13.3.2 The Principle of Equivalence

Newton’s law of gravity depends on a rather curious assumption. The concept of mass was introduced in Chapter 5 by considering the relationship between force and acceleration. The inertial mass of an object, which is the mass that appears in Newton’s second law, is found by measuring the object’s acceleration a in response to force F:

m_{inert} = inertial mass = \frac{F}{a}

(13.3)

Gravity plays no role in this definition of mass.

The quantities $m_{1}$ and $m_{2}$ in Newton’s law of gravity are being used in a very different way. Masses $m_{1}$ and $m_{2}$ govern the strength of the gravitational attraction between two objects. The mass used in Newton’s law of gravity is called the gravitational mass. The gravitational mass of an object can be determined by measuring the attractive force exerted on it by another mass M a distance r away:

m_{grav} = gravitational mass = \frac{r^{2} F_{M on m}}{G M}

(13.4)

Acceleration does not enter into the definition of the gravitational mass.

These are two very different concepts of mass. Yet Newton, in his theory of gravity, asserts that the inertial mass in his second law is the very same mass that governs the strength of the gravitational attraction between two objects. The assertion that $m_{grav} = m_{inert}$ is called the principle of equivalence. It says that inertial mass is equivalent to gravitational mass.

As a hypothesis about nature, the principle of equivalence is subject to experimental verification or disproof. Many exceptionally clever experiments have looked for any difference between the gravitational mass and the inertial mass, and they have shown that any difference, if it exists at all, is less than 10 parts in a trillion! As far as we know today, the gravitational mass and the inertial mass are exactly the same thing.

But why should a quantity associated with the dynamics of motion, relating force to acceleration, have anything at all to do with the gravitational attraction? This is a question that intrigued Einstein and eventually led to his general theory of relativity, the theory about curved spacetime and black holes. General relativity is beyond the scope of this textbook, but it explains the principle of equivalence as a property of space itself.