#!/usr/bin/env python
# coding: utf-8

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

# This [SageMath](https://www.sagemath.org/) 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](https://arxiv.org/abs/2506.09882). The notebook makes use of tools developed 
# developed through the [SageManifolds project](https://sagemanifolds.obspm.fr).

# In[1]:


sage.version.banner


# In[2]:


get_ipython().run_line_magic('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


# In[5]:


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


# 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()


# In[7]:


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


# ## 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()


# In[10]:


g.inverse().display()


# ## Levi-Civita connection

# In[11]:


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


# ## Conformal Killing-Yano 2-form $p$

# In[12]:


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


# $p$ is a closed 2-form:

# In[13]:


diff(p).display()


# The covariant derivative of $p$:

# In[14]:


nab_p = nabla(p)
nab_p.display()


# 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()


# 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)


# ## 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()


# 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()


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

# In[19]:


nab_f.antisymmetrize() == nab_f


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

# In[20]:


diff(f).display()


# ## 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()


# 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) 


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

# In[23]:


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