Proper Time

It is often helpful to write the invariant differential interval $ds^{\,2}$ in the form

$$ds^{\,2} = c^{\,2} \,d\tau^{\,2}.$$ (1714)

The quantity $d\tau$ is called the proper time. It follows that

$$d\tau^{\,2} = - \frac{dx^{\,2}+dy^{\,2}+dz^{\,2}}{c^{\,2}} + dt^{\,2}.$$ (1715)

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

$$d\tau^{\,2} = dt^{\,2}\left( -\frac{dx^{\,2}+dy^{\,2}+dz^{\,2}}{c^{\,2}\, dt^{\,2}} + 1\right) =dt^{\,2}\left(1-\frac{u^{\,2}}{c^{\,2}}\right),$$ (1716)

implying that

$$\frac{dt}{d\tau} = \gamma(u).$$ (1717)

It is clear that $dt=d\tau$ in the particle's rest frame. Thus, $d\tau$ 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 $\gamma(u)$ , in an inertial frame in which the particle is moving with velocity $u$ . 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

$$d^{\,4} x' = {\cal J}\, d^{\,4} x,$$ (1718)

where

$${\cal J} = \frac{\partial(x^{\,1'}, x^{\,2'}, x^{\,3'}, x^{\,4'})} {\partial (x^{\,1}, x^{\,2}, x^{\,3}, x^{\,4})}$$ (1719)

is the Jacobian of the transformation: that is, the determinant of the transformation matrix $p^{\,\mu'}_{\mu}$ . 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

$$d^{\,4} x' = d^{\,4} x$$ (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 $\gamma$ 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 $\gamma$ .