Notebook

Walker-Penrose Killing tensor in Kerr spacetime¶

This notebook demonstrates a few capabilities of SageMath in computations regarding Kerr spacetime. More precisely, it focuses on the Killing tensor $K$ found by Walker & Penrose [Commun. Math. Phys. 18, 265 (1970)]. This notebook makes use of SageMath 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 8.3, Release Date: 2018-08-03'

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

In [2]:

%display latex

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

In [3]:

Parallelism().set(nproc=8)

Spacetime manifold¶

We declare the Kerr spacetime (or more precisely the Boyer-Lindquist domain of Kerr spacetime) as a 4-dimensional Lorentzian manifold:

In [4]:

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
print(M)

4-dimensional Lorentzian manifold M

Let us declare the Boyer-Lindquist coordinates via the method chart(), the argument of which is a string expressing the coordinates names, their ranges (the default is $(-\infty,+\infty)$) and their LaTeX symbols:

In [5]:

BL.<t,r,th,ph> = M.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') 
print(BL) ; BL

Chart (M, (t, r, th, ph))

Out[5]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\mathcal{M},(t, r, {\theta}, {\phi})\right)$

In [6]:

BL[0], BL[1]

Out[6]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(t, r\right)$

Metric tensor

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

In [7]:

var('m, a', domain='real')

Out[7]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(m, a\right)$

We get the (yet undefined) spacetime metric by

In [8]:

g = M.metric()

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

In [9]:

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()

Out[9]:

$\newcommand{\Bold}[1]{\mathbf{#1}}g = \left( \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( a^{2} \cos\left({\theta}\right)^{2} + r^{2} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + {\left(\frac{2 \, a^{2} m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

A matrix view of the components with respect to the manifold's default vector frame:

In [10]:

g[:]

Out[10]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 & 0 & 0 & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ 0 & \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} & 0 & 0 \\ 0 & 0 & a^{2} \cos\left({\theta}\right)^{2} + r^{2} & 0 \\ -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 & 0 & {\left(\frac{2 \, a^{2} m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \end{array}\right)$

The list of the non-vanishing components:

In [11]:

g.display_comp()

Out[11]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} g_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \\ g_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} \\ g_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & a^{2} \cos\left({\theta}\right)^{2} + r^{2} \\ g_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & {\left(\frac{2 \, a^{2} m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \end{array}$

Levi-Civita Connection

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

In [12]:

nabla = g.connection() ; print(nabla)

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

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

In [13]:

nabla(g).display()

Out[13]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\nabla_{g} g = 0$

Killing vectors

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

In [14]:

M.default_frame() is BL.frame()

Out[14]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

In [15]:

BL.frame()

Out[15]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\mathcal{M}, \left(\frac{\partial}{\partial t },\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)$

Let us consider the first vector field of this frame:

In [16]:

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

Out[16]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\partial}{\partial t }$

In [17]:

print(xi)

Vector field d/dt on the 4-dimensional Lorentzian manifold M

The 1-form associated to it by metric duality is

In [18]:

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

Out[18]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left( -\frac{a^{2} \cos\left({\theta}\right)^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}$

Its covariant derivative is

In [19]:

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

Tensor field of type (0,2) on the 4-dimensional Lorentzian manifold M

Out[19]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left( \frac{a^{2} m \cos\left({\theta}\right)^{2} - m r^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} t\otimes \mathrm{d} r + \left( \frac{2 \, a^{2} m r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} t\otimes \mathrm{d} {\theta} + \left( -\frac{a^{2} m \cos\left({\theta}\right)^{2} - m r^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} r\otimes \mathrm{d} t + \left( \frac{{\left(a^{3} m \cos\left({\theta}\right)^{2} - a m r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} r\otimes \mathrm{d} {\phi} + \left( -\frac{2 \, a^{2} m r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} t + \left( \frac{2 \, {\left(a^{3} m r + a m r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\phi} + \left( -\frac{{\left(a^{3} m \cos\left({\theta}\right)^{2} - a m r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} r + \left( -\frac{2 \, {\left(a^{3} m r + a m r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}$

Let us check that the Killing equation is satisfied:

In [20]:

nab_xi.symmetrize() == 0

Out[20]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

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

In [21]:

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

Out[21]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\partial}{\partial {\phi} }$

In [22]:

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

Out[22]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

Principal null vectors¶

We introduce the principal null vectors $k$ and $\ell$ of Kerr spacetime (see e.g. Eqs. (12.3.5) and (12.3.6) of Wald's textbook General Relativity (1984)):

In [23]:

k = M.vector_field(name='k')
k[:] = [(r^2+a^2)/(2*rho2), -Delta/(2*rho2), 0, a/(2*rho2)]
k.display()

Out[23]:

$\newcommand{\Bold}[1]{\mathbf{#1}}k = \left( \frac{a^{2} + r^{2}}{2 \, {\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)}} \right) \frac{\partial}{\partial t } + \left( -\frac{a^{2} - 2 \, m r + r^{2}}{2 \, {\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)}} \right) \frac{\partial}{\partial r } + \frac{a}{2 \, {\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)}} \frac{\partial}{\partial {\phi} }$

In [24]:

el = M.vector_field(name='el', latex_name=r'\ell')
el[:] = [(r^2+a^2)/Delta, 1, 0, a/Delta]
el.display()

Out[24]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\ell = \left( \frac{a^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} \right) \frac{\partial}{\partial t } +\frac{\partial}{\partial r } + \left( \frac{a}{a^{2} - 2 \, m r + r^{2}} \right) \frac{\partial}{\partial {\phi} }$

Let us check that $k$ and $\ell$ are null vectors:

In [25]:

g(k,k).expr()

Out[25]:

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

In [26]:

g(el,el).expr()

Out[26]:

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

Their scalar product is $-1$:

In [27]:

g(k,el).expr()

Out[27]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-1$

Note that the scalar product (with respect to metric $g$) can also be computed by means of the method dot:

In [28]:

k.dot(el).expr()

Out[28]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-1$

Let us evaluate the "acceleration" of $k$, i.e. $\nabla_k k$:

In [29]:

acc_k = nabla(k).contract(k)
acc_k.display()

Out[29]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left( \frac{a^{4} m - m r^{4} - {\left(a^{4} m - a^{4} r + a^{2} m r^{2} - a^{2} r^{3}\right)} \sin\left({\theta}\right)^{2}}{2 \, {\left(a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}\right)}} \right) \frac{\partial}{\partial t } + \left( -\frac{a^{4} m - 2 \, a^{2} m^{2} r + 2 \, m^{2} r^{3} - m r^{4} - {\left(a^{4} m + 3 \, a^{2} m r^{2} - a^{2} r^{3} - {\left(a^{4} + 2 \, a^{2} m^{2}\right)} r\right)} \sin\left({\theta}\right)^{2}}{2 \, {\left(a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}\right)}} \right) \frac{\partial}{\partial r } + \left( \frac{a^{3} r - a m r^{2} + {\left(a^{3} m - a^{3} r\right)} \cos\left({\theta}\right)^{2}}{2 \, {\left(a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}\right)}} \right) \frac{\partial}{\partial {\phi} }$

We check that $k$ is a pregeodesic vector, i.e. that $\nabla_k k = \kappa_k k$ for some scalar field $\kappa_k$:

In [30]:

for i in [0,1,3]:
    show(acc_k[i] / k[i])

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{a^{2} m - m r^{2} - {\left(a^{2} m - a^{2} r\right)} \sin\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}}$

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{a^{2} r - m r^{2} + {\left(a^{2} m - a^{2} r\right)} \cos\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}}$

In [31]:

kappa_k = acc_k[[0]] / k[[0]]
kappa_k.display()

Out[31]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{a^{2} m - m r^{2} - {\left(a^{2} m - a^{2} r\right)} \sin\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \end{array}$

In [32]:

acc_k == kappa_k * k

Out[32]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

Similarly let us evaluate the "acceleration" of $\ell$:

In [33]:

acc_l = nabla(el).contract(el)
acc_l.display()

Out[33]:

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

Hence $\ell$ is a geodesic vector.

The check that $k$ and $\ell$ do define (repeated) principal null directions is performed in this notebook.

Walker-Penrose Killing tensor¶

We need the 1-forms associated to $k$ and $\ell$ by metric duality:

In [34]:

uk = k.down(g)
ul = el.down(g)

The Walker-Penrose Killing tensor $K$ is then formed as $$ K = \rho^2 (\underline{\ell}\otimes \underline{k} + (\underline{k}\otimes \underline{\ell}) + r^2 g $$

In [35]:

K = rho2*(ul*uk+ uk*ul) + r^2*g
K.set_name('K')
print(K)

Tensor field K of type (0,2) on the 4-dimensional Lorentzian manifold M

In [36]:

K.display_comp()

Out[36]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} K_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{a^{2} r^{2} + {\left(a^{4} - 2 \, a^{2} m r\right)} \cos\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ K_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & -\frac{2 \, a^{3} m r \sin\left({\theta}\right)^{4} - {\left(2 \, a^{3} m r - a^{3} r^{2} - a r^{4} - {\left(a^{5} + a^{3} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ K_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & -\frac{a^{4} \cos\left({\theta}\right)^{4} + a^{2} r^{2} \cos\left({\theta}\right)^{2}}{a^{2} - 2 \, m r + r^{2}} \\ K_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4} \\ K_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & -\frac{2 \, a^{3} m r \sin\left({\theta}\right)^{4} - {\left(2 \, a^{3} m r - a^{3} r^{2} - a r^{4} - {\left(a^{5} + a^{3} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ K_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & \frac{{\left(a^{8} - 2 \, a^{6} m r + a^{6} r^{2}\right)} \sin\left({\theta}\right)^{8} - {\left(2 \, a^{8} - 4 \, a^{6} m r + 3 \, a^{6} r^{2} - 2 \, a^{4} m r^{3} + a^{4} r^{4}\right)} \sin\left({\theta}\right)^{6} + {\left(a^{8} - 2 \, a^{6} m r + a^{6} r^{2} - 2 \, a^{4} m r^{3} - a^{4} r^{4} - a^{2} r^{6}\right)} \sin\left({\theta}\right)^{4} + {\left(a^{6} r^{2} + 3 \, a^{4} r^{4} + 3 \, a^{2} r^{6} + r^{8}\right)} \sin\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \end{array}$

In [37]:

DK = nabla(K)
print(DK)

Tensor field nabla_g(K) of type (0,3) on the 4-dimensional Lorentzian manifold M

In [38]:

DK.display_comp()

