with sage manifold i tried the folowing commands (in a sage 9.1 notebook)

M = Manifold(4, 'M', structure='Lorentzian')
structure='Lorentzian')
  
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Mani.<t,x,rh,ph> Mani.<t,X,rh,ph> = M.chart(r"t X:(-oo,+oo) rh:(0,+oo):\rho ph:(0,2*pi):\phi")ph:(0,2*pi):\phi")
  
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B = function('B')(X,rh,ph)
d_Ph_X = function('d_Ph_X')(X,rh,ph)
d_Ph_rh = function('d_Ph_rh')(X,rh,ph)
d_Ph_ph = function('d_Ph_ph')(X,rh,ph)function('d_Ph_ph')(X,rh,ph)
  
--------
 
g = M.metric()M.metric()
  
g[0,0] = (-1+B2* (-1+B**2*  ((d_Ph_X/B2)2 +(d_Ph_rh/B2)2 +((1/rhd_Ph_ph/B2)*2 ((d_Ph_X/B**2)**2 +(d_Ph_rh/B**2)**2 +((1/rh*d_Ph_ph/B**2)**2  ))))))

g[1,1] = B**2
g[2,2] = B**2
g[3,3] = rh**2*B**2

g[0,1] = B**2*(d_Ph_X/B**2)
g[0,2] = B**2*( d_Ph_rh/B**2)
g[0,3] = B**2*( d_Ph_ph/B**2)

g.christoffel_symbols_display()

g[1,1] = B2 g[2,2] = B2 g[3,3] = rh2*B2

g[0,1] = B2(d_Ph_X/B2) g[0,2] = B2( d_Ph_rh/B2) g[0,3] = B2*( d_Ph_ph/B2)

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g.christoffel_symbols_display()

the The last line "g.christoffel_symbols_display()" makes the notebook overflow... i really don't understand why..why.

the computation computation, even if it is rather tedious tedious, seems straightforward to me...

Any idea ? idea? Any help would be greatly appreciated. Have a good day,appreciated.