Definitions and Notation:
The Generalized BSSN variables are the conformal metric , the conformal factor , the trace of the extrinsic curvature , the trace-free part of the extrinsic curvature , and the conformal connection functions . The variables , , , and are related to the physical spatial metric and physical extrinsic curvature by
where is the determinant of the conformal metric and is the determinant of the physical metric. Note that is a type 0-2 tensor, is a scalar, is a scalar, and is a type 0-2 tensor.
The conformal connection functions are defined in terms of the conformal Christoffel symbols by
The lapse function is and the shift vector is .
Matter is defined by the mass density , the momentum density , and the spatial stress . Here, is the stress-energy-momentum tensor, is the future pointing unit normal to the spacelike hypersurfaces, and is the projection tensor for spacelike hypersurfaces.
Choose units with , where is Newton's constant.
Equations of motion:
In terms of the time derivative operator , the GBSSN equations are
Indices are raised and lowered with the conformal metric. The superscript TF denotes the trace-free part. The Ricci tensor is
where . The Lie derivatives contained in the operators on the left-hand sides of the equations of motion are:
The Lie derivative of is computed using the fact that Lie derivatives and partial derivatives commute.
The equations of motion are invariant under conformal transformations:
The ambiguity is removed from the equations of motion by specifying the evolution of . There are two natural choices. The Eulerian case is . The Lagrangian case is , which is equivalent to .
The GBSSN constraints in vacuum are:
The spatial curvature is built from the conformal metric only, so does not appear anywhere in . The constraint evolution system is
for any choice of .