SharedExtension of the metric - black hole.ipynbOpen in CoCalc
%display latex

Spacetime: black hole extension

M = Manifold(4, 'M', r'\mathcal{M}')
print M
4-dimensional differentiable manifold M
X.<v,t,th,ph> = M.chart(R' v:(-oo,+oo) t:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
X
(M,(v,t,θ,ϕ))\left(\mathcal{M},(v, t, {\theta}, {\phi})\right)

We introduce the new coordinate vv such that dv=dr+dtFdv=dr+\frac{dt}{F}. The metric thus takes the following form:

F=function('F')(t)
g = M.lorentzian_metric('g')
g[0,0] = F
g[1,1] = 0
g[1,0] = -1
g[2,2] = t^2
g[3,3] = (t*sin(th))^2
g.display()
g=F(t)dvdvdvdtdtdv+t2dθdθ+t2sin(θ)2dϕdϕg = F\left(t\right) \mathrm{d} v\otimes \mathrm{d} v -\mathrm{d} v\otimes \mathrm{d} t -\mathrm{d} t\otimes \mathrm{d} v + t^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + t^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}
gm1=g.inverse();
gm1.display()
g1=vttvF(t)tt+1t2θθ+1t2sin(θ)2ϕϕg^{-1} = -\frac{\partial}{\partial v }\otimes \frac{\partial}{\partial t }-\frac{\partial}{\partial t }\otimes \frac{\partial}{\partial v } -F\left(t\right) \frac{\partial}{\partial t }\otimes \frac{\partial}{\partial t } + \frac{1}{t^{2}} \frac{\partial}{\partial {\theta} }\otimes \frac{\partial}{\partial {\theta} } + \frac{1}{t^{2} \sin\left({\theta}\right)^{2}} \frac{\partial}{\partial {\phi} }\otimes \frac{\partial}{\partial {\phi} }

No coordinate singularity appears for F(t)=0F(t)=0 in the metric and inverse metric. It allows to cross both the outer and inner horizons, if they exist.

Radial null vectors

Outgoing null vector

k = M.vector_field(name='k')
k[0] = 1
k[1] = F/2 # outgoing since F/2>0 at infinity
k.display()
k=v+12F(t)tk = \frac{\partial}{\partial v } + \frac{1}{2} \, F\left(t\right) \frac{\partial}{\partial t }

Ingoing null vector

l = M.vector_field(name='l')
l[1] = -2
l.display()
l=2tl = -2 \frac{\partial}{\partial t }
kd=k.down(g)
ld=l.down(g)

Check that kk is a null vector

kk=kd['_a']*k['^a']
kk.display()
MR(v,t,θ,ϕ)0\begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}

Check that ll is a null vector

ll=ld['_a']*l['^a']
ll.display()
0:MR(v,t,θ,ϕ)0\begin{array}{llcl} 0:& \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}

Normalization

lk=ld['_a']*k['^a']
lk.display()
MR(v,t,θ,ϕ)2\begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & 2 \end{array}

Induced metric

h=g-1/2*(kd*ld+ld*kd)
hu=h.up(g)
h.display()
t2dθdθ+t2sin(θ)2dϕdϕt^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + t^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}

Expansion of a congruence of null geodesics

nab = g.connection() ; print nab
Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional differentiable manifold M

Check that the covariant derivative does not act on the metric:

nab(g).display()
gg=0\nabla_{g} g = 0
theta_outgoing=hu['^{ab}']*nab(kd)['_{ab}']
print theta_outgoing
Scalar field on the 4-dimensional differentiable manifold M
theta_outgoing.display()
MR(v,t,θ,ϕ)F(t)t\begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & \frac{F\left(t\right)}{t} \end{array}

The zeros of θoutgoing\theta_{outgoing}, hence of FF, correspond to the presence of horizons. The region with F(t)<0F(t)<0, where the expansion of outgoing null rays becomes negative, is a (future) trapped region.

theta_ingoing=theta=hu['^{ab}']*nab(ld)['_{ab}']
print theta_ingoing
Scalar field on the 4-dimensional differentiable manifold M
theta_ingoing.display()
MR(v,t,θ,ϕ)4t\begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & -\frac{4}{t} \end{array}

The expansion of ingoing radial null rays is strictly negative, which is expected in the presence of a black hole\textbf{black hole}.

Ricci's scalar

Rscal = g.ricci_scalar().expr()
Rscal.factor()
t22(t)2F(t)+4ttF(t)+2F(t)+2t2\frac{t^{2} \frac{\partial^{2}}{(\partial t)^{2}}F\left(t\right) + 4 \, t \frac{\partial}{\partial t}F\left(t\right) + 2 \, F\left(t\right) + 2}{t^{2}}

Kretschmann's scalar

R = g.riemann() ; print(R)
Tensor field Riem(g) of type (1,3) on the 4-dimensional differentiable manifold M
uR=R.up(g)
dR=R.down(g)
Kr_scalar = uR['^{abcd}']*dR['_{abcd}']
K=Kr_scalar.expr().factor()
K
t42(t)2F(t)2+4t2tF(t)2+4F(t)2+8F(t)+4t4\frac{t^{4} \frac{\partial^{2}}{(\partial t)^{2}}F\left(t\right)^{2} + 4 \, t^{2} \frac{\partial}{\partial t}F\left(t\right)^{2} + 4 \, F\left(t\right)^{2} + 8 \, F\left(t\right) + 4}{t^{4}}