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

# # The event horizon of Schwarzschild black hole as a degenerate submanifold

# This Jupyter notebook illustrates SageMath functionalities regarding smooth manifolds equipped with a degenerate metric on the concrete example of the event horizon of Schwarzschild black hole. The involved tools have been developed through the [SageManifolds](https://sagemanifolds.obspm.fr) project. 
# 
# *Author:* **Hans Fotsing Tetsing**

# A version of SageMath at least equal to 9.1 is required to run this notebook

# In[1]:


version()


# In[2]:


get_ipython().run_line_magic('display', 'latex')


# In[3]:


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


# We define now on $M$ the ingoing Eddington-Finkelstein coordinates

# In[4]:


X_M.<t, r, th, ph> = M.chart(r"t r:(0,oo) th:(0,pi):\theta ph:(0,2*pi):\phi") 


# In[5]:


var('m'); assume(m>0)


# In[6]:


Int = M.open_subset('Int', coord_def={X_M: r<2*m})
Ext = M.open_subset('Ext', coord_def={X_M: r>2*m})


# In[7]:


g = M.metric()
print(g)


# Below we set the component of $g$ to obtain the Schwarzschild metric in the ingoing Eddington-Finkelstein coordinates

# In[8]:


g[0,0], g[0,1], g[1,1], g[2,2], g[3,3] = -1+2*m/r, 2*m/r, 1+2*m/r, r^2, r^2*sin(th)^2
g.display()


# $M$ is now the Schwarzschild black hole. Its event horizon is the null hypersurface $H$ defined by $r=2m$.

# In[9]:


H = Manifold(3, 'H', ambient=M, structure='degenerate_metric')
print(H)


# In[10]:


X_H.<ht,hth,hph> = H.chart(r"ht:(-oo,oo):t hth:(0,pi):\theta hph:(0,2*pi):\phi")


# In[11]:


Phi = H.diff_map(M, {(X_H, X_M): [ht, 2*m, hth, hph]}, 
                 name='Phi', latex_name=r'\Phi')
Phi.display()


# In[12]:


Phi_inv = M.diff_map(H, {(X_M, X_H): [t, th, ph]}, 
                     name='Phi_inv', latex_name=r'\Phi^{-1}')
Phi_inv.display()


# In[13]:


phi_inv_t = M.scalar_field({X_M:r-2*m})


# In[14]:


H.set_immersion(Phi, inverse=Phi_inv)
H.declare_embedding()


# In[15]:


h = H.induced_metric()
print(h)


# In[16]:


h.display()


# In[17]:


v = M.vector_field()
v[:2] = [r,-r]
v.display()


# This vector field $v$ is everywhere neither tangent nor orthogonal to $H$ and then can be used as rigging to study the extrinsic geometry of $H$.

# In[18]:


H.set_transverse(rigging=v)


# In[19]:


xi = M.vector_field()
xi[0] =-1
xi.display()


# $\xi$ defines the normal tangent direction along $H$.

# In[20]:


e1 = M.vector_field()
e2 = M.vector_field()
e1[2] = 1
e2[3] = 1
e1.display(), e2.display()


# $S=Span(e1,e2)$ is a screen distribution for $H$, thus a complementary of $TH^\perp$ in $TH$.

# In[21]:


S = H.screen('S', [e1, e2], (xi))
print(S)


# The intergral surfaces of this screen distribution are the spheres $\{t=cste,r=2m\}$.

# In[22]:


T = H.adapted_frame(); T


# In[23]:


W = H.weingarten_map()
print(W)


# In[24]:


W.display(T)


# The Weingarten map of this event horizon does not vanish but is along the orthogonal tangent vector $\xi$. 
# 
# The Weingarten map can be decomposed as $$\nabla_U\xi=-A^\ast(U)-\tau(U)\xi,~~~~\forall U\in\mathfrak X(H).$$
# $A^\ast$ is the shape operator and $\tau$ the roration $1-$form. So, the hypersurface is totally geodesic and the rotation $1-$form $\tau$ vanishes on the screen distribution and $\tau(\xi)=\frac1{4m}$. We can confirm that $H$ is totally geodesic by computing the shape operator.

# In[25]:


SO = H.shape_operator()
print(SO)


# In[26]:


SO.display()