SageMath notebooks associated to the Black Hole Lectures (https://luth.obspm.fr/~luthier/gourgoulhon/bh16)
NHEK spacetime
This This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes. It explores the global Near-Horizon Extremal Kerr (NHEK) spacetime .
This notebook requires a version of SageMath at least equal to 9.0:
First we set up the notebook to display mathematical objects using LaTeX rendering, and, to speed up computations, we ask for running them in parallel on 8 threads:
Manifold
Global coordinate chart :
The coordinate 1-forms:
NHEK metric
The mass parameter appears in the the NHEK metric as an overall scale parameter . In what follows, we set for simplicity. We then declare as
The NHEK metric is a solution of the vacuum Einstein equation:
Inverse metric
Some simplifications are in order:
Hence we set:
and we get
Killing vectors
Two obvious Killing vectors are and , since the components of do not depend on or :
A third Killing vector arises from the isometry expressed by in Poincaré-type coordinates:
Finally a fourth Killing vector is of the Poincaré-type coordinates. We actually consider the linear combination since it has slightly simpler components in terms of the global coordinates :
Commutation relations
First of all we notice that commutes with all the three other Killing vectors:
To write easily the other commutation relations, we supplement by to make it a vector frame on :
Then we ask for the display of the commutators in that frame:
These commutation relations are not the standard ones for ; in order to get these, let us introduce the following linear combinations:
We have then
The above commutation relations are exactly those of in the standard matrix representation:
Komar angular momentum
The spacetime total angular momentum is computed from the axisymmetric Killing vector via the Komar integral: where
is the exterior derivative of the 1-form (the -dual to )
is the Hodge dual of with respect to
is a closed spacelike 2-surface
In vacuum, the Komar integral is independent of the choice of . Let us choose to be a sphere . We have then
The Killing vector :
The 1-form :
The 2-form :
The 2-form :
The component:
This can be simplified to :
Hence we have
If we restore the metric scale factor , we have , , and . The last property holds because in dimension 4, the Hodge dual is conformally invariant on 2-forms (as it can be easily checked from its definition). Hence we conclude that In other words, the angular momentum of the NHEK spacetime is identical to the angular momentum of the Kerr spacetime from which it derives.
For the record:
Conformal global coordinates
Let us introduce the "conformal" global coordinates such that :
Expression of the Killing vectors in terms of conformal coordinates:
Poincaré patch
Plot of the Killing vector
Plot of the Killing vector
Plot of Poincaré patches with and hypersurfaces
Adding some labels: