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Peeling at an infinity obtained from a generic spherically symmetric degenerate horizon through an extended Couch-Torrence inversion

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Kernel: SageMath 9.8

Peeling at an infinity obtained from generic spherically symmetric degenerate horizon through an extended Couch-Torrence inversion

This SageMath notebook accompanies the article Peeling at extreme black hole horizons by Jack Borthwick, Eric Gourgoulhon and Jean-Philippe Nicolas, arXIv:2303.14574. It involves differential geometry tools implemented in SageMath through the SageManifolds project. 

version()
'SageMath version 9.8, Release Date: 2023-02-11'
%display latex

We declare the spacetime manifold M\mathscr{M} as a 4-dimensional Lorentzian manifold, with the keyword signature='negative'to indicate that the metric signature is chosen to be (+,−,−,−)(+,-,-,-):

M = Manifold(4, 'M', latex_name=r'\mathscr{M}', structure='Lorentzian', 
             signature='negative')
print(M)
M
4-dimensional Lorentzian manifold M

M\displaystyle \mathscr{M}

The coordinates (u,R,θ,φ)(u,R,\theta,\varphi) on M\mathscr{M}:

CKC.<u,R,th,sph> = M.chart(r"u R th:(0,pi):\theta sph:(0,2*pi):\varphi") 
print(CKC); CKC
Chart (M, (u, R, th, sph))

(M,(u,R,θ,φ))\displaystyle \left(\mathscr{M},(u, R, {\theta}, {\varphi})\right)

The conformal metric gg (denoted g^\hat{g} in the article, cf. Eq. (5.7)):

g = M.metric()
f = function('f',nargs=1)
g[0,0] = f(R)*R^2
g[0,1] = -1
g[2,2] = -1
g[3,3] = -sin(th)^2
g.display()

g=R2f(R)du⊗du−du⊗dR−dR⊗du−dθ⊗dθ−sin⁡(θ)2dφ⊗dφ\displaystyle g = R^{2} f\left(R\right) \mathrm{d} u\otimes \mathrm{d} u -\mathrm{d} u\otimes \mathrm{d} R -\mathrm{d} R\otimes \mathrm{d} u -\mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -\sin\left({\theta}\right)^{2} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}

g.inverse().display()

g−1=−∂∂u⊗∂∂R−∂∂R⊗∂∂u−R2f(R)∂∂R⊗∂∂R−∂∂θ⊗∂∂θ−1sin⁡(θ)2∂∂φ⊗∂∂φ\displaystyle g^{-1} = -\frac{\partial}{\partial u }\otimes \frac{\partial}{\partial R }-\frac{\partial}{\partial R }\otimes \frac{\partial}{\partial u } -R^{2} f\left(R\right) \frac{\partial}{\partial R }\otimes \frac{\partial}{\partial R } -\frac{\partial}{\partial {\theta} }\otimes \frac{\partial}{\partial {\theta} } -\frac{1}{\sin\left({\theta}\right)^{2}} \frac{\partial}{\partial {\varphi} }\otimes \frac{\partial}{\partial {\varphi} }

Connection coefficients and curvature

The Levi-Civita connection:

nabla = g.connection()

g.christoffel_symbols_display()

Γ u u u u u u=12 R2∂ f∂R+Rf(R)Γ R u u R u u=12 R4f(R)∂ f∂R+R3f(R)2Γ R u R R u R=−12 R2∂ f∂R−Rf(R)Γ θ φ φ θ φ φ=−cos⁡(θ)sin⁡(θ)Γ φ θ φ φ θ φ=cos⁡(θ)sin⁡(θ)\displaystyle \begin{array}{lcl} \Gamma_{ \phantom{\, u} \, u \, u }^{ \, u \phantom{\, u} \phantom{\, u} } & = & \frac{1}{2} \, R^{2} \frac{\partial\,f}{\partial R} + R f\left(R\right) \\ \Gamma_{ \phantom{\, R} \, u \, u }^{ \, R \phantom{\, u} \phantom{\, u} } & = & \frac{1}{2} \, R^{4} f\left(R\right) \frac{\partial\,f}{\partial R} + R^{3} f\left(R\right)^{2} \\ \Gamma_{ \phantom{\, R} \, u \, R }^{ \, R \phantom{\, u} \phantom{\, R} } & = & -\frac{1}{2} \, R^{2} \frac{\partial\,f}{\partial R} - R f\left(R\right) \\ \Gamma_{ \phantom{\, {\theta}} \, {\varphi} \, {\varphi} }^{ \, {\theta} \phantom{\, {\varphi}} \phantom{\, {\varphi}} } & = & -\cos\left({\theta}\right) \sin\left({\theta}\right) \\ \Gamma_{ \phantom{\, {\varphi}} \, {\theta} \, {\varphi} }^{ \, {\varphi} \phantom{\, {\theta}} \phantom{\, {\varphi}} } & = & \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}

Riem = g.riemann()

Ric = g.ricci()
Ric.display()

Ric(g)=(12 R4f(R)∂2 f∂R2+2 R3f(R)∂ f∂R+R2f(R)2)du⊗du+(−12 R2∂2 f∂R2−2 R∂ f∂R−f(R))du⊗dR+(−12 R2∂2 f∂R2−2 R∂ f∂R−f(R))dR⊗du+dθ⊗dθ+sin⁡(θ)2dφ⊗dφ\displaystyle \mathrm{Ric}\left(g\right) = \left( \frac{1}{2} \, R^{4} f\left(R\right) \frac{\partial^2\,f}{\partial R ^ 2} + 2 \, R^{3} f\left(R\right) \frac{\partial\,f}{\partial R} + R^{2} f\left(R\right)^{2} \right) \mathrm{d} u\otimes \mathrm{d} u + \left( -\frac{1}{2} \, R^{2} \frac{\partial^2\,f}{\partial R ^ 2} - 2 \, R \frac{\partial\,f}{\partial R} - f\left(R\right) \right) \mathrm{d} u\otimes \mathrm{d} R + \left( -\frac{1}{2} \, R^{2} \frac{\partial^2\,f}{\partial R ^ 2} - 2 \, R \frac{\partial\,f}{\partial R} - f\left(R\right) \right) \mathrm{d} R\otimes \mathrm{d} u +\mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \sin\left({\theta}\right)^{2} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}

Scal = -g.ricci_scalar() # sign correction here
Scal.display()

−r(g):M⟶R(u,R,θ,φ)⟼−R2∂2 f∂R2−4 R∂ f∂R−2 f(R)+2\displaystyle \begin{array}{llcl} -\mathrm{r}\left(g\right):& \mathscr{M} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & -R^{2} \frac{\partial^2\,f}{\partial R ^ 2} - 4 \, R \frac{\partial\,f}{\partial R} - 2 \, f\left(R\right) + 2 \end{array}

Calculation of the d'Alembertian

phi = M.scalar_field(function('phi')(u,R,th,sph), 
                     name='phi', latex_name=r'\phi')
phi.display()

ϕ:M⟶R(u,R,θ,φ)⟼ϕ(u,R,θ,φ)\displaystyle \begin{array}{llcl} \phi:& \mathscr{M} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & \phi\left(u, R, {\theta}, {\varphi}\right) \end{array}

dal = phi.laplacian(g)
dal

Δg(ϕ)\displaystyle \Delta_{g}\left(\phi\right)

dal.display()

Δg(ϕ):M⟶R(u,R,θ,φ)⟼−R2f(R)sin⁡(θ)2∂2 ϕ∂R2+(R2∂ f∂R+2 Rf(R))sin⁡(θ)2∂ ϕ∂R+2 sin⁡(θ)2∂2 ϕ∂u∂R+cos⁡(θ)sin⁡(θ)∂ ϕ∂θ+sin⁡(θ)2∂2 ϕ∂θ2+∂2 ϕ∂φ2sin⁡(θ)2\displaystyle \begin{array}{llcl} \Delta_{g}\left(\phi\right):& \mathscr{M} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & -\frac{R^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \frac{\partial^2\,\phi}{\partial R ^ 2} + {\left(R^{2} \frac{\partial\,f}{\partial R} + 2 \, R f\left(R\right)\right)} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial R} + 2 \, \sin\left({\theta}\right)^{2} \frac{\partial^2\,\phi}{\partial u\partial R} + \cos\left({\theta}\right) \sin\left({\theta}\right) \frac{\partial\,\phi}{\partial {\theta}} + \sin\left({\theta}\right)^{2} \frac{\partial^2\,\phi}{\partial {\theta} ^ 2} + \frac{\partial^2\,\phi}{\partial {\varphi} ^ 2}}{\sin\left({\theta}\right)^{2}} \end{array}

We're now tackling the energy current

First we define the observer: the Morawetz vector field KK

K = M.vector_field(u^2, -2*(1+u*R), 0, 0, name='K')
K.display()

K=u2∂∂u+(−2 Ru−2)∂∂R\displaystyle K = u^{2} \frac{\partial}{\partial u } + \left( -2 \, R u - 2 \right) \frac{\partial}{\partial R }

KK's Killing form:

KilK = g.lie_derivative(K)
KilK.display()

(−2 R3u∂ f∂R−2 R2∂ f∂R−4 Rf(R)+4 R)du⊗du\displaystyle \left( -2 \, R^{3} u \frac{\partial\,f}{\partial R} - 2 \, R^{2} \frac{\partial\,f}{\partial R} - 4 \, R f\left(R\right) + 4 \, R \right) \mathrm{d} u\otimes \mathrm{d} u

Check of Eq. (5.10):

g(K, K).expr().factor()

(R2u2f(R)+4 Ru+4)u2\displaystyle {\left(R^{2} u^{2} f\left(R\right) + 4 \, R u + 4\right)} u^{2}

Then we define the Killing vector ξ=∂u\xi = \partial_u

xi = M.vector_field(1, 0, 0, 0, name='xi', latex_name=r'\xi')
xi.display()

ξ=∂∂u\displaystyle \xi = \frac{\partial}{\partial u }

Now the gradient and differential of a scalar field

phi.display()

ϕ:M⟶R(u,R,θ,φ)⟼ϕ(u,R,θ,φ)\displaystyle \begin{array}{llcl} \phi:& \mathscr{M} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & \phi\left(u, R, {\theta}, {\varphi}\right) \end{array}

from sage.manifolds.operators import grad

v = grad(phi)
v.display()

grad(ϕ)=−∂ ϕ∂R∂∂u+(−R2f(R)∂ ϕ∂R−∂ ϕ∂u)∂∂R−∂ ϕ∂θ∂∂θ−∂ ϕ∂φsin⁡(θ)2∂∂φ\displaystyle \mathrm{grad}\left(\phi\right) = -\frac{\partial\,\phi}{\partial R} \frac{\partial}{\partial u } + \left( -R^{2} f\left(R\right) \frac{\partial\,\phi}{\partial R} - \frac{\partial\,\phi}{\partial u} \right) \frac{\partial}{\partial R } -\frac{\partial\,\phi}{\partial {\theta}} \frac{\partial}{\partial {\theta} } -\frac{\frac{\partial\,\phi}{\partial {\varphi}}}{\sin\left({\theta}\right)^{2}} \frac{\partial}{\partial {\varphi} }

dphi = diff(phi)
dphi.display()

dϕ=∂ ϕ∂udu+∂ ϕ∂RdR+∂ ϕ∂θdθ+∂ ϕ∂φdφ\displaystyle \mathrm{d}\phi = \frac{\partial\,\phi}{\partial u} \mathrm{d} u + \frac{\partial\,\phi}{\partial R} \mathrm{d} R + \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} {\theta} + \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\varphi}

Calculation of g (Grav v , Grad v)

Sc = g(v,v)
Sc.display()

g(grad(ϕ),grad(ϕ)):M⟶R(u,R,θ,φ)⟼−R2f(R)sin⁡(θ)2(∂ ϕ∂R)2+2 sin⁡(θ)2∂ ϕ∂u∂ ϕ∂R+sin⁡(θ)2(∂ ϕ∂θ)2+(∂ ϕ∂φ)2sin⁡(θ)2\displaystyle \begin{array}{llcl} g\left(\mathrm{grad}\left(\phi\right),\mathrm{grad}\left(\phi\right)\right):& \mathscr{M} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & -\frac{R^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} + 2 \, \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} + \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{\sin\left({\theta}\right)^{2}} \end{array}

An equivalent way to get the same scalar field:

Sc0 = g.inverse()(dphi, dphi)
Sc0.display()

g−1(dϕ,dϕ):M⟶R(u,R,θ,φ)⟼−R2f(R)sin⁡(θ)2(∂ ϕ∂R)2+2 sin⁡(θ)2∂ ϕ∂u∂ ϕ∂R+sin⁡(θ)2(∂ ϕ∂θ)2+(∂ ϕ∂φ)2sin⁡(θ)2\displaystyle \begin{array}{llcl} g^{-1}\left(\mathrm{d}\phi,\mathrm{d}\phi\right):& \mathscr{M} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & -\frac{R^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} + 2 \, \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} + \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{\sin\left({\theta}\right)^{2}} \end{array}

Stress-energy tensor for the wave equation

T = dphi*dphi - (1/2)*Sc*g
T.set_name('T')
T.display()

T=(R4f(R)2sin⁡(θ)2(∂ ϕ∂R)2+2 R2f(R)sin⁡(θ)2∂ ϕ∂u∂ ϕ∂R+R2f(R)sin⁡(θ)2(∂ ϕ∂θ)2+R2f(R)(∂ ϕ∂φ)2+2 sin⁡(θ)2∂ ϕ∂u22 sin⁡(θ)2)du⊗du+(−R2f(R)sin⁡(θ)2(∂ ϕ∂R)2+sin⁡(θ)2(∂ ϕ∂θ)2+(∂ ϕ∂φ)22 sin⁡(θ)2)du⊗dR+∂ ϕ∂u∂ ϕ∂θdu⊗dθ+∂ ϕ∂u∂ ϕ∂φdu⊗dφ+(−R2f(R)sin⁡(θ)2(∂ ϕ∂R)2+sin⁡(θ)2(∂ ϕ∂θ)2+(∂ ϕ∂φ)22 sin⁡(θ)2)dR⊗du+(∂ ϕ∂R)2dR⊗dR+∂ ϕ∂R∂ ϕ∂θdR⊗dθ+∂ ϕ∂R∂ ϕ∂φdR⊗dφ+∂ ϕ∂u∂ ϕ∂θdθ⊗du+∂ ϕ∂R∂ ϕ∂θdθ⊗dR+(−R2f(R)sin⁡(θ)2(∂ ϕ∂R)2+2 sin⁡(θ)2∂ ϕ∂u∂ ϕ∂R−sin⁡(θ)2(∂ ϕ∂θ)2+(∂ ϕ∂φ)22 sin⁡(θ)2)dθ⊗dθ+∂ ϕ∂θ∂ ϕ∂φdθ⊗dφ+∂ ϕ∂u∂ ϕ∂φdφ⊗du+∂ ϕ∂R∂ ϕ∂φdφ⊗dR+∂ ϕ∂θ∂ ϕ∂φdφ⊗dθ+(−12 R2f(R)sin⁡(θ)2(∂ ϕ∂R)2−sin⁡(θ)2∂ ϕ∂u∂ ϕ∂R−12 sin⁡(θ)2(∂ ϕ∂θ)2+12 (∂ ϕ∂φ)2)dφ⊗dφ\displaystyle T = \left( \frac{R^{4} f\left(R\right)^{2} \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} + 2 \, R^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} + R^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + R^{2} f\left(R\right) \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2} + 2 \, \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u}^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} u\otimes \mathrm{d} u + \left( -\frac{R^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} + \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} u\otimes \mathrm{d} R + \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} u\otimes \mathrm{d} {\theta} + \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} u\otimes \mathrm{d} {\varphi} + \left( -\frac{R^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} + \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} R\otimes \mathrm{d} u + \left(\frac{\partial\,\phi}{\partial R}\right)^{2} \mathrm{d} R\otimes \mathrm{d} R + \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} R\otimes \mathrm{d} {\theta} + \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} R\otimes \mathrm{d} {\varphi} + \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} {\theta}\otimes \mathrm{d} u + \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} {\theta}\otimes \mathrm{d} R + \left( -\frac{R^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} + 2 \, \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} - \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \frac{\partial\,\phi}{\partial {\theta}} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\theta}\otimes \mathrm{d} {\varphi} + \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\varphi}\otimes \mathrm{d} u + \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\varphi}\otimes \mathrm{d} R + \frac{\partial\,\phi}{\partial {\theta}} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\varphi}\otimes \mathrm{d} {\theta} + \left( -\frac{1}{2} \, R^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} - \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} - \frac{1}{2} \, \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \frac{1}{2} \, \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2} \right) \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}

The energy current associated with our choice of observer

J = T.contract(K)
J.set_name('J')
J.display()

J=(2 R2u2f(R)sin⁡(θ)2∂ ϕ∂u∂ ϕ∂R+2 u2sin⁡(θ)2∂ ϕ∂u2+(R4u2f(R)2sin⁡(θ)2+2 R3uf(R)sin⁡(θ)2+2 R2f(R)sin⁡(θ)2)∂ ϕ∂R2+(R2u2f(R)sin⁡(θ)2+2 Rusin⁡(θ)2+2 sin⁡(θ)2)(∂ ϕ∂θ)2+(R2u2f(R)+2 Ru+2)(∂ ϕ∂φ)22 sin⁡(θ)2)du+(−u2sin⁡(θ)2(∂ ϕ∂θ)2+u2(∂ ϕ∂φ)2+(R2u2f(R)sin⁡(θ)2+4 Rusin⁡(θ)2+4 sin⁡(θ)2)(∂ ϕ∂R)22 sin⁡(θ)2)dR+(u2∂ ϕ∂u∂ ϕ∂θ−2 (Ru+1)∂ ϕ∂R∂ ϕ∂θ)dθ+(u2∂ ϕ∂u∂ ϕ∂φ−2 (Ru+1)∂ ϕ∂R∂ ϕ∂φ)dφ\displaystyle J = \left( \frac{2 \, R^{2} u^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} + 2 \, u^{2} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u}^{2} + {\left(R^{4} u^{2} f\left(R\right)^{2} \sin\left({\theta}\right)^{2} + 2 \, R^{3} u f\left(R\right) \sin\left({\theta}\right)^{2} + 2 \, R^{2} f\left(R\right) \sin\left({\theta}\right)^{2}\right)} \frac{\partial\,\phi}{\partial R}^{2} + {\left(R^{2} u^{2} f\left(R\right) \sin\left({\theta}\right)^{2} + 2 \, R u \sin\left({\theta}\right)^{2} + 2 \, \sin\left({\theta}\right)^{2}\right)} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + {\left(R^{2} u^{2} f\left(R\right) + 2 \, R u + 2\right)} \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} u + \left( -\frac{u^{2} \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + u^{2} \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2} + {\left(R^{2} u^{2} f\left(R\right) \sin\left({\theta}\right)^{2} + 4 \, R u \sin\left({\theta}\right)^{2} + 4 \, \sin\left({\theta}\right)^{2}\right)} \left(\frac{\partial\,\phi}{\partial R}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} R + \left( u^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\theta}} - 2 \, {\left(R u + 1\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\theta}} \right) \mathrm{d} {\theta} + \left( u^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\varphi}} - 2 \, {\left(R u + 1\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\varphi}} \right) \mathrm{d} {\varphi}

The energy density on a u=constu=\mathrm{const} slice

f0 = J(xi)
f0.display()

J(ξ):M⟶R(u,R,θ,φ)⟼2 R2u2f(R)sin⁡(θ)2∂ ϕ∂u∂ ϕ∂R+2 u2sin⁡(θ)2∂ ϕ∂u2+(R4u2f(R)2sin⁡(θ)2+2 R3uf(R)sin⁡(θ)2+2 R2f(R)sin⁡(θ)2)∂ ϕ∂R2+(R2u2f(R)sin⁡(θ)2+2 Rusin⁡(θ)2+2 sin⁡(θ)2)(∂ ϕ∂θ)2+(R2u2f(R)+2 Ru+2)(∂ ϕ∂φ)22 sin⁡(θ)2\displaystyle \begin{array}{llcl} J\left(\xi\right):& \mathscr{M} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & \frac{2 \, R^{2} u^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} + 2 \, u^{2} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u}^{2} + {\left(R^{4} u^{2} f\left(R\right)^{2} \sin\left({\theta}\right)^{2} + 2 \, R^{3} u f\left(R\right) \sin\left({\theta}\right)^{2} + 2 \, R^{2} f\left(R\right) \sin\left({\theta}\right)^{2}\right)} \frac{\partial\,\phi}{\partial R}^{2} + {\left(R^{2} u^{2} f\left(R\right) \sin\left({\theta}\right)^{2} + 2 \, R u \sin\left({\theta}\right)^{2} + 2 \, \sin\left({\theta}\right)^{2}\right)} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + {\left(R^{2} u^{2} f\left(R\right) + 2 \, R u + 2\right)} \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \end{array}

The energy density on an Hs,u0\mathcal{H}_{s,u_0} slice

A normal vector field to Hs,u0\mathcal{H}_{s,u_0}:

s = var('s', domain='real')

n = M.vector_field(1, (f(R)*R^2)*(s - 1)/s, 0, 0, name='n')
n.display()

n=∂∂u+R2(s−1)f(R)s∂∂R\displaystyle n = \frac{\partial}{\partial u } + \frac{R^{2} {\left(s - 1\right)} f\left(R\right)}{s} \frac{\partial}{\partial R }

f1 = J(n)
f1.display()

J(n):M⟶R(u,R,θ,φ)⟼2 R2su2f(R)sin⁡(θ)2∂ ϕ∂u∂ ϕ∂R+2 su2sin⁡(θ)2∂ ϕ∂u2+(R4u2f(R)2sin⁡(θ)2−2 (R3sf(R)−2 R3f(R))usin⁡(θ)2−2 (R2sf(R)−2 R2f(R))sin⁡(θ)2)∂ ϕ∂R2+(R2u2f(R)sin⁡(θ)2+2 Rsusin⁡(θ)2+2 ssin⁡(θ)2)(∂ ϕ∂θ)2+(R2u2f(R)+2 Rsu+2 s)(∂ ϕ∂φ)22 ssin⁡(θ)2\displaystyle \begin{array}{llcl} J\left(n\right):& \mathscr{M} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & \frac{2 \, R^{2} s u^{2} f\left(R\right) \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} + 2 \, s u^{2} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u}^{2} + {\left(R^{4} u^{2} f\left(R\right)^{2} \sin\left({\theta}\right)^{2} - 2 \, {\left(R^{3} s f\left(R\right) - 2 \, R^{3} f\left(R\right)\right)} u \sin\left({\theta}\right)^{2} - 2 \, {\left(R^{2} s f\left(R\right) - 2 \, R^{2} f\left(R\right)\right)} \sin\left({\theta}\right)^{2}\right)} \frac{\partial\,\phi}{\partial R}^{2} + {\left(R^{2} u^{2} f\left(R\right) \sin\left({\theta}\right)^{2} + 2 \, R s u \sin\left({\theta}\right)^{2} + 2 \, s \sin\left({\theta}\right)^{2}\right)} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + {\left(R^{2} u^{2} f\left(R\right) + 2 \, R s u + 2 \, s\right)} \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, s \sin\left({\theta}\right)^{2}} \end{array}

The 4-volume form and the 3-volume form on a u=constu=\mathrm{const} slice

eps = g.volume_form()
eps.display()

ϵg=sin⁡(θ)du∧dR∧dθ∧dφ\displaystyle \epsilon_{g} = \sin\left({\theta}\right) \mathrm{d} u\wedge \mathrm{d} R\wedge \mathrm{d} {\theta}\wedge \mathrm{d} {\varphi}

We have g(ξ,ξ)=R2f(R)g(\xi,\xi) = R^2f(R) so our vector v=1R2f(R)ξv = \frac{1}{R^2f(R)} \xi

g(xi, xi).expr()

R2f(R)\displaystyle R^{2} f\left(R\right)

v = M.vector_field(1/(R^2*f(R)), 0, 0, 0, name='v')
v.display()

v=1R2f(R)∂∂u\displaystyle v = \frac{1}{R^{2} f\left(R\right)} \frac{\partial}{\partial u }

g(xi, v).expr()

1\displaystyle 1

The volume 3-form:

tvol = v['^i']*eps['_{ijkl}']
tvol.display()

sin⁡(θ)R2f(R)dR∧dθ∧dφ\displaystyle \frac{\sin\left({\theta}\right)}{R^{2} f\left(R\right)} \mathrm{d} R\wedge \mathrm{d} {\theta}\wedge \mathrm{d} {\varphi}

The 3-volume form on an Hs,u0\mathcal{H}_{s,u_0} slice as well as I+\mathscr{I}^+

On Hs,u0\mathcal{H}_{s,u_0} we take l=−∂Rl = -\partial_R and we have g(n,l)=1g(n,l) = 1

l = M.vector_field(0, -1, 0, 0, name='l')
l.display()

l=−∂∂R\displaystyle l = -\frac{\partial}{\partial R }

g(n, l).expr()

1\displaystyle 1

scrvol = l['^i']*eps['_{ijkl}']
scrvol.display()

sin⁡(θ)du∧dθ∧dφ\displaystyle \sin\left({\theta}\right) \mathrm{d} u\wedge \mathrm{d} {\theta}\wedge \mathrm{d} {\varphi}

Error terms in the conservation law for J : ∇aJa=∇(aKb)Tab−16Scalgϕ∇Kϕ\nabla^a J_a = \nabla^{(a} K^{b)} T_{ab} - \frac{1}{6} \mathrm{Scal}_g \phi \nabla_{K} \phi

divJ = KilK.up(g)['^{ij}']*T['_{ij}'] - s*(Scal/6)*phi*dphi(K)
divJ.display()

M⟶R(u,R,θ,φ)⟼16 (R2∂2 f∂R2+4 R∂ f∂R+2 f(R)−2)su2ϕ(u,R,θ,φ)∂ ϕ∂u−13 ((R3∂2 f∂R2+4 R2∂ f∂R+2 Rf(R)−2 R)su+(R2∂2 f∂R2+4 R∂ f∂R+2 f(R)−2)s)ϕ(u,R,θ,φ)∂ ϕ∂R−2 (R3u∂ f∂R+R2∂ f∂R+2 Rf(R)−2 R)∂ ϕ∂R2\displaystyle \begin{array}{llcl} & \mathscr{M} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & \frac{1}{6} \, {\left(R^{2} \frac{\partial^2\,f}{\partial R ^ 2} + 4 \, R \frac{\partial\,f}{\partial R} + 2 \, f\left(R\right) - 2\right)} s u^{2} \phi\left(u, R, {\theta}, {\varphi}\right) \frac{\partial\,\phi}{\partial u} - \frac{1}{3} \, {\left({\left(R^{3} \frac{\partial^2\,f}{\partial R ^ 2} + 4 \, R^{2} \frac{\partial\,f}{\partial R} + 2 \, R f\left(R\right) - 2 \, R\right)} s u + {\left(R^{2} \frac{\partial^2\,f}{\partial R ^ 2} + 4 \, R \frac{\partial\,f}{\partial R} + 2 \, f\left(R\right) - 2\right)} s\right)} \phi\left(u, R, {\theta}, {\varphi}\right) \frac{\partial\,\phi}{\partial R} - 2 \, {\left(R^{3} u \frac{\partial\,f}{\partial R} + R^{2} \frac{\partial\,f}{\partial R} + 2 \, R f\left(R\right) - 2 \, R\right)} \frac{\partial\,\phi}{\partial R}^{2} \end{array}

This is all controlled by the energy density using uRuR bounded.