A Basic Field Equations
In the BSSN-puncture formalism, the typical variables to be evolved are
The typical basic equations are
where is the Christoffel symbol associated with , and
The gauges in the puncture formulation are
where is an auxiliary function and is a constant chosen to be of order with being
the total mass of the system.
A point in the BSSN formalism is to rewrite the Ricci tensor, e.g., as
where is the Ricci tensor with respect to and
Here, is the covariant derivative with respect to . Then, is written using , e.g., as
With this prescription, in the weak field limit, and thus, Equations (136) and (137
essentially constitute a Klein–Gordon-type wave equations for .
In the generalized harmonic GH formalism, the typical variables to be evolved are spacetime metric
The evolution equations in the GH formalism are written in first-order form. In contrast to the BSSN
formalism, there are several options for the basic equations, in particular, in the choice of the constraint
damping terms. The details of the formalism employed in the CCCW and LBPLI groups are described
in  and , respectively.
The GH gauge condition is explicitly written as
where is the Christoffel symbol associated with . A suitable choice for is still under
development, e.g., the CCCW group employs a rather phenomenological function , and LBPLI employs