Lifshitz-Vaidya-5D.sagews

%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

$\displaystyle \left(M,(v, x, {y_1}, {y_2}, r)\right)$

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()

$\displaystyle g = -e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \mathrm{d} v\otimes \mathrm{d} v + e^{\left({{\nu}} r\right)} \mathrm{d} v\otimes \mathrm{d} r + e^{\left(2 \, {{\nu}} r\right)} \mathrm{d} x\otimes \mathrm{d} x + e^{\left(2 \, r\right)} \mathrm{d} {y_1}\otimes \mathrm{d} {y_1} + e^{\left(2 \, r\right)} \mathrm{d} {y_2}\otimes \mathrm{d} {y_2} + e^{\left({{\nu}} r\right)} \mathrm{d} r\otimes \mathrm{d} v$

g[:]

$\displaystyle \left(\begin{array}{rrrrr} -e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) & 0 & 0 & 0 & e^{\left({{\nu}} r\right)} \\ 0 & e^{\left(2 \, {{\nu}} r\right)} & 0 & 0 & 0 \\ 0 & 0 & e^{\left(2 \, r\right)} & 0 & 0 \\ 0 & 0 & 0 & e^{\left(2 \, r\right)} & 0 \\ e^{\left({{\nu}} r\right)} & 0 & 0 & 0 & 0 \end{array}\right)$

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]

$\displaystyle -{{\nu}}^{2} e^{\left({{\nu}} r\right)} f\left(v, r\right) - \frac{3}{2} \, {{\nu}} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2}$

Ric = g.ricci()
print Ric

Field of symmetric bilinear forms Ric(g) on the 5-dimensional differentiable manifold M

Ric.display()

$\displaystyle \mathrm{Ric}\left(g\right) = \left( 2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right)^{2} + {\left(2 \, {{\nu}} + 1\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial\,f}{\partial r} - \frac{1}{2} \, {\left({{\nu}} + 2\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial v} + \frac{1}{2} \, e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial^2\,f}{\partial r^2} \right) \mathrm{d} v\otimes \mathrm{d} v + \left( -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left({{\nu}} r\right)} f\left(v, r\right) - {\left(2 \, {{\nu}} + 1\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2} \right) \mathrm{d} v\otimes \mathrm{d} r + \left( -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) - {{\nu}} e^{\left(2 \, {{\nu}} r\right)} \frac{\partial\,f}{\partial r} \right) \mathrm{d} x\otimes \mathrm{d} x + \left( -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r} \right) \mathrm{d} {y_1}\otimes \mathrm{d} {y_1} + \left( -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r} \right) \mathrm{d} {y_2}\otimes \mathrm{d} {y_2} + \left( -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left({{\nu}} r\right)} f\left(v, r\right) - {\left(2 \, {{\nu}} + 1\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2} \right) \mathrm{d} r\otimes \mathrm{d} v + \left( 2 \, {{\nu}} - 2 \right) \mathrm{d} r\otimes \mathrm{d} r$

Ric.display_comp()

$\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]

$\displaystyle 2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right)^{2} + {\left(2 \, {{\nu}} + 1\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial\,f}{\partial r} - \frac{1}{2} \, {\left({{\nu}} + 2\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial v} + \frac{1}{2} \, e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \frac{\partial^2\,f}{\partial r^2}$

Ric[0,4]

$\displaystyle -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left({{\nu}} r\right)} f\left(v, r\right) - {\left(2 \, {{\nu}} + 1\right)} e^{\left({{\nu}} r\right)} \frac{\partial\,f}{\partial r} - \frac{1}{2} \, e^{\left({{\nu}} r\right)} \frac{\partial^2\,f}{\partial r^2}$

Ric[1,1]

$\displaystyle -2 \, {\left({{\nu}}^{2} + {{\nu}}\right)} e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) - {{\nu}} e^{\left(2 \, {{\nu}} r\right)} \frac{\partial\,f}{\partial r}$

Ric[2,2]

$\displaystyle -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r}$

Ric[3,3]

$\displaystyle -2 \, {\left({{\nu}} + 1\right)} e^{\left(2 \, r\right)} f\left(v, r\right) - e^{\left(2 \, r\right)} \frac{\partial\,f}{\partial r}$

Ric[4,4]

$\displaystyle 2 \, {{\nu}} - 2$

g.display_comp()

$\displaystyle \begin{array}{lcl} g_{ \, v \, v }^{ \phantom{\, v } \phantom{\, v } } & = & -e^{\left(2 \, {{\nu}} r\right)} f\left(v, r\right) \\ g_{ \, v \, r }^{ \phantom{\, v } \phantom{\, r } } & = & e^{\left({{\nu}} r\right)} \\ g_{ \, x \, x }^{ \phantom{\, x } \phantom{\, x } } & = & e^{\left(2 \, {{\nu}} r\right)} \\ g_{ \, {y_1} \, {y_1} }^{ \phantom{\, {y_1} } \phantom{\, {y_1} } } & = & e^{\left(2 \, r\right)} \\ g_{ \, {y_2} \, {y_2} }^{ \phantom{\, {y_2} } \phantom{\, {y_2} } } & = & e^{\left(2 \, r\right)} \\ g_{ \, r \, v }^{ \phantom{\, r } \phantom{\, v } } & = & e^{\left({{\nu}} r\right)} \end{array}$

version()

SageMath Version 6.10, Release Date: 2015-12-18