Tensors and Relativity: Chapter 5

Mass in Newtonian theory

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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:

  • Inertial mass  tex2html_wrap_inline775 : This is a measure of its resistance to change in motion or inertia .
  • Passive gravitational mass  tex2html_wrap_inline777 : This is a measure of its reaction to a gravitational field.
  • Active gravitational mass  tex2html_wrap_inline779 : This is a measure of its source strength for producing a gravitational field.

Let us discuss each of these in turn. Inertial mass  tex2html_wrap_inline775 is the quantity occurring in Newton's second law [ tex2html_wrap_inline783 ]. It is a measure of a body's inertia. Note that as far as Newtonian theory, this mass has nothing to do with gravitation. The other two masses however do.

Passive gravitational mass  tex2html_wrap_inline777 measures a body's response to being placed in a gravitational field. Let the gravitational potential at some point be tex2html_wrap_inline787 , then if tex2html_wrap_inline777 is placed at this point, it will experience a force on it given by


On the other hand active gravitational mass  tex2html_wrap_inline779 measures the strength of the gravitational field produced by the body itself. If tex2html_wrap_inline779 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 tex2html_wrap_inline797 and tex2html_wrap_inline799 and passive gravitational mass tex2html_wrap_inline801 and tex2html_wrap_inline803 are dropped from the same height in a gravitational field. We have:


The observational result is tex2html_wrap_inline805 from which we get on dividing


Repeating this experiment with other bodies, we see that this ratio is equal to a universal constant tex2html_wrap_inline807 say. By a suitable choice of units we can take tex2html_wrap_inline809 , from which we obtain the result:

  • inertial mass = passive gravitational mass.

This equality is one of the best test results in physics and has been verified to 1 part in tex2html_wrap_inline815 .

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 tex2html_wrap_inline821 , and so we conclude that


and using the same argument as before, we see that

  • Passive gravitational mass = active gravitational mass.

That is why in Newtonian theory  we can simply refer to the mass m of a body where


This may see obvious to you, but it has very deep significance and Einstein used it as the central pillar for his equivalence principle .

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Next: The principle of equivalence Up: Title page Previous: Non- existence of an