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

(1419) 
The quantity is called the proper time. It follows that

(1420) 
Consider a series of events on the worldline of some material
particle. If the particle has speed then

(1421) 
implying that

(1422) 
It is clear that in the particle's
rest frame. Thus, corresponds to the
time difference between two neighbouring events on the particle's worldline,
as measured by a clock attached to the particle (hence, the name proper
time). According to Eq. (1422), 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 4dimensional volume element
in spacetime transforms under
a general Lorentz transformation. We have

(1423) 
where

(1424) 
is the Jacobian of the transformation: i.e., 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 wellknown that the Jacobian of the latter two
transformations is unity, since they are both volume preserving transformations
which do not affect time. Likewise, it is easily seen
[e.g.,
by taking the determinant of the transformation matrix (1401)]
that the Jacobian of a standard Lorentz transformation is also unity.
It follows that

(1425) 
for a general Lorentz transformation. In other words, a general Lorentz
transformation preserves the volume of spacetime. Since time is dilated by
a factor in
a moving frame, the volume of spacetime
can only be preserved if the volume of
ordinary 3space is reduced by the same factor. As is wellknown, this
is achieved by length contraction along the
direction of motion by a factor .
Next: 4velocity and 4acceleration
Up: Relativity and electromagnetism
Previous: Spacetime
Richard Fitzpatrick
20060202