Running the black hole example

Physical background

Black holes are key objects in GR! The isotropic form of the black hole metric was studied in an exercise in The ADM decomposition, so you should revise that if you are not familiar with it. It represents a wormhole spacetime where the region within the horizon is a (poorly resolved) mirror universe with spatial infinity mapping to the coordinate $r=0$.

The spacetime will evolve to the "trumpet" solution of the puncture gauge over time. This penetrates the horizon, but terminates at a fixed distance from the singularity (that is, the coordinate $r=0$ maps to a sphere of finite proper radius around the singularity once stationary).

How to run

You should be able to execute the sections of the workbook by pressing shift and then enter. At each stage, try to understand what the code is doing - setting up initial data, integrating the solution, or plotting outputs.

Key things to observe

If you plot the lapse you will see the "collapse of the lapse" that indicates the formation of the puncture gauge.

You can also look at the shift to see how it becomes positive over time, indicating that the timeline observers move outwards relative to the normal observers. Looking at K will tell you that the normal observers are converging (K is positive) and so the shift maintains them at a fixed separation once a stable gauge is reached.

If you plot the BSSN variables, you will see some evolution. But this is not physical, but only gauge evolution since the solution provided is of course stationary. What you observe is the gauge evolving from the isotropic one in which the initial data is solved for, and the "puncture gauge" type slicing that the code uses.

Some exercises to try

1. Review the features mentioned here. Find where are they implemented in the code and confirm they work as described.
2. Look at the evolution of each of the variables and try to think of what physically is happening (refer to the lecture slides ADMtoBSSN.pdf to remind yourself what each quantity is).
3. The main "diagnostic" is the Hamiltonian constraint. Why is it so bad in the centre at the initial time?
4. How small a constraint error is ok? The important thing is that the magnitude of the error should reduce as the resolution is increased - check that this does happens, both for the initial conditions and the evolution.
5. Change the gauge evolution - change the value of the damping parameter eta and see its effect on the evolution of the shift vector. Set just the shift to zero (for all time - think about initial conditions and evolution). Does the simulation remain stable?
6. What is happening at line 50-61 of the rhsevolution.py code? Comment this part out and see if the simulation remains stable.

Further Extensions

You could convince yourself that this is only a gauge evolution by finding and plotting the horizon area and showing that is stays constant over time (exercise coming soon).

You can add a scalar field to the black hole to study how it behaves, as suggested in Exercises to extend the code: level 2.