Out[38]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \nabla_{g} K_{ \, t \, r \, {\phi} }^{ \phantom{\, t}\phantom{\, r}\phantom{\, {\phi}} } & = & a r \sin\left({\theta}\right)^{2} \\ \nabla_{g} K_{ \, t \, {\theta} \, {\phi} }^{ \phantom{\, t}\phantom{\, {\theta}}\phantom{\, {\phi}} } & = & {\left(a^{3} - 2 \, a m r + a r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right) \\ \nabla_{g} K_{ \, t \, {\phi} \, r }^{ \phantom{\, t}\phantom{\, {\phi}}\phantom{\, r} } & = & -a r \sin\left({\theta}\right)^{2} \\ \nabla_{g} K_{ \, t \, {\phi} \, {\theta} }^{ \phantom{\, t}\phantom{\, {\phi}}\phantom{\, {\theta}} } & = & -{\left(a^{3} - 2 \, a m r + a r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right) \\ \nabla_{g} K_{ \, r \, t \, {\phi} }^{ \phantom{\, r}\phantom{\, t}\phantom{\, {\phi}} } & = & -a r \cos\left({\theta}\right)^{2} + a r \\ \nabla_{g} K_{ \, r \, r \, {\theta} }^{ \phantom{\, r}\phantom{\, r}\phantom{\, {\theta}} } & = & \frac{2 \, {\left(a^{4} \cos\left({\theta}\right)^{3} + a^{2} r^{2} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{a^{2} - 2 \, m r + r^{2}} \\ \nabla_{g} K_{ \, r \, {\theta} \, r }^{ \phantom{\, r}\phantom{\, {\theta}}\phantom{\, r} } & = & -\frac{{\left(a^{4} \cos\left({\theta}\right)^{3} + a^{2} r^{2} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{a^{2} - 2 \, m r + r^{2}} \\ \nabla_{g} K_{ \, r \, {\theta} \, {\theta} }^{ \phantom{\, r}\phantom{\, {\theta}}\phantom{\, {\theta}} } & = & -a^{2} r \cos\left({\theta}\right)^{2} - r^{3} \\ \nabla_{g} K_{ \, r \, {\phi} \, {\phi} }^{ \phantom{\, r}\phantom{\, {\phi}}\phantom{\, {\phi}} } & = & -a^{2} r - r^{3} + {\left(a^{2} r + r^{3}\right)} \cos\left({\theta}\right)^{2} \\ \nabla_{g} K_{ \, {\theta} \, t \, {\phi} }^{ \phantom{\, {\theta}}\phantom{\, t}\phantom{\, {\phi}} } & = & {\left(a^{3} - 2 \, a m r + a r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right) \\ \nabla_{g} K_{ \, {\theta} \, r \, r }^{ \phantom{\, {\theta}}\phantom{\, r}\phantom{\, r} } & = & -\frac{{\left(a^{4} \cos\left({\theta}\right)^{3} + a^{2} r^{2} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{a^{2} - 2 \, m r + r^{2}} \\ \nabla_{g} K_{ \, {\theta} \, r \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, r}\phantom{\, {\theta}} } & = & -a^{2} r \cos\left({\theta}\right)^{2} - r^{3} \\ \nabla_{g} K_{ \, {\theta} \, {\theta} \, r }^{ \phantom{\, {\theta}}\phantom{\, {\theta}}\phantom{\, r} } & = & 2 \, a^{2} r \cos\left({\theta}\right)^{2} + 2 \, r^{3} \\ \nabla_{g} K_{ \, {\theta} \, {\phi} \, {\phi} }^{ \phantom{\, {\theta}}\phantom{\, {\phi}}\phantom{\, {\phi}} } & = & -{\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{3} \\ \nabla_{g} K_{ \, {\phi} \, t \, r }^{ \phantom{\, {\phi}}\phantom{\, t}\phantom{\, r} } & = & -a r \sin\left({\theta}\right)^{2} \\ \nabla_{g} K_{ \, {\phi} \, t \, {\theta} }^{ \phantom{\, {\phi}}\phantom{\, t}\phantom{\, {\theta}} } & = & -{\left(a^{3} - 2 \, a m r + a r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right) \\ \nabla_{g} K_{ \, {\phi} \, r \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, r}\phantom{\, {\phi}} } & = & -{\left(a^{2} r + r^{3}\right)} \sin\left({\theta}\right)^{2} \\ \nabla_{g} K_{ \, {\phi} \, {\theta} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\theta}}\phantom{\, {\phi}} } & = & -{\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{3} \\ \nabla_{g} K_{ \, {\phi} \, {\phi} \, r }^{ \phantom{\, {\phi}}\phantom{\, {\phi}}\phantom{\, r} } & = & 2 \, {\left(a^{2} r + r^{3}\right)} \sin\left({\theta}\right)^{2} \\ \nabla_{g} K_{ \, {\phi} \, {\phi} \, {\theta} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}}\phantom{\, {\theta}} } & = & 2 \, {\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{3} \end{array}$

Let us check that $K$ is a Killing tensor:

In [39]:

DK.symmetrize().display()

Out[39]:

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

Equivalently, we may write, using index notation:

In [40]:

DK['_(abc)'].display()

Out[40]:

$\newcommand{\Bold}[1]{\mathbf{#1}}0$