The BSSNOK formulation of the 3+1 evolution equations described in Section 2.8 is based on a conformal rescaling of the metric and extrinsic curvature. However, in its standard version this formulation uses a rescaling of the extrinsic curvature that is in fact not the most natural rescaling for this quantity. Here I will present a version of the BSSNOK equations that uses the natural rescaling for a tracefree symmetric tensor.
We start by summarizing the standard BSSNOK equations. We perfom a conformal transformation of the spatial metric and extrinsic curvature of the form
where the conformal factor is taken to be
, so that the conformal metric
has unit determinant. In the previous expression K is the trace of K ij, which implies that the tensor Ã ij is traceless. We also introduces the auxiliary variables
where the second equality follows from the fact that
has unit determinant.
The full system of evolution equations then becomes
where TF denotes the tracefree part of the expression inside the brackets, and where we have used the Hamiltonian constraint to eliminate the Ricci scalar from the evolution equation for K
, and the momentum constraints to eliminate the divergence of Ã ij
from the evolution equation for
. In the previous expressions we have
the Lie derivative with respect to the shift that must be calculated for tensor densities: ψ, a scalar density of weight 1/6, and
and Ã ij
, tensor densities with weight −2/3. Also, even though the
is strictly speaking not a vector, its Lie derivative is understood as that corresponding to a vector density of weight 2/3. Finally, the Ricci tensor is separated into two contributions in the following way:
is the Ricci tensor associated with the conformal metric
is given by φ:
the covariant derivative associated with the conformal metric.
The Hamiltonian and momentum constraints also take the form
Now, in Chapter 3 it was shown that the natural conformal transformation for the tracefree extrinsic curvature is in fact Āij=ψ10 A ij (Ā ij=ψ2 A ij), with φ=ln ψ. We will then consider the following conformal transformation
We will furthermore introduce the densitized lapse
With this new conformal scaling, the evolution equations become instead (do notice that in some places there is
and in others just α)
and the constraints become
Notice that with this new rescaling, a term involving derivatives of the conformal factor φ has disappeared from both the evolution equation for
and the momentum constraints.