# BSSN

### From GRwiki

#### 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

Equivalently,

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

where .

The Lie derivatives contained in the operators on the left-hand sides of the equations of motion are:

#### Constraints:

The BSSN constraints are: