Walker-Penrose Killing tensor in Kerr spacetime

This notebook demonstrates a few capabilities of SageMath in computations regarding Kerr spacetime. More precisely, it focuses of the Killing tensor $K$ found by Walker & Penrose [Commun. Math. Phys. 18, 265 (1970)]. The employed differential geometry tools have been developed within the SageManifolds project (version 1.3, as included in SageMath 8.3).

Click here to download the notebook 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 8.2 is required to run this notebook:

In [1]:
version()
Out[1]:
'SageMath version 8.3, Release Date: 2018-08-03'

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

In [2]:
%display latex

To speed up the computations, we ask for running them in parallel on 8 cores:

In [3]:
Parallelism().set(nproc=8)

Spacetime manifold

We declare the Kerr spacetime (or more precisely the Boyer-Lindquist domain of Kerr spacetime) as a 4-dimensional Lorentzian manifold:

In [4]:
M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
print(M)
4-dimensional Lorentzian manifold M

Let us declare the Boyer-Lindquist coordinates 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 [5]:
BL.<t,r,th,ph> = M.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') 
print(BL) ; BL
Chart (M, (t, r, th, ph))
Out[5]:
In [6]:
BL[0], BL[1]
Out[6]:

Metric tensor

The 2 parameters $m$ and $a$ of the Kerr spacetime are declared as symbolic variables:

In [7]:
var('m, a', domain='real')
Out[7]:

We get the (yet undefined) spacetime metric by

In [8]:
g = M.metric()

The metric is set by its components in the coordinate frame associated with Boyer-Lindquist coordinates, which is the current manifold's default frame:

In [9]:
rho2 = r^2 + (a*cos(th))^2
Delta = r^2 -2*m*r + a^2
g[0,0] = -(1-2*m*r/rho2)
g[0,3] = -2*a*m*r*sin(th)^2/rho2
g[1,1], g[2,2] = rho2/Delta, rho2
g[3,3] = (r^2+a^2+2*m*r*(a*sin(th))^2/rho2)*sin(th)^2
g.display()
Out[9]:

A matrix view of the components with respect to the manifold's default vector frame:

In [10]:
g[:]
Out[10]:

The list of the non-vanishing components:

In [11]:
g.display_comp()
Out[11]:

Levi-Civita Connection

The Levi-Civita connection $\nabla$ associated with $g$:

In [12]:
nabla = g.connection() ; print(nabla)
Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional Lorentzian manifold M

Let us verify that the covariant derivative of $g$ with respect to $\nabla$ vanishes identically:

In [13]:
nabla(g).display()
Out[13]:

Killing vectors

The default vector frame on the spacetime manifold is the coordinate basis associated with Boyer-Lindquist coordinates:

In [14]:
M.default_frame() is BL.frame()
Out[14]:
In [15]:
BL.frame()
Out[15]:

Let us consider the first vector field of this frame:

In [16]:
xi = BL.frame()[0] ; xi
Out[16]:
In [17]:
print(xi)
Vector field d/dt on the 4-dimensional Lorentzian manifold M

The 1-form associated to it by metric duality is

In [18]:
xi_form = xi.down(g) ; xi_form.display()
Out[18]:

Its covariant derivative is

In [19]:
nab_xi = nabla(xi_form) ; print(nab_xi) ; nab_xi.display()
Tensor field of type (0,2) on the 4-dimensional Lorentzian manifold M
Out[19]:

Let us check that the Killing equation is satisfied:

In [20]:
nab_xi.symmetrize() == 0
Out[20]:

Similarly, let us check that $\frac{\partial}{\partial\phi}$ is a Killing vector:

In [21]:
chi = BL.frame()[3] ; chi
Out[21]:
In [22]:
nabla(chi.down(g)).symmetrize() == 0
Out[22]:

Principal null vectors

We introduce the principal null vectors $k$ and $\ell$ of Kerr spacetime:

In [23]:
k = M.vector_field(name='k')
k[:] = [(r^2+a^2)/(2*rho2), -Delta/(2*rho2), 0, a/(2*rho2)]
k.display()
Out[23]:
In [24]:
el = M.vector_field(name='el', latex_name=r'\ell')
el[:] = [(r^2+a^2)/Delta, 1, 0, a/Delta]
el.display()
Out[24]:

Let us check that $k$ and $\ell$ are null vectors:

In [25]:
g(k,k).expr()
Out[25]:
In [26]:
g(el,el).expr()
Out[26]:

Their scalar product is $-1$:

In [27]:
g(k,el).expr()
Out[27]:

Note that the scalar product (with respect to metric $g$) can also be computed by means of the method dot:

In [28]:
k.dot(el).expr()
Out[28]:

Let us evaluate the "acceleration" of $k$, i.e. $\nabla_k k$:

In [29]:
acc_k = nabla(k).contract(k)
acc_k.display()
Out[29]:

We check that $k$ is a pregeodesic vector, i.e. that $\nabla_k k = \kappa_k k$ for some scalar field $\kappa_k$:

In [30]:
for i in [0,1,3]:
    show(acc_k[i] / k[i])
In [31]:
kappa_k = acc_k[[0]] / k[[0]]
kappa_k.display()
Out[31]:
In [32]:
acc_k == kappa_k * k
Out[32]:

Similarly let us evaluate the "acceleration" of $\ell$:

In [33]:
acc_l = nabla(el).contract(el)
acc_l.display()
Out[33]:

Hence $\ell$ is a geodesic vector.

Walker-Penrose Killing tensor

We need the 1-forms associated to $k$ and $\ell$ by metric duality:

In [34]:
uk = k.down(g)
ul = el.down(g)

The Walker-Penrose Killing tensor $K$ is then formed as $$ K = \rho^2 (\underline{\ell}\otimes \underline{k} + (\underline{k}\otimes \underline{\ell}) + r^2 g $$

In [35]:
K = rho2*(ul*uk+ uk*ul) + r^2*g
K.set_name('K')
print(K)
Tensor field K of type (0,2) on the 4-dimensional Lorentzian manifold M
In [36]:
K.display_comp()
Out[36]:
In [37]:
DK = nabla(K)
print(DK)
Tensor field nabla_g(K) of type (0,3) on the 4-dimensional Lorentzian manifold M
In [38]:
DK.display_comp()
Out[38]:

Let us check that $K$ is a Killing tensor:

In [39]:
DK.symmetrize().display()
Out[39]:

Equivalently, we may write, using index notation:

In [40]:
DK['_(abc)'].display()
Out[40]: