Checking that Kerr metric is a solution of Einstein's equation

This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.

The computations make use of tools developed through the SageManifolds project.

NB: a version of SageMath at least equal to 8.2 is required to run this notebook:

In [1]:
version()
Out[1]:
'SageMath version 9.5.rc2, Release Date: 2022-01-16'

First we set up the notebook to display mathematical objects using LaTeX formatting:

In [2]:
%display latex

To speed up computations, we ask for running them in parallel on 8 threads:

In [3]:
Parallelism().set(nproc=8)

Spacetime

We declare the spacetime manifold $M$:

In [4]:
M = Manifold(4, 'M', structure='Lorentzian')
print(M)
4-dimensional Lorentzian manifold M

and the Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ as a chart on $M$:

In [5]:
XBL.<t,r,th,ph> = M.chart(r't r th:(0,pi):\theta ph:(0,2*pi):\phi')
XBL
Out[5]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(M,(t, r, {\theta}, {\phi})\right)\]
In [6]:
XBL.coord_range()
Out[6]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}t :\ \left( -\infty, +\infty \right) ;\quad r :\ \left( -\infty, +\infty \right) ;\quad {\theta} :\ \left( 0 , \pi \right) ;\quad {\phi} :\ \left( 0 , 2 \, \pi \right)\]

Kerr metric

We define the metric $g$ by its components w.r.t. the Boyer-Lindquist coordinates:

In [7]:
g = M.metric()
m, a = var('m a')
rho2 = r^2 + (a*cos(th))^2
Delta = r^2 - 2*m*r + a^2
g[0,0] = -(1 - 2*m*r/rho2)
g[0,3] = -2*a*m*r*sin(th)^2/rho2
g[1,1] = rho2/Delta
g[2,2] = rho2
g[3,3] = (r^2 + a^2 + 2*m*r*(a*sin(th))^2/rho2)*sin(th)^2
g.display()
Out[7]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}g = \left( \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( a^{2} \cos\left({\theta}\right)^{2} + r^{2} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + {\left(\frac{2 \, a^{2} m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\]
In [8]:
g.display_comp()
Out[8]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} g_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \\ g_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} \\ g_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & a^{2} \cos\left({\theta}\right)^{2} + r^{2} \\ g_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & {\left(\frac{2 \, a^{2} m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \end{array}\]

The inverse metric:

In [9]:
g.inverse()[:]
Out[9]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -\frac{2 \, a^{2} m r \sin\left({\theta}\right)^{2} + a^{2} r^{2} + r^{4} + {\left(a^{4} + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{2} - 2 \, m r^{3} + r^{4} + {\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}} & 0 & 0 & -\frac{2 \, a m r}{a^{2} r^{2} - 2 \, m r^{3} + r^{4} + {\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}} \\ 0 & \frac{a^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 & 0 \\ 0 & 0 & \frac{1}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 \\ -\frac{2 \, a m r}{a^{2} r^{2} - 2 \, m r^{3} + r^{4} + {\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}} & 0 & 0 & \frac{a^{2} \cos\left({\theta}\right)^{2} - 2 \, m r + r^{2}}{2 \, a^{2} m r \sin\left({\theta}\right)^{4} - {\left(2 \, a^{2} m r - a^{2} r^{2} + 2 \, m r^{3} - r^{4} - {\left(a^{4} + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}} \end{array}\right)\]

The Christoffel symbols:

