(1714) |

The quantity is called the

(1715) |

Consider a series of events on the world-line of some material particle. If the particle has speed then

(1716) |

implying that

It is clear that in the particle's rest frame. Thus, corresponds to the time difference between two neighboring events on the particle's world-line, as measured by a clock attached to the particle (hence, the name ``proper time''). According to Equation (1719), the particle's clock appears to run slow, by a factor , in an inertial frame in which the particle is moving with velocity . This is the celebrated

Let us consider how a small 4-dimensional volume element in space-time transforms under a general Lorentz transformation. We have

(1718) |

where

(1719) |

is the Jacobian of the transformation: that is, the determinant of the transformation matrix . A general Lorentz transformation is made up of a standard Lorentz transformation plus a displacement and a rotation. Thus, the transformation matrix is the product of that for a standard Lorentz transformation, a translation, and a rotation. It follows that the Jacobian of a general Lorentz transformation is the product of that for a standard Lorentz transformation, a translation, and a rotation. It is well known that the Jacobians of the latter two transformations are unity, because they are both volume preserving transformations that do not affect time. Likewise, it is easily seen [e.g., by taking the determinant of the transformation matrix (1698)] that the Jacobian of a standard Lorentz transformation is also unity. It follows that

(1720) |

for a general Lorentz transformation. In other words, a general Lorentz transformation preserves the volume of space-time. Because time is dilated by a factor in a moving frame, the volume of space-time can only be preserved if the volume of ordinary 3-space is reduced by the same factor. As is well-known, this is achieved by