Definitions and Notation:
The 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 conformal metric has unit determinant, . The variables , , , and are defined in terms of the physical spatial metric and physical extrinsic curvature by
where is the determinant of the physical metric. Note that is a type 0-2 tensor density of weight -2/3, is the logarithm of a scalar density of weight 1/6, is a scalar, and is a type 0-2 tensor density of weight -2/3.
The conformal connection functions are defined in terms of the "conformal Christoffel symbols" by
The lapse function is and the shift vector is .
Equations of motion:
In terms of the time derivative operator , the BSSN equations are
where and are the covariant derivative and Ricci tensor built from the physical metric. The indices on are raised with the conformal metric. Indices on are raised with the physical metric. The superscript TF denotes the trace-free part.
In terms of BSSN variables, the physical Ricci tensor is
The Lie derivatives contained in the operators on the left-hand sides of the equations of motion are:
The BSSN constraints are: