Walker-Penrose Killing tensor in Kerr spacetime

This worksheet demonstrates a few capabilities of SageManifolds (version 1.0, as included in SageMath 7.5) in computations regarding Kerr spacetime. More precisely, it focuses of the Killing tensor $K$ found by Walker & Penrose [Commun. Math. Phys. 18, 265 (1970)].

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:

In [1]:
version()
Out[1]:
'SageMath version 8.1.beta5, Release Date: 2017-09-11'

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)

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

In [4]:
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 diffentiable manifold:

In [5]:
M = Manifold(4, 'M', r'\mathcal{M}')
print(M)
4-dimensional differentiable 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 [6]:
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[6]:
In [7]:
BL[0], BL[1]
Out[7]:

Metric tensor

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

In [8]:
var('m, a', domain='real')
Out[8]:

Let us introduce the spacetime metric:

In [9]:
g = M.lorentzian_metric('g')

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 [10]:
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[10]:

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

In [11]:
g[:]
Out[11]:

The list of the non-vanishing components:

In [12]:
g.display_comp()
Out[12]:

Levi-Civita Connection

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

In [13]:
nabla = g.connection() ; print(nabla)
Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional differentiable manifold M

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

In [14]:
nabla(g).display()
Out[14]:

Killing vectors

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

In [15]:
M.default_frame() is BL.frame()
Out[15]:
In [16]:
BL.frame()
Out[16]:

Let us consider the first vector field of this frame:

In [17]:
xi = BL.frame()[0] ; xi
Out[17]:
In [18]:
print(xi)
Vector field d/dt on the 4-dimensional differentiable manifold M

The 1-form associated to it by metric duality is

In [19]:
xi_form = xi.down(g) ; xi_form.display()
Out[19]:

Its covariant derivative is

In [20]:
nab_xi = nabla(xi_form) ; print(nab_xi) ; nab_xi.display()
Tensor field of type (0,2) on the 4-dimensional differentiable manifold M
Out[20]:

Let us check that the Killing equation is satisfied:

In [21]:
nab_xi.symmetrize() == 0
Out[21]:

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

In [22]:
chi = BL.frame()[3] ; chi
Out[22]:
In [23]:
nabla(chi.down(g)).symmetrize() == 0
Out[23]:

Principal null vectors

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

In [24]:
k = M.vector_field(name='k')
k[:] = [(r^2+a^2)/(2*rho2), -Delta/(2*rho2), 0, a/(2*rho2)]
k.display()
Out[24]:
In [25]:
el = M.vector_field(name='el', latex_name=r'\ell')
el[:] = [(r^2+a^2)/Delta, 1, 0, a/Delta]
el.display()
Out[25]:

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

In [26]:
g(k,k).expr()
Out[26]:
In [27]:
g(el,el).expr()
Out[27]:

Their scalar product is $-1$:

In [28]:
g(k,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 [43]:
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 differentiable manifold M
In [44]:
K.display_comp()
Out[44]:
In [45]:
DK = nabla(K)
print(DK)
Tensor field nabla_g(K) of type (0,3) on the 4-dimensional differentiable manifold M
In [46]:
DK.display_comp()
Out[46]:

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

In [40]:
DK.symmetrize().display()
Out[40]:

Equivalently, we may write, using index notation:

In [41]:
DK['_(abc)'].display()
Out[41]:
In [ ]: