Kerr Spacetime.ipynb

Kernel: SageMath 8.2

%display latex

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian'); M

var('a', domain='real')
assume(a > 0)
assume(a < 1)

BL.<t,r,th,ph> = M.chart(r't r:(1+sqrt(1-a^2),+oo) th:(0,pi):\theta ph:(0,2*pi):\phi'); BL

g = M.metric()

Sigma = r^2 + (a*cos(th))^2
Delta = r^2 - 2*r + a^2
g[0,0] = -(1-2*r/Sigma)
g[0,3] = -2*a*r*sin(th)^2/Sigma
g[1,1] = Sigma/Delta
g[2,2] = Sigma
g[3,3] = (r^2 + a^2 + 2*r*(a*sin(th))^2/Sigma)*sin(th)^2
g.display()

g.christoffel_symbols_display()