Solving Einstein equation to get Kottler solution
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 declare the chart of spherical coordinates :
The static and spherically symmetric metric ansatz, with the unknown functions and :
The Christoffel symbols of , with respect to the default chart:
Einstein equation
The cosmological constant:
The Einstein equation:
Simplifying and rearranging the equations
Solving Einstein equation
The following combination of eq0 and eq1 is particularly simple:
The solution is , where is a constant:
Let us rename the constant to :
We replace by the above value in the remaining equations:
Let us solve eq5 for . Note that we are using eq5.expr()
to get a symbolic expression, as expected by the function desolve
, while eq5
is a coordinate function.
We rename the constant to and set the value of constant to :
Let us check that eq6
is fulfilled by the found value of :
Final expression of the metric
We have got the Kottler metric:
which reduces to Schwarzschild metric as soons as the cosmological constant vanishes.
Let us check that Einstein equation is satisfied by the above metric:
The Ricci scalar is constant for this solution:
The Ricci tensor is proportional to the metric tensor: