100
7 Classical Gravity and Geometry
such forces may also be turned off by going to a different lab or coordinate system;
moreover, and very importantly, such fictitious forces are represented by connection
terms in the classical equations of motion (5.65). Since gravity is similar to fictitious
forces in that it is proportional to mass, and can be transformed away, might it then
be represented by connection terms in equations of motion and thereby be thought
of as a geometric effect? The answer is of course “yes” as we will show in the next
section.
An important caveat is associated with the equivalence principle. We again emphasize that the lab must be considered so small that the gravitational field is effectively
uniform over it. In a larger lab there is an obvious difference between the earth
lab and the accelerated lab: two balls falling in the earth lab will converge ever so
slightly as they fall toward the center of the earth, and in the rocket lab they will
not (see Fig. 7.4). The slight convergence is due to the fact that the earth lab has a
gravitational field with a gradient and consequent tidal forces. These tidal forces are
the intrinisic signature of the gravitational field, not the acceleration of a test body.
Indeed, this fact is crucially important; in relativistic gravity we will see that the
Riemann curvature tensor is the analog of Newtonian tidal forces and is the signature of the gravitational field or spacetime curvature. We will study and make further
use of this fact in Chap. 8.
Einstein elevated the principle of equivalence from an observation about
mechanics to a general principle of physics; he assumed that not only mechanical
effects like those we mentioned above but all physical effects will be the same in a
gravitational field as in the equivalent accelerating system (Kenyon 1990; Will 1993).
This is often called the Einstein equivalence principle. For example, one consequence
is that light must be deflected in a gravitational field, because in the equivalent accelerating lab a beam of light waves sent across the lab will clearly be seen to curve
downward.
Fig. 7.4 The equivalence principle does not apply if the lab is large enough that nonuniformity in
the gravitational field is detectable. The balls are seen to converge toward the center of the earth
7 Classical Gravity and Geometry
such forces may also be turned off by going to a different lab or coordinate system;
moreover, and very importantly, such fictitious forces are represented by connection
terms in the classical equations of motion (5.65). Since gravity is similar to fictitious
forces in that it is proportional to mass, and can be transformed away, might it then
be represented by connection terms in equations of motion and thereby be thought
of as a geometric effect? The answer is of course “yes” as we will show in the next
section.
An important caveat is associated with the equivalence principle. We again emphasize that the lab must be considered so small that the gravitational field is effectively
uniform over it. In a larger lab there is an obvious difference between the earth
lab and the accelerated lab: two balls falling in the earth lab will converge ever so
slightly as they fall toward the center of the earth, and in the rocket lab they will
not (see Fig. 7.4). The slight convergence is due to the fact that the earth lab has a
gravitational field with a gradient and consequent tidal forces. These tidal forces are
the intrinisic signature of the gravitational field, not the acceleration of a test body.
Indeed, this fact is crucially important; in relativistic gravity we will see that the
Riemann curvature tensor is the analog of Newtonian tidal forces and is the signature of the gravitational field or spacetime curvature. We will study and make further
use of this fact in Chap. 8.
Einstein elevated the principle of equivalence from an observation about
mechanics to a general principle of physics; he assumed that not only mechanical
effects like those we mentioned above but all physical effects will be the same in a
gravitational field as in the equivalent accelerating system (Kenyon 1990; Will 1993).
This is often called the Einstein equivalence principle. For example, one consequence
is that light must be deflected in a gravitational field, because in the equivalent accelerating lab a beam of light waves sent across the lab will clearly be seen to curve
downward.
Fig. 7.4 The equivalence principle does not apply if the lab is large enough that nonuniformity in
the gravitational field is detectable. The balls are seen to converge toward the center of the earth