Oppenheimer_Snyder_hist.ipynb

Kernel: SageMath 7.5.beta3

%display latex

M = Manifold(4, 'M')
print(M)

4-dimensional differentiable manifold M

X.<t,x,th,ph> = M.chart(r't:\tau x:(0,+oo):\chi th:(0,pi):\theta ph:(0,2*pi):\phi')
X

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.display_comp()

Ric = g.ricci()
print(Ric)

Field of symmetric bilinear forms Ric(g) on the 4-dimensional differentiable manifold M

Ric.display_comp()

Ric.display()

Ric[0,0]

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

G.display_comp()

u = M.vector_field('u')
u[0] = 1
u.display()

g(u,u).display()

u_form = u.down(g)
print(u_form)

1-form on the 4-dimensional differentiable manifold M

u_form.display()

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

T.display()

E = G - 8*pi*T
E.set_name('E')
print(E)

Field of symmetric bilinear forms E on the 4-dimensional differentiable manifold M

E.display()

E.display_comp()

E[0,0]

eq = (-E[0,0]/4).expr().numerator() == 0
eq

solve(eq, rho(t,x))