General Relativity and Cosmology
393
respect to accelerations, there appears to be an absolute or a preferred frame of
reference in which the inertial forces are zero.
Mach’s Principle
The inertial frames may be determined by considering the inertial forces on the
surface of the earth. For example, the rate of rotation of the earth with respect to
the inertial frames may be estimated by the measurement of the centrifugal and
coriolis forces on the earth. The rate of rotation thus obtained is found to be
approximately the same as the rate of the earth’s rotation with respect to distant
matter, e.g. the distant galaxies. This leads to an important result that the average
motion of distant galaxies with respect to the inertial frame is zero.
According to Mach (1872), the above result is not an accident. It suggests
that the inertial frame is not an absolute frame but is related to the distribution
of matter in the universe. In fact, Mach asserted that the concept of inertia (and
the inertial frame) can be given meaning only in terms of background stars and
galaxies. This is known as Mach’s principle. It implies that if there were no
matter in the universe except for a given body, there would be no inertia or
inertial forces and it is meaningless to ask whether it is accelerating with respect
to an inertial frame.
A theory of the universe which incorporates Mach’s principle cannot include
an inertial frame without reference to the distribution of matter in the universe.
Principle of Equivalence
An important input of Einstein’s theory of general relativity is the observation
that the effect of gravity can be simulated locally by a noninertial frame.
Consider the motion of an object of inertial mass m I , in a region where the
gravitational acceleration g is approximately constant. If there is a
nongravitational force F acting on it, its motion is given by
m I a = m g g + F
(11.1)
where the gravitational mass m g , in principle, may be different from the inertial
mass m I . Alternatively, consider a frame of reference without a gravitational
force, but moving with acceleration – g. If the acceleration of mass m I in this
frame is a, the corresponding acceleration in the inertial frame is a – g so that
the equation of motion is
m I (a – g) = F
(11.2)
or
m I a = m I g + F
(11.3)
Experimentally is observed to a high accuracy (about 1 part in 10
11
) that
m I = m g , so that Eqs. (11.1) and (11.3) describe the same motion. This result
was generalized by Einstein into what is known as the principle of equivalence:
393
respect to accelerations, there appears to be an absolute or a preferred frame of
reference in which the inertial forces are zero.
Mach’s Principle
The inertial frames may be determined by considering the inertial forces on the
surface of the earth. For example, the rate of rotation of the earth with respect to
the inertial frames may be estimated by the measurement of the centrifugal and
coriolis forces on the earth. The rate of rotation thus obtained is found to be
approximately the same as the rate of the earth’s rotation with respect to distant
matter, e.g. the distant galaxies. This leads to an important result that the average
motion of distant galaxies with respect to the inertial frame is zero.
According to Mach (1872), the above result is not an accident. It suggests
that the inertial frame is not an absolute frame but is related to the distribution
of matter in the universe. In fact, Mach asserted that the concept of inertia (and
the inertial frame) can be given meaning only in terms of background stars and
galaxies. This is known as Mach’s principle. It implies that if there were no
matter in the universe except for a given body, there would be no inertia or
inertial forces and it is meaningless to ask whether it is accelerating with respect
to an inertial frame.
A theory of the universe which incorporates Mach’s principle cannot include
an inertial frame without reference to the distribution of matter in the universe.
Principle of Equivalence
An important input of Einstein’s theory of general relativity is the observation
that the effect of gravity can be simulated locally by a noninertial frame.
Consider the motion of an object of inertial mass m I , in a region where the
gravitational acceleration g is approximately constant. If there is a
nongravitational force F acting on it, its motion is given by
m I a = m g g + F
(11.1)
where the gravitational mass m g , in principle, may be different from the inertial
mass m I . Alternatively, consider a frame of reference without a gravitational
force, but moving with acceleration – g. If the acceleration of mass m I in this
frame is a, the corresponding acceleration in the inertial frame is a – g so that
the equation of motion is
m I (a – g) = F
(11.2)
or
m I a = m I g + F
(11.3)
Experimentally is observed to a high accuracy (about 1 part in 10
11
) that
m I = m g , so that Eqs. (11.1) and (11.3) describe the same motion. This result
was generalized by Einstein into what is known as the principle of equivalence: