52 General Relativity
FFigure 3.2Tidal force퐹acting between two test bodies falling freely towards the surface of a
gravitating body. On the right a spherical cluster of small bodies is seen to become ellipsoidal
on approaching the body.
An interesting example is offered by a sphere of freely falling particles. Since the
strength of the gravitational field increases in the direction of fall, the particles in the
front of the sphere will fall faster than those in the rear. At the same time the lateral
cross-section of the sphere will shrink due to the tidal effect. As a result, the sphere
will befocusedinto an ellipsoid with the same volume. This effect is responsible for
the gravitational breakup of very nearby massive stars.
If the tidal effect is too small to be observable, the laboratory can be considered
to be local. On a larger scale the gravitational field is clearly quite nonuniform, so if
we make use of the equivalence principle to replace this field everywhere by locally
flat frames, we get a patchwork of frames which describe a curved space. Since the
inhomogeneity of the field is caused by the inhomogeneous distribution of gravitating
matter, Einstein realized that the space we live in had to be curved, and the curvature
had to be related to the distribution of matter.
But Einstein had already seen the necessity of introducing a four-dimensional
space-time, thus it was not enough to describe space-time in a nonuniform gravi-
tational field by a curved space, time also had to be curved. When moving over a
patchwork of local and spatially distinct frames, the local time would also have to
be adjusted from frame to frame. In each frame the strong equivalence principle
requires that measurements of time would be independent of the strength of the
gravitational field.
Falling Photons. Let us return once more to the passenger in the Einstein lift for
a demonstration of the relation between gravitation and the curvature of space-time.
Let the lift be in free fall; the passenger would consider that no gravitational field
is present. Standing by one wall and shining a pocket lamp horizontally across the
lift, she sees that light travels in a straight path, a geodesic in flat space-time. This
is illustrated in Figure 3.3. Thus she concludes that in the absence of a gravitational
field space-time is flat.