Kerr_Killing_tensor.ipynb

Kernel: SageMath 9.1.rc1

Walker-Penrose Killing tensor in Kerr spacetime

This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.

It focuses on the Killing tensor $K$ found by Walker & Penrose [Commun. Math. Phys. 18, 265 (1970)].

The involved computations are based on tools developed through the SageManifolds project.

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

version()

'SageMath version 9.1.rc1, Release Date: 2020-04-22'

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

%display latex

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

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:

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:

BL.<t,r,th,ph> = M.chart(r't r th:(0,pi):\theta ph:(0,2*pi):\phi') 
print(BL) ; BL

Chart (M, (t, r, th, ph))

BL[0], BL[1]

BL.coord_range()

Metric tensor

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

var('m, a', domain='real')

We get the (yet undefined) spacetime metric by

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:

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()

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

g[:]

The list of the non-vanishing components:

g.display_comp()

Levi-Civita Connection

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

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:

nabla(g).display()

Killing vectors

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

M.default_frame() is BL.frame()

BL.frame()

Let us consider the first vector field of this frame:

xi = BL.frame()[0] ; xi

print(xi)

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

The 1-form associated to it by metric duality is

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

Its covariant derivative is

nab_xi = nabla(xi_form) ; print(nab_xi) ; nab_xi.display()

Tensor field of type (0,2) on the 4-dimensional Lorentzian manifold M

Let us check that the Killing equation is satisfied:

nab_xi.symmetrize() == 0

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

eta = BL.frame()[3] ; eta

nabla(eta.down(g)).symmetrize() == 0

Principal null vectors

Let $k$ be the null vector tangent to the ingoing principal null geodesics associated with their affine parameter $-r$; the expression of $k$ in term of the 3+1 ingoing Kerr coordinates $(\tilde{t}, r, \theta, \tilde\phi)$ is $$k = \frac{\partial}{\partial\tilde{t}} - \frac{\partial}{\partial\tilde{r}}$$ The expression of $k$ in terms of the Boyer-Lindquist coordinates is

k = M.vector_field(name='k')
k[:] = [(r^2 + a^2)/Delta, -1, 0, a/Delta]
k.display()

Regarding the null vector tangent to the outgoing principal null geodesics, we select one associated with a (non-affine) parameter $\lambda$ that is regular accross the two Killing horizons:

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

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

g(k,k).expr()

g(el,el).expr()

Their scalar product is $-\rho^2/(r^2 + a^2)$:

g(k,el).expr()

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

k.dot(el).expr()

Let us check that $k$ is a geodesic vector, i.e. that it obeys $\nabla_k k = 0$:

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

We check that $\ell$ is a pregeodesic vector, i.e. that $\nabla_\ell \ell = \kappa_\ell \ell$ for some scalar field $\kappa_\ell$:

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

kappa_l = acc_l[[0]] / el[[0]]
kappa_l.display()

acc_l == kappa_l * el

Walker-Penrose Killing tensor

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

uk = k.down(g)
ul = el.down(g)

uk.display()

uk[3]

uk[3].factor()

uk[3] == a*sin(th)^2

uk[3] = a*sin(th)^2

uk.display()

ul.display()

ul[3].factor()

ul[3] == a*sin(th)^2*(a^2 + r^2 - 2*m*r)/(2*(a^2 + r^2))

ul[3] = a*sin(th)^2*(a^2 + r^2 - 2*m*r)/(2*(a^2 + r^2))

ul.display()

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

K = 2*((r^2 + a^2)*(ul*uk)).symmetrize() + r^2*g
K.set_name('K')
print(K)

Field of symmetric bilinear forms K on the 4-dimensional Lorentzian manifold M

The non-vanishing components of $K$:

K.display_comp()

We may simplify a little bit some components:

K[0,3].factor()

K[0,3] == - a*sin(th)^2*(r^2 + a^2 - 2*m*r*a^2*cos(th)^2/rho2)

K[0, 3] = - a*sin(th)^2*(r^2 + a^2 - 2*m*r*a^2*cos(th)^2/rho2)

K[1,1].factor()

K[3,3] == sin(th)^2 * (r^2*(r^2 + a^2) 
                       + a^2*sin(th)^2*(r^2 + a^2 - 2*m*r/rho2*a^2*cos(th)^2))

K[3,3] = sin(th)^2 * (r^2*(r^2 + a^2) 
                       + a^2*sin(th)^2*(r^2 + a^2 - 2*m*r/rho2*a^2*cos(th)^2))

K.display_comp()

DK = nabla(K)
print(DK)

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

DK.display_comp()

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

DK.symmetrize().display()

Equivalently, we may write, using index notation:

DK['_(abc)'].display()

pt = var('pt', latex_name=r'p^t')
pr = var('pr', latex_name=r'p^r')
pth = var('pth', latex_name=r'p^\theta')
pph = var('pph', latex_name=r'p^\phi')
p = M.vector_field(name='p')
p[:] = [pt, pr, pth, pph]
p.display()

K(p, p).expr()

s = K(p, p).expr()
s

Metric dual of the Killing tensor ($K^{ab}$)

uK = K.up(g)
uK.set_name('uK', latex_name=r'\hat{K}')
print(uK)

Tensor field uK of type (2,0) on the 4-dimensional Lorentzian manifold M

uK.display_comp()

uK[0,0] == a^2/Delta*(r^2 + a^2 - 2*m*r^3*sin(th)^2/rho2)

uK[0,0] = a^2/Delta*(r^2 + a^2 - 2*m*r^3*sin(th)^2/rho2)

uK[0,3] == a/Delta*(r^2 + a^2 - 2*m*r^3/rho2)

uK[0,3] = a/Delta*(r^2 + a^2 - 2*m*r^3/rho2)

uK[1,1] == - Delta*a^2*cos(th)^2/rho2

uK[1,1] =  - Delta*a^2*cos(th)^2/rho2

uK[3,3] == (a^2 + r^2/sin(th)^2*(1 - 2*m*r/rho2))/Delta

uK[3,3] = (a^2 + r^2/sin(th)^2*(1 - 2*m*r/rho2))/Delta

uK.display_comp()

Killing tensor leading directly to Carter constant Q

We introduce first the Killing vector $V := \eta + a \xi$:

V = eta + a*xi
V.display()

V_form = V.down(g)
V_form.display()

and form the Killing tensor $$\tilde{K} = K - \underline{V}\otimes\underline{V}$$

K2 = K - V_form*V_form
K2.set_name('tK', latex_name=r'\tilde{K}')
K2.display_comp()

Let us check that $\tilde{K}$ is indeed a Killing tensor:

nabla(K2).symmetrize().display()

We check that $$\tilde{K}^{\alpha\beta} = - a^2 \cos^2 \theta \, g^{\alpha\beta} + \mathrm{diag}(-a^2\cos^2\theta, 0, 1, \tan^{-2}\theta)^{\alpha\beta}$$

T = K2 + (a^2*cos(th)^2)*g
uT = T.up(g)
uT[:]