This worksheet demonstrates a few capabilities of SageManifolds (version 1.0, as included in SageMath 7.5) in computations regarding the 3+1 slicing of Kerr spacetime.
Click here to download the worksheet file (ipynb format). To run it, you must start SageMath within the Jupyter notebook, via the command sage -n jupyter
NB: a version of SageMath at least equal to 7.5 is required to run this worksheet:
version()
'SageMath version 7.5.1, Release Date: 2017-01-15'
First we set up the notebook to display mathematical objects using LaTeX rendering:
%display latex
Since some computations are quite long, we ask for running them in parallel on 8 cores:
Parallelism().set(nproc=8)
We consider some hypersurface $\Sigma$ of a spacelike foliation $(\Sigma_t)_{t\in\mathbb{R}}$ of Kerr spacetime; we declare $\Sigma_t$ as a 3-dimensional manifold:
Sig = Manifold(3, 'Sigma', r'\Sigma', start_index=1)
print(Sig)
3-dimensional differentiable manifold Sigma
On $\Sigma$, we consider the "rational-polynomial" coordinates $(r,y,\phi)$ inheritated from the standard Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ of Kerr spacetime, via $y=\cos\theta$:
X.<r,y,ph> = Sig.chart(r'r:(1,+oo) y:(-1,1) ph:(0,2*pi):\phi')
print(X) ; X
Chart (Sigma, (r, y, ph))
First the two Kerr parameters:
var('m, a', domain='real')
assume(m>0)
assume(a>0)
assumptions()
For dealing with extreme Kerr, the following must be uncommented:
# m = 1 ; a = 1
Some shortcut notations:
rho2 = r^2 + a^2*y^2
Del = r^2 -2*m*r + a^2
AA2 = rho2*(r^2 + a^2) + 2*a^2*m*r*(1-y^2)
BB2 = r^2 + a^2 + 2*a^2*m*r*(1-y^2)/rho2
The metric $\gamma$ induced by the spacetime metric $g$ on $\Sigma$:
gam = Sig.riemannian_metric('gam', latex_name=r'\gamma')
gam[1,1] = rho2/Del
gam[2,2] = rho2/(1-y^2)
gam[3,3] = BB2*(1-y^2)
gam.display()
A matrix view of the components w.r.t. coordinates $(r,y,\phi)$:
gam[:]
N = Sig.scalar_field(sqrt(Del / BB2), name='N')
print(N)
N.display()
Scalar field N on the 3-dimensional differentiable manifold Sigma
b = Sig.vector_field('beta', latex_name=r'\beta')
b[3] = -2*m*r*a/AA2
# unset components are zero
b.display()
We use the formula $$ K_{ij} = \frac{1}{2N} \mathcal{L}_{\beta} \gamma_{ij} $$ which is valid for any stationary spacetime:
K = gam.lie_der(b) / (2*N)
K.set_name('K')
print(K) ; K.display()
Field of symmetric bilinear forms K on the 3-dimensional differentiable manifold Sigma
Check (comparison with known formulas):
Krp = a*m*(1-y^2)*(3*r^4+a^2*r^2+a^2*(r^2-a^2)*y^2) / rho2^2/sqrt(Del*BB2)
Krp
K[1,3] - Krp
Kyp = 2*m*r*a^3*(1-y^2)*y*sqrt(Del)/rho2^2/sqrt(BB2)
Kyp
K[2,3] - Kyp
For now on, we use the expressions Krp and Kyp above for $K_{r\phi}$ and $K_{ry}$, respectively:
K1 = Sig.sym_bilin_form_field('K')
K1[1,3] = Krp
K1[2,3] = Kyp
K = K1
K.display()
K.display_comp()
The type-(1,1) tensor $K^\sharp$ of components $K^i_{\ \, j} = \gamma^{ik} K_{kj}$:
Ku = K.up(gam, 0)
print(Ku) ; Ku.display()
Tensor field of type (1,1) on the 3-dimensional differentiable manifold Sigma
We may check that the hypersurface $\Sigma$ is maximal, i.e. that $K^k_{\ \, k} = 0$:
trK = Ku.trace()
print(trK)
trK.display()
Scalar field zero on the 3-dimensional differentiable manifold Sigma
Let us call $D$ the Levi-Civita connection associated with $\gamma$:
D = gam.connection(name='D')
print(D) ; D
Levi-Civita connection D associated with the Riemannian metric gam on the 3-dimensional differentiable manifold Sigma
The Ricci tensor associated with $\gamma$:
Ric = gam.ricci()
print(Ric) ; Ric
Field of symmetric bilinear forms Ric(gam) on the 3-dimensional differentiable manifold Sigma
Ric.display_comp(only_nonredundant=True)
Ric[1,1]
Ric[1,2]
Ric[1,3]
Ric[2,2]
Ric[2,3]
Ric[3,3]
The scalar curvature $R = \gamma^{ij} R_{ij}$:
R = gam.ricci_scalar(name='R')
print(R)
R.display()
Scalar field R on the 3-dimensional differentiable manifold Sigma
Let us check that the vacuum 3+1 Einstein equations are satisfied.
