Vaidya spacetime
This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.
The computations make use of tools developed through the SageManifolds project.
First we set up the notebook to display mathematical objects using LaTeX formatting:
Spacetime
We declare the spacetime manifold :
We introduce coordinates analogous to the ingoing null Eddington-Finkelstein coordinates in Schwarzschild spacetime, i.e. such that is constant along ingoing radial null geodesics:
Metric tensor
The metric tensor corresponding to the Vaidya solution is:
Curvature
The Ricci tensor is
It has zero trace, i.e. the Ricci scalar vanishes:
The Riemann tensor:
The Kretschmann scalar :
Wave vector
Check that 𝑘 is a null vector:
Check that is a geodesic vector field, i.e. fulfils :
Homethetic Killing vector for :
If is a linear function, the above result is identically zero, showing that in that case, i.e. that is a homethetic Killing vector.
Ingoing Eddington-Finkelstein coordinates
Let us introduce a new chart such that the advanced time is : ; this is the analog of ingoing Eddington-Finkelstein (IEF) coordinates in Schwarzschild spacetime.
We declare the transition map between the and coordinates:
Expression of the metric tensor in the IEF coordinates:
From now on, we set the IEF chart X
to be the default one on :
Then g.display(X)
can be substituted by g.display()
:
Einstein equation
The Ricci tensor in terms of the IEF coordinates:
The notation to denote is quite unfortunate (this shall be improved in a future version). The display of the corresponding symbolic expression is slightly better, standing for the derivative of function with respect to its first (index ) and unique argument, i.e. :
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 is a pregeodesic vector field, i.e. it obeys :
local/var/lib/sage/venv-python3.8/lib/python3.8/site-packages/scipy/integrate/## Integration of the outgoing radial null geodesics
The outgoing radial null geodesics are the field lines of ; they thus obey to . Hence the value of :
Choice of function
Let us choose a simple function , based on one of the following smoothstep functions:
NB: we don't use Sage's predefined heaviside
function, since it is incompatible with SciPy numerical integrators.
Case of singularity entirely hidden under the event horizon
The singularity is entirely hidden under the event horizon for large energy densities of the radiation shell, i.e. for small values of the shell width . The precise criterion is for . We select here :
Numerical integration of the outgoing radial null geodesics
We plug the function into the expression of along the outgoing radial null geodesics found above:
and we perform a numerical integration:
Plot parameters:
Outgoing radial null geodesics from the Minkowski region:
Drawing of the radiation region (yellow rays):
Adding the outgoing radial null geodesics:
The ingoing null geodesics:
The curvature singularity (in orange):
The event horizon (in black):
Check of the determination of by comparison with the analytic formula for :
The trapping horizon (in red):
The vectors and at some point :
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.
Case of a naked singularity
The naked singularity is obtained for small energy densities of the radiation shell, i.e. for large values of the shell width . The precise criterion is for . We select here :
Outgoing radial null geodesics from the Minkowski region:
Outgoing radial null geodesics in the black hole region:
The Cauchy horizon (in blue):
Outgoing radial null geodesics emerging from the initial singularity:
Drawing of the radiation region (yellow rays):
Adding the outgoing radial null geodesics:
The ingoing null geodesics:
The curvature singularity (in orange):
The event horizon (in black):
The trapping horizon (in red):
A zoom on the initial singularity: