SM_Kerr.sagews

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Kerr spacetime

This worksheet demonstrates a few capabilities of SageManifolds (version 0.8) in computations regarding Kerr spacetime.

It is released under the GNU General Public License version 3.

(c) Eric Gourgoulhon, Michal Bejger (2015)

The corresponding worksheet file can be downloaded from here.

 

Spacetime manifold

We can then declare the Kerr spacetime as a 4-dimensional diffentiable manifold:

M = Manifold(4, 'M', r'\mathcal{M}')

Let us use the standard Boyer-Lindquist coordinates on it, by first introducing the part $\mathcal{M}_0$ covered by these coordinates

M0 = M.open_subset('M0', r'\mathcal{M}_0')
# BL = Boyer-Lindquist
BL.<t,r,th,ph> = M0.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') 
print BL ; BL

chart (M0, (t, r, th, ph))

$\left(\mathcal{M}_0,(t, r, {\theta}, {\phi})\right)$

BL[0], BL[1]

($t$, $r$)

Metric tensor

The 2 parameters $m$ and $a$ of the Kerr spacetime are declared as symbolic variables:

var('m, a')

($m$, $a$)

Let us introduce the spacetime metric:

g = M.lorentz_metric('g')

The metric is set by its components in the coordinate frame associated with Boyer-Lindquist coordinates, which is the current manifold's default frame:

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], g[2,2] = rho2/Delta, rho2
g[3,3] = (r^2+a^2+2*m*r*(a*sin(th))^2/rho2)*sin(th)^2
g.display()

$g = \left( -\frac{a^{2} \cos\left({\theta}\right)^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( a^{2} \cos\left({\theta}\right)^{2} + r^{2} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + \left( \frac{2 \, a^{2} m r \sin\left({\theta}\right)^{4} + {\left(a^{2} r^{2} + r^{4} + {\left(a^{4} + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

g[:]  # matrix view of all the components in the manifold's default vector frame

$\left(\begin{array}{rrrr} -\frac{a^{2} \cos\left({\theta}\right)^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 & 0 & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ 0 & \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} & 0 & 0 \\ 0 & 0 & a^{2} \cos\left({\theta}\right)^{2} + r^{2} & 0 \\ -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 & 0 & \frac{2 \, a^{2} m r \sin\left({\theta}\right)^{4} + {\left(a^{2} r^{2} + r^{4} + {\left(a^{4} + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \end{array}\right)$

g.display_comp()

$\begin{array}{lcl} g_{ \, t \, t }^{ \phantom{\, t } \phantom{\, t } } & = & -\frac{a^{2} \cos\left({\theta}\right)^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, t \, {\phi} }^{ \phantom{\, t } \phantom{\, {\phi} } } & = & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, r \, r }^{ \phantom{\, r } \phantom{\, r } } & = & \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} \\ g_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta} } \phantom{\, {\theta} } } & = & a^{2} \cos\left({\theta}\right)^{2} + r^{2} \\ g_{ \, {\phi} \, t }^{ \phantom{\, {\phi} } \phantom{\, t } } & = & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi} } \phantom{\, {\phi} } } & = & \frac{2 \, a^{2} m r \sin\left({\theta}\right)^{4} + {\left(a^{2} r^{2} + r^{4} + {\left(a^{4} + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \end{array}$

Levi-Civita Connection

The Levi-Civita connection $\nabla$ associated with $g$:

nab = g.connection() ; print nab

Levi-Civita connection 'nabla_g' associated with the Lorentzian metric 'g' on the 4-dimensional manifold 'M'

Let us verify that the covariant derivative of $g$ with respect to $\nabla$ vanishes identically:

nab(g) == 0

$\mathrm{True}$

nab(g).display() # another view of the above property

$\nabla_{g} g = 0$

The nonzero Christoffel symbols (skipping those that can be deduced by symmetry of the last two indices):

g.christoffel_symbols_display()

$\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} m - m r^{2} - {\left(a^{2} m - a^{2} r\right)} \sin\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)^{5} - 2 \, {\left(a^{2} m r - a^{2} r^{2}\right)} \cos\left({\theta}\right)^{3} + {\left(2 \, a^{2} m r + 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}$

Killing vectors

The default vector frame on the spacetime manifold is the coordinate basis associated with Boyer-Lindquist coordinates:

M.default_frame() is BL.frame()

$\mathrm{True}$

BL.frame()

$\left(\mathcal{M}_0 ,\left(\frac{\partial}{\partial t },\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)$

Let us consider the first vector field of this frame:

xi = BL.frame()[0] ; xi

$\frac{\partial}{\partial t }$

print xi

vector field 'd/dt' on the open subset 'M0' of the 4-dimensional manifold 'M'

The 1-form associated to it by metric duality is

xi_form = xi.down(g) ; xi_form.display()

$\left( -\frac{a^{2} \cos\left({\theta}\right)^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}$

Its covariant derivative is

nab_xi = nab(xi_form) ; print nab_xi ; nab_xi.display()

tensor field of type (0,2) on the open subset 'M0' of the 4-dimensional manifold 'M'

$\left( \frac{a^{2} m \cos\left({\theta}\right)^{2} - m r^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} t\otimes \mathrm{d} r + \left( \frac{2 \, a^{2} m r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} t\otimes \mathrm{d} {\theta} + \left( -\frac{a^{2} m \cos\left({\theta}\right)^{2} - m r^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} r\otimes \mathrm{d} t + \left( \frac{{\left(a^{3} m \cos\left({\theta}\right)^{2} - a m r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} r\otimes \mathrm{d} {\phi} + \left( -\frac{2 \, a^{2} m r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} t + \left( \frac{2 \, {\left(a^{3} m r + a m r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\phi} + \left( -\frac{{\left(a^{3} m \cos\left({\theta}\right)^{2} - a m r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} r + \left( -\frac{2 \, {\left(a^{3} m r + a m r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}$

Let us check that the Killing equation is satisfied:

nab_xi.symmetrize() == 0

$\mathrm{True}$

Similarly, let us check that $\frac{\partial}{\partial\phi}$ is a Killing vector:

chi = BL.frame()[3] ; chi

$\frac{\partial}{\partial {\phi} }$

nab(chi.down(g)).symmetrize() == 0

$\mathrm{True}$

Curvature

The Ricci tensor associated with $g$:

Ric = g.ricci() ; print Ric

field of symmetric bilinear forms 'Ric(g)' on the 4-dimensional manifold 'M'

Let us check that Kerr metric is a solution of the vacuum Einstein equation:

Ric == 0

$\mathrm{True}$

Ric.display() # another view of the above property

$\mathrm{Ric}\left(g\right) = 0$

The Riemann curvature tensor associated with $g$:

R = g.riemann() ; print R

tensor field 'Riem(g)' of type (1,3) on the 4-dimensional manifold 'M'

Contrary to the Ricci tensor, the Riemann tensor does not vanish; for instance, the component $R^0_{\ \, 123}$ is

R[0,1,2,3]

$-\frac{{\left(a^{7} m - 2 \, a^{5} m^{2} r + a^{5} m r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{5} + {\left(a^{7} m + 2 \, a^{5} m^{2} r + 6 \, a^{5} m r^{2} - 6 \, a^{3} m^{2} r^{3} + 5 \, a^{3} m r^{4}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{3} - 2 \, {\left(a^{7} m - a^{5} m r^{2} - 5 \, a^{3} m r^{4} - 3 \, a m r^{6}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} r^{6} - 2 \, m r^{7} + r^{8} + {\left(a^{8} - 2 \, a^{6} m r + a^{6} r^{2}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{6} r^{2} - 2 \, a^{4} m r^{3} + a^{4} r^{4}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{4} r^{4} - 2 \, a^{2} m r^{5} + a^{2} r^{6}\right)} \cos\left({\theta}\right)^{2}}$

Bianchi identity

Let us check the Bianchi identity $\nabla_p R^i_{\ \, j kl} + \nabla_k R^i_{\ \, jlp} + \nabla_l R^i_{\ \, jpk} = 0$:

DR = nab(R) ; print DR  #long (takes a while)

for i in M.irange():
    for j in M.irange():
        for k in M.irange():
            for l in M.irange():
                for p in M.irange():
                    print DR[i,j,k,l,p] + DR[i,j,l,p,k] + DR[i,j,p,k,l] ,

If the last sign in the Bianchi identity is changed to minus, the identity does no longer hold:

DR[0,1,2,3,1] + DR[0,1,3,1,2] + DR[0,1,1,2,3] # should be zero (Bianchi identity)

DR[0,1,2,3,1] + DR[0,1,3,1,2] - DR[0,1,1,2,3] # note the change of the second + to -

Kretschmann scalar

The tensor $R^\flat$, of components $R_{ijkl} = g_{ip} R^p_{\ \, jkl}$:

dR = R.down(g) ; print dR

The tensor $R^\sharp$, of components $R^{ijkl} = g^{jp} g^{kq} g^{lr} R^i_{\ \, pqr}$:

uR = R.up(g) ; print uR

The Kretschmann scalar $K := R^{ijkl} R_{ijkl}$:

Kr_scalar = uR['^{ijkl}']*dR['_{ijkl}']
Kr_scalar.display()

A variant of this expression can be obtained by invoking the factor() method:

Kr = Kr_scalar.function_chart()  # the coordinate function representing the scalar field in the default chart
Kr.factor()

As a check, we can compare Kr to the formula given by R. Conn Henry, Astrophys. J. 535, 350 (2000):

Kr == 48*m^2*(r^6 - 15*r^4*(a*cos(th))^2 + 15*r^2*(a*cos(th))^4 - (a*cos(th))^6) / (r^2+(a*cos(th))^2)^6

The Schwarzschild value of the Kretschmann scalar is recovered by setting $a=0$:

Kr.expr().subs(a=0)

K1 = Kr.expr().subs(m=1, a=0.9)

plot3d(K1, (r,1,3), (th, 0, pi))