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# Proper Time

It is often helpful to write the invariant differential interval in the form

 (1714)

The quantity is called the proper time. It follows that

 (1715)

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

 (1716)

implying that

 (1717)

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 time dilation effect.

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 length contraction along the direction of motion by a factor .

Next: 4-Velocity and 4-Acceleration Up: Relativity and Electromagnetism Previous: Space-Time
Richard Fitzpatrick 2014-06-27