# Shift Conditions

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## Contents |

#### Notation

For the minimal strain and minimal distortion conditions, denotes the spatial metric. For the Gamma freezing and Gamma driver conditions, denotes the conformal metric. The spatial covariant derivative is , the extrinsic curvature is , the shift vector is and the lapse function is .

#### Minimal Strain

Extremize with respect to the shift, where . This gives the condition:

#### Minimal Distortion

Extremize with respect to the shift, where is the trace-free part of . This gives the condition:

#### Gamma freezing

The Gamma freezing condition is where are the conformal connection functions and is the conformal metric with unit determinant, . Explicitly,

where the physical metric is defined by and the physical extrinsic curvature is defined by . (See the BSSN system of evolution equations.)

#### Gamma driver

The Gamma driver condition is

where are the conformal connection functions built from the conformal metric . The conformal metric has unit determinant, . The numerical parameter is usually set to a value between zero and . The term is given by

where the physical metric is defined by and the physical extrinsic curvature is defined by . (See the BSSN system of evolution equations.)

Variants of the Gamma driver condition are obtained by dropping one or more of the advection terms , , or .