NB: a version of SageMath at least equal to 7.5 is required to run this worksheet:
version()
'SageMath version 8.0.beta12, Release Date: 2017-06-22'
First we set up the notebook to display mathematical objects using LaTeX rendering:
%display latex
We also define a viewer for 3D plots (use 'threejs'
or 'jmol'
for interactive 3D graphics):
viewer3D = 'threejs' # must be 'threejs', jmol', 'tachyon' or None (default)
Since some computations are quite long, we ask for running them in parallel on 8 cores:
Parallelism().set(nproc=8)
We declare the Kerr-AdS spacetime as a 5-dimensional diffentiable manifold:
M = Manifold(5, 'M', r'\mathcal{M}')
print(M)
5-dimensional differentiable manifold M
Let us define Boyer-Lindquist-type coordinates (rational polynomial version) on $\mathcal{M}$, via the method chart()
, the argument of which is a string expressing the coordinates names, their ranges (the default is $(-\infty,+\infty)$) and their LaTeX symbols:
BL.<t,r,mu,ph,ps> = M.chart(r't r:(0,+oo) mu:(-1,1):\mu ph:(0,2*pi):\phi ps:(0,2*pi):\psi')
BL
Note that $\mu$ is related to the standard Boyer-Lindquist coordinate $\theta$ by $$ \mu = \cos\theta$$
The 4 parameters $m$, $a$, $b$ and $\ell$ of the Kerr-AdS spacetime are declared as symbolic variables, $a$ and $b$ being the two angular momentum parameters and $\ell$ being related to the cosmological constant by $\Lambda = - 6 \ell^2$:
var('m a b', domain='real')
var('l', domain='real', latex_name=r'\ell')
# Particular cases
# a = 0
# m = 0
# b = a
Some auxiliary functions:
sig = (1+r^2*l^2)/r^2
Delta = (r^2+a^2)*(r^2+b^2)*sig - 2*m
sinth2 = 1-mu^2
Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2
rho2 = r^2 + a^2*mu^2 + b^2*sinth2
Xi_a = 1 - a^2*l^2
Xi_b = 1 - b^2*l^2
The metric is set by its components in the coordinate frame associated with the Boyer-Lindquist-type coordinates, which is the current manifold's default frame. These components are given by Eq. (5.22) of the article S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D 59, 064005 (1999):
g = M.lorentzian_metric('g')
tmp = 1/rho2*( -Delta + Delta_th*(a^2*sinth2 + b^2*mu^2) + a^2*b^2*sig )
g[0,0] = tmp.simplify_full()
tmp = a*sinth2/(rho2*Xi_a)*( Delta - (r^2+a^2)*(Delta_th + b^2*sig) )
g[0,3] = tmp.simplify_full()
tmp = b*mu^2/(rho2*Xi_b)*( Delta - (r^2+b^2)*(Delta_th + a^2*sig) )
g[0,4] = tmp.simplify_full()
g[1,1] = (rho2/Delta).simplify_full()
g[2,2] = (rho2/Delta_th/(1-mu^2)).simplify_full()
tmp = sinth2/(rho2*Xi_a^2)*( - Delta*a^2*sinth2 + (r^2+a^2)^2*(Delta_th + sig*b^2*sinth2) )
g[3,3] = tmp.simplify_full()
tmp = a*b*sinth2*mu^2/(rho2*Xi_a*Xi_b)*( - Delta + sig*(r^2+a^2)*(r^2+b^2) )
g[3,4] = tmp.simplify_full()
tmp = mu^2/(rho2*Xi_b^2)*( - Delta*b^2*mu^2 + (r^2+b^2)^2*(Delta_th + sig*a^2*mu^2) )
g[4,4] = tmp.simplify_full()
g.display()
g.display_comp(only_nonredundant=True)
The Ricci tensor of $g$ is
Ric = g.ricci()
print(Ric)
Field of symmetric bilinear forms Ric(g) on the 5-dimensional differentiable manifold M
Ric.display_comp(only_nonredundant=True)
Let us check that $g$ is a solution of the vacuum Einstein equation with the cosmological constant $\Lambda = - 6 \ell^2$:
Lambda = -6*l^2
Ric == 2/3*Lambda*g