr⊗dv+(2 ν−2)dr⊗dr\displaystyle \mathrm{Ric}\left(g\right) = \left( 2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right)^{2} + {\left(2 \, {{\nu}} + 1\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial\,f}{\partial r} - \frac{1}{2} \, {\left({{\nu}} + 2\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial v} + \frac{1}{2} \, e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial^2\,f}{\partial r^2} \right) \mathrm{d} v\otimes \mathrm{d} v + \left( -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left({{\nu}} r\right)} f\left(v, r\right) - {\left(2 \, {{\nu}} + 1\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2} \right) \mathrm{d} v\otimes \mathrm{d} r + \left( -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) - {{\nu}} e^{\left(2 \, {{\nu}} r\right)} \frac{\partial\,f}{\partial r} \right) \mathrm{d} x\otimes \mathrm{d} x + \left( -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r} \right) \mathrm{d} {y_1}\otimes \mathrm{d} {y_1} + \left( -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r} \right) \mathrm{d} {y_2}\otimes \mathrm{d} {y_2} + \left( -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left({{\nu}} r\right)} f\left(v, r\right) - {\left(2 \, {{\nu}} + 1\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2} \right) \mathrm{d} r\otimes \mathrm{d} v + \left( 2 \, {{\nu}} - 2 \right) \mathrm{d} r\otimes \mathrm{d} r
Ric.display_comp()
Ric(g) v v v v=2 (ν2+ν)e(2 νr)f(v,r)2+(2 ν+1)e(2 νr)f(v,r)∂ f∂r−12 (ν+2)e(νr)∂ f∂v+12 e(2 νr)f(v,r)∂2 f∂r2Ric(g) v r v r=−2 (ν2+ν)e(νr)f(v,r)−(2 ν+1)e(νr)∂ f∂r−12 e(νr)∂2 f∂r2Ric(g) x x x x=−2 (ν2+ν)e(2 νr)f(v,r)−νe(2 νr)∂ f∂rRic(g) y1 y1 y1 y1=−2 (ν+1)e(2 r)f(v,r)−e(2 r)∂ f∂rRic(g) y2 y2 y2 y2=−2 (ν+1)e(2 r)f(v,r)−e(2 r)∂ f∂rRic(g) r v r v=−2 (ν2+ν)e(νr)f(v,r)−(2 ν+1)e(νr)∂ f∂r−12 e(νr)∂2 f∂r2Ric(g) r r r r=2 ν−2\displaystyle \begin{array}{lcl} \mathrm{Ric}\left(g\right)_{ \, v \, v }^{ \phantom{\, v } \phantom{\, v } } & = & 2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right)^{2} + {\left(2 \, {{\nu}} + 1\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial\,f}{\partial r} - \frac{1}{2} \, {\left({{\nu}} + 2\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial v} + \frac{1}{2} \, e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial^2\,f}{\partial r^2} \\ \mathrm{Ric}\left(g\right)_{ \, v \, r }^{ \phantom{\, v } \phantom{\, r } } & = & -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left({{\nu}} r\right)} f\left(v, r\right) - {\left(2 \, {{\nu}} + 1\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2} \\ \mathrm{Ric}\left(g\right)_{ \, x \, x }^{ \phantom{\, x } \phantom{\, x } } & = & -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) - {{\nu}} e^{\left(2 \, {{\nu}} r\right)} \frac{\partial\,f}{\partial r} \\ \mathrm{Ric}\left(g\right)_{ \, {y_1} \, {y_1} }^{ \phantom{\, {y_1} } \phantom{\, {y_1} } } & = & -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r} \\ \mathrm{Ric}\left(g\right)_{ \, {y_2} \, {y_2} }^{ \phantom{\, {y_2} } \phantom{\, {y_2} } } & = & -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r} \\ \mathrm{Ric}\left(g\right)_{ \, r \, v }^{ \phantom{\, r } \phantom{\, v } } & = & -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left({{\nu}} r\right)} f\left(v, r\right) - {\left(2 \, {{\nu}} + 1\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2} \\ \mathrm{Ric}\left(g\right)_{ \, r \, r }^{ \phantom{\, r } \phantom{\, r } } & = & 2 \, {{\nu}} - 2 \end{array}
Ric[0,0]
2 (ν2+ν)e(2 νr)f(v,r)2+(2 ν+1)e(2 νr)f(v,r)∂ f∂r−12 (ν+2)e(νr)∂ f∂v+12 e(2 νr)f(v,r)∂2 f∂r2\displaystyle 2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right)^{2} + {\left(2 \, {{\nu}} + 1\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial\,f}{\partial r} - \frac{1}{2} \, {\left({{\nu}} + 2\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial v} + \frac{1}{2} \, e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial^2\,f}{\partial r^2}
Ric[0,4]
−2 (ν2+ν)e(νr)f(v,r)−(2 ν+1)e(νr)∂ f∂r−12 e(νr)∂2 f∂r2\displaystyle -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left({{\nu}} r\right)} f\left(v, r\right) - {\left(2 \, {{\nu}} + 1\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2}
Ric[1,1]
−2 (ν2+ν)e(2 νr)f(v,r)−νe(2 νr)∂ f∂r\displaystyle -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) - {{\nu}} e^{\left(2 \, {{\nu}} r\right)} \frac{\partial\,f}{\partial r}
Ric[2,2]
−2 (ν+1)e(2 r)f(v,r)−e(2 r)∂ f∂r\displaystyle -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r}
Ric[3,3]
−2 (ν+1)e(2 r)f(v,r)−e(2 r)∂ f∂r\displaystyle -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r}
Ric[4,4]
2 ν−2\displaystyle 2 \, {{\nu}} - 2
g.display_comp()
g v v v v=−e(2 νr)f(v,r)g v r v r=e(νr)g x x x x=e(2 νr)g y1 y1 y1 y1=e(2 r)g y2 y2 y2 y2=e(2 r)g r v r v=e(νr)\displaystyle \begin{array}{lcl} g_{ \, v \, v }^{ \phantom{\, v } \phantom{\, v } } & = & -e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \\ g_{ \, v \, r }^{ \phantom{\, v } \phantom{\, r } } & = & e^{\left({{\nu}} r\right)} \\ g_{ \, x \, x }^{ \phantom{\, x } \phantom{\, x } } & = & e^{\left(2 \, {{\nu}} r\right)} \\ g_{ \, {y_1} \, {y_1} }^{ \phantom{\, {y_1} } \phantom{\, {y_1} } } & = & e^{\left(2 \, r\right)} \\ g_{ \, {y_2} \, {y_2} }^{ \phantom{\, {y_2} } \phantom{\, {y_2} } } & = & e^{\left(2 \, r\right)} \\ g_{ \, r \, v }^{ \phantom{\, r } \phantom{\, v } } & = & e^{\left({{\nu}} r\right)} \end{array}




version()
SageMath Version 6.10, Release Date: 2015-12-18