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It is often helpful to write the invariant differential interval
in
the form
![$\displaystyle ds^{\,2} = c^{\,2} \,d\tau^{\,2}.$](img3688.png) |
(1714) |
The quantity
is called the proper time. It follows that
![$\displaystyle d\tau^{\,2} = - \frac{dx^{\,2}+dy^{\,2}+dz^{\,2}}{c^{\,2}} + dt^{\,2}.$](img3690.png) |
(1715) |
Consider a series of events on the world-line of some material
particle. If the particle has speed
then
![$\displaystyle d\tau^{\,2} = dt^{\,2}\left( -\frac{dx^{\,2}+dy^{\,2}+dz^{\,2}}{c^{\,2}\, dt^{\,2}} + 1\right) =dt^{\,2}\left(1-\frac{u^{\,2}}{c^{\,2}}\right),$](img3691.png) |
(1716) |
implying that
![$\displaystyle \frac{dt}{d\tau} = \gamma(u).$](img3692.png) |
(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
![$\displaystyle d^{\,4} x' = {\cal J}\, d^{\,4} x,$](img3695.png) |
(1718) |
where
![$\displaystyle {\cal J} = \frac{\partial(x^{\,1'}, x^{\,2'}, x^{\,3'}, x^{\,4'})} {\partial (x^{\,1}, x^{\,2}, x^{\,3}, x^{\,4})}$](img3696.png) |
(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
![$\displaystyle d^{\,4} x' = d^{\,4} x$](img3698.png) |
(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