Event and trapping horizons in Vaidya spacetime
This Jupyter/SageMath worksheet is relative to the lectures Geometry and physics of black holes
These computations are based on SageManifolds (v0.9)
Click here to download the worksheet file (ipynb format). To run it, you must start SageMath with the Jupyter notebook, with the command sage -n jupyter
First we set up the notebook to display mathematical objects using LaTeX formatting:
Spacetime
We declare the spacetime manifold :
We use coordinates analogous to the 3+1 Eddington-Finkelstein coordinates in Schwarzschild spacetime, i.e. coordinates such that the advanced time is constant on the ingoing radial null geodesics:
Metric tensor
The metric tensor corresponding to the Vaidya solution is:
Einstein equation
Let us compute the Ricci tensor associated with the metric :
The Ricci scalar is vanishing:
The energy-momentum vector ensuring that the Einstein equation is fulfilled is then:
Since , we have :
The derivative of the function :
The future-directed null vector along the ingoing null geodesics:
Outgoing radial null geodesics
Let us consider the vector field:
It is a null vector:
Moreover it is a pregeodesic vector field:
Integration of the outgoing radial null geodesics
The outgoing radial null geodesics are the field lines of ; they obey thus to . Hence the value of :
Let us choose a simple function :
We plug this function into the expression of found above:
and we perform a numerical integration for
The event horizon:
The trapping horizon:
A zoom on the trapping horizon in its dynamical part: notice that the "outgoing" null geodesics cross it with a vertical tangent, in agreement with the cross-sections of the trapping horizon being marginally trapped surfaces.