In [10]:
g.christoffel_symbols_display()
Out[10]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \Gamma_{ \phantom{\, t} \, t \, r }^{ \, t \phantom{\, t} \phantom{\, r} } & = & -\frac{a^{4} m - m r^{4} - {\left(a^{4} m + a^{2} m r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} r^{4} - 2 \, m r^{5} + r^{6} + {\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, t} \, t \, {\theta} }^{ \, t \phantom{\, t} \phantom{\, {\theta}} } & = & -\frac{2 \, a^{2} m r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \\ \Gamma_{ \phantom{\, t} \, r \, {\phi} }^{ \, t \phantom{\, r} \phantom{\, {\phi}} } & = & -\frac{{\left(a^{3} m r^{2} + 3 \, a m r^{4} - {\left(a^{5} m - a^{3} m r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} r^{4} - 2 \, m r^{5} + r^{6} + {\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, t} \, {\theta} \, {\phi} }^{ \, t \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & -\frac{2 \, {\left(a^{5} m r \cos\left({\theta}\right) \sin\left({\theta}\right)^{5} - {\left(a^{5} m r + a^{3} m r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{3}\right)}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, r} \, t \, t }^{ \, r \phantom{\, t} \phantom{\, t} } & = & \frac{a^{2} m r^{2} - 2 \, m^{2} r^{3} + m r^{4} - {\left(a^{4} m - 2 \, a^{2} m^{2} r + a^{2} m r^{2}\right)} \cos\left({\theta}\right)^{2}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, r} \, t \, {\phi} }^{ \, r \phantom{\, t} \phantom{\, {\phi}} } & = & -\frac{{\left(a^{3} m r^{2} - 2 \, a m^{2} r^{3} + a m r^{4} - {\left(a^{5} m - 2 \, a^{3} m^{2} r + a^{3} m r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, r} \, r \, r }^{ \, r \phantom{\, r} \phantom{\, r} } & = & \frac{a^{2} r - m r^{2} + {\left(a^{2} m - a^{2} r\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{2} - 2 \, m r^{3} + r^{4} + {\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, r} \, r \, {\theta} }^{ \, r \phantom{\, r} \phantom{\, {\theta}} } & = & -\frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, r} \, {\theta} \, {\theta} }^{ \, r \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{a^{2} r - 2 \, m r^{2} + r^{3}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, r} \, {\phi} \, {\phi} }^{ \, r \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & \frac{{\left(a^{4} m r^{2} - 2 \, a^{2} m^{2} r^{3} + a^{2} m r^{4} - {\left(a^{6} m - 2 \, a^{4} m^{2} r + a^{4} m r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{2} r^{5} - 2 \, m r^{6} + r^{7} + {\left(a^{6} r - 2 \, a^{4} m r^{2} + a^{4} r^{3}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{3} - 2 \, a^{2} m r^{4} + a^{2} r^{5}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\theta}} \, t \, t }^{ \, {\theta} \phantom{\, t} \phantom{\, t} } & = & -\frac{2 \, a^{2} m r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\theta}} \, t \, {\phi} }^{ \, {\theta} \phantom{\, t} \phantom{\, {\phi}} } & = & \frac{2 \, {\left(a^{3} m r + a m r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, r }^{ \, {\theta} \phantom{\, r} \phantom{\, r} } & = & \frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} r^{2} - 2 \, m r^{3} + r^{4} + {\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, {\theta} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\theta}} } & = & \frac{r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, {\theta} \, {\theta} }^{ \, {\theta} \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, {\phi} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\frac{{\left({\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{5} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{3} + {\left(2 \, a^{4} m r + 4 \, a^{2} m r^{3} + a^{2} r^{4} + r^{6}\right)} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\phi}} \, t \, r }^{ \, {\phi} \phantom{\, t} \phantom{\, r} } & = & -\frac{a^{3} m \cos\left({\theta}\right)^{2} - a m r^{2}}{a^{2} r^{4} - 2 \, m r^{5} + r^{6} + {\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, t \, {\theta} }^{ \, {\phi} \phantom{\, t} \phantom{\, {\theta}} } & = & -\frac{2 \, a m r \cos\left({\theta}\right)}{{\left(a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}\right)} \sin\left({\theta}\right)} \\ \Gamma_{ \phantom{\, {\phi}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } & = & -\frac{a^{2} m r^{2} + 2 \, m r^{4} - r^{5} + {\left(a^{4} m - a^{4} r\right)} \cos\left({\theta}\right)^{4} - {\left(a^{4} m - a^{2} m r^{2} + 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{4} - 2 \, m r^{5} + r^{6} + {\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{a^{4} \cos\left({\theta}\right) \sin\left({\theta}\right)^{4} - 2 \, {\left(a^{4} - a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{2} + {\left(a^{4} + 2 \, a^{2} r^{2} + r^{4}\right)} \cos\left({\theta}\right)}{{\left(a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}\right)} \sin\left({\theta}\right)} \end{array}\]

