# 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 project.

Author: Hans Fotsing Tetsing

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

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version()

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'SageMath version 9.1, Release Date: 2020-05-20'
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%display latex

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

4-dimensional Lorentzian manifold M


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

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

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

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

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g = M.metric()
print(g)

Lorentzian metric g on the 4-dimensional Lorentzian manifold M


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

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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()

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

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

degenerate hypersurface H embedded in 4-dimensional differentiable manifold M

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

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

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

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

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

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h = H.induced_metric()
print(h)

degenerate metric gamma on the degenerate hypersurface H embedded in 4-dimensional differentiable manifold M

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h.display()

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

Out:

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 $H$.

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H.set_transverse(rigging=v)

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xi = M.vector_field()
xi =-1
xi.display()

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

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e1 = M.vector_field()
e2 = M.vector_field()
e1 = 1
e2 = 1
e1.display(), e2.display()

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

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

screen distribution S along the degenerate hypersurface H embedded in 4-dimensional differentiable manifold M mapped into the 4-dimensional Lorentzian manifold M


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

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T = H.adapted_frame(); T

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W = H.weingarten_map()
print(W)

Tensor field nabla_g(xi)|X(H) of type (1,1) along the degenerate hypersurface H embedded in 4-dimensional differentiable manifold M with values on the 4-dimensional Lorentzian manifold M

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W.display(T)

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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.

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SO = H.shape_operator()
print(SO)

Tensor field A^* of type (1,1) along the degenerate hypersurface H embedded in 4-dimensional differentiable manifold M with values on the 4-dimensional Lorentzian manifold M

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SO.display()

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