The General Theory of Relativity 145
the spatial location of the event while the fourth component, t, describes
the time when the event occurs. These co-ordinates are defined with
respect to a stationary frame of reference. With respect to some other
frame of reference moving with a constant velocity, v, with respect to the
original frame, another set of co-ordinates x′, y′, z′ and t′ are defined.
Einstein showed that the co-ordinates x′, y′, z′ and t′ are related to those
of x, y, z, and t by the following formulae when the velocity, v, is along
the x-axis:
x = (x′ + vt′)/ 1 −v^2 /c^2 y = y′ z = z′t = (t′ + vx′/c^2 )/ 1 −v^2 /c^2.
The details of these equations are not important for the purposes of
our discussion so don’t let their apparent complication disturb you if you
are not mathematically inclined. The important thing to notice about
these equations, known as the Lorentz transformation, is that both the
position, x, and the time, t, in the stationary frame, depends on both
x′ and t′ in the moving frame. In Newtonian physics, the time in the
moving frame would be identical to the time in the stationary frame,
i.e. t = t′ and x = x′ + vt.
We see, in relativistic physics, that space and time are interwoven.
The contraction of length and the slowing down of moving clocks that
the observer in the stationary frame observes occurring in the moving
frame, may be described as rotations in the four-dimensional space-time
continuum. From the Minkowski point of view, an interval of time may
be regarded as a length in the four-dimensional space-time continuum
lying in the t-direction. The time interval is multiplied by c so that it has
the same dimension as a length. Let us reconsider the example of the
lightning striking the two ends of a train that we discussed in the
beginning of the last chapter. From the point of view of the observer at
rest, the lightning strikes the front of the train vLo/c^2 seconds before it
hits the end of the train, as can be verified from Fig. 14.1. The spatial
length of the train, on the other hand, is contracted according to the
stationary observer and appears to have the length Lo 1 −v^2 /c^2. The
separation of the two events, which to the moving observer was purely
spatial, become both spatial and temporal to the stationary observer,
demonstrating the equivalence of space and time. The observation of the
stationary observer may be obtained from the moving observer’s
observations by rotating in the space-time continuum about an axis