So far we have been rather vague about what we mean by the mass of a body. Even in Newtonian theory we can ascribe three masses to any body which describe quite different properties:
Passive gravitational mass measures a body's response to being placed in a gravitational field. Let the gravitational potential at some point be , then if is placed at this point, it will experience a force on it given by
On the other hand active gravitational mass measures the strength of the gravitational field produced by the body itself. If is placed at the origin, then the gravitational potential at any point a distance r from the origin is given by
We will now see how these three masses are related in the Newtonian framework.
Galileo discovered in his famous Pisa experiments [ see Figure 5.3 ] that when two bodies are dropped from the same height, they reach the ground together irrespective of their internal composition.
Figure 5.3: The Galileo Piza experiment.
Let's assume that two particles of inertial mass and and passive gravitational mass and are dropped from the same height in a gravitational field. We have:
The observational result is from which we get on dividing
Repeating this experiment with other bodies, we see that this ratio is equal to a universal constant say. By a suitable choice of units we can take , from which we obtain the result:
In order to relate passive gravitational mass to active gravitational mass, we make use of the observation that nothing can be shielded from a gravitational field. Consider two isolated bodies situated at points Q and R moving under their mutual gravitational attraction. The gravitational potential due to each body is
The force which each body experiences is
If we taken the origin to be Q then the gradient operators are
so that
But by Newton's third law , and so we conclude that
and using the same argument as before, we see that
This may see obvious to you, but it has very deep significance and Einstein used it as the central pillar for his equivalence principle .