# affine parameter

Given a geodesic curve, an *affine parameterization* for that curve is a parameterization by a parameter $t$ such that the parametric equations for the curve satisfy the geodesic equation.

Put another way, if one picks a parameterization of a geodesic curve by an arbitrary parameter $s$ and sets ${u}^{\mu}=d{x}^{\mu}/ds$, then we have

$${u}^{\mu}{\nabla}_{\mu}{u}^{\nu}=f(s){u}^{\nu}$$ |

for some function $f$. In general, the right hand side of this equation does not equal zero — it is only zero in the special case where $t$ is an affine parameter.

The reason for the name “affine parameter” is that, if ${t}_{1}$ and ${t}_{2}$ are affine parameters for the same geodesic curve, then they are related by an affine transform, i.e. there exist constants $a$ and $b$ such that

$${t}_{1}=a{t}_{2}+b$$ |

Conversely, if $t$ is an affine parameter, then $at+b$ is also an affine parameter.

From this it follows that an affine parameter $t$ is uniquely determined if we specify its value at two points on the geodesic or if we specify both its value and the value of $d{x}^{\mu}/dt$ at a single point of the geodesic.

Title | affine parameter |
---|---|

Canonical name | AffineParameter |

Date of creation | 2013-03-22 14:35:47 |

Last modified on | 2013-03-22 14:35:47 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 53C22 |

Defines | affinely-parameterized |