We start by the contraint equations:
Let us first evaluate the term $K_{ij} K^{ij}$:
Kuu = Ku.up(gam, 1)
trKK = K['_ij']*Kuu['^ij']
print(trKK) ; trKK.display()
Scalar field on the 3-dimensional differentiable manifold Sigma
The vacuum Hamiltonian constraint equation is $$ R + K^2 -K_{ij} K^{ij} = 0 $$
Ham = R + trK^2 - trKK
print(Ham) ; Ham.display()
Scalar field zero on the 3-dimensional differentiable manifold Sigma
In vaccum, the momentum constraint is $$ D_j K^j_{\ \, i} - D_i K = 0 $$
mom = D(Ku).trace(0,2) - D(trK)
print(mom)
mom.display()
1-form on the 3-dimensional differentiable manifold Sigma
Let us first evaluate the symmetric bilinear form $k_{ij} := K_{ik} K^k_{\ \, j}$:
KK = K['_ik']*Ku['^k_j']
print(KK)
Tensor field of type (0,2) on the 3-dimensional differentiable manifold Sigma
KK1 = KK.symmetrize()
KK == KK1
KK = KK1
print(KK)
Field of symmetric bilinear forms on the 3-dimensional differentiable manifold Sigma
KK.set_name('(KK)')
KK.display_comp()
In vacuum and for stationary spacetimes, the dynamical Einstein equations are $$ \mathcal{L}_\beta K_{ij} - D_i D_j N + N \left( R_{ij} + K K_{ij} - 2 K_{ik} K^k_{\ \, j}\right) = 0 $$
dyn = K.lie_der(b) - D(D(N)) + N*( Ric + trK*K - 2*KK )
print(dyn)
dyn.display()
Tensor field of type (0,2) on the 3-dimensional differentiable manifold Sigma
The electric part is the bilinear form $E$ given by $$ E_{ij} = R_{ij} + K K_{ij} - K_{ik} K^k_{\ \, j} $$
E = Ric + trK*K - KK
print(E)
Field of symmetric bilinear forms +Ric(gam)-(KK) on the 3-dimensional differentiable manifold Sigma
E.set_name('E')
E.display_comp(only_nonzero=False)
The magnetic part is the bilinear form $B$ defined by
$$ B_{ij} = \epsilon^k_{\ \, i l} D_k K^l_{\ \, j}, $$
where $\epsilon^k_{\ \, i l}$ are the components of the type-(1,2) tensor $\epsilon^\sharp$, related to the Levi-Civita alternating tensor $\epsilon$ associated with $\gamma$ by $\epsilon^k_{\ \, i l} = \gamma^{km} \epsilon_{m i l}$. In SageManifolds, $\epsilon$ is obtained by the command volume_form()
and $\epsilon^\sharp$ by the command volume_form(1)
(1
= one index raised):
eps = gam.volume_form()
print(eps) ; eps.display()
3-form eps_gam on the 3-dimensional differentiable manifold Sigma
epsu = gam.volume_form(1)
print(epsu) ; epsu.display()
Tensor field of type (1,2) on the 3-dimensional differentiable manifold Sigma
DKu = D(Ku)
B = epsu['^k_il']*DKu['^l_jk']
print(B)
Tensor field of type (0,2) on the 3-dimensional differentiable manifold Sigma
Let us check that $B$ is symmetric:
B1 = B.symmetrize()
B == B1
Accordingly, we set
B = B1
B.set_name('B')
print(B)
Field of symmetric bilinear forms B on the 3-dimensional differentiable manifold Sigma
B.display_comp(only_nonzero=False)