Introduction to Cosmology

The Principle of Equivalence 51

that the lift is accelerating upwards, tests inside the lift reveal that the objects acquire
an acceleration larger than푔, and vice versa when the lift is accelerating downwards.
In the limit of free fall (unpleasant to the passenger) the objects appear weightless,
corresponding to zero acceleration.
Let us now replace the lift with a spacecraft with the engines turned off, located
at some neutral point in space where all gravitational pulls cancel or are negligible:
a good place is theLagrange point, where the terrestrial and solar gravitational fields
cancel. All objects, including the pilot, would appear weightless there.
Now turning on the engines by remote radio control, the spacecraft could be accel-
erated upwards so that objects on board would acquire an acceleration푔towards the
floor. The pilot would then rightly conclude that

gravitational pull and local acceleration are equivalent

and indistinguishable if no outside information is available and if푚=푚G.Thiscon-
clusion forms theweak equivalence principle (WEP), which states that an observer in
a gravitational field will not experience free fall as a gravitational effect, but as being
at rest in a locally accelerated frame.
A passenger in the lift measuring푔could well decide from his local observations
that Earth’s gravitation actually does not exist, but that the lift is accelerating radially
outwards from Earth. This interpretation does not come into conflict with that of
another observer on the opposite side of Earth whose frame would accelerate in the
opposite direction, because that frame is only local to him/her.
The weak equivalence principle is already embodied in theGalilean equivalence
principlein mechanics between motion in a uniform gravitational field and a uni-
formly accelerated frame of reference. What Einstein did was to generalize this to all
of physics, in particular phenomena involving light.
The more general formulation is the importantstrong equivalence principle (SEP),
which states that

to an observer in free fall in a gravitational field the results of all local exper-

iments are completely independent of the magnitude of the field.

In a suitably small lift or spacecraft, curved space-time can always be approximated
by flat Minkowski space-time. In the gravitational field of Earth the gravitational accel-
eration is directed toward its center. Thus the two test bodies in Figure 3.2 with a
space-like separation do not actually fall along parallels, but along different radii, so
that their separation decreases with time. This phenomenon is called thetidal effect,
or sometimes the tidal force, since the test bodies move as if an attractive exchange
force acted upon them. The classic example is the tide caused by the Moon on the
oceans. The force experienced by a body of mass푚and diameter푑in gravitational
interaction with a body of mass푀at a distance푟is proportional to the differential of
the force of attraction [Equation (1.28])] with respect to푟. Neglecting the geometrical
shapes of the bodies, the tidal force is

퐹tidal≈GMmd∕푟^3.

Thus parts of푚located at smaller distances푟feel a stronger force.