Notebook

Killing-Yano form and Killing tensor in Kerr off shell¶

This SageMath notebook accompanies the article Black hole photon ring beyond General Relativity: an integrable parametrization by J. Ben Achour, E. Gourgoulhon and H. Roussille, arXiv:2506.09882. The notebook makes use of tools developed developed through the SageManifolds project.

In [1]:

sage.version.banner

Out[1]:

'SageMath version 10.6, Release Date: 2025-03-31'

In [2]:

%display latex

Spacetime manifold¶

In [3]:

M = Manifold(4, 'M', structure='Lorentzian')

In [4]:

X.<tau,r,y,phi> = M.chart(r'tau:\tau r y phi:(0,2*pi):\varphi')
X

Out[4]:

$\displaystyle \left(M,({\tau}, r, y, {\varphi})\right)$

In [5]:

dtau, dr, dy, dphi = X.coframe()[:]
dtau, dr, dy, dphi

Out[5]:

$\displaystyle \left(\mathrm{d} {\tau}, \mathrm{d} r, \mathrm{d} y, \mathrm{d} {\varphi}\right)$

Let us introduce two basic 1-forms:

In [6]:

om1 = dtau + y^2*dphi
om1.set_name('om1', latex_name=r'\omega_1')
om1.display()

Out[6]:

$\displaystyle \omega_1 = \mathrm{d} {\tau} + y^{2} \mathrm{d} {\varphi}$

In [7]:

om2 = dtau - r^2*dphi
om2.set_name('om2', latex_name=r'\omega_2')
om2.display()

Out[7]:

$\displaystyle \omega_2 = \mathrm{d} {\tau} -r^{2} \mathrm{d} {\varphi}$

Metric tensor¶

In [8]:

Sigma = r^2 + y^2
Dr = function('Dr', latex_name=r'\Delta_r')
Dy = function('Dy', latex_name=r'\Delta_y')

In [9]:

g = M.metric()
g.set( - Dr(r)/Sigma*(om1*om1)
       + Dy(y)/Sigma*(om2*om2) 
       + Sigma/Dr(r)*(dr*dr)
       + Sigma/Dy(y)*(dy*dy) )
g.display()

Out[9]:

$\displaystyle g = \left( -\frac{\Delta_r\left(r\right) - \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\tau}\otimes \mathrm{d} {\tau} + \left( -\frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\tau}\otimes \mathrm{d} {\varphi} + \left( \frac{r^{2} + y^{2}}{\Delta_r\left(r\right)} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{r^{2} + y^{2}}{\Delta_y\left(y\right)} \right) \mathrm{d} y\otimes \mathrm{d} y + \left( -\frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\varphi}\otimes \mathrm{d} {\tau} + \left( -\frac{y^{4} \Delta_r\left(r\right) - r^{4} \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

In [10]:

g.inverse().display()

Out[10]:

\(\displaystyle g^{-1} = \left( \frac{y^{4} \Delta_r\left(r\right) - r^{4} \Delta_y\left(y\right)}{{\left(r^{2} \Delta_r\left(r\right) + y^{2} \Delta_r\left(r\right)\right)} \Delta_y\left(y\right)} \right) \frac{\partial}{\partial {\tau} }\otimes \frac{\partial}{\partial {\tau} } + \left( -\frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{{\left(r^{2} \Delta_r\left(r\right) + y^{2} \Delta_r\left(r\right)\right)} \Delta_y\left(y\right)} \right) \frac{\partial}{\partial {\tau} }\otimes \frac{\partial}{\partial {\varphi} } + \left( \frac{\Delta_r\left(r\right)}{r^{2} + y^{2}} \right) \frac{\partial}{\partial r }\otimes \frac{\partial}{\partial r } + \left( \frac{\Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \frac{\partial}{\partial y }\otimes \frac{\partial}{\partial y } + \left( -\frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{{\left(r^{2} \Delta_r\left(r\right) + y^{2} \Delta_r\left(r\right)\right)} \Delta_y\left(y\right)} \right) \frac{\partial}{\partial {\varphi} }\otimes \frac{\partial}{\partial {\tau} } + \left( \frac{\Delta_r\left(r\right) - \Delta_y\left(y\right)}{{\left(r^{2} \Delta_r\left(r\right) + y^{2} \Delta_r\left(r\right)\right)} \Delta_y\left(y\right)} \right) \frac{\partial}{\partial {\varphi} }\otimes \frac{\partial}{\partial {\varphi} }\)

Levi-Civita connection¶

In [11]:

nabla = g.connection()
nabla.display(only_nonredundant=True)

Out[11]:

\(\displaystyle \begin{array}{lcl} \Gamma_{ \phantom{\, {\tau}} \, {\tau} \, r }^{ \, {\tau} \phantom{\, {\tau}} \phantom{\, r} } & = & \frac{r^{4} \frac{\partial\,\Delta_r}{\partial r} + r^{2} y^{2} \frac{\partial\,\Delta_r}{\partial r} - 2 \, r^{3} \Delta_r\left(r\right) + 2 \, r^{3} \Delta_y\left(y\right)}{2 \, {\left(r^{4} \Delta_r\left(r\right) + 2 \, r^{2} y^{2} \Delta_r\left(r\right) + y^{4} \Delta_r\left(r\right)\right)}} \\ \Gamma_{ \phantom{\, {\tau}} \, {\tau} \, y }^{ \, {\tau} \phantom{\, {\tau}} \phantom{\, y} } & = & \frac{2 \, y^{3} \Delta_r\left(r\right) - 2 \, y^{3} \Delta_y\left(y\right) + {\left(r^{2} y^{2} + y^{4}\right)} \frac{\partial\,\Delta_y}{\partial y}}{2 \, {\left(r^{4} + 2 \, r^{2} y^{2} + y^{4}\right)} \Delta_y\left(y\right)} \\ \Gamma_{ \phantom{\, {\tau}} \, r \, {\varphi} }^{ \, {\tau} \phantom{\, r} \phantom{\, {\varphi}} } & = & -\frac{2 \, r^{5} \Delta_y\left(y\right) - {\left(r^{2} \frac{\partial\,\Delta_r}{\partial r} - 2 \, r \Delta_r\left(r\right)\right)} y^{4} - {\left(r^{4} \frac{\partial\,\Delta_r}{\partial r} - 4 \, r^{3} \Delta_r\left(r\right)\right)} y^{2}}{2 \, {\left(r^{4} \Delta_r\left(r\right) + 2 \, r^{2} y^{2} \Delta_r\left(r\right) + y^{4} \Delta_r\left(r\right)\right)}} \\ \Gamma_{ \phantom{\, {\tau}} \, y \, {\varphi} }^{ \, {\tau} \phantom{\, y} \phantom{\, {\varphi}} } & = & \frac{2 \, y^{5} \Delta_r\left(r\right) + 2 \, {\left(r^{4} y + 2 \, r^{2} y^{3}\right)} \Delta_y\left(y\right) - {\left(r^{4} y^{2} + r^{2} y^{4}\right)} \frac{\partial\,\Delta_y}{\partial y}}{2 \, {\left(r^{4} + 2 \, r^{2} y^{2} + y^{4}\right)} \Delta_y\left(y\right)} \\ \Gamma_{ \phantom{\, r} \, {\tau} \, {\tau} }^{ \, r \phantom{\, {\tau}} \phantom{\, {\tau}} } & = & \frac{r^{2} \Delta_r\left(r\right) \frac{\partial\,\Delta_r}{\partial r} + y^{2} \Delta_r\left(r\right) \frac{\partial\,\Delta_r}{\partial r} - 2 \, r \Delta_r\left(r\right)^{2} + 2 \, r \Delta_r\left(r\right) \Delta_y\left(y\right)}{2 \, {\left(r^{6} + 3 \, r^{4} y^{2} + 3 \, r^{2} y^{4} + y^{6}\right)}} \\ \Gamma_{ \phantom{\, r} \, {\tau} \, {\varphi} }^{ \, r \phantom{\, {\tau}} \phantom{\, {\varphi}} } & = & \frac{y^{4} \Delta_r\left(r\right) \frac{\partial\,\Delta_r}{\partial r} + 2 \, r y^{2} \Delta_r\left(r\right) \Delta_y\left(y\right) + {\left(r^{2} \Delta_r\left(r\right) \frac{\partial\,\Delta_r}{\partial r} - 2 \, r \Delta_r\left(r\right)^{2}\right)} y^{2}}{2 \, {\left(r^{6} + 3 \, r^{4} y^{2} + 3 \, r^{2} y^{4} + y^{6}\right)}} \\ \Gamma_{ \phantom{\, r} \, r \, r }^{ \, r \phantom{\, r} \phantom{\, r} } & = & -\frac{r^{2} \frac{\partial\,\Delta_r}{\partial r} + y^{2} \frac{\partial\,\Delta_r}{\partial r} - 2 \, r \Delta_r\left(r\right)}{2 \, {\left(r^{2} \Delta_r\left(r\right) + y^{2} \Delta_r\left(r\right)\right)}} \\ \Gamma_{ \phantom{\, r} \, r \, y }^{ \, r \phantom{\, r} \phantom{\, y} } & = & \frac{y}{r^{2} + y^{2}} \\ \Gamma_{ \phantom{\, r} \, y \, y }^{ \, r \phantom{\, y} \phantom{\, y} } & = & -\frac{r \Delta_r\left(r\right)}{{\left(r^{2} + y^{2}\right)} \Delta_y\left(y\right)} \\ \Gamma_{ \phantom{\, r} \, {\varphi} \, {\varphi} }^{ \, r \phantom{\, {\varphi}} \phantom{\, {\varphi}} } & = & \frac{y^{6} \Delta_r\left(r\right) \frac{\partial\,\Delta_r}{\partial r} + {\left(r^{2} \Delta_r\left(r\right) \frac{\partial\,\Delta_r}{\partial r} - 2 \, r \Delta_r\left(r\right)^{2}\right)} y^{4} - 2 \, {\left(r^{5} \Delta_r\left(r\right) + 2 \, r^{3} y^{2} \Delta_r\left(r\right)\right)} \Delta_y\left(y\right)}{2 \, {\left(r^{6} + 3 \, r^{4} y^{2} + 3 \, r^{2} y^{4} + y^{6}\right)}} \\ \Gamma_{ \phantom{\, y} \, {\tau} \, {\tau} }^{ \, y \phantom{\, {\tau}} \phantom{\, {\tau}} } & = & -\frac{2 \, y \Delta_r\left(r\right) \Delta_y\left(y\right) - 2 \, y \Delta_y\left(y\right)^{2} + {\left(r^{2} + y^{2}\right)} \Delta_y\left(y\right) \frac{\partial\,\Delta_y}{\partial y}}{2 \, {\left(r^{6} + 3 \, r^{4} y^{2} + 3 \, r^{2} y^{4} + y^{6}\right)}} \\ \Gamma_{ \phantom{\, y} \, {\tau} \, {\varphi} }^{ \, y \phantom{\, {\tau}} \phantom{\, {\varphi}} } & = & \frac{2 \, r^{2} y \Delta_r\left(r\right) \Delta_y\left(y\right) - 2 \, r^{2} y \Delta_y\left(y\right)^{2} + {\left(r^{4} + r^{2} y^{2}\right)} \Delta_y\left(y\right) \frac{\partial\,\Delta_y}{\partial y}}{2 \, {\left(r^{6} + 3 \, r^{4} y^{2} + 3 \, r^{2} y^{4} + y^{6}\right)}} \\ \Gamma_{ \phantom{\, y} \, r \, r }^{ \, y \phantom{\, r} \phantom{\, r} } & = & -\frac{y \Delta_y\left(y\right)}{r^{2} \Delta_r\left(r\right) + y^{2} \Delta_r\left(r\right)} \\ \Gamma_{ \phantom{\, y} \, r \, y }^{ \, y \phantom{\, r} \phantom{\, y} } & = & \frac{r}{r^{2} + y^{2}} \\ \Gamma_{ \phantom{\, y} \, y \, y }^{ \, y \phantom{\, y} \phantom{\, y} } & = & \frac{2 \, y \Delta_y\left(y\right) - {\left(r^{2} + y^{2}\right)} \frac{\partial\,\Delta_y}{\partial y}}{2 \, {\left(r^{2} + y^{2}\right)} \Delta_y\left(y\right)} \\ \Gamma_{ \phantom{\, y} \, {\varphi} \, {\varphi} }^{ \, y \phantom{\, {\varphi}} \phantom{\, {\varphi}} } & = & \frac{2 \, r^{4} y \Delta_y\left(y\right)^{2} - {\left(r^{6} + r^{4} y^{2}\right)} \Delta_y\left(y\right) \frac{\partial\,\Delta_y}{\partial y} + 2 \, {\left(2 \, r^{2} y^{3} \Delta_r\left(r\right) + y^{5} \Delta_r\left(r\right)\right)} \Delta_y\left(y\right)}{2 \, {\left(r^{6} + 3 \, r^{4} y^{2} + 3 \, r^{2} y^{4} + y^{6}\right)}} \\ \Gamma_{ \phantom{\, {\varphi}} \, {\tau} \, r }^{ \, {\varphi} \phantom{\, {\tau}} \phantom{\, r} } & = & \frac{r^{2} \frac{\partial\,\Delta_r}{\partial r} + y^{2} \frac{\partial\,\Delta_r}{\partial r} - 2 \, r \Delta_r\left(r\right) + 2 \, r \Delta_y\left(y\right)}{2 \, {\left(r^{4} \Delta_r\left(r\right) + 2 \, r^{2} y^{2} \Delta_r\left(r\right) + y^{4} \Delta_r\left(r\right)\right)}} \\ \Gamma_{ \phantom{\, {\varphi}} \, {\tau} \, y }^{ \, {\varphi} \phantom{\, {\tau}} \phantom{\, y} } & = & -\frac{2 \, y \Delta_r\left(r\right) - 2 \, y \Delta_y\left(y\right) + {\left(r^{2} + y^{2}\right)} \frac{\partial\,\Delta_y}{\partial y}}{2 \, {\left(r^{4} + 2 \, r^{2} y^{2} + y^{4}\right)} \Delta_y\left(y\right)} \\ \Gamma_{ \phantom{\, {\varphi}} \, r \, {\varphi} }^{ \, {\varphi} \phantom{\, r} \phantom{\, {\varphi}} } & = & \frac{r^{2} y^{2} \frac{\partial\,\Delta_r}{\partial r} + y^{4} \frac{\partial\,\Delta_r}{\partial r} + 2 \, r^{3} \Delta_r\left(r\right) - 2 \, r^{3} \Delta_y\left(y\right)}{2 \, {\left(r^{4} \Delta_r\left(r\right) + 2 \, r^{2} y^{2} \Delta_r\left(r\right) + y^{4} \Delta_r\left(r\right)\right)}} \\ \Gamma_{ \phantom{\, {\varphi}} \, y \, {\varphi} }^{ \, {\varphi} \phantom{\, y} \phantom{\, {\varphi}} } & = & -\frac{2 \, y^{3} \Delta_r\left(r\right) - 2 \, y^{3} \Delta_y\left(y\right) - {\left(r^{4} + r^{2} y^{2}\right)} \frac{\partial\,\Delta_y}{\partial y}}{2 \, {\left(r^{4} + 2 \, r^{2} y^{2} + y^{4}\right)} \Delta_y\left(y\right)} \end{array}\)

Conformal Killing-Yano 2-form $p$¶

In [12]:

p = y*dy.wedge(om2) - r*dr.wedge(om1)
p.set_name('p')
p.display()

Out[12]:

$\displaystyle p = r \mathrm{d} {\tau}\wedge \mathrm{d} r -y \mathrm{d} {\tau}\wedge \mathrm{d} y -r y^{2} \mathrm{d} r\wedge \mathrm{d} {\varphi} -r^{2} y \mathrm{d} y\wedge \mathrm{d} {\varphi}$

$p$ is a closed 2-form:

In [13]:

diff(p).display()

Out[13]:

$\displaystyle \mathrm{d}p = 0$

The covariant derivative of $p$:

In [14]:

nab_p = nabla(p)
nab_p.display()

Out[14]:

\(\displaystyle \nabla_{g} p = \left( \frac{\Delta_r\left(r\right) - \Delta_y\left(y\right)}{\Delta_r\left(r\right)} \right) \mathrm{d} {\tau}\otimes \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{\Delta_r\left(r\right) - \Delta_y\left(y\right)}{\Delta_y\left(y\right)} \right) \mathrm{d} {\tau}\otimes \mathrm{d} y\otimes \mathrm{d} y + \Delta_r\left(r\right) \Delta_y\left(y\right) \mathrm{d} {\tau}\otimes \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi} + \left( -\frac{\Delta_r\left(r\right) - \Delta_y\left(y\right)}{\Delta_r\left(r\right)} \right) \mathrm{d} r\otimes \mathrm{d} {\tau}\otimes \mathrm{d} r + \left( -\frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{\Delta_r\left(r\right)} \right) \mathrm{d} r\otimes \mathrm{d} {\varphi}\otimes \mathrm{d} r + \left( -\frac{\Delta_r\left(r\right) - \Delta_y\left(y\right)}{\Delta_y\left(y\right)} \right) \mathrm{d} y\otimes \mathrm{d} {\tau}\otimes \mathrm{d} y + \left( -\frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{\Delta_y\left(y\right)} \right) \mathrm{d} y\otimes \mathrm{d} {\varphi}\otimes \mathrm{d} y -\Delta_r\left(r\right) \Delta_y\left(y\right) \mathrm{d} {\varphi}\otimes \mathrm{d} {\tau}\otimes \mathrm{d} {\varphi} + \left( \frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{\Delta_r\left(r\right)} \right) \mathrm{d} {\varphi}\otimes \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{\Delta_y\left(y\right)} \right) \mathrm{d} {\varphi}\otimes \mathrm{d} y\otimes \mathrm{d} y\)

The 1-form $h_a = \frac{1}{3} \nabla_b p^b_{\ \, a}$ (note that we taking the trace on the indices 0 and 2, i.e. first and last, because $\nabla_c p^b_{\ \, a} = (\nabla p^\sharp)^b_{\ \, ac}$):

In [15]:

h = 1/3*nab_p.up(g,0).trace(0,2)
h.set_name('h')
h.display()

Out[15]:

$\displaystyle h = \left( -\frac{\Delta_r\left(r\right) - \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\tau} + \left( -\frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\varphi}$

Let us check that $p$ satisfies the conformal Killing-Yano equation, taking into account that $\nabla_c p_{ab} =(\nabla p)_{abc}$,

In [16]:

nab_p == -2*(h*g).antisymmetrize(0,1)

Out[16]:

$\displaystyle \mathrm{True}$

Killing-Yano 2-form $f$¶

$f$ is obtained as the Hodge dual of $p$:

In [17]:

f = p.hodge_dual(g)
f.set_name('f')
f.display()

Out[17]:

$\displaystyle f = -y \mathrm{d} {\tau}\wedge \mathrm{d} r -r \mathrm{d} {\tau}\wedge \mathrm{d} y + y^{3} \mathrm{d} r\wedge \mathrm{d} {\varphi} -r^{3} \mathrm{d} y\wedge \mathrm{d} {\varphi}$

Check that $f$ obeys the Killing-Yano equation, i.e. $\nabla_{(a} f_{b)c} = 0$

In [18]:

nab_f = nabla(f)
nab_f.symmetrize(0,2).display()

Out[18]:

$\displaystyle 0$

Equivalently, we have $\nabla_{[a} f_{bc]} = \nabla_{a} f_{bc}$:

In [19]:

nab_f.antisymmetrize() == nab_f

Out[19]:

$\displaystyle \mathrm{True}$

Note that, contrary to $p$, $f$ is not a closed 2-form:

In [20]:

diff(f).display()

Out[20]:

$\displaystyle \mathrm{d}f = \left( -3 \, r^{2} - 3 \, y^{2} \right) \mathrm{d} r\wedge \mathrm{d} y\wedge \mathrm{d} {\varphi}$

Killing tensor $K$¶

We define $K$ as the square of $f$, i.e. $K_{ab} = f_{ac} f_b^{\ \, c}$:

In [21]:

K = f['_ac']*f.up(g,1)['_b^c']
K.set_name('K')
K.display()

Out[21]:

$\displaystyle K = \left( \frac{y^{2} \Delta_r\left(r\right) + r^{2} \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\tau}\otimes \mathrm{d} {\tau} + \left( \frac{y^{4} \Delta_r\left(r\right) - r^{4} \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\tau}\otimes \mathrm{d} {\varphi} + \left( -\frac{r^{2} y^{2} + y^{4}}{\Delta_r\left(r\right)} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{r^{4} + r^{2} y^{2}}{\Delta_y\left(y\right)} \right) \mathrm{d} y\otimes \mathrm{d} y + \left( \frac{y^{4} \Delta_r\left(r\right) - r^{4} \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\varphi}\otimes \mathrm{d} {\tau} + \left( \frac{y^{6} \Delta_r\left(r\right) + r^{6} \Delta_y\left(y\right)}{r^{2} + y^{2}} \right) \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

Check of Eq. (1.9) of the article:

In [22]:

K == (y^2/Sigma*Dr(r)*om1*om1 + r^2/Sigma*Dy(y)*om2*om2 + Sigma*r^2/Dy(y)*dy*dy 
      - Sigma*y^2/Dr(r)*dr*dr) 

Out[22]:

$\displaystyle \mathrm{True}$

Let us check that $K$ obeys the Killing equation, namely $$ \nabla_{(a} K_{bc)} = 0$$

In [23]:

nabla(K).symmetrize().display()

Out[23]:

$\displaystyle 0$