98
7 Classical Gravity and Geometry
special relativity the fundamental equation (7.1) must be modified so that gravitational effects propagate at c or less. The situation is rather remarkable: the theory
must be changed despite a lack of any experimental evidence that it is wrong.
Problem (2), the equality of the inertial and gravitational masses, is curious in
that it leads to fundamentally strange consequences, which are quite well-known
and familiar. From the above definitions we may write Newton’s second law for the
acceleration of a test body in a gravitational field
F = −m∇φ = m
a, so
d
2 x
j
dt 2 = −φ , j .
(7.10)
Because of the equality of inertial and gravitational mass the mass of the test body
cancels from the equation for the acceleration and the acceleration is independent of
it. Thus, for example, if two objects of different mass in the earth’s field begin at the
same position with the same velocity they will follow the same trajectory. Similarly,
the paths of planets around the sun are independent of the planet mass. Astronauts
inside a spacecraft in orbit follow the same trajectory as the spacecraft and therefore
float freely inside the craft. We call this free-fall.
The fact that different bodies fall at the same rate in a gravitational field is often
referred to as the universality of free fall or the weak equivalence principle. We will
say more about it in the following section when we further discuss the equivalence
principle and when we study the intrinsic signature of gravity in general relativity in
Sect. 8.5.
It is worth emphasizing that the equality of inertial and gravitational mass is
subject to experimental test of very high accuracy. Eotvos in the early twentieth
century showed that the two are equal within about a part in 10
8 , while more recently
Dicke et al. have increased the accuracy to better than a part in about 10
12 (Eotvos
1922; Will 2014). There are presently proposals to test the equality in a spacecraft
with an accuracy of about a part in 10
17 (Will 2014).
In the context of Newtonian theory the question of why such an extraordinary
situation should occur is a deep mystery. By contrast it follows easily and naturally
from a geometrical viewpoint, and was one of the main guides used by Einstein in
developing the relativistic theory of gravity (Zee 1989).
7.2 The Equivalence Principle
Let us follow Einstein in his reasoning concerning the equivalence principle (EP)
using gedanken experiments. We begin by putting one observer in a lab on the earth
and one in an identical lab in a rocket ship in space accelerating at g, as shown in
Fig. 7.2. (Einstein used an elevator rather than a rocket.)
In the two labs we then do various mechanics experiments, like dropping balls,
weighing objects, setting up levers and inclined plane systems etc. In the earth lab a
ball accelerates downward due to the force of gravity, and independent of its mass. In
7 Classical Gravity and Geometry
special relativity the fundamental equation (7.1) must be modified so that gravitational effects propagate at c or less. The situation is rather remarkable: the theory
must be changed despite a lack of any experimental evidence that it is wrong.
Problem (2), the equality of the inertial and gravitational masses, is curious in
that it leads to fundamentally strange consequences, which are quite well-known
and familiar. From the above definitions we may write Newton’s second law for the
acceleration of a test body in a gravitational field
F = −m∇φ = m
a, so
d
2 x
j
dt 2 = −φ , j .
(7.10)
Because of the equality of inertial and gravitational mass the mass of the test body
cancels from the equation for the acceleration and the acceleration is independent of
it. Thus, for example, if two objects of different mass in the earth’s field begin at the
same position with the same velocity they will follow the same trajectory. Similarly,
the paths of planets around the sun are independent of the planet mass. Astronauts
inside a spacecraft in orbit follow the same trajectory as the spacecraft and therefore
float freely inside the craft. We call this free-fall.
The fact that different bodies fall at the same rate in a gravitational field is often
referred to as the universality of free fall or the weak equivalence principle. We will
say more about it in the following section when we further discuss the equivalence
principle and when we study the intrinsic signature of gravity in general relativity in
Sect. 8.5.
It is worth emphasizing that the equality of inertial and gravitational mass is
subject to experimental test of very high accuracy. Eotvos in the early twentieth
century showed that the two are equal within about a part in 10
8 , while more recently
Dicke et al. have increased the accuracy to better than a part in about 10
12 (Eotvos
1922; Will 2014). There are presently proposals to test the equality in a spacecraft
with an accuracy of about a part in 10
17 (Will 2014).
In the context of Newtonian theory the question of why such an extraordinary
situation should occur is a deep mystery. By contrast it follows easily and naturally
from a geometrical viewpoint, and was one of the main guides used by Einstein in
developing the relativistic theory of gravity (Zee 1989).
7.2 The Equivalence Principle
Let us follow Einstein in his reasoning concerning the equivalence principle (EP)
using gedanken experiments. We begin by putting one observer in a lab on the earth
and one in an identical lab in a rocket ship in space accelerating at g, as shown in
Fig. 7.2. (Einstein used an elevator rather than a rocket.)
In the two labs we then do various mechanics experiments, like dropping balls,
weighing objects, setting up levers and inclined plane systems etc. In the earth lab a
ball accelerates downward due to the force of gravity, and independent of its mass. In