One often hears about “the equivalence of inertial mass and gravitational mass”. We shall see that all masses are equivalent, but some masses are more equivalent than others. In this document we will explore what this means, and what inequivalence might look like, theoretically and experimentally.
Let us consider a rigid body of mass m. In accordance with Newton’s law of universal gravitation, the gravitational force on the body is
where F is a vector and the local field-strength g is a vector with dimensions of acceleration.
In accordance with the three laws of motion, the force required to accelerate the body is
If we write this as a = F/m we see that m is the resistance to acceleration, i.e. the inertia.
The structure of equation 1 parallels the structure of equation 2. We can interpret this as a type of equivalence. The mass gives rise to a gravitational force (equation 1 in the same way as it gives rise to inertia (equation 2).
There are of course other ways of interpreting the equations. It makes at least as much sense to say that the mass is the mass, by definition, and to interpret the equations as expressing the equivalence of gravitation (equation 1) and acceleration (equation 2).
For a body of nonzero size, the total force is
which means the total inertia is:
where we have used the fact that the body is rigid to bring a outside the integral. This means we can speak of ∫dm as the total inertia. For a rigid body, the inertia depends only on the zeroth moment of the mass distribution.
Let’s try to play the same game with equation 1.
So we have already learned something about the equivalence of gravitation and acceleration: We get into trouble if the gravitational field is nonuniform. We can salvage the idea of equivalence if we say there is local equivalence, which is of course an approximation because nothing of interest is ever completely local.
The inertia depends only on the zeroth moment of the mass distribution, but the gravitational field couples to all moments from zero on up. This leads to practical applications. You can build a gravity gradiometer i.e. an instrument to measure non-uniformities in the gravitational field. Sensitive gradiometers are used by prospectors and geologists to map out mineral deposits. Submarines use gradiometers to verify their location relative to undersea mountains and trenches.
Another type of inequivalence arises when we consider rotational motion (as opposed to straight-line motion). In analogy to equation 4, for a rigid body, the total rotational inertia about the origin is given by the moment of inertia aka tensor of inertia:
where r is the usual position vector, and ⊗ denotes the outer product. Also ⟨⋯| denotes a row vector while |⋯⟩ denotes a column vector.
As a consequence of equation 6, I is necessarily a symmetric second-rank tensor. In three-dimensional Cartesian coordinates equation 6 reduces to
Meanwhile, in analogy to equation 5, the total gravitational torque on the body is:
where the local field-strength g is (as always) a vector, and ∧ denotes the wedge product. The torque τg is a bivector.
Experiments have been done to check for various types of inequivalence. These are sometimes called Eötvös experiments, in honor of the early, careful work of Loránd von Eötvös ... although he was not the first and certainly not the last to address this issue.
Einstein assumed a certain type of equivalence and used it as a guide to the development of general relativity. Hence the name, Einstein’s equivalence principle. People who think about gravitation in terms of general relativity – in a qualitative way, without looking at the details of the equations – sometimes conclude that inertial mass and gravitational mass simply must be equivalent, although (as discussed in section 1) this is going quite a bit too far.