5D Kerr-AdS spacetime: general solution with 2 angular momenta

NB: a version of SageMath at least equal to 7.5 is required to run this worksheet:

In [1]:
version()
Out[1]:
'SageMath version 8.0.beta12, Release Date: 2017-06-22'

First we set up the notebook to display mathematical objects using LaTeX rendering:

In [2]:
%display latex

We also define a viewer for 3D plots (use 'threejs' or 'jmol' for interactive 3D graphics):

In [3]:
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:

In [4]:
Parallelism().set(nproc=8)

Spacetime manifold

We declare the Kerr-AdS spacetime as a 5-dimensional diffentiable manifold:

In [5]:
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:

In [6]:
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
Out[6]:

Note that $\mu$ is related to the standard Boyer-Lindquist coordinate $\theta$ by $$ \mu = \cos\theta$$

Metric tensor

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 $\Lambda = - 6 \ell^2$:

In [7]:
var('m a b', domain='real')
Out[7]:
In [8]:
var('l', domain='real', latex_name=r'\ell')
Out[8]:
In [9]:
# Particular cases
# a = 0
# m = 0
# b = a

Some auxiliary functions:

In [10]:
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):

In [11]:
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()
Out[11]:
In [12]:
g.display_comp(only_nonredundant=True)
Out[12]:

Einstein equation

The Ricci tensor of $g$ is

In [13]:
Ric = g.ricci()
print(Ric)
Field of symmetric bilinear forms Ric(g) on the 5-dimensional differentiable manifold M
In [14]:
Ric.display_comp(only_nonredundant=True)
Out[14]:

Let us check that $g$ is a solution of the vacuum Einstein equation with the cosmological constant $\Lambda = - 6 \ell^2$:

In [15]:
Lambda = -6*l^2
Ric == 2/3*Lambda*g
Out[15]:
In [ ]: