# Walker-Penrose Killing tensor in Kerr spacetime¶

This notebook demonstrates a few capabilities of SageMath in computations regarding Kerr spacetime. More precisely, it focuses on the Killing tensor $K$ found by Walker & Penrose [Commun. Math. Phys. 18, 265 (1970)]. This notebook makes use of SageMath tools developed through the SageManifolds project.

NB: a version of SageMath at least equal to 8.2 is required to run this notebook:

In :
version()

Out:
'SageMath version 8.3, Release Date: 2018-08-03'

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

In :
%display latex


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

In :
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 :
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 :
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:
In :
BL, BL

Out:

## Metric tensor

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

In :
var('m, a', domain='real')

Out:

We get the (yet undefined) spacetime metric by

In :
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 :
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:

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

In :
g[:]

Out:

The list of the non-vanishing components:

In :
g.display_comp()

Out:

## Levi-Civita Connection

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

In :
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 :
nabla(g).display()

Out:

## Killing vectors

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

In :
M.default_frame() is BL.frame()

Out:
In :
BL.frame()

Out:

Let us consider the first vector field of this frame:

In :
xi = BL.frame() ; xi

Out:
In :
print(xi)

Vector field d/dt on the 4-dimensional Lorentzian manifold M


The 1-form associated to it by metric duality is

In :
xi_form = xi.down(g) ; xi_form.display()

Out:

Its covariant derivative is

In :
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:

Let us check that the Killing equation is satisfied:

In :
nab_xi.symmetrize() == 0

Out:

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

In :
chi = BL.frame() ; chi

Out:
In :
nabla(chi.down(g)).symmetrize() == 0

Out:

## Principal null vectors¶

We introduce the principal null vectors $k$ and $\ell$ of Kerr spacetime (see e.g. Eqs. (12.3.5) and (12.3.6) of Wald's textbook General Relativity (1984)):

In :
k = M.vector_field(name='k')
k[:] = [(r^2+a^2)/(2*rho2), -Delta/(2*rho2), 0, a/(2*rho2)]
k.display()

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

Out:

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

In :
g(k,k).expr()

Out:
In :
g(el,el).expr()

Out:

Their scalar product is $-1$:

In :
g(k,el).expr()

Out:

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

In :
k.dot(el).expr()

Out:

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

In :
acc_k = nabla(k).contract(k)
acc_k.display()

Out:

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

In :
for i in [0,1,3]:
show(acc_k[i] / k[i])

In :
kappa_k = acc_k[] / k[]
kappa_k.display()

Out:
In :
acc_k == kappa_k * k

Out:

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

In :
acc_l = nabla(el).contract(el)
acc_l.display()

Out:

Hence $\ell$ is a geodesic vector.

The check that $k$ and $\ell$ do define (repeated) principal null directions is performed in this notebook.

## Walker-Penrose Killing tensor¶

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

In :
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 :
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 :
K.display_comp()

Out:
In :
DK = nabla(K)
print(DK)

Tensor field nabla_g(K) of type (0,3) on the 4-dimensional Lorentzian manifold M

In :
DK.display_comp()

Out:

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

In :
DK.symmetrize().display()

Out:

Equivalently, we may write, using index notation:

In :
DK['_(abc)'].display()

Out: