Proper time

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

$$ds^2 = c^2 d\tau^2.$$ (1419)

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

$$d\tau^2 = - \frac{dx^2+dy^2+dz^2}{c^2} + dt^2.$$ (1420)

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

$$d\tau^2 = dt^2\left[ -\frac{dx^2+dy^2+dz^2}{c^2 dt^2} + 1\right] =dt^2\left(1-\frac{u^2}{c^2}\right),$$ (1421)

implying that

$$\frac{dt}{d\tau} = \gamma(u).$$ (1422)

It is clear that $dt=d\tau$ in the particle's rest frame. Thus, $d\tau$ corresponds to the time difference between two neighbouring events on the particle's world-line, 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 $\gamma(u)$, 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

$$d^4 x' = {\cal J} d^4 x,$$ (1423)

where

$${\cal J} = \frac{\partial(x^{1'}, x^{2'}, x^{3'}, x^{4'})} {\partial (x^{1}, x^{2}, x^{3}, x^{4})}$$ (1424)

is the Jacobian of the transformation: i.e., 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 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

$$d^4 x' = d^4 x$$ (1425)

for a general Lorentz transformation. In other words, a general Lorentz transformation preserves the volume of space-time. Since 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 $\gamma$.