Einstein's equation

Let us check that the Ricci tensor of $g$ vanishes identically, which is equivalent to Einstein's equation in vacuum:

In [11]:
g.ricci().display()
Out[11]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{Ric}\left(g\right) = 0\]

On the contrary, the Riemann tensor is not zero:

In [12]:
R = g.riemann()
print(R)
Tensor field Riem(g) of type (1,3) on the 4-dimensional Lorentzian manifold M
In [13]:
R[0,1,2,3]
Out[13]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left({\left(a^{7} m - 2 \, a^{5} m^{2} r + a^{5} m r^{2}\right)} \cos\left({\theta}\right)^{5} - {\left(3 \, a^{7} m - 2 \, a^{5} m^{2} r + 8 \, a^{5} m r^{2} - 6 \, a^{3} m^{2} r^{3} + 5 \, a^{3} m r^{4}\right)} \cos\left({\theta}\right)^{3} + 3 \, {\left(3 \, a^{5} m r^{2} - 2 \, a^{3} m^{2} r^{3} + 5 \, a^{3} m r^{4} + 2 \, a m r^{6}\right)} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{a^{2} r^{6} - 2 \, m r^{7} + r^{8} + {\left(a^{8} - 2 \, a^{6} m r + a^{6} r^{2}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{6} r^{2} - 2 \, a^{4} m r^{3} + a^{4} r^{4}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{4} r^{4} - 2 \, a^{2} m r^{5} + a^{2} r^{6}\right)} \cos\left({\theta}\right)^{2}}\]

The Kretschmann scalar

The Kretschmann scalar is the following square of the Riemann tensor: $$ K = R_{abcd} R^{abcd} $$ We compute first the tensors $R_{abcd}$ and $R^{abcd}$ by respectively lowering and raising the indices of $R$ with the metric $g$:

In [14]:
dR = R.down(g)
print(dR)
Tensor field of type (0,4) on the 4-dimensional Lorentzian manifold M
In [15]:
uR = R.up(g)
print(uR)
Tensor field of type (4,0) on the 4-dimensional Lorentzian manifold M

Then we perform the contraction:

In [16]:
K = dR['_{abcd}']*uR['^{abcd}']
print(K)
K.display()
Scalar field on the 4-dimensional Lorentzian manifold M
Out[16]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{48 \, {\left(a^{6} m^{2} \cos\left({\theta}\right)^{6} - 15 \, a^{4} m^{2} r^{2} \cos\left({\theta}\right)^{4} + 15 \, a^{2} m^{2} r^{4} \cos\left({\theta}\right)^{2} - m^{2} r^{6}\right)}}{a^{12} \cos\left({\theta}\right)^{12} + 6 \, a^{10} r^{2} \cos\left({\theta}\right)^{10} + 15 \, a^{8} r^{4} \cos\left({\theta}\right)^{8} + 20 \, a^{6} r^{6} \cos\left({\theta}\right)^{6} + 15 \, a^{4} r^{8} \cos\left({\theta}\right)^{4} + 6 \, a^{2} r^{10} \cos\left({\theta}\right)^{2} + r^{12}} \end{array}\]

A variant of this expression can be obtained by invoking the factor() method on the coordinate function representing the scalar field in the manifold's default chart:

In [17]:
Kr = K.expr().factor()
Kr
Out[17]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{48 \, {\left(a^{2} \cos\left({\theta}\right)^{2} + 4 \, a r \cos\left({\theta}\right) + r^{2}\right)} {\left(a^{2} \cos\left({\theta}\right)^{2} - 4 \, a r \cos\left({\theta}\right) + r^{2}\right)} {\left(a \cos\left({\theta}\right) + r\right)} {\left(a \cos\left({\theta}\right) - r\right)} m^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)}^{6}}\]

The Schwarzschild value of the Kretschmann scalar is recovered for $a=0$

In [18]:
Kr.subs(a=0)
Out[18]:
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{48 \, m^{2}}{r^{6}}\]

For a more detailed Kerr notebook (Killing vectors, Bianchi identity, etc.) see here.

In [ ]: