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

$\left(\mathcal{M},(v, t, {\theta}, {\phi})\right)$

We introduce the new coordinate $v$ such that $dv=dr+\frac{dt}{F}$. 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()

$g = F\left(t\right) \mathrm{d} v\otimes \mathrm{d} v -\mathrm{d} v\otimes \mathrm{d} t -\mathrm{d} t\otimes \mathrm{d} v + t^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + t^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$
gm1=g.inverse();
gm1.display()

$g^{-1} = -\frac{\partial}{\partial v }\otimes \frac{\partial}{\partial t }-\frac{\partial}{\partial t }\otimes \frac{\partial}{\partial v } -F\left(t\right) \frac{\partial}{\partial t }\otimes \frac{\partial}{\partial t } + \frac{1}{t^{2}} \frac{\partial}{\partial {\theta} }\otimes \frac{\partial}{\partial {\theta} } + \frac{1}{t^{2} \sin\left({\theta}\right)^{2}} \frac{\partial}{\partial {\phi} }\otimes \frac{\partial}{\partial {\phi} }$

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

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()

$k = \frac{\partial}{\partial v } + \frac{1}{2} \, F\left(t\right) \frac{\partial}{\partial t }$

Ingoing null vector

l = M.vector_field(name='l')
l[1] = -2
l.display()

$l = -2 \frac{\partial}{\partial t }$
kd=k.down(g)
ld=l.down(g)


Check that $k$ is a null vector

kk=kd['_a']*k['^a']
kk.display()

$\begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}$

Check that $l$ is a null vector

ll=ld['_a']*l['^a']
ll.display()

$\begin{array}{llcl} 0:& \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}$

Normalization

lk=ld['_a']*k['^a']
lk.display()

$\begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & 2 \end{array}$

# Induced metric¶

h=g-1/2*(kd*ld+ld*kd)

hu=h.up(g)

h.display()

$t^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + t^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

# 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()

$\nabla_{g} g = 0$
theta_outgoing=hu['^{ab}']*nab(kd)['_{ab}']
print theta_outgoing

Scalar field on the 4-dimensional differentiable manifold M
theta_outgoing.display()

$\begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & \frac{F\left(t\right)}{t} \end{array}$

The zeros of $\theta_{outgoing}$, hence of $F$, correspond to the presence of horizons. The region with $F(t)<0$, 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()

$\begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(v, t, {\theta}, {\phi}\right) & \longmapsto & -\frac{4}{t} \end{array}$

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

# Ricci's scalar¶

Rscal = g.ricci_scalar().expr()
Rscal.factor()

$\frac{t^{2} \frac{\partial^{2}}{(\partial t)^{2}}F\left(t\right) + 4 \, t \frac{\partial}{\partial t}F\left(t\right) + 2 \, F\left(t\right) + 2}{t^{2}}$

# 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

$\frac{t^{4} \frac{\partial^{2}}{(\partial t)^{2}}F\left(t\right)^{2} + 4 \, t^{2} \frac{\partial}{\partial t}F\left(t\right)^{2} + 4 \, F\left(t\right)^{2} + 8 \, F\left(t\right) + 4}{t^{4}}$